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Full text of "The Mathematics of Investment"

. 

THE MATHEMATICS OF 

' INVESTMENT 



BY 

WILLIAM L. HART, Pn.D, 

AS3O01ATH PBOTESSOR OF MATHEMATICS I2T THH 
OF MINWBSOTA 




D. 0. HEATH AND COMPANY 
BOSTON NEW YORK CHICAGO LONDON 



COPYRIGHT, 1934, 

BY D. C. HEATH AND COMPANY 

2B0 



PBINTBD D? U.S.A. 



PREFACE 

THIS book provides an elementary course in the theory and the 
application of annuities certain and in the mathematical aspects of 
life insurance. The book is particularly adapted to the needs of 
students in colleges of business administration, but it is also fitted 
for study by college students of mathematics who are not specializing 
in business. Annuities certain and their applications are considered 
in Part I, life insurance is treated in Part II, and a treatment of 
logarithms and of progressions IB given in Part III. The prerequisites 
for the study -of the book are three semesters of high school algebra 
and an acquaintance with progressions and logarithmic computation. 
Very complete preparation would be furnished by three semesters of 
\ high school algebra and a course in college algebra including logarithms. 
The material in the book has been thoroughly tested by the author 
through the teaching of it, in mimeographed form, for two years in 
; classes at the University of Minnesota. It has been aimed in this 

book to present the subject in such a way that its beautiful simplicity 

& and great' usefulness will be thoroughly appreciated by all of the 

\ students to whom it is taught. Features of the book which will 

appeal to teachers of the subject are as follows : 

1. Illustrative examples are consistently used throughout to 
illuminate new theory, to illustrate new methods, and to supply models 
for the solution of problems by the students. 

2. Large groups of problems are supplied to illustrate each topic, 
and, in addition, sets of miscellaneous problems are given at the close 
of each important chapter, while a review set is placed at the end of 
6ach of the major parts of the book. 

3. Flexibility in the length, of the course is provided for ; the teacher 
can conveniently choose from this book either a one or a two semester, 
three-hour course, on account of the latitude afforded by (a) the large 
number of problems, (6) the segregation of optional methods and 
difficult topics into Supplementary Sections whose omission does not 
break the continuity of the remainder of the book, and (c) the possi-' 
bility of the omission of all of Part II, where life insurance is con- 
sidered. 



ui 



IV PREFACE 

4. The concept of an equation of value is emphasized as a unifying 
principle throughout. 

6. Formulas are simplified and reduced to as small a number as 
the author considers possible, if the classical notation of the subject 
is to be preserved. In Part I, a simplification is effected by the use 
of the interest period instead of the year as a time unit in a final pair 
of formulas for the amount and for the present value of an annuity 
certain. By use of these two formulas, the present values and the 
amounts of most annuities met in practice can be conveniently com- 
puted with the aid of the standard tables. In the applications of 
annuities certain, very few new formulas are introduced. The 
student is called upon to recognize all usual problems involving 
amortization, sinking funds, bonds, etc., as merely different instances 
of a single algebraic problem; that is, the finding of one unknown 
quantity by the solution of one of the fundamental pair of annuity 
formulas. 

6. Interpolation methods are used to a very great extent and then 1 
logical and practical completeness is emphasized. Some problems 
solved by interpolation are treated by other methods, as well, and such 
optional methods are found segregated into Supplementary Sections. 

7. Practical aspects of the subject are emphasized throughout. 

8. Very complete tables are provided, including a five-place table 
of logarithms, the values of the interest and annuity functions for 
twenty-five interest rates, tables of the most essential insurance 
functions, and a table of squares, square roots, and reciprocals. The 
tables may be obtained either bound with the book or bound separately. . 

9. In the discussion of life annuities and life insurance, the em- 
phasis is placed on methods and on principles rather than on manip- 
ulative proficiency. It is aimed to give the student a clear conception 
of the mathematical foundations of the subject. No attempt is made 
to prepare the student as an insurance actuary, but the treatment in 
this book is an excellent introduction to more advanced courses in 
actuarial science. 

The interest and annuity tables prepared in connection with this 
book make possible the solution of most problems accurately to cents, 
if ordinary arithmetic is used. Results can be obtained with sufficient 
accuracy for most class purposes if the five-place table of logarithms 
is used in the computations. If the teacher considers it desirable to 
use seven-place logarithms, the author recommends the use of Glover's 
Tables from Applied Mathematics. These excellent tables contain the 



PREFACE V 

values, and the seven-place logarithms of the values, of the interest and 
annuity functions, a standard seven-place table of 'logarithms, and a 
variety of other useful tables dealing with insurance and statistics. 

The author acknowledges his indebtedness to Professor James 
Glover for his permission to publish in the tables of this book certain 
extracts from Glover's Tables which were published, for the first time, 
in that book. 

UNIVERSITY OP MINNESOTA, 
January 1, 1924. 



CONTENTS 

PART I ANNUITIES CERTAIN 

CHAPTER I 
SIMPLE INTEREST AND SIMPLE DISCOUNT 

IEOTION ' PAQH 

1. Definition of interest 1 

2. Simple interest 1 

3. Ordinary and exact interest 3 

4. Algebraic problems in simple interest 5 

5. Simple discount 6 

6. Banking use of simple discount ...... 9 

7. Discounting notes 10 

CHAPTER II 
COMPOUND INTEREST 



8. 


Definition of compound interest . .. . * . 


. 14 


9. 


Compound interest formula 


. 15 


10. 


Nominal and effective rates 


. 18 


11. 


Interest for parts of a period 


. 20 


12. 


Graphical comparison of simple and compound interest . 


. 23* 


13, 


Values of obligations 


. 24 


14. 


Equations of value 


. 27 


15. 


Interpolation of methods 


. 29 


16.* 


Logarithmic methods . . , 


. 32 


17.* 


Equated time ...-. 


. 33 


18.* 


Interest, converted continuously 


. 35 




Miscellaneous problems 


. 36 




* Supplementary section. 





vu 



viii CONTENTS 

CHAPTER III 
ANNUITIES CERTAIN 

SECTION FAGH 

19. Definitions 39 

20. Special cases 41 

21. Formulas in the most simple case 42 

22. Annuities paid p times per interest period .... 45 

23. Most general formulas 49 

24. Summary of annuity formulas .50 

25. Annuities due 56 

26. Deferred annuities 59 

27.* Continuous annuities , .62 

28.* Computations of high accuracy 63 

Miscellaneous problems 64 

CHAPTER IV 

PROBLEMS IN ANNUITIES 

.29. Outline of problems 67 

30. Determination of payment 67 

31. Determination of term 70 

32. Determination of interest rate 71 

33.* Difficult cases and exact methods 74 

Miscellaneous problems 76 

CHAPTER V 

PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 

34. Amortization of a debt 78 

35. Amortization of a bonded debt 81 

36. Problems where the payment is known 83 

37. Sinking fund method 85 

38. Comparison of the amortization and the sinking fund methods 88 
Miscellaneous problems 89 

39.* Funds invested with building and loan associations . . 92 

40.* Retirement of loans made by building and loan associations 94 

* Supplementary section. 



CONTENTS k 

CHAPTER VI 
DEPRECIATION, PERPETUITIES, AND CAPITALIZED COST 



PAOH 



41. Depreciation, sinking fund plan ...... 96 

42. Straight line method ........ 98 

43. Composite life .......... 99 

44. Valuation of a mine ........ 101 

45. Perpetuities .......... 103 

46. Capitalized cost ......... 105 

47.* Difficult cases under perpetuities .' ..... 108 

48.* Constant percentage method of depreciation .... 109 

Miscellaneous problems ....... Ill 

CHAPTER Vn 

BONDS 

49. Terminology .......... 113 

50. Meaning of the investment rate ...... 113 

51. Purchase price of a bond at a given yield .... 114 

52. Changes in book value ........ 117 

53. Price at a given yield between interest dates .... 121 

54. Professional practices in bond transactions .... 124 

55. Approximate bond yields ....... 126 

66. Yield on a dividend date by interpolation .... 128 

57. Special types of bonds ........ 130 

58.* Yield of a bond bought between interest dates . . . .131 

Miscellaneous problems ........ 133 

Review problems on Part I ....... 135 

PART II LIFE INSURANCE 
CHAPTER 



LIFE ANNUITIES 

59. Probability .......... 147 

60. Mortality Table ......... 148 

61. Formulas for probabilities of living and dying ... 150 

* Supplementary section. 



X CONTENTS 

SECTION PAGE 

62. Mathematical expectation ....... 152 

63. Present value of pure endowment 153 

64. Whole life annuity 155 

65. Commutation symbols 157 

66. Temporary and deferred life annuities 159 

67. Annuities due 162 

Miscellaneous problems 163 

CHAPTER IX 

LIFE INSURANCE 

68. Terminology 165 

69. Net single premium, whole life insurance .... 166 

70. Term insurance 168 

71. Endowment insurance 170 

72. Annual premiums .171 

73.* Net single premiums as present values of expectations . . 175 

74.* Policies of irregular type 176 

CHAPTER X 

POLICY RESERVES 

75. Policy reserves 178 

76. Computation of the reserve 180 

Supplementary Exercise 183 

Miscellaneous Problems on Insurance 184 

PART III AUXILIARY SUBJECTS 
CHAPTER XI 

LOGARITHMS 

77. Definition of logarithms ... ..... 187 

78. Properties of logarithms .188 

79. Common logarithms 190 

80. Properties of the mantissa and the characteristic , . .191 

81. Use of tables of mantissas 193 

82. Logarithms of numbers with five significant digits , 194 

* Supplementary section. 



CONTENTS xi 

SECTION . PAGH 

83. To find the number when the logarithm is given . . . 196 

84. Computation of products and of quotients .... 197 
86. Computation of powers and of roots 198 

86. Problems in computation 200 

87. Exponential equations 201 

88.* Logarithms to bases different from 10 203 

CHAPTER XII 
PROGRESSIONS 

89. Arithmetical progressions 204 

90. Geometrical progressions 205 

91. Infinite geometrical progressions 207 

APPENDIX 

Note 1. Approximate determination of the time to double money 

at compound interest 211 

Note 2. Approximate determination of the equated time . . 211 

Note 3. Solution of equations by interpolation . ". . . 212 

Note 4. Abridged multiplication 213 

Note 5. Accuracy of interpolation for finding the time, under com- 
pound interest 214 

Note 6. Accuracy of interpolation for finding the term of an annuity 214 

INDEX 217 

* Supplementary section. 



MATHEMATICS OF INVESTMENT 

PART I ANNUITIES CERTAIN 

CHAPTER I 
SIMPLE INTEREST AND SIMPLE DISCOUNT 

1. Definition of interest. Interest is the income received 
from invested capital. The invested capital is called the prin- 
cipal ; at any time after the investment of the principal, the sum 
of the principal plus the interest due is called the amount. The 
interest charge is usually stated as a rate per cent of the principal 
per year. If $P is the principal, r the rate expressed as a decimal, 
and i the interest for 1 year, then by definition i = Pr, or 

-i en 



Thus, if $1000 earns $36.60 interest in one year, r - ^^ = .0366, or the 

1000 

rate is 3.66%. Also, if P = $1 in equation 1, then r = i, or the rate r equals 
the interest on $1 for 1 year. 

2. Simple interest. If interest is computed on the original 
principal during the whole life of a transaction, simple interest 
is being charged. The simple interest on a principal P is propor- 
tional to the time P is invested. Thus, if the simple interest for 
1 year is $1000, the interest for 5.7 years is $5700, 

Let / be the interest earned by P in t years, and let A be the 
amount due at the end of t years ; since amount = principal plus 
interest, A = P + I. (2) 

Since the interest earned by P in 1 year is Pr, the interest earned 
in t years is <(Pr) or j _ p r f (3) 

Hence P + I = P + Prt, so that, from equation 2, 

I 



2 MATHEMATICS OF INVESTMENT 

It is important to realize that equation 4 relates two sums, P 
and A, which are equally desirable if money can be invested at, 
or is worth, the simple interest rate r. The possession of P at 
any instant is as desirable as the possession of A at a time t years 
later, because if P is invested at the rate r, it will grow to the 
amount A in t years. We shall call P the present value or pres- 
ent worth of the amount A, due at the end of t years. 

3. Ordinary and exact interest. Simple interest is computed 
by equation 3, where t is the time in years. If the time is given 
in days, there are two varieties of interest used, ordinary and exact 
simple interest. In computing ordinary interest we assume one 
year to have 360 days, while for exact interest we assume 365 days. 

Example 1. Find the ordinary and the exact interest at 5% on 
$5000 for 59 days. 

Solution. For the ordinary interest I use equation 3 with f = ^, and 
for the exact interest I e use t = -gfc. 

I - 5000(.Q5) jfc = 840.97; I, = 5000(.05) Jft = $40.41. 

A relation exists between the ordinary interest I a and the exact 
interest I e on a principal P for D days at the rate r. 



From equations 5 it is seen that PrD = 360 I = 365 I e or 

Io _ 73 ,n\ 

J.-T? (6) 

which shows that ordinary interest is greater than exact interest. 
From equation 6, *, T 



'-^'-'-w- (8) 

Thus, if the exact interest 7. =* $40.41, we obtain from equation 7, 
/= 40.41 +^=$40.97. 



SIMPLE INTEREST AND SIMPLE DISCOUNT 



3 



EXERCISE I 

la the first five problems find the interest by use of equation 3. 



PBOB. 


PRINCIPAL 


RATH 


Turn 


INTHBBST 


1. 


$ 6,570. 


3.5% 


75 days 


exact 


2. 


8,000. 


.045 


93 days 


ordinary 


3. 


115,380. 


.0626 


80 days 


exact 


4. 


4,838.70 


7.5% 


35 days 


ordinary 


5. 


2,500. 


.055 


27 days 


exact 



s ' 6. The exact interest on a certain principal for a certain number 
of days is $60.45. Find the ordinary interest for the same period 
of time. 

7. The ordinary interest on a certain principal for a certain number 
of days ia $35.67. Find the exact interest for the same period.- 

8. In problem 1 find the ordinary interest by use of the result of 
problem 1. 

When the rate is 6%, the ordinary interest for 60 days is (.01)P, 
which is obtained by moving the decimal point in P two places 
to the left, while the interest for 6 days is (.001) P. These facts 
are the basis of the 6% method for computing ordinary interest 
at 6% or at rates conveniently related to 6%. 

'Example 2. By the 6% method find the ordinary interest on 
$1389.20 for 83 days at 6%, and at 4.5%. 
Solution. $13.892 is interest at 6% for 60 days 

4.168 ia interest at 6% for 18 days (3 times 6 days) 
1.158 is interest at 6% for 5 days (^ of 60 days) 
$19.218 is interest at 6% for 83 days ' 

4.805 is interest for 83 days at 1.5% (i of 6%) 
$14.413 is interest for 83 days at 4.5% 

- The extensive interest tables used in banks make it unnecessary 
to perform multiplications or divisions in computing simple in- 
terest. Table IV in this book makes it unnecessary to perform 
divisions. 

Example 3. Find the exact interest at 5% on $8578 for 96 days. 

Solution. From Table IV the interest at 1% on $10,000 for 96 days is 
$26.3013699. The interest on $8578 is (.8578) (5) (26.3014) - $112.81. 



4 MATHEMATICS OF INVESTMENT 

To find the time between two dates, it is sometimes assumed 
that each month has 30 days. For example* 

February 23, 1922, is 1922 : 2 : 23 or 1921 : 14 : 23 (9) 

June 3, 1921, is 1921: 6: 3 

Elapsed time - : 8 : 20 - 260 days 

The exact time can be found from Table III. February 23, 1922, 
is the 54th day of 1922 or the 419th day from January 1, 1921. 
June 3, 1921, is the 154th day from January 1, 1921. The 
elapsed time is (419 154) = 265 days. 

NOTE. In this book, for the sake of uniformity, proceed as follows, unless 
otherwise directed, in problems involving simple interest : (a) use ordinary 
interest if the time interval is given in days ; (b) in computing the number 
of days between dates, find the exact number of days; (c) if the time is 
given in months, reduce it to a fraction of a year on the basis of 12 months 
to the year, without changing to days. Methods in the business world are 
lacking in uniformity in these respects, and, in any practical application, ex- 
plicit information should be obtained as to the procedure to be followed. 

EXERCISE n 

Find the ordinary interest in the first five problems by use of the 
6% method. 

1. P = $3957.50, t = 170 days, r - .06. 

2. P = $3957.50, t = 170 days, r = .07. 

3. P = $4893.75, t = 63 days, r - .04. 

4. P = $13,468.60, t = 41 days, r = .03. 
6. P = $9836.80, t = 134 days, r = .05. 

6. Find the exact interest in problem 4 by use of Table IV. 

7. Find the ordinary interest on $8500 at 6% from August 11, 
1921, to March 13, 1922. Use the approximate number of days, as 
in expression 9 above. 

8. Find the ordinary interest in problem 7, but use the exact num- 
ber of days. 

9. Find the ordinary and exact interest on $1750 at 5% from April 
3, 1921, to October 13, 1921, using the exact number of days. 

10. (a) Find the ordinary and exact interest in problem 9, using 
the approximate number of days. (&) Which of the four methods 
of problems 9 and 10 is the most favorable to a creditor? 



SIMPLE INTEREST AND SIMPLE DISCOUNT 5 

4. Algebraic problems. If a sufficient number of the quantities 
(A, P, I, r, f) are given, the others can be determined by equations 
2, 3, and 4, When the rate r, or the time t, is unknown, equation 
3 should be used. When the present value P is unknown, equa- 
tion 4 is most useful. 

Example 1. If a $1000 principal increases to $1250- when in- 
vested at simple interest for 3 years, what is the interest rate ? 

Solution. P= $1000, A = $1250, t = 3. The interest 7 - $260. 
From 7 = Prt, 250 - 3000 r, or r = .0833. 

Example 2. What principal invested at 5.5% simple interest will 
amount to $1150 after 2 years, 6 months? 

Solution. Use equation 4. 1150 = P[l +2.5(.055)] = 1.1375 P; P 

_ 1150 _ J1010.99. An equivalent statement of Example 2 would be, 

1.1375 

" Find the present value of $1150, due at the end of 2 years, 6 months, if 
money is worth 5.5% simple interest." 



EXERCISE DI 

Find the missing quantities in the table below : 



PROS. 


A 


P 


/ 


RATE 


TIME 


1. 




$ 750. 




.04 


3 yr., 6 mo. 


2. 


$3500. 






.055 


2yr. 


3. 




3500. 




.058 


2yr. 


4. 






$150 


.075 


6 mo. 


6. 


2500. 






.035 


2 yr., 3 mo. 


6. 


1200. 






.06 


2 yr., 6 mo. 


7. 




1800.60 


300 


.055 




8. 






650 


.03 


3 yr;, 9 mo. 


9. 




1680. 




.0375 


11 mo. 


10. 


9850.50 






.0725 


1 yr., 6 mo. 



11. Find the present value of $6000, due after 8 months, if money 
is worth 9%. 

12. W borrowed $360 from B and agreed to repay it at the end of 
8 months, with simple interest at the rate 5.25%. What did W pay 
at the end of 8 months? 



6 MATHEMATICS OF INVESTMENT 

13. Find the present worth of $1350, due at the end of 2 months, 
if money is worth 5% simple interest. 

^ U. At the end of 3 months I must pay $1800 to B. To cancel 
this obligation immediately, what should I pay B if he is willing to 
accept payment and is able to reinvest money at 6% simple interest? 

Examples. A merchant is offered a $50 discount for cash payment. 
of a $1200 bill due after 60 days. If he pays cash, at what rate may he 
consider his money to be earning interest for the next 60 days? 

Solution. He would pay $1150 now in place of $1200 at the end of 60 days. 
To find 'the interest rate under which $1150 is the present value of $1200, due 



in 60 days, use I = Prf; 60 = , or r = .26087. Hia money earns in- 

6 

terest at the rate 26.087% ; he could afford to borrow at any smaller rate in 
order to be able to pay cash. 

15. A merchant is offered a $30 discount for cash payment of a 
$1000 bill due at the end of 30 days. What is the largest rate at 
which he could afford to borrow money in order to pay cash? 

16. A 3% discount is offered for cash payment of a $2500 bill, due at 
the end of 90 days. At what rate is interest earned over the 90 days if 
cash payment is made ? 

17. The terms of payment for a certain debt are : net cash in 90 days or 
2% discount for cash in 30 days. At what rate is interest earned if the 
discount is taken advantage of? 

HINT. For a $100 bill, $100 paid after 90 days, or $98 at the end of 30 
days, would cancel the debt. 

5. Simple discount. The process of finding the present value 
P of an amount A, due at the end of t years, is called discounting 
A. The difference between A and its present value P, or A P, 
is called the discount on A* From A = P + I, I A P; 
thus I, which is the interest on P, also is the discount on A, If 
$1150 is the discounted value of $1250, due at the end of 7 months, 
the discount on the $1250 is $100 ; the interest on $1150 for 7 
months is $100. 

In considering I as the interest on a known principal P we com- 
puted I at a certain rate per cent of P, In considering 7 as the 
discount on a known amount A, it is convenient to compute / 
at a certain rate per cent per, year, of A, If i is the discount on 




SIMPLE INTEREST AND SIMPLE DISCOUNT 7 

A, due at the end of 1 year, and if djs_thajiiscoiint-,rate,expreflafid^ 
as ajdecimal, then by (definition i = Ad, or 



Simple discount, like simple interest, is discount which is pro- 
portional to the time. If Ad is the discount on A, due in 1 year, 
the discount on an amount A due at the end of t years is t(Ad), or 

I=Adt. (11) 

From P = A- I, P = A- Adt, or 

P = A(l - df). (12) 

If the time is given in days, we may use either exact or ordinary 
simple discount, according as we use one year as equal to 365 or 
to 360 days, in finding the value of t. 

NOTB. In simple discount problems in this book, for the sake of uni- 
formity, proceed as follows unless otherwise directed : (a) if the time is given 
.orrlinnrjr dincouatj (ft) in computing the number of days between 



dates, findjthe^ exact number of dasa ; (c) if the time is given in months, 
duce'it to a fraction ojLa year^on the jgasisjDf J2^LQnEKB~to~tn"e yea Business 
practices are not uniform, and hence in any practical application of discount 
one should obtain explicit information as to the procedure to follow. 

Example 1 . Find the discount rate if $340 is the present value of $350, 
due after 60 days. 



' Solution. Prom I - Adt, 10 = ; d = .17143, or 17.143%. 

6 

Example 2. If the discount rate is 6%, find the present value of $300, 
due at the end of 3 months. 

Solution. From /. = Adt, I = 300 (- 06 ) = 4.60. P = A - I - 300 
- 4.60 = $295.50. 

NOTE, If A is known and P is unknown, it is easier to find P when the 
discount rate is given than when the interest rate is known. To appreciate 
this fact compare the solution of Example 2 above 'with that of Example 2 of . 
Section 4, where a quotient had to be commuted. This simplifying property of 
discount rates is responsible for their wide use in banking and 'business. 

The use of a discount rate in finding the present value P of a 
known amount A is equivalent to the use of some interest rate, 
which is always different from the discount rate. 



8 



MATHEMATICS OF INVESTMENT 



Example 3. (a) If a 6% discount rate is charged in discounting 
amounts due after one year, what equivalent interest rate could be used? 
(6) What would be the interest rate if amounts due after 3 months were 
being discounted? 

Solution. (a) Suppose A = $100, due after 1 year. Then / = 100(.06) 
= $6,. and P = 594. Let r be the equivalent interest rate, and use I = Prt. 

6 = 94r; r -.063830. (&) If A 

98.60 r . 



$100, due after 3 months, I = 1QO (- 06 ) 

4 



$1.60, and P = 898.60. From J = Prt, 1.50 



.060914. 



Compare the results of Example 3. When a discount rate is 
being used, the equivalent interest rate is larger for long-term than 
for short-term transactions and in both cases is larger than the 
discount rate. 

The brief methods available for computing simple interest apply 
as well to the computation of simple discount because both opera- 
tions involve multiplication by a small decimal. Thus, we may 
use the 6% method for computing discount, and simple interest 
tables may be used as simple discount tables. 

EXERCISE IV 

1. Find the discount rate if the discount on $1500, due after one year, 
is $82.50. 

2. Find the discount rate if the present value of $1250, due after 
8 months, is $1193.75. 

Find the missing quantities in the table by use of equations 11 and 12. 



PROS. 


DlSOOTFNTBD 

V A.IJ-CIH, P 


AMOUNT, A 


A IB DUB AJTBE 


/, DISCOUNT ON 
A OB INT. ON P 


DISCOUNT 
RATE, d 


3. 




$1200 


lyr. 




.05 


4. 


$145.50 


150 


6mb. 






6. 






3 mo. 


$250 


.07 


6. 




' 2000 


90 da. 




.045 


7. 




. 800 


120 da. 




.06 


8. 


357.75 


375 


9 mo. 






9. 


s * 




5 mo. 


300 


.08 


10. 


750. 




72 da. 




.0625 


11. 




- 1500. 


6 mo. 


35 




12. 




7500 




100 


.06 



SIMPLE INTEREST AND SIMPLE DISCOUNT 9 

13. Write in words problems equivalent to problems 3, 4, 10, and 12 
above. 

14. (a) What simple interest rate would be equivalent to the charge 
made in problem 3 ; (&) in problem 7? 

15. If d = .045, (a) what is the equivalent interest rate for a 1-year 
transaction ; (6) for one whose term is 4 months? 

r / 16. What discount rate would be equivalent to the use of a 6% interest 
rate in a 1-year transaction? HINT. Let P = $100; find A and I, and 
use I = Adt. , 

17. What discount rate would be equivalent to the use of a 6% interest 
rate in a 6-month transaction? 

STTPPLEMENTABY NOTE. Formulas carx be obtained relating the discount 
rate d and the equivalent interest rate r on 1-year transactions. Suppose 
that $1 is due at the end of 1 year. Then, in equations 11 and 12, A = $1, 
I - d, and P = 1 - d. From I = Prt with t = 1, d - (1 d)r, or 

' "rV as) 

From equation 13, r rd *= d, or r = d(l + r), so that 

d= iT7 . a*) 

It must be remembered that equations 13 and 14 connect the discount and the 
interest rates only in the case of 1-year transactions. 

SUPPLEMENTARY EXERCISE V 

1. By use of equations 13 and 14, solve problems 14, 15, 16, and 17 of . 
Exercise IV. 

6. Banking use of simple discount in lending money. If X 
asks for a $1000 loan for six months from a bank B which is charg- 
ing 6% discount, B will cause X to sign a note promising to pay 
B $1000 at the end of 6 months. Then, B will give X the present 
value of the $1000 which he has promised to pay. The bank 
computes this present value by use of its discount rate. 
P = 1000 30 = $970, which X receives. The transaction is - . 
equivalent to B lending X $970 for 6 months. The interest rate 
which X is paying is that which is equivalent to the 6% discount ; 
rate. The banker would tell X that he is paying 6% interest in \ I 
advance, but this would merely be a colloquial manner "of stating - V 
that the discount rate is 6%< In this book the phrase interest in 
advance is always used in this customary colloquial sense. 



10 



MATHEMATICS OF INVESTMENT 



Example 1. X requests a loan of $9000 for 3 months from a bank B 
charging 5% discount. Find the immediate proceeds of the loan and the 
interest rate which X is paying. 

Solution. X promises to pay $9000 at the end of 3 months. Discount on 
$9000 for 3 mouths at 5% is $112.50. Immediate proceeds, which X receives, 
are $9000 - $112.50 = $8887.50. To determine the interest rate, use I = Prt. 
112.6 = 8887.6 r(*), or r = .060833. 

Example 2. X wishes to receive $9000 as the immediate proceeds of a 
90-day loan from a bank B which is charging 5% interest payable in ad- 
vance. For what sum will X draw the note which he will give to B? 

Solution. P = $9000, t = 90 days, d = .05, and A is unknown. From 
equation 12, 9000 = A(l - .0126). A = $9113.92, for which X will draw 
the note. 

EXERCISE VI 

Determine how much X receives from the .bank B. In the first three 
problems, also determine the interest rate which X is paying. 



PBOB. 


LOAN REQUESTED BY X 


FOB 


DISCOUNT RATH OF B 


vl. 


$5,000 


6 months 


.065 


-A 2. 


1,760 


75 days 


.07' 


3. 


3,570 


90 days 


.06 


4. 


190 


30 days 


.05 


5. 


7,500 


45 days 


.055 


6. 


3,800 


3 months 


.0625 



Determine the size of the loan which X would request from B if X 
desired the immediate proceeds given in the table. 



PBOB. 


IMMEDIATE PHOOBHDB 


THBM OF LOAN 


DISCOUNT RATH OF B 


7. 


$ 3,500 


30 days 


.06 


8. 


8,000 


4 months 


.05 


9. 


1,300 


3 months 


.07 


10. 


150,000 


60 days 


.055 


11. 


4,300 


90 days 


.045 


12. 


9,350 


6 months 


.05 



7. Discounting notes. The discounting of promissory notes 
gives rise to problems similar to those of Exercise VI. .Consider 
the following notes : <, / 



oiij-iiiiliiiiii 



SIMPLE INTEREST AND SIMPLE DISCOUNT 11 

NOTE (a) 



Minneapolis, June 1, 
Six months after date I promise to pay to Y or order $5000 
without interest. Value received. Signed X. 



NOTE (6) 



Chicago, June 1, 1922. 

One hundred and eighty days after date I promise to pay to Y or 
order $5000 together with simple interest from date at the rate 7%. 
Value received. Signed X. 



On August I, Y sells note (a) to a bank B. The sale is ac- 
complished by Y indorsing the note, transferring his rights to B, 
who -will receive the $5000 on the maturity date. The transaction 
is called discounting the note because B pays Y the present of 
discounted value of $5000, due on December 1, 1922. 

Example 1. If B discounts notes at 5%, what will Y receive on 
August 1? 

Solution. B is using the discount rate 5% in computing present values. 
The discount on $5000, due after 4 months, is $83.33 ; B will pay Y $4916.67. 

Example 2. On July 31, Y discounts note (6) at the bank B of Ex- 
ample 1. What are Y's proceeds from the sale of the note? . 

Solution. B first computes the maturity value of the note, or what X will 
pay on the maturity date, which is November 28. The transaction is equiva- 
lent to discounting this maturity value. Term of the discount is 120 days 
(July 31 to November 28). From equation 4, the maturity value of the note 
is 5000(1 + .036) = $5175.00. Discount for 120 days at 5% on $5175 is 
$86.25. Proceeds = $5175 - $86.25 = $5088.75. 

EXERCISE VH 
1. X paid Y for an order of goods with the following note : 



Chicago, June 1, 
Sixty days after date I promise to pay to Y or order $375 at the 
Continental Trust Company. Value received. Signed X. 



12 



MATHEMATICS OF INVESTMENT 



Thirty days later, Y discounted this note at a bank charging 5.5% 
discount. Find Y's proceeds from the sale of the note. 

2. Find the proceeds in problem 1, if the discount rate is 6%. 

3. The bank B of problem 1, after buying the note from Y, immediately 
rediscounted it at a Federal Reserve Bank : whose rediscount rate, was 
.035. What did B receive for the note? 

4. The holder of a non-interest-bearing note dated October. 1, 1911, 
payable 4 months, after date, discounted it at a bank on October 1, at the 
rate 4%. The bank's discount on the note was $20. What was the face 
of the note? 

5. X owes a firm Y $800, due immediately. In payment X draws a 90- 
day non-interest-bearing note for such a sum that, if Y immediately dis- 
counts it at a bank charging 6% discount, the proceeds will be $800. What 
is the face value of the note? 

HINT. In equation 12, P ** $800 and A is unknown. 

6. X draws a 60-day non-interest-bearing note in payment of a bill for 
$875, due now. What should be the face of the note so that the immediate 
proceeds to the creditor will be $875 if he discounts it immediately at a 
bank whose discount rate is 6.5%? 

Find the proceeds from the sale of the following notes : 



PBOD. 


FA.OD or 
NOTH 


DATE OF 
Norn 


THHM 


NOTHBlABfl 

INT. At 


SOLD OK 


Disc. RAfan 
. OF BUYER 


7. 


$ 450 


6/10/17 


30 days 


.06 


6/20/17 


.07 


8. 


1200 


6/12/18 ' 


120 days 


.05 


6/26/18 


, .06 


9. 


. 376 


3/25/20 


90 days 


.07 


4/24/20 


.04 


10. 


470 


11/20/21 


60 days 


.00 


12/ 5/21 


.065 


11." 


325 


4/30/20 


3 months 


.06 


6/ 1/20 


.08 


12." 


3000 


8/14/19 


6 months 


.08 


12/16/19 


.06 



1 A Federal Reserve Bank discounts commercial notes brought to it by banks 
belonging to the Federal Reserve System. The rate of the Federal Reserve Bank is 
called a rediscount rate because all notes discounted by it have been discounted 
previously by some other bank. This previous discounting has no effect so far as 
the computation of the present value by the Federal Reserve Bank is concerned. 

3 The note is due on 7/31/20, the last day of the third month from April. Find 
the exact number of days between 6/1/20 and 7/31/20. 

1 Find the exaot number of days between 12/16/19 and 2/14/20. 



SIMPLE INTEREST AND SIMPLE DISCOUNT 13 

J 

13. Y owes W $6000 due now. In payment Y draws a 45-day non- 
interest-bearing note, which W discounts immediately at a bank charging 
6% interest in advance. What is the face of the note if W's proceeds are 
$5000? 

14. W desires $2500 as the immediate proceeds of a 6-month loan from 
a bank which charges 7% interest in advance. What loan .will W re- 
quest? 

15. A bank B used the rate 6% in discounting a 90-day note for $1000. 
The note was immediately rediscounted by B at a Federal Reserve Bank 
whose rate was 4%. Find B's profit on the transaction. 



CHAPTER II 
COMPOUND INTEREST 

8. Definition of compound interest. If, at stated intervals 
during the term of an investment, the interest due is added to 
the principal and thereafter earns interest, the sum by which the 
original principal has increased by the end of the term of the in- 
vestment is called compound interest. At the end of the term, 
the total amount due, which consists of the original principal plus 
the compound interest, is called the compound amount. 

We speak of interest being compounded, or payable, or con- 
verted into principal. The time between successive conversions 
of interest into principal is called the conversion period. In a 
compound interest transaction we must know (a) the conversion 
period and (&) the rate at which interest is earned during a con- 
version period. Thus, if the rate is 6%, compounded quarterly, 
the conversion period is 3 months and interest is earned at the 
rate 6% per year during each period, or at the rate 1.5% per con- 
version period. 

Example 1. Find the compound amount after 1 year if $100 is in- 
vested at the rate 8%, compounded quarterly. 

Solution. The rate per conversion period is ,02. Original principal is $100. 

At end of 3 mo. $2.000 interest is due j new principal is $102.000. 

At end of 6 nxo. $2.040 interest is due ; new principal is $104.040. 

At end of 9 mo. $2.081 interest is due ; new principal is $106.121. 

At end of 1 yr, $2.122 interest is due; new principal is $108.243. 

The compound interest earned in 1 year is $8.243. The rate of increase of 

O 0/tD 

principal per year is ^J22. = .08243, or 8.243%. 

EXERCISE Vm 

1. By the method of Example 1 find the compound amount after 1 year 
if $100 is invested at the rate 6%, payable quarterly. What was the 
compound amount after 6 months? At what rate per year does principal 
increase in this case? 

14 



COMPOUND INTEREST 15 

2. Find the annual rate of growth of principal under the rate .04, con- 
verted quarterly. 

NOTE. Hereafter, the unqualified word interest will always refer to 
compound interest. If a transaction extends over more than 1 year, compound 
interest should be used. If the tune involved is less than 1 year, simple 
interest generally is used. 

9. The compound interest formula. Let the interest rate 
per conversion period be r, expressed as a decimal. Let P be the 
original principal and let A be the compound amount to which P 
accumulates by the end of k conversion periods. Then,. we shall 
prove that A = P(l + r)*. (15) 

The method of Example 1, Section 8, applies in estabhshing 
equation 15. 

Original principal invested is P. 

Interest due at end of let period is Pr. 

New principal at end of 1st period is P + Pr = P(l + r) . 

Interest due at end of 2d period is P(l + r)r. 

New principal at end of 2d period is P(l + r) + P(l + r)r = P(l + r) 2 . 

By the end of each period, the principal on hand at the beginning 
of the period has been multiplied by (1 + r). Hence, by the end 
of k periods, the original principal P has been multiplied k suc- 
cessive times by (1 + r) or by (1 + r)*. Therefore, the compound 
amount after k periods is P(l + r)*. 

If money can be invested at the rate r per period, the sums 
P .and A } connected by equation 15, are equally desirable, be- 
cause if P is invested now it will grow to the value A by the end 
of k periods. We shall call P the present value of A, due at the 
end of k periods. 

The fundamental problems under compound interest are the 
following : 

(a) The accumulation problem, or the determination of the 
amount A when we know the principal P, the interest rate, and the 
time for which P is invested. To accumulate P, means to find 
the compound amount A resulting from the investment of P. 

(6) The discount problem, or the determination of the present 
value P of a known amount A, when we know the interest rate and 



16 MATHEMATICS OF INVESTMENT 

the date on which A is due. To discount A means to find its 
present value P. The discount on A is (A P). 
The accumulation problem is solved by equation 15. 

Exampk 1. Find the compound amount after 9 years and 3 months 
on a principal P = $3000, if the rate is 6%, compounded quarterly. 

Solution. The rate per period is r = .015 ; the number of periods is 
fc = 4(9.25) = 37. 

A = 3000(1.016)" = 3000(1.73477663) = $5204.33. (Using Table V) 
The compound interest earned is $6204.33 - $3000 - $2204.33. 

NOTE. If interest is converted m times per year, find the number fc of 
conversion periods in n years from the equation fc = mn. 

To solve the discount problem we first solve equation 15 for P, ob- 
taining A 

p - - = A 



Exampk 2. Find the present value of $5000, due at the end of 4 years 
and 6 months, if money earns 4%, converted semi-annually. 

Solution. Rate per period is r = .02; number of periods is fc = 2(4.5) = 9. 
P >= 6000(1.02)-' = 5000083675527) - $4183.78. (Using Table VI) 
The discount on A is $5000 $4183.78 = $816.22. 

NOTE. Recognize that Example 2 involves the formation of a product 
when solved by Table VI. A problem is solved incorrectly if available tables 
are not used to simplify the work. Since products are easier to compute than 
quotients, the following solution of Example 2 should be considered incorrect, 
although mathematically flawless, because a quotient is computed. 

F ~ 

In describing interest rates in the future, a standard abbrevia- 
tion will be used. When we state the rate to be (.05, m = 2), 
the " m = 2" signifies that interest is compounded twice per year, 
or semi-annually. The rate (.08, m = 1) means 8%, compounded 
annually; (.07, m = 12) means 7%, converted monthly; (.06, 
m = 4) means 6%, compounded quarterly. 

NOTE. The quantity (1 + r) in equation 16 is sometimes called the 
accumulation factor, while the quantity - or (1 + r)" 1 in equation 16 is 

called the discount factor. In many books the letter v is used to denote the 
discount factor, or v = (1 + r)" 1 . Thus, at the rate 7%, u 4 = (1.07)" 4 , 



COMPOUND INTEREST 
EXERCISE IX 



17 



1. By use of the binomial theorem verify all digits of the entry for 
(1.02) 4 in Table V. 

2. In Table VI verify all digits of the entry for (1.02)- 6 by using the 
entry for (1.02) in Table V and by completing the long division involved. 



3. Find the compound amount on $3,000,000 after 16 years and 
3 months, if the rate is (.06, m = 4). 

4. Accumulate a $40,000 principal for 15 years under the rate (.05, 
m = 4). What compound interest is earned? 

6. Find the present value of $6000, due after 4J years, if money can 
earn interest at the rate (.08, m = 4). What is. the discount on the 
$6000? 

6. Discount $5000 for 19 years and 6 months, at the rate (.05, m = 2). 

In the table below, find that quantity, P or A, which is not given. In 
the first four problems, before doing the numerical work, write equivalent 
problems in words. 



PROS. 


PRINCIPAL, 
P 


AMOUNT, 
A 


P ACCUMULATES FOB, 
OR A IB DUB APTHB 


RATE 


7. 




$4000 


6 yr., 6 mo. 


.04, m = 2 


8. 


$1000. 




10 yr., 3 mo. 


.07, m = 4 


9. 


3000. 




12 yr. 


.06, m = 1 


10. 




6000 


Syr.,. 6 mo. 


.03, 77i = 4 


11. 


2600. 




13 yr., 9 mo. 


.08, m = 4 


12. 




1600 


7 yr., 6 mo. 


.06, 77i = 2 


13. 


576.60 




3 yr., 6 mo. 


.06, m - 12 


14. 


1398.60 




16 yr., 3 mo. 


.05, m = 4 


15. 




8300 


14 yr., 6 mo. 


;056, 771 = 2 


16. 




0500 


5yr. 


.045, TO = 1 


17. 


1300. 




2 yr., 9 mo. 


.03, m = 4 


18. 


1. 




76 yr. 


.05, m = 1 


19. 




100 


100 yr. 


.035, m = 1 


20. 1 


100. 




176 yr. 


.045, m = 1 


21. 1 




100 


173 yr. 


.065, m = 1 


22. 




1 


30 int. periods 


.04, per period 


23. 


1. 




36 int. periods 


.06, per period 



1 In problem 20 use (1.046) 1 " - (1.045) 1 (1.045). In. problem 21 use 



18 MATHEMATICS OF INVESTMENT 

24. (a) If the rate is i, compounded annually, and if the original 
principal is P, derive the formula for the compound amount after 10 years. 
(6) After n years. 

26. If $100 had been invested in the year 1800 A.D. at the rate 
(.03, m = 1), what would be the compound amount now? 

26. (a) If the rate is j, compounded m times per year, derive a formula 
for the compound amount of a principal P after 10 years. (&) After n 
years. _,i(, _ .._-;-' . , ? , - 

10. Nominal and effective rates. Under a given type of 
compound interest _the rate per year at which principal grows is 
called the effective rate. The per cent quoted in stating a type 
of compound interest is called the nominal rate ; it is the rate per '/ 
year at which money earns interest during a conversion period. * 
In the illustrative Example 1 of Section 8 it was seen that, when 
the_"nommdl rg,te.,was 8%, converted quarterly, the effective rate 
was 8.24%. We shall say " tiitTrate is (j, m) " to abbreviate " the 
nominal rate is j, converted m times per year." Let i represent 
the effective rate. 

The effective rate i corresponding to the nominal rate j, converted 
m times per year, satisfies the equation 

+ ff> ( &T) 

To prove this, consider investing P = $1 for 1 year at the rate 

(j, m). The rate per period is -^ and the number of periods in 

m 

1 year is m. The amount A after 1 year and the interest I earned 
in that time are 



m 

The rate of increase of principal per year is i = = /, because 
P = $l. Hence 



i 
J 



Transpose the 1 in equation 18 and equation 17 is obtained. 

Example 1. What is the effective rate corresponding to the rate 
(.06, m- 4)? ,/ 



COMPOUND INTEREST Id 

Solution. Use equation 17. 1 + i = (1.0125) 4 = 1.05094634. 

i = .05094534. 

Example 2. What nominal rate, if converted 4 times per year, will 
. yield the effective rate 6%? 

i Solution. From equation 17, 1.06 = 

1 + i = (1.06)* = 1.01467385, from Table X. (19) 

j = 4(.01467385) = .05869640." 
Table XI furniahes an easier solution. From equation 19, 

I - (1.06)* - 1 ; j = 4[(1.06)* - 1] - .05869538. (Table XI) 

Example 3. What nominal rate, converted quarterly, will give the 
same yield as (.05, m = 2) ? 

Solution. Let j be the unknown nominal rate. The effective rate i cor- 
responding to (.05, m = 2) must equal the effective rate corresponding to the 
nominal rate j, compounded quarterly. From equation 17, 

(1.025)'; 1 + i = (l + 0*. ... (1.025) 2 = (l + {)* 
(1.025)* = 1.01242284. j = 4(.01242284) = .04969136. 



1 + i 
1 + 



EXERCISE X 

1. (a) In Table X verify the entry for (1.05)* by use of Table II. 
(6) In Table XI verify the entry f or p = 4 and i = .05, by using Table X. 

2. Find .the effective rates corresponding to the rates (.06, m 2) 
and (.06, m = 4). 

3. Find the nominal rate which, if converted semi-annually, yields the 
effective rate .05. (a) Solve by Table X. (6) Read the result out of 
Table XI. 

Solve the problems in the table orally by the aid of Tables V and XI. 
State equivalent problems in words. 



PROB. 


i ' 


m 


i- 


FqOB. 


3 


m 


i 


4. 


.07 


2 




10. 




' 12 


.04 


5. 




2 


.07 


11. 


.03 


2 




6. 


.10 


4 




12. 




2 


.0275 


7. 




2 


.035 


13. 




4 


.026 


* 8. 


.09 


3 




14. 


.05 


1 




9. 


.09 


4 




15. 




1 


.06- 



12Q) 



MATHEMATICS OF INVESTMENT 



16. Derive a formula for the nominal rate which, if converted p times 
per year, gives the effective rate i. NOTE. The resulting value of j is 
denoted by the notation j p , as in Table XI. 

17. Which interest rate is the better, 5% compounded monthly or 
6.5% compounded semi-annually? 

, 18. Which rate is the better, (.062, m = 1) or (.06, m = 2) ? 

19. Determine the nominal rate which, if converted semi-annually, ./ 
may be used in place of the rate 5%, compounded quarterly. >f St>& ***S 

20. What nominal rate compounded quarterly could equitably replace 
the rate (.04, m = 2) ? 

NOTE. When interest is compounded annually, the nominal and the 
effective rates are equal because in this case both represent the rate of 
increase of principal per year. This equality is seen also by placing m = 1 
in equation 17. Hence, to say that money is worth the effective rate 5% is 
equivalent to saying that money is worth the nominal rate 5%, compounded 
annually. 

If m, the number of conversion periods per year of. a nomi- 
nal rate, is increased, the corresponding effective rate is also in- 
creased. If j = .06, the effective- rates i for different values of 
m are: 



m = 


1 


2 


4 


12 


52 


365 


i f= 


.06000 


.06090 


.06136 


.06168 


.06180 


.06183 



We could consider interest converted every day or every moment 
or every second, or, as a limiting case, converted continuously 
(m = infinity). The more frequent the conversions, the more 
just is the interest method from the standpoint of a lender, so 
that interest, converted continuously, is theoretically the most 
ideal. The effective rate does not increase enormously as we 
increase the frequency of compounding. When j = .06, in the 
extreme case of continuous conversion (see Section 18, below), 
i = .06184, only slightly in excess of .06183, which is the-effective 
rate when m 365. 

11. Interest for fractional parts of a period. In deriving 
equation 15 we assumed fc to be an integer. Let us agree as a new 



COMPOUND INTEREST 21 

definition that^the compound a^ojjuil J <l.shaILbe_given Jby equation^ 
15 also wEen k is not an integer. 

/' Exampk 1. Accumulate $1000 for 2 years and 2 months, at the rate 
'(.08, in = 4). ,- - ^ 

Solution. The rate per period is r = .02, and k =, 4(2J) = 8f . T 

A = 1000(1.02)'* = 1000(1.02)8(1.02)1- = 1000^:imS94JtL02)* 
; f log 1.02 - 0.005734. 

| log 1171.66 - 3.068801. 

log A = 3.074635; A - $1187.23. 

Example 2. Find the present value of $3500, due at the end of 2 years 
and 10 months, if money is worth (.07, m = 2). / -^ "- 1 ,?-^, -- -- , - 
Solution. The rate per period is r = .035, and k = 2(2|) = 5f . 
P = 3500(1.035)^ - 3500(1.035)-(1.035)*. 
P = 3500(.81360064) (1.01163314) = $2880.09.1 (Tables VI and X) 

The methods of Examples 1 and 2 are complicated unless k is a 
convenient number. Approximate practical methods are described 
below. 

, Rule 1. To find the compound amount after k periods when 
k is not an integer, (a) compute the compound amount after the 
largest number of whole conversion periods contained in the given 
time. (6) Accumulate this amount for the remaining time at 
simple interest at the given nominal, rate, 

Example 3. Find the amount in Example 1 by use of Rule 1. 

Solution. Compound amount after 2 years is 1000(1.02)" = $1171.66. 
Simple interest at the rate 8% for 2 months on $1171.66 is $15.62. Amount 
at end of 2 years and. 2 months is 1171.66 + 15,62 = $1187.28, slightly 
greater than the result of Example 1. The use of Rule 1 is always slightly in 
favor of the creditor. 

Rule 2. To find the present value of A, due at the end of 
k periods, when k is not an integer, (a) discount the amount A for 
the smallest number of whole periods containing the given time. 
This gives the discounted value of A at a time before the present. 

1 If five-place logarithms are used in multiplications or divisions, the results will 
be accurate to only four significant figures. Hence, in Example 2, if we desire P 
accurately to cents, ordinary multiplication must be used (unless logarithm tables 
with seven or more places are available). In performing the ordinary multiplica- 
tion, as in finding P in Example 2, the student is advised to use the abridged 
method described in the Appendix, Note 4. 



22 



MATHEMATICS OF INVESTMENT 



(&) Accumulate this result up to the present time at simple 
interest at the given nominal rate. 

Example 4. Solve Example 2 by use of Rule 2. 

Sol-uOon. The smallest number of conversion periods containing 2 years 
and 10 months is 6 periods, or 3 years. Discounted value of $3500, 3 years 
before due, or 2 months before the present, is SSOOfl.OSS)- 6 = $2847.25. 
Simple interest on $2847.25 at 7% for 2 months is $33.22. Present value is 
2847.25 + 33.22 = $2880.47, which is greater than the result of Example 2. 
Results computed by use of Rule 2 are always slightly larger than the true 
present values as found from equation 16. 

NOTE. Unless otherwise directed, use the methods of Examples 1 and 
2 when fc is not an integer. Compute the time between dates approximately, 
as in expression 9, of Chapter I, and reduce to years on the basis of 360 days to 
the year. 

EXERCISE XI 

Find P or A, whichever is not given. Use Table X whenever possible. 



"Pi* (TO 


PSBSHNT 


AMOUNT, 


P ACCUMULATES FOB, 


INTEREST 


STJMJJ3* 


VALUE, P 


A 


OR A IS DUE AFTER 


RATH 


1. 


$2000 




3 years, 3 mo. 


.06, m => 2 


2. 




1000 


6 years, 1 mo. 


.07, m = 4 


8. 


8000 




16 years, 8 mo. 


.05, m = 1 


4. 


4000 




13 years, 7 mo. 


.08, m - 4 


5. 




5000 


11 years, 5 mo. 


.04, m = 2 


6. 


1000 




6 years, 4 mo. 


.05, m - 2 


7. 




1500 


7 years, 10 mo. 


.06, m <= 4 


8. 


1500 




7 years, 10 mo. 


.06, m = 4 



9. Find the amount in problem 1 by use of Rule 1. 

10. Find the present value in problem 5 by use of Rule 2. 

11. Find the amount in problem 4 by use of Rule 1. 

12. Find the present value in problem 7 by use of Rule 2. 

^ia. On June 1, 1920, X borrows $2000, from Y and.agrees to pay the 
compound amount on whatever date ne settles his account, By use of 
Rule 1, determine what X shouldpay on August 1, 1922, if interest is at the* 
rate 6%, compounded quarterly? fl, 4, y t $ 

14, At the end of 5 years and 3 mouths, $10,000 is due. Discount it to 
the present time if money is worth (.05, m = 2). Use Rule 2. 



COMPOUND INTEREST 



23 



12, Graphical comparison of simple and compound interest. 
In Figure 1 the straight 
line EF is the graph of the 
equation 

A = 1 + .06 t, 

the amount after t years if 
$1 is invested at simple in- 
terest at the rate .06. The 
curved line EH is the graph 
of the equation 

A = (1.06)*, 

the amount after t years if 
$1 is invested at the rate A 
.06 compounded annually. 
This curve was sketched 
through the points corre- 
sponding to the following 
table of values : FIG. i 




A 


1 


1.0147 


1.0296 


1.06 


1.124 


1.338 


1.791 


t 





i 


* 


1 


2 


5 


10 



A=l,06 



The entries f or t = % and t = are from Table X. In Figure 2, 
that part of the curves for which t = to t = 1 has been magnified 

(and distorted vertically, for em- 
phasis). Figure 2 shows that, 
when the time is less than one 
conversion period, the amount 
at simple interest is greater than 
the amount at compound inter- 
est. The two amounts are the 
same when t = 1, and, thereafter, 
the compound amount rapidly grows greater than the amount at 
simple interest* The ratio, (compound amount) -f- (amount at 
simple interest), approaches infinity as t approaches infinity. 




Fia, 2 



24 MATHEMATICS OF INVESTMENT 

EXERCISE Xn 

1. (a) Draw graphs on the same coordinate system, of the amount at 
simple interest, rate 5%, and of the amount at (.05, m = 1), for a principal 
of $1, from t = O'to t = 10 years. (6) Draw a second graph of that 
part of the curves for which t = to t = 1 with your original scales magni- 
fied 10 times. 

13. Values of obligations. A financial obligation is a promise 
to pay, or, an obligation is equivalent to a promissory note. Con- 
sider the following obligations or notes : 

(a) Three years and 9 months after date, X promises to pay 
$1000 to Y or order. 

(&) Three years and 9 months after date, X promises to pay 
$1000 together with all accumulated interest at the rate 6%, com- 
pounded quarterly, to Y or order. 

Exampk 1. One year after date of note (a), what does Y receive on 
discounting it with a banker B to whom money is worth (.05, m = 4) ? 

Solution. B pays the present value of $1000, due after 2 years and 
9 months, or 1000 (1.0 125) "" = $872.28. 

Example 2. One year after date of note (&), what is its value to a man 
W to whom money is worth (.07, m 4) ? 

Solution. Maturity value of obligation (6) is 1000(1.016) ll! - $1250.23. 
Its value to W, 2 years and 9 months before due, is 1250.23(1.0175)"" 
= $1033.03. 

Under a stipulated rate of interest, the value of an obligation, 
n years after its maturity date, is the compound amount which 
would be on hand if the maturity value had been invested for 
n years at the stipulated interest rate, 

Example 3. Note (6) was not paid when due. What should X pay at 
the end of 5 years to cancel the obligation if money is worth (.07, m =? 4) 
toY? 

Solution. Maturity value of note is lOOO(l.Olfi) 1 * - $1250.23. Value 
at the rate (.07, m = 4) to be paid by X, 1 year and 3 months after maturity 
date, is 1250.23(1.0175)* - $1363,52. 

NOTB. In all succeeding problems in compound interest, reckon elapsed 
time between dates approximately, as in expression 9 of Chapter I, If it is 
stated that a sum is due on & certain* date, the sum is understood to be due 




COMPOUND INTEREST %5 

v -. . 

without interest. If a sum is dite with accumulated interest, this fact will be 
mentioned explicitly. 

EXERCISE 



1. If money is worth (.07, m = 2) to W, what would he pay to Y for 
note (a), above, 3 months after date of the note? 

2. If money is worth (.06, m = 2) to W, what should he pay to Y for 
note (&), above, 3 months after date of the note? 

3. X borrows $1500 from Y and gives him the following note : 



BOSTON, July 15, 19SS, 

Three years and 6 months after date, I promise to pay to Y or order 
at the First National Bank, $1600 together with accumulated interest 
at the rate (.07, m = 2) 
Value received. Signed 5. 



On January 15, 1923, what does Y receive on selling this note to a 
bank which uses the rate (.06, m = 2) in discounting? 

4. What would Y receive for the note in problem 3 if he discounted it 
on July. 15, 1922, at a bank using the rate (.055, m = 2) ? 

6. X owes $300, due with accumulated interest at the rate (.04, m = 4) 
at the end of 5 years -and 3 months. What is the value of this obligation 
two years before it is due to a man to whom money is worth (.06, m = 1) ? 

6. At the end of 4$ years, $7000 is due, together with accumulated 
interest at. the rate (.045, m = 2). (a) Find the value of this obligation 
2i years before it is due if money is worth (.05, m = 2). (6) What is 
its value then under the rate (.045, m = 2) ? 

^ 7. On May 15, 1918, $10,000 was borrowed. It was to be repaid on 
August 15, 1921, with accumulated interest at the rate (.08, m = 4). 
No payment was made until August 15, 1923. What was due then if 
money was considered worth (.07, m = 2) after August 15, 1921 T*/ 

8.' On May 15, 1922, what was the value of the obligation of problem 7 
if money was worth (.07, m = 4) after August 15, 1921 ? 

9. Find the value of the obligation of problem 7 on November 15, 
1923, if money is worth (.05, m = 4), commencing on August 15, 1921. 

10. X owes Y (a) $2000, due in 2 years, and (6) $1000, due in 3 years 
with accumulated interest at the rate (.05, m = 2). At the end of one 
year what should X pay to cancel the obligations if money is worth 
(.04, m =* 2) to Y? 

HINT. X should pay the. sum of the values of his obligations. 



26 MATHEMATICS OF INVESTMENT 

11. At the end of 3 years and 3 months, $10,000 is due with accumulated 
interest at (.05, m = 4). (a) What is the value of this obligation at the 
end of 5 years if money is worth (.07, m = 4) ? (6) What is its value 
then if money is worth (.04, m = 4) ? 

12. The note of problem 3 is sold by Y on October 16, 1924, to a banker 
to whom money is worth 6%, effective. By use of Rule 2 of Section 11 
find the amount the banker will pay. 

The value of an obligation depends on when it is due. Hence, 
to compare two obligations, due on different dates, the values of 
the obligations must be compared on some common date. 

Example 4. If money is worth (.05, m - 1), which is the more 
valuable obligation, (a) $1200 due at the end of 2 years, or (&) $1000 due at 
the end of 4 years with accumulated interest at (.06, m = 2) ? 

Solution. Compare values at the end of 4 years under the rate (.05, 
m = 1). The value of (a) after 4 years is 1200(1.05)" = $1323.00. Tho 
value of (b) after 4 years is 1000(1.03)* = $1266.77. Hence, (a) is tho more 
valuable. 

NOTE. The value of an obligation on any date, the present for example, 
is the sum of money which if possessed to-day is as desirable as the payment 
promised in the obligation, If the present values of two obligations are the 
same, their values at any future time must likewise be equal, because these 
future values are the compound amounts of the two equal present values. 
Similarly, if the present values are equal, the values at any previous date must 
have been equal, because these former values would be the results obtained 
on discounting the two equal present values to the previous date. Hence, any 
comparison date may be used in comparing the values of two obligations, be- 
cause if their values are equal on one date they are equal on all other dates, 
both past and future. If the value of one obligation is greater than that of 
another on one date, it will be the greater on all dates. For instance, in 
Example 4 above, on comparing values at the end of 3 years, the value of (a) is 
1200(1.05) = $1260.00; the value of (b) is 1000(1.03)(1.06)" a - $1206.45. 
Hence, as in the original solution, (a) is seen to be tho more valuable. The 
comparison date should be selected so as to minimize the computation required. 
Therefore, the original solution of Example 4 was the most desirable. 

EXERCISE XIV 

1. If money ia worth (.04, m = 2), which obligation is the more valu- 
able : (a) $1400 due after 2 years, or (6) $1500 due after 3 years? 

2. If money is worth (.05, m = 2), which obligation is the more valu- 
able: (a) $1400 due after 5 years, or (&} $1000 due after 4 yeajs with 



COMPOUND INTEREST 27 

accumulated interest at (.07, m = 2) ? Use 4 years from now as the 
comparison date. 

3. Solve problem 2 with 6 years from now as the comparison date. 

4. If money is worth (.06, m = 2), compare the value of (a) $6000 
due after 4 years with (&) an obligation to pay $4000 after 3 years with 
accumulated interest at (.05, m = 1). 

6. Compare the set of obligations (a) with set (&) if money is worth 
(.06, m = 2) : 

(a) $1600 due after 3 years ; $1000 due after 2 years with accumu- 
lated interest at the rate (.04, m = 2). 

(&) $1200 due after 2 years ; $1400 due after 2 years with accumu- 
lated interest at the rate (.05, m = 2). 

6. Which obligation is the more valuable if money is worth (.06, 
m = 4) : (a) $8000 due after 3 years with accumulated interest at 
(.05, m = 4), or (6) $8500 due after 3 years? 

14. Equations of value. An equation of value is an equation 
stating that the sum of the values, on a certain comparison date, 
of one set of obligations equals the sum of the values on this date 
of another set. Equations of value are the most powerful tools 
available for solving problems throughout the mathematics of 
investment. 

NOTE. In writing an equation of value, the comparison date must be 
explicitly mentioned, and every term in the equation must represent the value 
of some obligation on this date. To avoid errors, make preliminary lists of 
the sets of obligations being compared. 

Exampk 1. W owes Y (a) $1000 due after 10 years, (&) $2000 due 
after 5 years with accumulated interest at (.05, m = 2), and (c) $3000 due 
after 4 years with accumulated interest at (.04, m = 1). W wishes to 
pay in full by making two equal payments at the ends of the 3d and 4th 
years. If money is worth (.06, m = 2) to Y, find the siae of Ws payments. 

Solution. Let $tc be the payment. W wishes to replace his old obligations 
by two new ones. Let 4 years from now be the comparison date. 



OLD OBLIGATIONS 


NEW OBLIGATIONS 


(a) $1000 due in 10 years. 
(6) 2000(1.025) 10 due in 6 years, 
(c) 3000(1,04)* due in 4 years. 


$z due in 3 years. 
$z due in 4 years. 



2$ MATHEMATICS OF INVESTMENT 

In the following equation of value the left member is the sum of the values 
of the old obligations on the comparison date. This sum must equal the sum 
of the values of the new obligations given in the right member. 

1000(1.03)~ w + 2000(1.025) 10 (1.03)~ a + 3000(1.04) 4 = 3(1.03)' + as. (20) 
6624.16 - z(1.0609) + a? = 2.0609 as. 

x = $3214.21. 

If 5 years from the present were used as the comparison date, the equation 
would be 

1000(1.03)" 10 + 2000(1.025) 10 + 3000(1.04)*(1.03) 3 = 3(1.03)* + aj(1.03) a , (21) 
from which, of course, the same value of re is obtained because equation 21 
could be obtained by multiplying both sides of equation 20 by (1.03) 8 . All 
obligations were accumulated for one more year in writing, equation 21 as 
compared with equation 20. 

EXERCISE XV 

Solve each problem by writing an equation of value. List the obliga- 
tions being compared. 

1. W owes Y $1000 due after 4 years and $2000 due after 3 years and 

3 months. "What sum paid now will discharge these debts if money is 
worth (.08, m = 4) to Y? 

2. W desires to discharge his obligations in problem 1 by two equal 
payments made at the ends of 1 year and of 1 year and 6 months, respec- 
tively. Find the payments if money is worth (.06, m = 4) to Y. 

3. "W 'desires to pay his obligations in problem 1 by three equal pay- 
ments made after 1, 2, and 3 years. Find the payments if money is worth 
(.06, m = 4) to Y. 

4. What payment made at the end of 2 years will discharge the fol- 
lowing obligations jf money is worth (.05, m 2) : (a) $10,000 due after 

4 years, and (6) $2000 due after 3J years with accumulated interest ait 
(.07,m = 2)? 

5. If money is worth (.06, m = 2), determine the size of the equal pay- 
ments which, if made at the ends of the 1st and 2d years, will discharge the 
obligations of problem 4, 

6. What sum, paid at the end of 2 years, will complete payment of the 
obligations of problem 4 if twice that sum was previously paid at the end 



pf the first year? Money is worth (.08, m = 2). 

interest at (.06, m >- 2) after 4J years. ' W,paid $1500 after 2 years, v 



of thi 
'I 7. 



W owed Y $1000 due after 3 years, and $3000 due with accumulated 



What should he pay at the end of 3 years to cancel his debts if money is 
worth 7%, compounded semVannually, to Y?jJ( 




COMPOUND INTEREST 



29 



8. A man, owing the obligations (a) and (6) of problem 4, paid $8000 at 
the end of 3 years. What single additional payment should he make at 
the end of 5 years to cancel his obligations if money is worth (.04, m = 2) 
to his creditor? 

9. Determine whether it would be to the creditor's advantage in 
problem 8 to stipulate that money is worth (.05, m = 2) to him. 

16. Interpolation methods. The usual problem in compound 
interest, where the rate or the time is the only unknown quantity, 
may be solved approximately by interpolating in Table V. The 
method is the same as that used in finding a number N from a 
logarithm table when log N is known. 

Example 1. Find the nominal rate under which $2350 will accumulate 
to $3500 by the end of 4 years and 9 months, if interest is compounded 
quarterly. 

Solution. Let r be the unknown rate per period. The nominal rate will 
be 4 r. From equation 16, 

3600 = 2360(1 + r) 19 ; (1 + r) 19 - jj? = 1.4894. 



i 


(1 + i) 1B 


.02 

i = r 
.0225 


1.4568 
1.4894 
1.5262 



1.4668 to 1.5262. 

Hence r = .02 + |H(-0025) = .0212. 



The first and third entries in the table are from the 
row in Table V for n = 19. In finding r by interpola- 
tion we assume that r is the same proportion of the way 
from .02 to .0225 as 1.4894 is of the way from 1.4568 to 
1.5262. Since 1.5262 - 1.4568 = .0694, and 1.4894 
- 1.4568 = .0326, then 1.4894 is f$$ of the way from 
The distance from .02 to .0225 is .0226 .02 = .0025. 
The nominal rate is 4 r = .0848. 



Interest rates per period determined as above are usually in error by not 
more than 1 J'fr.of the difference between the table rates used. Thus, the 
value of r above is probably in error by not more than ^(.0025) or about 
.0001. The error happens to be much less, because a solution by exact 
methods gives r = .02119. Results obtained by interpolation should be 
computed to one more than the number of decimal places wfiich are ex- 
pected to be accurate. * r 

NOTB. When interpolating, it is sufficient to use only four decimal places 
of the entries in Table V. Use of more places does not increase the accuracy of 
the final results and causes unnecessary computation. 

1 The author gives no theoretical justification for this statement. He has 
verified its truth for- numerous examples distributed over the complete range of 

T,bi,v. 8*-. 4 



30 MATHEMATICS OF INVESTMENT 

Example 2. How long will it take $5250 to accumulate to $7375 if 
invested at (.06, m = 4) ? 
Solution. Let k be the necessary number of interest periods. 

7375 - 5250(1.015)*; (1.015)* - JJ^ - 1.4048. 

The first and third entries in the table are from Table V. 
Since 1.4084 - 1.3876 - .0208, and 1.4048 - 1.3876 
= .0172,. then k is tff of the way from 22 to 23, or 
k = 22 + tfft = 22.83 periods of 3 months. The time 



n 



22 

n = Te 
23 



(1.015)" 



1.3876 
1.4048 
1.4084 



is = 5.71 years. A value of k obtained as above 

4 

is in error by not more than J of the interest rate J per period. The error hi 
Example 2 is much less, because an exact solution of the problem gives 
k = 22.831. 

Example 3. X owes Y $1000 due after 1 year, and $2000 due after 3 
years with accumulated interest at (.05, m = 2) . When would the pay- 
ment of $4000 balance X'B account if money is worth (.06, m = 4) to Y? 

Solution. Let k be the number of conversion periods of the rate (.06, 
m = 4) between the present and the date when $4000 should be paid. With 
the present as a comparison date, the equation of value for the obligations is 
4000(1.015)7* = 1000(1.015)"* + 2000(1.026)'(1.015)~ u = 2882.09. 

(1.015)~* = .72052. 

From interpolation in Table VI, k = 22 + yfrfo. = 22.02. X should pay $4000 
after 22^ - 5.60 years. 

Example 4. How long will it take for money to double itself if left to 
accumulate at (.06, m = 2) ? 

Solution. Let P = $1 and A = $2. If k represents the necessary num- 
ber of conversion periods, a solution by interpolation gives k * 23.44; the 
time is 11.72 years. Another approximate method is furnished by the follow- 
ing rule. 

Rule I. 2 To determine the time necessary for money to double 
itself at compound interest : (a) Divide .693 by the rate per period. 
(6) Add .35 to this result. The sum is the time in conversion 
periods. The error of this approximate result generally is less than 
a few hundredths of a period. 

On solving Example 4 by this rule, Jb- ^ + .36 = 23.45. 

.03 

1 For justification of this statement BQO Appendix, Note 5. A knowledge of the 
calculus is necessary in reading this note. 

* For a proof of this rule see Appendix, Note 1, 



COMPOUND INTEREST 



EXERCISE XVI 1 

Solve all problems by interpolation unless otherwise directed. In each 
problem in the table, find the missing quantity. 



PBOB. 


AMOUNT, 
A 


PRINCIPAL, 
P 


P AOOUMTTLA.THB FOB, 

OB A IB DUB AFTEB 


NOMINAI 
RATE 


CONVBHSIONB 
PHB YBAB 


1. 


$2735 


$1500 




.05 


1 


2. 


2500 


2000 




.06 


2 


3. 


2 


1 


15 years 




1 


4. 


1000 


750 


3 years, 9 mo. 




4 


5. 


5010 


4250 




.07 


2 


6. 


6575 


4270 


7 years, 6 mo. 




2 


7. 


3000 


1000 




.05 


2 



^ 8. Find the nominal rate under which $3500 -is the present value 
of $5000, due -at the end of 12J years. Interest is compounded semi- 
annually. j? * V V *) * : j 

^ 9. How long will it take for money to quadruple itself if invested at 
(.06, TO = 2)? % 3- H-$ f^ 

10. (a) At what nominal rate compounded annually will money double 
in 14 years? (&) Solve by use of Rule 1. 

' 11. If money is worth (.07, m = 1), when will the payment of $4000 
cancel the obligations (a) $2000 due after 3 years, and (&) $2000 due after 
7 years? 

12. If money is worth (.05, m = 2), when will the payment of $3000 
cancel the obligations (a) $1500 due after 3 years, and (&) $1000 due at the 
end of 2 years with accumulated interest at (.06, m = 4) ? 

15. By use of Rule 1, determine how long it takes" for money to double 
itself under each of the following rates : (a) (.06, m = 4) ; (&) (.04, m = 2) ; 
(c) (.06, TO -2); (d) (.03, m = 1). 

14. By use of the results of problem 13, determine how long it takes for 
money to quadruple itself under each of the four rates in problem 13. 

16. If money is worth (.04, m = 2), when will the payment of $3500 
cancel the liabilities (a) $1000 due after 18 months, and (6) $2000 due 
after 2J years? 

i The Miscellaneous Problems at the end of the chapter may be taken up im- 
mediately after the completion of Exercise XVI. 



32 MATHEMATICS OF INVESTMENT 

SUPPLEMENTARY MATERIAL 

16. Logarithmic methods. Problems may arise to which the 
tables at hand do not apply, or in which more accuracy is desired 
than is obtainable by interpolation methods. Logarithmic 
methods are available in such cases. 

Example 1. Find the present value of $350.75, due at the end of 6 
years and 6 months, if interest is at the rate (.0374, m = 2). 

Solution. P = 350.75(1.0187)-" = 



log 350.75 = 2.54500 
13 log 1.0187 = 13 (.0080463) = 0.10460 (Using Table H) 

(subtract) log P = 2.44040. P = $275.68. 

If Table I were used in obtaining log (1.0187) 131 , 13 log 1.0187 = 13(.00804) 
= 0.10452, in error by.. 00008. 

Example 2. If interest is converted quarterly, find the nominal rate 
under which $2350 is the present value of $2750, due after 4 years and 9 
months. 

Solution. Let r be the unknown rate per period ; the nominal rate is 4 r. 



2750 - 2.350(1 + r) ; 1 + r 

log 2750 =2.43933 

log 2360 =2.37107 

log quot. = 0.06826. ^ log quot. = 0.00359. 

.-. 1 + r = 1.0083, r = .0083. The nominal rate is 4 r = .0332, converted 
quarterly. 

Exampk 3, How long will it take for $3500 to accumulate to $4708 if 
interest is at the rate (.08, m = 4) ? 

Solution. Let k be the necessary number of conversion periods. 



4708 - 3500(1.02)*; (1.02)* - JJ& ..% fc log 1.02 = log 

log 4708 - 3.67284 log 1.02 = 0.0086002 

log 3600 - 3.54407 
log quot. = 0.12877. .-. fc(0.0086002) = .12877. 

fc - 12877 lo g 12877 = 4.10982 

860.02' log 860.02 - 2.93451 

(subtract) log fe= 1.17531 

The time is k = 14.973 periods of 3 months, or 3.743 years, , , 



COMPOUND INTEREST LJ ' 

EXERCISE XVH 

Use exact logarithmic methods in all problems on this page. Use 
Table II whenever advisable. 

1. Find the compound amount after 3 years and 3 months, if $3500 is 
invested at the rate (.063, m - 4). 

2. Find the present value of $3500 which is due at the end of 8 years 
and 6 months, if money is worth (.078, m = 2). 

3. At what nominal rate, converted quarterly, is $5000 the present 
value of $7300, due at the end of 2 years and 9 months? 

4. Find the length of time necessary for a principal of $2000 to ac- 
cumulate to $3600, if interest is at the rate (.05, m = 1). 

6. Solve problems 2 and 5 of Exercise XVI by exact methods. 
6. Solve problems 3 and 4 of Exercise XYI by exact methods. 
V 7. Find the nominal rate which, if converted semi-annually, yields the 
effective rate .0725. '/ * / $- */ % 

8. Find the nominal rate which, if converted semi-annually, is equiva- 
lent to the rate .068, compounded quarterly. 

9. (a) Determine how long it will take for money to double itself at 
the rate (.06, m - 1). (6) Compare your answer with the result you 
obtain on using Hule 1 of Section 15. 

10. One dollar is allowed to accumulate at (.03, m - 2). A second 
dollar accumulates at (.06, m = 1). When will the compound amount 
on the second dollar be three times that on the first? 

HINT. Take the logarithm of both aides of the equation obtained. 

17. The equated time. The equated date for a set of 

obligations is the date on which they could be discharged by a 
single payment equal to the sum of the maturity values of the 
obligations. The time between the present and the equated 
date is called the equated time, and it is found by solving an 
equation of value. 

Example 1. If money is worth (.05, m = 2), find the equated time 
for the payment of the obligations (a) $2000 due after 3 years, and 
(6) $1000 due after 2 years with accumulated interest at the rate 
(.04, m = 2). 

Solution, The sum of the maturity values of (a) and (b) is 2000 
+ 1000(1.02)* - $3082.43. Let the equated time be k conversion periods oi 



34 MATHEMATICS OF INVESTMENT 

the rate (.05, m = 2). The value of the obligation $3082.43, due after k periods 
(on the equated date), must be equal to the sum of the values of the given 
obligations. With the present as the comparison date, the corresponding 
equation of value is 

3082.4(1.025}-* = 2000(1.026)- 8 + 1000(1.02)(1.026)-* = 2705.2. 



(1.025)* = = 1.1394. 



By the method of Section 16, k = 5.286 six-month periods or, the equated time 
is 2.643 years. By the interpolation method, k = 5.28. The present was 
used as the comparison date above to avoid having k appear on both sides of 
the equation. 

To obtain the equated time approximately, the following rule 
is usually used. 

Rule I. 1 Multiply the maturity value of each obligation by the 
time in years (or months, or days) to elapse before it is due. Add 
these products and divide by the sum of the maturity values to 
obtain the equated time. 

On using this rule in Example 1 above, we obtain 

equated time = 3(2(100)^2(1082.4) - 2.65 years. 

Rule 1 is always used in finding the equated date for short-term 
commercial accounts. The equated date for an account is also 
called the average date and the process of finding the average date 
is called averaging the account. Since Rule 1 does not involve 
the interest rate, it is unnecessary to state the rate when asking for 
the equated date for an account. 

NOTE. Results obtained by use of Rule 1 are always a little too large, so 
that a debtor is favored by its use. The accuracy of the rule is greater when 
the interest rate is low than when it is high. The accuracy is greater for short- 
term than for long-term obligations. 

EXERCISE XVm 

1. If money is worth (.05, m 1), find the equated time for the pay- 
ment of (a) $1000 due after 3 years, and (6) $2000 due after 4 years. 
Solve by Rule 1. 

2. Solve problem 1 by the exact method of Example 1 above. 

1 For derivation of the rule see Appendix. Note 2. 



COMPOUND INTEREST 

3. (I) If money is worth (.07, m = 2), find the equated time for the 
payment of (a) $1000 due after 3 years and (&) $2000 due after 4 years with 
accumulated interest at (.05, m = 2). Solve by the exact method 
(II) Solve by Rule 1. 

* 4. Find the equated time for an account requiring the payment of $55 
after 3 months, $170 after 9 months, and $135 after 7 months. Use Rule 1 . 

6. (a) A man owes four 180-day, non-interest-bearing notes dated as 
follows: March 9, for $400; May 24, for $250; August 13, for $525; 
August 30, for $500. By use of Rule 1 find the equated time and the 
equated date for the payment of the notes, considering for convenience 
that March 9 is the present. (&) How much must be paid on the 
equated date to cancel these obligations, if no other payment is made? 

6. If money is worth 6%, simple interest, what should be paid 30 days 
after the equated date in problem 5 in order to balance the account, if no 
other payment is made? 

18. Continuously convertible interest ; the force of interest. 1 
The compound amount on $1 at the end of one year, if inter- 
est is at the nominal rate j, converted m times per year, is 

A <= (\ + 3-\ - It was seen at the end of Section 10 that, as m 
\ m/ 

increases, the amount A increases. As m increases without bound, 
or in other words, as m approaches infinity, the amount A does not 
increase without bound but approaches a limiting value e', where 
e = 2,7182818 + is the base of the Naperian, or natural, system 
of logarithms. To prove this we use the theory of limits. 



lim A = lim (l + iy = lim |7l +^77 = flim (l + 

tn-oo m =fo\ m/ m=ooL\ m/ J l_m=co\ W 

It is known that 2 lim (l + ^V = e - Therefore, 
m=A m/ 

. (22) 



It is customary to say that this limiting value e 1 ' is the com- 
pound amount on $1 at the end of one year in the ideal case where 

1 A knowledge of the theory of limits is advisable in reading this section. 
1 See Granville's Calculus, Revised Edition, page 22. 



3 ; MATHEMATICS OF INVESTMENT 

interest is converted continuously. For every value of w, the 
effective rate i corresponding to the nominal rate j is given by 

i = |Yi _|_ 1Y"_ i~l Hence, in the limiting case where interest 

is converted continuously, it follows from equation 22 that 

i = lim ( 1 + iV" - 1 - & - 1. 
7n=co \ m/ 

J + z = ei. (23) 

Example 1. Find the effective rate if the nominal rate is .05, con- 
verted continuously. 

Solution. 1 + i = e- OH . log (1 + i) - -05 log e, where e - 2.71828. 

.05 log e = .05(0.43429) = 0.02171 = log (1 + i). 

.-. 1+i- 1.0513; i- .0513. 

The force of interest, corresponding to a given effective rate i, 
is the nominal rate which, if converted continuously, will yield the 
effective rate i. Hence, if 5 represents the force of interest, the 
value j = 8 must satisfy equation 23, or 

1 + 1 - e. . (24) 

Example 2. Find the force of interest if the effective rate is .06. 

Solution. 1.06 = e. .'. 5 log e = log 1.06. 
log 1.06 .0253059 



loge .43429 

Under the effective rate i, the compound amount of a principal 
P at the end of n years is A = P(l + i) n . If the nominal rate is 
j, converted continuously, (1 + i} = e*', hence A P(e*) n , or 

A = Pert. 
To compute A we use logarithms ; log A = log P + nj log e. 

EXERCISE XIX 

1. Find the effective rate if the nominal rate is .06, converted con- 
tinuously. - 9 - 

J 2. Find the force of interest if the effective rate is .05. "* ' / J 

3. (a) Find the amount after 20 years if $2000 is invested at the rate 
.07, converted continuously, (6) Compare your answer with the com- 
pound amount in case the rate is (.07, m => 4) . 



F * <t 



COMPOUND INTEREST 37 

MISCELLANEOUS PROBLEMS 

1. A man, in buying a house, is offered the option of paying $1000 cash 
and $1000 annually for the next 4 years, or $650 cash and $1100 annually 
for the next 4 years. If money is worth (.06, m = 1), which method is the 
better from the purchaser's standpoint? 

2. A merchant desires to obtain $6000 from his banker, (a) If the 
loan is to be for 90 days and if the banker charges 6% interest in advance, 
for what sum will the merchant make out the note which he will give to 
the banker? (&) What simple interest rate is the man paying? 

3. A merchant who originally invested $6000 has $8000 capital at the 
end of 6 years. What has been the annual rate of growth of his capital if 
the rate is assumed to have been uniform through the 6 years? 

4. If gasoline consumption is to increase at the rate of 5% per year, 
when will the consumption be double what it is now? Solve by two 
methods. 

5. When will the payment of $5000 cancel the obligations $2000 due 
after 3 years, and $2500 due after 6 years? Money is worth (.05, m = 2) . 

6. A certain life insurance company lends money to policy holders at 
6% interest, payable in advance, and allows repayment of all or part of the 
loan at any time. Six months before the maturity of a $2000 loan, the 
policy holder A sends a check for $800 to apply on his loan. What 
additional sum will A pay at maturity? 

HINT. First find the sum, due in 6 months, of which $800 is the present value. 

7. A must pay B $2000 after 2 years, and $1000 after 3 years and 
6 months. At the end of 1 year A paid B $1500. If money is worth 
(.05, m = 2), what additional .equal payments at the ends of 2 years and 
6 months and of 3 years will cancel A's liability? 

8. (a) If you were a creditor, would you specify that money is worth 
a high or a low rate of interest to you, if one of your debtors desired to 
pay the value of an obligation on a date before it is due? Justify your 
answer in one sentence. (6) If a debtor desires to discharge an obligation 
by making payment on a date after it was due, what rate, high or low, 
should the creditor specify as the worth of money? 

9. At the end of 4 years and 7 months, $3000 is due. Find its present 
value by the practical rule if money is worth (.08, m = 4). 

10. A man has his money invested in bonds which yield 5%, payable 
semi-annually. If he desires to reinvest his money, what is the lowest 
rate, payable quarterly, which his new securities should yield? 



< 38 MATHEMATICS OF INVESTMENT 

11. One dollar is invested at simple interest, rate 5%. A second dollar 
." is invested at (.05, m = 1). When will the compound amount on the 
1 second dollar be double the simple interest amount on the first? 

j HINT. Solve the equation by interpolation; see illustrative Example 1 in 

< Appendix, Note 3. 

12. After how long a time will the compound amount on $1 at the rate 
(.06, m = 2) be double the amount on a second $1 at the rate (.035, 
m = l)? 

HINT. Use either interpolation or logarithmic methods. See problem 11. 

13. If $100 is invested now, what will be the compound amount after 
20 years if the effective rate of interest for the first 5 years will be 6%, 
whereas interest will be at the rate (.04, m = 2) for the last 15 years? 

14. A man owes $2000, due at the end of 10 years. Find its present 
value if it is assumed money will be worth 4% effective, for the first 5 years, 
and 6% effective, for the last 5 years. 

IB. If $100 is due at the end of 5 years, discount it to the present 
time, (a) under the rate 5%, compounded annually ; (6) under the simple 
interest rate 6% ; (c) under the simple discount rate 5%. 



CHAPTER III 
ANNUITIES CERTAIN 

19. Definitions. An annuity is a sequence of periodic pay- 
ments. An annuity certain is one whose payments extend over a 
fixed term of years. For instance, the monthly payments made 
in purchasing a house on the instalment plan, form an annuity 
certain. A contingent annuity is one whose payments -last for a 
period of time which depends on events whose dates of occurrence 
cannot be accurately foretold. For instance, a sequence of pay- 
ments (such as the premiums on an insurance policy) which ends 
at the death of some individual form a contingent annuity. In 
Part I of this book we consider only annuities certain. 

The sum of the payments of an annuity made in one year is 
called the antwl rent. The time between successive payment 
dates is the payment interval. The time between the beginning of 
the first 'payment interval and the end of the last, is called the 
term of the annuity. Unless otherwise stated, all payments of an 
annuity are equal, and they are due at the ends of the. payment inter- 
vals; the first payment is due at the end of the first interval, and 
the last is due at the end of the term. Thus, for an annuity of 
$50 per month for 15 years, the payment interval is 1 month, the 
annual rent is $600, and the term is 15 years ; the first payment 
is due after 1 month, and the last, after 15 years. 

Under a specified rate of interest, the present value of an annuity 
is the sum of the .present values of all .payments of the annuity. 
The amount of an annuity is the sum of the compound amounts 
that would be on hand at the end of the term if all payments 
should accumulate at interest until then from the dates on which 
they are due. 

NOOTB 1. Consider an annuity of $100, payable annually for 5 years, with 
interest at the rate 4%, effective. We obtain the present value A of this 
annuity by adding the 2d column in the table below, and the amount S by 
adding the 3d column. ^ -"" 

39 



Li 




40 



MATHEMATICS OF INVESTMENT 



PAYMENT OF $100 
DUB AT END OF 


PRESENT VALUE OP PAYMENT 


COMPOUND AMOUNT AT END OF 
TERM IF PAYMENT is LBFT TO 

ACCUMULATE AT INTEREST 


1 year 
2 years 
3 years 
4 years 
6 years 


100C1.04)" 1 = 96.16385 
100(1. 04) ""* = 92.46562 
100(1.04) "* = 88.89964 
100(1. 04) ^ = 86.48042 
100(1.04)~ 5 - 82.19271 


100(1.04)* = 116.98586 
100(1.04) s = 112.48640 
100(1.04) 9 = 108.16000 
100(1.04) = 104.00000 
100 = 100.00000 




(add) A = $445.18224 


(add) S = $641.63226 



The present value A = $446.18 is as desirable as the future possession of all 
payments of the annuity. The amount S = $541,63, possessed at the end of 
5 years, is as desirable as all of the payments. Hence, A should be the present 
value of S, due at the end of the term, or we should have S = A (1.04) 5 . This 
relation is verified to hold ; 

A(1.04)= = 446.182 X 1.21665290 = $541.632 = S. (25) 

NOTE 2. In the table below it is verified that, if a fund is formed by 
investing $445.182 at 4% effective, this fund will provide for all payments of 
the annuity of Note 1 and, in so doing, will become exactly exhausted at the 
time of the last payment. This result can be foreseen theoretically because 
$445.182 is the -sum of the present values of all of the payments. 



YEAR 


IN FUND AT 
BEGINNING 
OF YEAS 


INT. AT 4% ' 
Dux AT END 
OF YEAH 


IN FUND XT END 
OF YEAB BEFORE 
PAYMENT IB MADE 


PAYMENT 
AT END 
OF YEAR 


1 


$445.182 


$17.807 


$462.989 


$100. 


2 


362.989 


14.520 


377.509 


100 


3 


277.509 


11.100 


288.609 


100 


4 


188.609 


7.544 


196.153 


100 


5 


96.153 


3.846 


99.999 


100 



EXERCISE XX 

1. (a) Form a table as in Note 1 above in order to find the present 
value and the amount of an annuity which pays $1000 at the end of each 
6 months for 3 years. Money is worth 6%, compounded semi-annually. 
(6) Verify as in equation 25, that A is the present value of S, due at the 
end of the term, (c) Form a table as in Note 2, to verify that the present 
value A, if invested at (.06, m = 2), creates a fund exactly sufficient to 
provide the payments of the annuity. 



ANNUITIES CERTAIN 



41 



20. The examples below 1 illustrate methods used later to 
obtain fundamental annuity formulas. 

Example 1. If money is worth (.06, m = 4), find the present value A 
and the amount S of an annuity whose annual rent is $200, payable seml- 
annually for 15 years. 

Solution. Each payment is $100. The entries in the 2d and 4th columns 
below are verified by the principles of compound interest. 



PAYMENT OF 
$1OO Dms AT 
THE END OF 


PHHSENT VALUE 
OP PAYMENT 


TIME FROM DATE 
OF PAYMENT TO 
END OF TERM 


COMP. AMT. AT END op 
THHM IF PAYT. ifl LEFT 
TO ACCUMULATE AT INT. 


6 months 
1 year 
etc. 
14 yr., 6 mo. 
15 years 


100(1.016)~ J 
100(1.015) * 
etc. 
100(1.015)~ 88 
100(1.015) H10 


14 yr., 6 mo. 
14 years 
etc. 
6 months 
months 


100(1.015)" 
100(1.015)" 
etc. 
100(1.015) a 
100 


- 


Sum = A 




Sum = S 



Hence, S = 100[1 + (1.015) 3 + etc. + (1.015) + (1.015)"]. 

The bracket contains a geometrical progression of 30 terms for which the 
ratio is w - (1.015) 2 , the first term a = 1, and the last term L = (1.015) 68 . 
By the formula for the sum of a geometrical progression, 2 

.2.44321978 - 1 



100 : 



84774.918. 



1.03022500 - 1 
On adding the 2d column in the table we obtain the present value 

A = lOOUl.QlS)^ + (1.015)- 58 + etc. + (1.015)-* + (1.015).-*]. 

The geometrical progression, in .the bracket has the ratio w = (1.015) 3 , while 
a = (LOIS)- 80 and L = (1.015)-*. Since wL - 1, 

L l - (1.015)-* = 1rtn 1 - .40929597 _ 



1Q0 



1.03022500 - 1 



(1.015) 1 - 1 

The present value of $4774.918, due at the end of 16 years, should equal A, 
orS = A(1.016) M . We verify that 

A (1.015)8 = (1954.356) (2.44321978) - $4774.920. (26) 

" l Geometrical Progressions in Part III, Chapter XII, should be studied if the 
student has not met them previously. Section 20 may be omitted without dis- 
turbing the continuity of the succeeding sections, but geometrical progressions are 
needed in Sections 21 and 22, 
> See Part III, Section 90. 



42 



MATHEMATICS OF INVESTMENT 



Example 2. Find the present value A of an annuity of $100 pei 
month for 3 years and 6 months, if money is worth (.05, m = 2) . 

Solution. A is the sum of the entries in the 2d row below. 



Payment of 
$100 due after 


1 month 


2 months 


etc. 


3 yr., 5 mo. 


3 yr.j 6 mo. 


Present value 
of payment 


100(1.025)'^ 


100(1 .025)'* 


etc. 


100(1.025)-^ 


100(1.025)-^ 



A - 100 [(1.025)-^ + (1.026)-* + - etc. - + (1.025)-* + (1.025)-*]. 
The ratio of the geometrical progression is w = (1.025)*; a = (1.025)~ T am 
L = (1.025)-*. 'Since wL ~ a = 1 - (1.025)- 7 , "~ 

1QQ 1 - (1.025)-* 1Q0 ^- .84J.26S24 , ^^ (TablesVIajldX 



(1.025)i - 



1.00412392 - I 



EXERCISE XXI 



In each problem derive formulas for A and S for the annuity describee 
using the method of Examples 1 and 2 above. 
*fr\ 



An annuity whose annual rent is $200, payable quarterly for 1 
years. Money is worth (.08, m = 4) . Compute the formulas for A and 
and verify as in equation 26 that A is the present value of S, due at tt 
end of the term. /\ ~ S & * LrJL fi ~Jy a tf ^ "^ ^ ^ S . tf 6 ^ i 

2. Fifteen successive annual payments of $1000, the first due aft< 
1 year. Money is worth (.05, m = 2). Compute A and S and veril 
that A is the present value of S, due at the end of the term. 

3. Payments of $100, made at the end of each 3 months for 15 year 
Money is worth (.05, m = 4). 

4. (a) The annual rent of the annuity is $2000, the payment interv 
is 3 months, and the term is 12^ years. Money is worth (.06, m = 1 
(6) Solve the problem if money is worth (.06, m = 2). 

5. An annuity which pays $100 at the end of each interest period f 
10 interest periods. Money is worth .045, per interest period. 

21, Formulas for A and S in the most simple case. Co: 

sider the annuity paying $1 at the end of each year for n year 
Let (a/n\ at i) be the present value, and (Sn\ at i) be the amount 
this annuity when money is worth the rate i compounded annuall 
The entries in the table below are easily verified. 



ANNUITIES CERTAIN 



PAYMENT OP SI 
DUB AT THE 
END OP 


PHHSBNT VALUB OP 
THB PAYMENT 


TIME FBOM DATE OP 
PAYMENT TO END 

OFTBBM 


COMP. AMT. AT END OP 
THEM n- PAYT. IB LEFT 

TO ACCUMULATE AT INT. 


1 year 
2 years 
etc. 


etc. 


(n. 1) yr. 
(n - 2) yr. 
etc. 


(1 + i)"- J 
etc. 


(n - 1) yr. 
n years 


d + tr^ 1 - 


1 year 
years. 


(1+0 
1 




Sum = (an\at i) 




Sum = (&n\ati) 



Hence, 

(85! 0*1) = 1 + (1 + i) + ..- etc. - + (1+ i)^ 2 + (1 + t)*~ x . 
This is a geometrical progression where the ratio 10= (1 + i), the 
first term a = 1, and the last term L = (1 + i) n-1 . Since 

(wL - a) = (1 + i)* - 1, and (w - 1) = i, the formula wL ~ a 

iw 1 

= v + y - *. (27) 



On adding the 2d column of the table we obtain 

(o^oti) = (i+;r n +u+;r n+1 + - etc. . 

which is a geometrical progression with the ratio w = (1 + i), 
a = (1 + i)" n , and L = (1 + i)" 1 . Since 

(wL - a) = [(1 + i)(l + i)- 1 - - (1 + iT"], and (w - 1) = i, 



cfi) 



(28) 



If each payment of the annuity had been $# instead of $1, the 
present value A and the amount S would have been A = R(a^ati} 
and S = R(8n\at i). 

It is important to realize that formulas 27 and 28 may be used 
whenever the payment interval of the annuity equals the con- 
version period of the interest rate, In deriving the formulas, the 
interest period was called 1 year, merely for concreteness. Hence, 
if i is the interest rate per period, then R(Sn\ at i) represents the 
amount and R(a^ at i) the present value of an annuity which pays 
$R at the end of each interest period for n periods. Thus, 



44 



MATHEMATICS OF INVESTMENT 



at .025) is the present value of an annuity paying $100 
at the end of each interest period for 18 periods if money is worth 
the rate .025 per period. 

Example 1. Find the amount and the present value of an annuity 
paying 1150 at the end of each 3 months for 15 years and 6 months, if 
money is worth 6%, compounded quarterly. 

Solution. Since the payment interval equals the interest period, formulas 
27 and 28 apply with the number of payments n = 62, and with i = .016. 

Amount = 160(s m at .016) = 150(101.13773966) = $15170.66. 
Pr. val. = 150(0^ at .015) - 150( 40.18080408) = $6027.12. 

The value of s^ is from Table VII and that of ian is bom Table VIII. 

NOTE. - Recognize that the solution above makes' no use of the explicit 
expressions for 05^1 and s^ because their values are tabulated. The use of the 
explicit formulas for o^ or a^ in such a case would be a complicated, and 
therefore an incorrect method. 



EXERCISE XXE 

1. (a) In Table VII verify the entry for (^ at .02) = (1 ' 2) J! ~ 1 

.02 

by use of Table V. (&) Verify the entry for (0^ at .04) in Table VTH by 
use of Table VI. 

2. Find the present value and the amount of an annuity which pays 
$500 at the end of each year for 20 years, if money is worth (.05, m => 1). 

3. If money is worth (.05, m = 2), find the present value and the amount 
of an. annuity whose annual rent is $240, payable semi-annually for 13 
years and 6 months. 

Find the present values and the amounts of the annuities below. 



PBOB. 


EACH 
PAYMENT 


PAYMENT 
INTERVAL 


THEM 


ANNUAL RUNT 


INTEREST RATH 


4. 


$ 50 


3 mo. 


14 yr., 9 mo. 




.06, m = 4 


5. 


10,000 


1 yr. 


18 yr. 




.065, m = 1 


6. 


500 


6 mo. 


19 yr., 6 mo. 




.07, m = 2 


7. 




6 mo. 


15 yr. 


$1000 


.055, m = 2 


8. 


300 


, 1 vr. 


25 yr. 




.04, m - 1 


&, 




6 mo. 


23 yr. 


2000 


.03, m <- 2 


10. 




1 mo. 


7yr. 


2400 


.06, TO 12 



ANNUITIES CERTAIN 

In purchasing a house a man agrees to pay $1000 cash and 
at the end of each 6 months for the next 6 years. If money is worth 
(.07,. m - 2), what would be an equivalent cash valuation for the house? 

HINT. The cash price is the sum of the present values of all payments. 
The present value of the first payment is $1000. The remaining 12 payments 
come at the ends of the payment intervals and hence form a standard annuity 
.whose present value is W00(aj^ at .035). 

12. The man of problem 11 has just paid the installment due at the end 
of 4 years and 6 months. What additional payment, if made immediately, 
would cancel his remaining indebtedness if money is worth (.08, m = 2) ? 

HINT. His remaining indebtedness at any time, or the principal out- 
standing, is the present value of all remaining payments. 

13. If you deposit $50 at the end of each 3 months in a savings bank 
which pays interest quarterly at the rate 3%, how much will be to your 
credit after 20 years and 6 months, if you make no withdrawals? 

14. A man in buying a house has agreed to pay $1000 at the beginning 
of each 6 months until 29 installments have been paid. If money is worth 
6%, compounded semi-annually, what is an equivalent cash price for the 
house? 

15. At the end of each year a corporation places $5000 in a depreciation 
fund which is to provide for plant replacement at the end of 12 years, 
(a) What sum will be in the fund at the end of 12 years if it accumulates 
at the effective rate 7% ? (&) What sum is in the fund at the beginning of 
the seventh year? 

16. A man desires to deposit with a trust company a sufficient sum to 
provide his family with $500 at the end of each 3 months for the next 15 
years. If the trust company credits interest at the rate 6%, quarterly, on 
all funds, what should the man deposit? 

HINT. See the table of Note 2 of Section 19. 

22. Further annuity formulas. Consider the annuity whose 
annual.rent is $1, payable p times per year for n years. Each of 
the np payments is ^ ; the first is due at the end of - years, and 
the others are due at intervals of - years for the rest of the term. 

If money is worth the rate i, compounded annually, let (s at i} 
represent the amount of the annuity and (og, at i) its present 



46 



MATHEMATICS OF INVESTMENT 

we form the table 



value. To derive formulas for a^ and 
below. 



SI 
PAYMENT OF 

DUE AT THE 
END OF 


PRESENT VALUE OF 
THE PAYMENT 


Too, FBOM DATE OF 
PAYMENT TO END 
OF TERM 


COMF. AMT. AT END OF 
THBM IF PAYT. IB LEFT 
TO ACCUMULATE AT 
INTBBBST 


1 

- years 

P 


P 


(n-^yr. 


P 


2 

-years 

P 


P 


(o\ 
n ) yr. 
P' 


P 


etc. 


etc. 


etc. 


etc. 


('-;)" 


P 


- years 
P 


P 


n years 


P 


years 


P 



Hence, on adding the fourth column we obtain 



The progression in the bracket has the ratio w = (1 + i)p, the 
first term a = 1, and the last term L = (1 + i) n ~5. Since wL a 
= (1+ i) n - 1, andw - 1 = (1 + i^ - 1, 



= ^ + V ~ ' . (29) 

P[(l + 1)3 - JT] 

The denominator of the last fraction is the expression we have 
previously called 1 (j p at ) . On multiplying numerator and denom- 
inator of the last fraction by i, we obtain 



Since, by formula 27, the last fraction is (s^ at i) t 



(30) 



1 See Exercise X, Problem 16. Also see heading of Table XI. The fact that 
O'n at t) is the nominal rate which, if converted p times per year, yields the effective 
rate -i, is of importance in the applications of equation 29. We use j f merely as a 
convenient abbreviation for its complicated algebraic expression. 



ANNUITIES CERTAIN 47 

From the second column of the table we obtain 

(off ati) = %l + t)- + (1 + i^* 1 * + - etc. - + (1 + i)-Jj.' 
p J 

The ratio of the geometrical progression in the bracket is 

w = (\ + 0*i the first term a = (1 + 0"*, and L = (1 -f 0~X 
Since wL - a = 1 - (1 + i)~*, 



From this expression we derive, as in formula 30, 

(<#>af lO-lfo of i). (32) 

fa 

If the sum of the payments made in 1 year, or the annual rent, 
had been $12 instead of $1, the present value of the annuity would 
have been R(a$ at i) and the amount, R(&j\ at {). 

In the discussion above, money was worth the rate i, compounded 
once per year, while the annuity was payable p times per year for 
n years and the sum of the payments made in 1 year was $1. The 
word year was used in this statement and in the proof of formulas 
29 to 32 for the sake of concreteness. All of the reasoning remains 
valid if the word year is changed throughout to interest period. 
Thus, when money is worth the rate i, per interest period, if an 
annuity is payable p times per interest period for a term of n 
interest periods, and if the sum of the payments made in one in- 
terest period is $R, the present value A and the amount S are 
given by 



? (33) 

S = R(s^ati) =R4-(s^ati). 
Jp 

Example 1. If money is worth (.05, m = 2), find A and S for an 
annuity of fifty quarterly payments of $100 each, the first due at the end 
of three months. 



48 MATHEMATICS OF INVESTMENT 

Solution. Payments occur twice in each interest period. Hence, use 
formulas 33 with the data listed below. Tables XII, VIH, and VII are used in 
computing. 



n = 2(12.6) 25 int. periods, 
p =2, R - $200, i t = .025. 



A - 200(0^ at .025) = 200 ^(o^ .025), 
A= 200(1 .00621142) (18.42437642), 



A - $3707.76. 



-025 / 



B - 200(a^ at .025) - 200^(8^ of .026) = 200(1.00621142) (34.15776393), 

S = $6873.99. 

Example 2. If money is worth (.06, m = 4), find A and 5 for an 
annuity whose annual rent is $1000, payable monthly for 12 years and 
3 months. 

Solution, Use formulas 33, because the payments occur three times in 
each interest period. 



4(12^) = 49 int. periods, 
3, R = $250, i - .015. 



A 250(ogai .015) = 250 ^(a^ at .015), 
A =250(1.00498346) (34.52468339). 
8 - 260(g, of .015) - 250^(s^ of .015) - 250(1.00498346) (71.60869758). 

NOTHJ. When p = I, formulas 29 and 31 reduce to formulas 27 and 28, or 
(s^j* at i} = (s^ at i) and (aj^ erf i) = (o^ oi i) . We may obtain the same 
results on placing p = 1 in formulas 30 and 32, because (j\ at i) 1[ (1 + ^) 

- 1] - i and ^-= 1. These results could have been foretold because, when 

Ji 

p = 1, the payment interval equals the interest period, and hence formulas 27 
and 28 apply as well as formulas 29 and 31. In the future think of (s^ erf i) as 
(a~? at i), with the value of j> left off and understood to be p = 1 (just as we 
omit the exponent 1 in algebra when we write x instead of x l ). Thus, formulas 
30 and 32 express the present value and the amount of an annuity payable 
p tunes per interest period in terms of the present value and the amount of an 
annuity payable once per interest period. 

EXERCISE XXm 

1. Verify the entry in Table XII for i = .06 and p = 2. 
HINT. ^ =s '^i. . from Table XI. Complete the division. 

2, If money is w.orth (.06, m =2), find the present value and the 
amount of an annuity whose term is 9 years and 6 months, and whose 

rent is $1200, payable monthly. 



ANNUITIES CERTAIN (49O 

Compute the present values and the amounts of the annuities below. 



PHOB. 


ANNUAL 
RENT 


EACH 
PAYMENT 


PAYMENT 
INTERVAL 


TEBM 


INTEREST BATE 


3. 


$1000 




6 mo. 


15 yr. t 


.05, TO = 1 


4. 


6000 




1 mo. 


12 yr. 


.06, m = 1 


6. 




$500 


6 mo. 


9 yr., 6 mo. 


.07, m - 2 


6. 




226 


3 mo. 


19 yr. 


.05, TO = 1 


7. 




- 200 


3 mo. 


8 yr., 6 mo. 


.08, TO = 2 


8. 


2000 




3 mo. 


10 yr., 6 mo. 


.055, TO = 2 


9. 


600 




1 mo. 


6 yr., 3 mo. 


.06, m = 4 


10. 




760 


3 mo. 


Syr. 


.04, m = 1 



i 11. In buying a farm it has been agreed to pay $100 at the end of each 
month for tHe next 25 years. If money is worth the effective rate 7%, 
what would be an equivalent cash valuation for the farm?*/^ <*/-_'?, * 
'"1 12. If $50 is deposited in a bank at the end of every month for the next 
15 years and is, left to accumulate, what will be on hand at the end of 15 
years if the bank pays 6%, compounded annually on deposits f )Jfa $ **-'. 6> 

13. A sinking fund is being accumulated by payments of $1000, made 
at the end of each 3 months. Just after the 48th payment to the fund has 
been made, how much is in the fund if it accumulates at (.045, m 1) ? 

14. An investment yields $50 at the end of each 3 months, and pay- 
ments will continue for 25i years. What is a fair valuation for the 
project if money is worth (.05, m 2) ? 

16. How much could a railroad company afford to pay to eliminate a 
dangerous crossing requiring the attention of two watchmen, each re- 
ceiving $75 per month, if money is worth (.04, m = 1) ? Assume that 
the crossing will be used for 50 years. 

16. Prove from formula 29, that (dfi at i) = -?-. Thus - is the sum 

which, if paid at the end of 1 year, is equivalent to p payments of 
made at equal intervals during the year. 

23 , The most general annuity formulas . Consider the annuity 
whose annual rent is $1, payable p times per year for n years. To 
find the present value and the amount of this annuity when money 
is worth the nominal rate j, compounded m times per year, we 
might first compute the corresponding effective rate i and then 
use formulas 29 and 31. It is better to use equation 17 to obtain 



50 MATHEMATICS OF INVESTMENT 

entirely new formulas in terms of the given quantities j and m. 
From equation 17, 



(i + i) = (l + } ; (l + i)-" = (l + 

\ m/ \ m 

On substituting these expressions in formulas 29 and 31 we obtain 



(34) 



If the annual rent of the annuity above were $R instead of $1, 
the present value would be R(a$ atj, m) and the amount would 
beRdJjgatj, m). 

NOTE. Formulas 34 include all previous formulas as special cases, because 
wnenwt - landj = i, formulas 34 reduce to formulas 29 and 31, from which we 
started. Thus, think of ( at $ as being ($> at j = i, m = 1) with the value 

of m left out and understood to be m - 1. Likewise, (a-, at i) = (a^? at 

3 = Vm = 1). ^i ' x ^l 

24. Summaiy. For an annuity under Case 1 below we usually 
may compute the present value A and the amount 8 by means of 
our tables. For an annuity under Case 2, the explicit formulas 
for A and S must be computed with much less aid from the tables. 
Case 1. The annuity is payable p times per interest period 
where p is an integer. The method of Section 22 applies, with 
additional simplification when p 1. If 

P = the number of payments per interest period, 
n = the term, expressed in interest periods, 
i = the rate, per interest period, and 

$R = the sum of the payments made in one interest period, then 
A = Z(a% at f) = R ati) S = *(,> at i) = R- at f. I 



ANNUITIES CERTAIN 51 

When p = 1, $R is the annuity payment, n is the number of pay- 
ments, and 

ti). (H) 



The values of A and S in I and II can usually be computed by 
Tables VII, VIII, and XII. 

NOTE. One or more of Tables VII, VIII, and XII will not apply if i is not a 
table interest rate, or if n is not an integer. In that case the explicit formulas 
29 and 31 for (a^ at i) and (^ at i) must be computed. 

Case 2. The annuity is not payable an integral number of 
times per interest period. The general formulas 34 must be used, 
and if 

n = the term in years, p = the number of payments per year, 
$R = the annual rent, j = the nominal rate, and 
m = the number of conversion periods per year, then 



S = tf atj, m) = 

*[(' + ) J - 'J 

STJPPLHMENTAHY NOTE. From formulas II and III it can be proved that 



/ 

\ 



m 

These formulas can be used to simplify the computation of the present 
values and the amounts of many annuities coming under Case 2. Other sim- 
plifying formulas could be derived but they would not be of sufficiently gen- 
eral application to justify their consideration. 

Example 1. An annuity will pay $500 semi-annually for 8 years. 
Find the present value A if money is worth (.06, m = 4). 

Solution. The annuity comes under Case 2. 

A - 1000(a^ at .06, m - 4), 



Case 2 

n = 8 years, p 2, 

j - .06, m *= 4, JB - $1000. 



52 MATHEMATICS OF INVESTMENT 

Exampk 2. In buying a house a man has agreed to pay $1000 cash, 
and $200 at the end of each month for 4 years and 3 months. If money 
is worth (.06, m =.2), what would be an equivalent cash price for the 
property? 

Solution. First disregard the cash payment. The other payments form 
an annuity under Case 1 whose present value is 



Casel 

7i = 8.5 int. periods, 
p = 6, i = .03, R - $1200. 



A = 1200(a^ at .03) = 1200 ^(Og^ a< -03), 



A = 1200(1.01242816) 



1 

A ~ 



(1.03)-"- B = (1.03)~ 9 (1.03)* = (.76641673) (1.01488916) = .7778280. 
A = 1200(1.01242816) (1 - .7778280) _ jg 997 33 

.03 
The equivalent cash price is $1000 + $8997.30 = $9997.33. 

Example 3. At the end of each 3 months a man deposits $60 with a 
building and loan association. What sum is to his credit at the end of 
4 years if interest is accumulating at the rate (.075, m = 2), from the date 
of each deposit? 

Solution. The amount on hand is the amount of an annuity which comes 
under Case 1. 



Casel 

n 8 int. periods, 
p = 2, i - .0375, R = $100. 



S = 1000$' at .0375), 
S = 100 (1-0375) 8 - 1 . (Formula 29) 
2[(1.0375)* - 1] 



} log (1.0375) = 0.0079940, from Table II. S 

(1.0375)* = 1.018577, from Table II. 

8 log (1.0375) = 8C0159881) = 0.12790. S = = $921.8. 

.037154 
(1.0375) 8 = 1.3425, from Table I. 

The answer is not stated to five digits because the numerator 34.25 was obtain- 
able only to four digits from Table I. 

NOTE. In every problem where the present value or amount of an annuity 
is to be computed, first list the case and the elements of the annuity as in the 
examples above. 

NOTE. To find A and S for an annuity we could always proceed as under 
Case 2, even though the annuity comes under Case 1. Thus, for the annuity of 
Example 3 above, the term is n = 4 years, the annual rent is R = $200, 
payable p = 4 times per year, j = .075, and m = 2. Hence, from formulas 
in of Case 2, 

S = 2000$ at .075, m = 2) = 200 P- 0375 ) 8 ~ 1 , 

4[(1.0376) - 1] 



ANNUITIES CERTAIN 



53 



which is the same as obtained above. The only difference in method is that, 
under Case 2, the fundamental time unit is the year, whereas under Case 1 it is 
the interest period. The classification of annuity computations under two 
cases would not be advisable if we were always to compute A and S by the 
explicit formulas, as is necessary in Example 3. But, if we used the general 
formulas of Case 2, with the year as a time unit, in problems under Case 1 to 
which Tables VII, VIII, and XII apply, unnecessary computational confusion 
would result and other inconvenient auxiliary formulas would have to be 
derived. Hence, use the method of Case 1 whenever possible. 

EXERCISE XXIV 

Compute A and S for each annuity in the table. Use Table II when 
it is an aid to accuracy. 



PHOB. 


ANNUAL 
RENT 


EACH 
PAYMENT 


PAYMENT 
INTERVAL 


TERM 


INTEREST RATE 


1. 


$10,000 




1 month 


15 years 


.05, m = 4 


2. 




$ 400 


1 month 


12 years 


.06, m = 1 


3. 




2500 


6 months 


19 yr., 6 mo. 


.05, m = 4 


4. 


500 




8 months 


7 yr., 6 mo. 


.05, m = 2 


6. 


240 




3 months 


11 yr., 6 mo. 


.04, m = 4 


6. 




150 


1 year 


18 years 


.09, m = 4 


7. 


100 




6 months 


28 years 


.05, m = 1 


8. 


5,000 




3 months 


6 yr., 9 mo. 


.07, m =2 


9. 




125 


6 months 


10 years 


.0626, m = 1 


10. 


2,000 




1 year 


15 years 


.05, m = 2 


11. 


900 




3 months 


9 yr., 3 mo. 


.08, m = 4 


12. 




700 


6 months 


20 years 


.005, m = 1 


13. 




50 


3 months 


30 years 


.048, m =2 


14. 




100 


4 months 


9 years 


.04, m = 2 


16. 


3,000 




3 months 


12 years 


.04, m = 2 


16. 


500 




1 year 


35 years 


.07, m =2 


17. 




200 


6 months 


12 years 


.055, m = 2 


18. 


1,200 




6 months 


15 yr., 6 mo. 


.03, m = 4 


19. 




150 


3 months 


6 yr., 3 mo. 


.06, m = 2 


20. 




250 


4 months 


9 years 


.04, m = 2 


21. 


' 500 




6 months 


10 years 


.045, m = 2 


22. 




25 


1 month 


17 years 


.06, m = 1 



23. To provide for the retirement of a bond issue at the end of 20 years, 
a city will place $100,000 in a sinking fund at the end of each 6 months, 
(a) If the fund accumulates at the rate (.05, m = 2), what sum will be 
available at the end of 20 years? (&) What sum is in the fund at the 
beginning of the 12th year? 



54 MATHEMATICS OF INVESTMENT 

24. An investment will yield $50 at the end of each month for the 
next 15 years. If money is worth (.05, m = 4), what would be a fair 
present valuation for the project? 

25. A depreciation fund is being accumulated by semi-annual deposits 
of $250 in a bank which pays 5%, compounded quarterly. How much will 
be in the fund just after the 30th deposit? 

26. A will decrees that X shall receive $1000 at the beginning of each 
6 months until 10 payments have been made. If money is worth (.06, 
m = 2), on what sum should X's inheritance tax be computed, assuming 
that the payments will certainly be made ? 

27. A certain bond has attached coupons for $5 each, payable at the 
end of each year for the next 25 years. If money is worth 5% effective, 
find the present value of the coupons. 

28. The bond of problem 27 will be redeemed for $100 by the issuing 
corporation at the end of 25 years. What should an investor pay for the 
bond if he desires 5% effective on his investment? 

HINT. He should pay the present value of the coupons plus the present 
value of the redemption price. 

29. A farm is to be paid for by 10 successive annual installments of 
$5000 in addition to a cash payment of $15,000. What is an equivalent 
cash price for the farm if money is worth (.05, m = 2) ? 

SO. (a) At the end of the 5th year in problem 29, after the payment 
due has been made, the debtor wishes to make an additional payment 
immediately which will cancel his remaining liability. The creditor is 
willing to accept payment if money is considered worth 4% effective. 
What does the debtor pay? (6) Why should the creditor specify the 
rate 4% effective instead of a higher rate, (.05, m = 1) for instance? 

HINT. Find the present value of the remaining payments. 

81. A man has been placing $100 in a bank at the end of each month 
' for the last 12 years. What is to his credit if his savings have been ac- 
cumulating at the rate 6%, compounded semi-annually from their dates 
\ of deposit? 

x 32. A man wishes to donate immediately to a university sufficient 
money to provide for the erection and the maintenance, for the next 50 
years, of a building which will cost $500,000 to erect and will require $1000 
at the end of each month to maintain. How much should he donate if 
the university is able to invest its funds at 5%, converted semi-annually ? 



ANNUITIES CERTAIN 55 

33. A certain bond has attached coupons for $5 each, payable semi- 
annually for the next 10 years. At the end of 10 years the bond will be 
redeemed for $125. What should an investor pay for the bond if he 
desires 6%, compounded semi-annually, on his investment? 

HINT. See problem 28. 

34. A man, who borrowed a sum of money, is to discharge the liability 
by paying $500 at the end of each 3 months for the next 8 years. What 
sum did he borrow if the creditor's interest rate is (.055, m 2) ? 

36. (a) In problem 34, at the end of 4 years, just after the installment 
due has been paid, what additional payment would cancel the remaining 
liability if money is still worth (.055, m = 2) to the creditor? (&) What 
would be the payment if money is worth (.04, m = 4) to the creditor? 

36. A man X agreed to pay $1000 to his creditor at the end of each 
6 months for 15 years, but defaulted on his first 7 payments, (a) What 
should X pay at the end of 4 years, if money is worth (.06, m = 2) to his 
creditor? (6) What should he pay if money is worth (.05, m = 2) ? . 

37. In problem 36, at the end of 4 years, X desires to make a single 
payment which will cancel his liability due to his previous failure to pay, 
and also will discharge the liability of the payments due in the future. 
(a) What should he pay if money is worth (.06, m = 2) to his creditor? 

. (6) Find the payment if the rate is (.05, m = 2). 

38. A certain bond has attached coupons of $2 each, payable quarterly 
for the next 20 years, and at the end of that time the bond itself will be 
redeemed for $110. What should a man pay for the bond if he considers 
money worth 6%, -effective? 

39. Prove by use of formulas 34 that the present value of an annuity, 
accumulated at the rate (j, m) for n years, will equal the amount of the 
annuity ; that is, prove algebraically that 



atj, m)l +- = B( atj, m) = S. 



Another statement of this result would 'be that "A is the present value of 
S, due at the end of the term of the annuity," 

40. What is the amount of an annuity whose term is 14 years, and whose 
present value is $1575, if interest is at the rate (.06, m = 2) ? 

HINT. Use the result of problem 39. 

41. What is the effective rate of interest in use if the present value of 
an annuity is $2500, the amount $3750, and the term 10 years? 



56 MATHEMATICS OP INVESTMENT 

42. A will bequeaths to a boy who is now 10 years old, $20,000 worth 
of bonds which pay 6% interest semi-annually. The will requires that 
half of the interest shall be deposited in a savings bank which pays 4%, 
compounded quarterly. The accumulation of the savings account, and 
the bonds themselves, are to be given to the boy on his 25th birthday. 
Find the value of the property received by him on that date. 

43. A man desires to deposit with a trust company a sufficient sum to 
provide his family an annuity of $200 per month for 10 years. What 
should he deposit if the trust company will credit interest at the rate 
5% compounded quarterly, on the unexpended balance of the fund? 

44. If you can invest money at (.03, m = 2), what is the least sum you 
would take at the present time in return for a contract on your part to pay 
$100 at the end of each 6 months for the next 15 years? 

45. If money is worth (.04, m = 1), is it more profitable to pay $100 
at the end of each month for 3 years as rent on a motor truck, or to buy one 
for $3000, assuming that the truck will be useless after 3 years? Assume 
in both cases that you would have to pay the upkeep. 

25. Annuities due. The payments of the standard annuities 
considered previously were made at the ends of the payment 
intervals. An annuity due is one whose payments occur at the 
beginning of each, interval, so that the first payment is due im- 
mediately. The definitions of the amount and of the present value 
of an annuity as given in Section 19 apply without change of word- 
ing to an annuity due. It must be noticed, however, that the 
last payment of an annuity due occurs at the beginning of the last 
interval, whereas the end of the term is the end of this interval. 
Hence, the amount of an annuity due is the sum of the compound 
amounts of the payments one interval after the last payment is 
,made. For an annuity due whose annual rent is $100, payable 
quarterly for 6 years, the last $25 payment is made at the end of 
5 years and 9 months. The amount of this annuity is the sum of 
the compound amounts of the payments at the end of 6 years, the 
end of the term. 

For the treatment of annuities due and for other purposes in 
the future, it is essential to recognize that, regardless of when a 
sequence of periodic payments start, they will form an ordinary 
annuity if judged from a date one payment interval before the 
first payment. Hence, one interval before the first payment, the 



ANNUITIES CERTAIN 57 

sum of the discounted values of the payments is the present value 
of the ordinary annuity they form. Moreover, the sum of the 
accumulated values of the payments on the last payment date is 
the amount of this ordinary annuity. 

Example 1. If money is worth (.05, m = 2), find the present value A 
and the amount S of an annuity due whose annual rent is $100, payable 
quarterly for 6 years. 

Q > increasing time . 

1 X X X K X X X X X X X X X )( X X X X )( )( )( ?( )( X -I 

N L 

represents 3 months N represents the present 

FIG. 3 

Solution, Consider the time scale in Figure 3, where X represents a payment 
date, T is the end of the term, 6 years from the present, and L is the last pay- 
ment date, 3 months before T. Q is 3 months before the present. Considered 
from Q the payments form an ordinary annuity whose term ends at L and 
whose present value A' and amount S' are 



Case 1 
12 int. periods, 



A' = 50(0^0^.025), 



Since A' is the sum of the discounted values of the payments at Q, 3 months 
before the present, we accumulate A' for 3 months to find A, the present value 
of the annuity due. 

A - ^'(1.025)* - 50(og,ai .025) (1.025)*. 

Since S' is the sum of the accumulated values of the payments at L, we ac- 
cumulate S' for 3 months to find S, which is the sum of the values at time T. 

S = S'(1.Q25)* = 50(3^ at .025) (1.025)*. 
Tables VII, VIII, X, and XII would be used to compute A and S. 

Second solution. The first $25 payment is cash and the remaining pay- 
ments form an ordinary annuity, as judged from the present. Its present 

value A ' is 



Case 1 

n = 11.5 int. periods, 
p = 2, i - .025, R = $50. 



4' = 50(0^ of .025). 
Hence, A = 25 + 60(ogL at .025). 



To find S, first consider a new annuity consisting of all payments of the an- 
nuity due, with an additional $25 due at tune T. Since T is the last payment 
date of the new sequence of payments, the sum of their values at time T is the 
amount S f of an ordinary annuity, or 



58 



MATHEMATICS OF INVESTMENT 



Case 1 

n = 12.5 int. periods, 
p - 2, i = .026, R = $50. 



fl' - SOCsgj, a* .025). 

The value of the additional $25 payment at 
time T is included in S', or S' = S + 25. 
S - 60(8^, at .025) - 25. To find the nu- 
merical values of A and S, a^-Q and fij^ must be computed from formulas 
29 and 31. Hence, the first solution was less complicated numerically. In 
some problems, however, the second solution would be the least complicated. 

Two rules may be stated corresponding, respectively, to the 
two methods of solution considered above. 

Rule 1. To find A and S for an annuity due, first find the 
present value A' and the amount S' of an ordinary annuity having 
the same term, a.nmifl.1 rent, and payment interval. Then : 
(a) A is the compound amount on A' after one payment interval. 
(6) S is the compound amount on S' after one payment interval. 

Rule 2. To find A for an annuity due, first find A', the 'present 
value of all payments, omitting the first. Then, if W is the annuity 
payment, A = A' + W. To find S first obtain S 1 , the amount 
of the ordinary annuity having a payment at the end of tho 
term in addition to the payments of the annuity due. Then, 
8-S'-W. 

NOTE. It is customary in actuarial textbooks to use black roman type 
to indicate amounts and present values of annuities duo. Thus (s^i at j, m) 
represents the amount, and (a^ at j, m) the present value of an annuity duo 
whose annual rent is $1, payable p times per year for n years, if money is worth 
0', ) 

EXERCISE XXV 

In each problem draw a figure similar to Figure 3. Find A and S for 
each annuity due in the table, by use of the specified rule. 



PHOB. 


TEBM 


PAYMENT 

iNTBHVALi 


ANNUAL 
RENT 


INTHRHBT 
RATH 


BUIiM 


1. 


10 yr. 


3 mo. 


$ 300 


.06, m - 4 


2 


2, 


7 yr., 6 mo. 


6 mo. 


500 


.05, m " 2 


2 


3. 


12 yr., 6 mo. 


6 mo, 


3600 


.03, m - 1 


2 


4. 


12 yr. 


3 mo. 


1000 


.06, m 4 


1 



ANNUITIES CERTAIN 

6. Carry through, the solution of problem 2 by Rule 1 far enough to 
be able to state why it is inconvenient. 

6. A man deposited $100 in a bank at the beginning of each 3 months 
for 10 years, (a) What sum is to his credit at the end of 10 years if the 
bank credits 6% interest quarterly from the date of deposit? (6) What 
sum is to his credit at the end of 9 years and 9 months, after the deposit 
hag been made at that time? 

J 7. In purchasing a house, a man has agreed to pay $100 at the beginning 
of each month for the next 5 years, (a) If money is worth 6% effective, 
find the present value of the payments. (6) If money is worth (.06, 
TO = 12), find their present value. { *\ l f / 7, ', ;', av V if $ * > 

8. A man was loaned $75 on the 1st of each month, for 12 months each 
year, during the four years of his college course, (a) If his creditor con- 
siders money worth 3% effective, what is the liability of the debt at the 
end of the 4 years? (6) If the debtor makes no payment until four 
years after he graduates, .what should he pay then to settle in full? 

9. If money is worth the effective rate i, prove that (a_, at i), the 
present value, and (s-, at i), the amount of an annuity due of $1 payable 
annually for n years, are given by 

(aj] at t) = l + (o^ at i), (s^ at t) = (^ at t) - 1. 
10. Prove that 
(sjjy at j, w) = (1 + i)'(aj| at j, m) ; (a, at j, w) = (1 + t)(a^ atj, m). 

26. Deferred annuities. A deferred annuity is one whose term 
does not begin until the expiration of a certain length of time. 
Thus, an annuity whose term is 6 years, deferred 4 years, and whose 
annual rent is $1000, payable semi-annually, consists of 12 pay- 
ments of $500, the first due after (4 years -+ 6 months) and the last, 
after (4 years + 6 years). 

Example 1. If money is worth (.05, m = 1), find A and S for the de- 
ferred annuity of the last paragraph. 
2f J3 T 

1 o o o o o o o o >( x x x x >< xx x x x 

> increasing time " " represents C months 

FIG. 4 

In Fig. 4, X represents a payment date of the deferred annuity, N, the 
present, T, the end of 10 years, the end of the term, and B, the beginning of 
the term, 



60 



MATHEMATICS OF INVESTMENT 



Solution. Consider the time scale in Figure 4. The payments form an 
ordinary annuity when judged from B, 6 months before the first payment. 
S', the amount, and A', the present value (at B), of this ordinary annuity are 



p 


n = 
-2,t 


Case 1 
6 int. periods, 
= .05, R = $1000. 



(2) 



S'-1000(BJjfoi.05). 
A' = 1000 (a^at .05). 



S' is the sum of accumulated values at time T. Since 8, the amount of the 
deferred annuity, is also equal to the sum of values at T, 

S = S' = 1000 (sjj at .05). 

Since A' is the sum of the discounted values at B, we must discount A' 
for 4 years to obtain the present value A. 

A - 



Second solution for A. Consider a new annuity having payments of $500 at 
the end of each 6 months for the first 4 years as well as for the last 6. The 
new payment dates are indicated by circles in Figure 4. The present value A 
of the deferred annuity equals the present value A ' of the new annuity over the 
whole 10 years minus the present value A" of the payments over the first 
4 years, which are not to be received. Both A.' and A" are the present values 
of ordinary annuities. 



Case 1 

n = 10, and 4, int. periods, 
p = 2, i = .05, R = $1000. 



A = A' -A 1 



1000[(o^, 
1000 ^[( 

Jt 



A' = 1000(ogj at .05). 
^" = 1000(0^ at .05). 

a .05) - (aj? at .05)], 
ojol a* -05) - (aji at .05)]. 



From Example 1 it is clear that the amount of a deferred annuity 
equals the amount of an ordinary annuity having the same term. 

Corresponding to the two methods used above in obtaining A, we 
state the two rules below. 

Rule 1. To obtain A for an annuity whose term is deferred 
w years, first find A', the present value of the ordinary annuity 
having the same term. Then, A equals the value of A' discounted 
for w years. 

Rule 2. If the term of the annuity is n years, deferred w years, 
then 



ANNUITIES CERTAIN 



A = [present value of an ordinary annuity with term (w + n) years] 
(present value of an ordinary annuity with term w years), 

where these new annuities have the same annual rent and payment 
interval as the deferred annuity. 

NOTE. The present values and the amounts of deferred annuities are 
indicated in actuarial writings by the symbols for ordinary annuities with a 
number prefixed showing the time for which the term is deferred. Thus 

( a-, atj, m) and ( a-, atj, m} 
x n\ " ' w n\ *" 

represent the present value and the amount when the term is deferred w years. 

EXERCISE XXVI 1 

In each problem draw a figure similar to Figure 4. Find the present 
value of each deferred annuity in the table, by use of the specified rule. 



PBOH. 


TERM 


TERM DEFERRED 


PAYMENT 
INTERVAL 


ANNUAL 
RENT 


INTEREST RATE 


RULE 


1. 


6yr. 


10 yr., 6 mo. 


3 mo. 


$1000 


.05, m => 4 


2 


2. 


7yr. 


8 yr., 6 mo. 


1 mo. 


200 


.06, m = 2 


1 


3. 


9 yr. 


12 yr. 


1 yr. 


300 


.07, m = 1 


2 


4. 


13 yr. 


10 yr., 6 mo. 


1 mo. 


1200 


.05, m = 1 


1 



5. Carry through the solution of problem 4 by Rule 2 until you are 
able to state why it is inconvenient. 
I 6. Solve problem 3 if money is worth (.07, m = 2). 

7. A man will receive a pension of $50 at the end of each month 

10 years, first payment to occur 1 month after he is 65 years old. Assum- 
ing that he will live to receive all payments, find the present value of his 
expectation if money is worth (.04, m = 1), and if he is now 50 years old. 

8. A certain mine will yield a semi-annual profit of $50,000, the first 
payment to come at the end of 7 years, and the last after 42 years, at which 
time the mine will become worthless. What is a fair valuation for the 
mine if money is worth 5%, effective? 

9. A recently paved road will require no upkeep until the end of 3 
years, at which time $3000 will be needed for repairs. After that, $3000 
will be used for repairs at the end of each 6 months for 15 years. Find 
the present value of all future upkeep if money is worth (.05, m = 2). 

1 The Miscellaneous Problems at the end of the chapter may be taken up im- 
mediately after the completion of Exercise XXVI. 



62 MATHEMATICS OF INVESTMENT 

10. By use of Rules 1 and 2 prove the relations below, for an annuity 
whose annual rent is $1, payable p times per year, and whose term is n 
years, deferred w years. 



. 
SUPPLEMENTARY MATERIAL :/ >."~' 

27. 1 Continuous annuities. If money is worth the effective 
rate i, the present value of an annuity whose annual rent is $1, 
payable p times per year, is 

ti) = 1 (a-, ati). 
JP 

We may consider an annuity payable weekly, p = 52, or daily, 
p = 365, and we may ask what value does (o^| at i) approach as 
p becomes large without bound? We obtain 

lim (a-, at i) = i(a-, at i) lim ; 
P=OO v i w| ' z*-o (j p at i) 

But, from Section 18 we have lim j p = d, the force of interest 

p=00 

corresponding to the effective rate i. Hence 

,. , o , .v i(a-] at i) 
hm fe at t) = aL -- 

P-03 "' ' 5 

In the same way it can be shown that 



g 

As p = oo, the annuity approaches the ideal case of an annuity 
whose annual rent is payable continuously. If we let (a^ at i) 
represent the present value and (s- at <) the amount of a contin- 
uous annuity, the results above show that 

/- , -N ^C a i at *) /- ^ ^ ^( s l at 

(a^, at i) = -i-aL - , (^ ai t ) -- -L -- (35) 



Recall that 1 + i - (* so that 6 = lo g-^ + ^- Since log e 

loge 

log 2.7182818 = 0.4342945, we obtain from equation 35, 

1 Section 18 is a prerequisite for the reading of this section. 



ANNUITIES CERTAIN 63 

i(a-i ati) (.4342945) tfa of a) (.4342945) 

(a-, ai t) = ^j - ,., , .. - , (s n a i) = , ,., , . N - . 
n| ' log (1 + 1) ' v nl log(l + &) 

The present value and the amount of an annuity, which is 
payable continuously, differ but slightly from the corresponding 
quantities for an annuity which is payable a very large number of 
times per year (see problem 1 below). Hence we may use o^ and 

I-| as approximations for a^ and s^ if p is very large. 

EXERCISE XXV3I 

1. (a) If money is worth 6%, effective, find the present value and the 
amount of an annuity whose annual rent is $100, and whose term is 10 
years, if the annuity is payable continuously. (6) Solve the problem if 
the annuity is payable monthly. 

2. If one year equals 360 days, find approximately the amount of an 
annuity of $1 per day for 20 years if money is worth 4%, effective. 

HINT. Use a continuous annuity as an appropriation. 

3. A member of a labor union has agreed to contribute $.20 per day to 
a benefit fund for 3 years. Under the rate (.05, m = 1), find approxi- 
mately the present value of his agreement if a year has 365 days. 

4. An industrial insurance policy for $100 calls for a premium of 10 
cents at the end of each week. Find approximately, by use of a contin- 
uous annuity, the equivalent premium which could be paid at the end 
of the year if money is worth 3$%. 

NOTE. If the conversion period of an interest rate is not stated, assume 

it to be 1 year. *-\ l*> 

k^^V*-*-"* 
28. 1 Computations of high accuracy. The binomial theorem 

can be used in interest computations to which the tables do not 
apply. As a special case of the binomial theorem, 2 we have 



21 o ! 

- 1) (n - r + l)s r + . . . 

When 7i is a positive integer, series 36 contains (n + 1) terms, 
the last of which is x n . If n is a negative integer or a fraction, 
the series contains infinitely many terms. In this case, if x lies 

1 A knowledge of the binomial theorem ifl needed in this section. 
1 See page 93 in Bietz and Crathprne's College Algebra. 



64 MATHEMATICS OP INVESTMENT 

between 1 and + 1, the infinite series converges, and, if x is 
very small, the sum of the first few terms gives a good approxima- 
tion to the value of (1 + x) n . The proof of this statement is too 
difficult for an elementary treatment. 

Example 1. Find the value of (a^ at .033) accurately to six signifi- 
cant figures. 

Solution. From formula 31, 

(o .033)= l-ay>-. (37) 



From equation (36) with n = % and x = .033, 

(1.033)4= 1 +(.033) -TM-OSS^ + ^COS^-Te^OOSS)' + . - (38) 
4=1(1.033)^ l] = .03300000- .00040838+. 00000786 -.00000018= .03269930. 
The next term in series 38, beyond the last one computed, is negligible in the 
8th decimal place, and hence our result is accurate to the 7th decimal place 
with only a slight doubt as to the 8th. To compute (LOSS)"*, first compute 
(1.033) and then take the reciprocal; (1.033) 8 = 1.2150718; (1.033)- 
.8229966. Hence, 

"" 



The final 9 is not dependable because the final in the denominator was 
doubtful. 

EXERCISE XXVm 

/n\ /n\ 

1. Compute (s^, at .0325) and (a^ at .0325) accurately to five sig- 
nificant figures. 

2. Compute (3, at .02) accurately to the 7th decimal place. 

3 . A man pays $50 to a building and loan association at the end of each 
week. If the deposits accumulate at the rate (.06, m = 2), how much 
will be to his credit at the end of 3,,years? Obtain the result correct to 
four significant figures. 

MISCELLANEOUS PROBLEMS 

1. If $1750 is the present value of an annuity whose term is 12 years, 
what is its amount if money is worth (.05, m = 4) ? 

2. If $100 is deposited in a hank at the beginning of each month for 
10 years, what is the accumulated amount at the time of the last payment 
if interest is at the rate 6% compounded HRrni-n.nTinfl.ny on all money from 
the date of deposit? 



ANNUITIES CERTAIN 65 

3. In problem 2, what is the amount at the end of 10 years? 

4. A man W has occupied a farm for 5 years and, pending decision of 
a case in court, has paid no rent to B to whom the farm is finally awarded. 
What should W pay at the end of 5 years if rent of $100 should have 
been paid monthly, in advance, and if money is worth (.06, m = 12) ? 

5. What should W pay in problem 4 if the rent is considered due at 
the end of each month and if money is worth 6%, compounded annually? 

6. A building will cost $500,000. It will require, at the beginning 
of each year, $5000 for heat and light, $5000 for janitor service , and, at the 
end of each year, $3000 for small repairs. It is to be completely renovated, 
at a cost of $20,000 at the end of each 15 years. If the cost of the annual 
repairs is included in the cost of renovation at the end of each 16 years, 
and if the building is to be renovated at the end of 90 years, what present 
sum will provide for the erection of the building and for its upkeep for 
the next 90 years, if money is worth 6% effective ? 

7. A young man, just starting a four-year college course, estimates his 
future earning power, in excess of living expenses, at $100 per month for the 
first 3 years after graduation from college, $200 per month for the next 
7 years, and $300 per month for the next 20 years. What is the present 
value of this earning power, if money is worth 4%, effective? 

8. If the man in problem 7 should place his surplus earnings in a 
bank, what will he have at the end of his working life if his savings earn 
interest at the rate (.05, m = 2), for the first 10 years, and at the rate 
(.04, m = 1) for the balance of the time? 

9. In purchasing a homestead from the government, a war veteran 
has agreed to pay $100 at the end of 5 years, and monthly thereafter until 
the last payment occurs at the end of 9 years. What is the present value 
of his agreement if money is worth (.045, m = 1) ? 

10. At the end of 3 years, the man of problem 9 decides to pay off his 
obligation to the government immediately. What should he pay if money 
is worth (.045, m = 1) ? 

11. A savings bank accepts deposits of $1 at the beginning of each 
week during the year from small depositors who are creating Christmas 
funds. Just after the 52d payment, what will each fund amount to if 
the bank accumulates the savings at 6%, effective? 

12. A contract provides for the payment of $1000 at the end of each 
6 months for the next 25 years. What is the present value of the contract 
if the future liabilities are discounted at (.06, m = 2) over the last 15 years 
of the life of the transaction and at (.05, m = 2) over the first 10 years? 



66 MATHEMATICS OF INVESTMENT 

13. In purchasing a house it was agreed to pay $50 at the end of each 
month for a certain time. The purchaser desires to change to annual 
payments. What should he pay at the end of each year if money is con- 
sidered worth (.05, m = 2) ? 

14. At what rate of interest compounded semi-annually will $1650 
be the present value and $3500 the amount of an annuity whose term is 
14 years, if the annuity is payable weekly? Would the result be any 
different if the annuity were payable annually ? 

16. If money is worth 5%, effective, what is the least sum which you 
would accept now in return for a contract on your part to pay $50 at the 
end of each month for 20 years, first payment to occur at the end of 10 
years and 1 month? 



CHAPTER IV 
PROBLEMS IN ANNUITIES 

29. In every problem below, the payment interval of the 
annuity and the conversion period of the interest rate will be 
given or, in other words, the p and m of equations I and III of 
Section 24 will always be known. For an annuity under Case 1 
there remain for consideration the five quantities (A, S, R, i, ri). 
If three of (S, R, i, ri) are known, we use S = R(s%\ at i} to find 

the fourth ; while, if three of (A, R, i, ri) are known, we use 
A = R(a ( Q at i). If an annuity comes under Case 2, similar 

remarks apply to the four quantities (S, R, j, ri) and to (A, R, j, ri). 
Problems in which A and S were unknown were treated in Chap- 
ter III. 

30. Determination of the payment. For future convenience 
it is essential to know that 

I J -T\ ' /. _j '\ I * ~~ 7~ _J \ N""/ 



(s ; at i) (a, at i) (s at i) (a, at i) 

N n| ' x n| ' v n| ' x n| ' 

From formulas 27 and 28 we obtain 

i -!-.-* + *'d + fl" ~ * 




)V t \ 

) n U - (1 + f)-*y 



(as] ai i) (1 + i) n - 1 
and hence relation 39 is true. 
Exampk 1. Find the value of 



F) at - 
67 



68 MATHEMATICS OF INVESTMENT 

Solution. From Table IX, * ^ rN = .09634229. Therefore, from 

at .05) 



relation 39, - - - . = .09634229 - .05 = ,04634229. On doing this sub- 



traction mentally, we are able to read the result .04634229 directly from 
Table IX. 

NOTB. From Example 1 we see that, because of relation 39, Table IX 
gives the values of - - ^ as well as those of - - r It would be equally 



- - ^ - - r 

(a-, at i) (a^ at i) 

convenient to have a table of the values of - - r from which we could 



obtain those of - by adding the interest rate i. 

(a\ at i) 

Example 2. What annuity, payable quarterly for 20 years and 6 
months, could be purchased for $5000, if money is worth (.05, m = 4) ? 

Solution. The present value of the annuity is $5000. Let $x be the 
quarterly payment. 



Casel 

n 82 int. periods, 

p = 1, t = .0125, 

A = $5000, R = $x. 



5000 



8a| 



x - 5000 1 MM . - 5000(.01956437) 

(ag2l at .0125) 

x = $97.822. The annual rent is 4 x = $391.29. 

Exam-pie 3. If money is worth (.06, m = 2), find the annual rent of an 
annuity, payable quarterly for Hi years, if its amount is $10,000. 
Solution. Let $x be the sum of the payments in one interest period. 



Case 1 
n = 23 int. periods, 
p = 2, i = .03, 
S - $10,000, R = $x. 


10000 = x(.crf .03) - x^ (jjj, c 
x 100000'j at .03) 100000'j at .03) 


it .03), 
1 


03(^0* 


.03) .03 ( 


s^ at .03) 



C03081390) . 



.03 
Tables IX and XI were used. The annual rent is 2 x = $611.72. 

NOTE. Recognize that the solutions above wjere arranged so as to avoid 
computing quotients, except for the easy division by .03. 

Example 4. Find the annual rent if $3500 is the present value of an 
annuity which is payable semi-annually for 8 years. Interest is at the 
rate 5 %, compounded quarterly. 



PROBLEMS IN ANNUITIES 
Solution. Let $x be the B.nmifl.1 rent. 



Case 2 

n = 8 yr., p = 2, 

j = .05, m = 4, 

R = $x, A = $3500. 



3500 



3500 = x -32801593 
2(.02515625) 



(Tables V and VI) 



EXERCISE XXIX 



1. Compute 



a< .05) 10.379658 
Find the annual rents of the annuities below. 



to verify the entry in Table IX. 



PEOD. 


PATMHNT 
INTERVAL 


INTEREST 
RATH 


THEM 


AMOUNT 


PRESENT 
VALUE 


2. 


3 mo. 


.06, m = 4 


12 yr., 3 mo. 




$ 6,500 


3. 


6 mo. 


.05, m = 2 


17 yr., 6 mo. 


{ 8,500 




4. 


lyr. 


.04, m = 1 


15 yr. 




3,000 


5. 


6 mo. 


.05, m = 1 


Syr. 




15,000 


6. 


3 mo. 


.05, m = 2 


6 yr., 6 mo. 


3,750 




7. 


3 mo. 


.06, w = 1 


15 yr. 




4,000 


8. 


lyr. 


.05, m = 1 


17 yr. 




7,000 


9. 


6 mo. 


.05, m = 4 


12 yr., 6 mo. 


10 3 000 




10. 


1 yr. 


.07, m = 2 


9yr. 




2,500 


11. 


6 mo. 


.07, m = 2 


9yr. 


2,500 





' J 12. If money is worth 6%, effective, find the annuity, payable annually 
for 25 years, which may be purchased for $1000. 

13. If money is worth 6%, effective, find the annuity, payable annually 
for 10 years, whose amount is $1. 

14. If money is worth 4%, compounded annually, what annuity, pay- 
able annually for 15 years, may be purchased for $1 ? 

16. If money is worth the effective rate i, derive a formula for the 
payment of the annuity, payable annually for n years, which may be 
purchased for $1. 

16. If money is worth the effective rate i, derive a formula for the pay- 
ment of the annuity, payable annually for n years, whose amount is $1. 

17. In order to create a fund of $2000 by the end of 10 years, what 
must a man deposit at the end of each 6 months in a bank which credits 
interest semi-annually at the rate 3%? 



70 



MATHEMATICS OF INVESTMENT 



18. The present liability of a debt is $12,000. If money is worth 5.5%, 
compounded semi-annually, what should be paid at the end of each year 
for 10 years to discharge the liability in full? 

31. Determination of the term. If the term of an annuity is 
unknown, interpolation methods furnish the solution of the 
problem with sufficient accuracy for practical purposes. 

Example 1. For how long must a man deposit $175 at the end of 
each 3 months in a bank in order to accumulate a fund of $7500, if the 
bank credits interest quarterly at the rate G%? 

Solution. Tho deposits form an annuity whose amount is $7600. Let 
the unknown term be k interest periods. 



Case 1 

n = k int. periods, 

p = 1, i => .015, 

R - $175, 8 - $7500. 



7600 = 175(8^.015), 



at .015) = 



7500 
175 



42.857. 



n 


fag, at .015) 


33 


42.299 


k 


42.857 


34 


43.933 



From the column in Table VII for i = .015, we obtain 
the first and third entries at the left. . By interpolation, 



k - 33 + 



658 
1634 



33.341 int. periods. 



k 
The term is - = 8.34 years. However, since an annuity whose term is not on 

integral number of payment intervals has not been defined, this result is useful 
only because it permits us to make tho following statement : The $175 pay- 
ments must continue for 8.5 years to create a fund of at least $7500 ; when the 
33d payment occurs at tho end of 8.25 years, the fund amounts to less than 
$7500. 

Exampk 2. Find the term of an annuity whose present value is 
$8500 and whose annual rent is $2000, payable quarterly. Interest is at 
the rate (.06, m = 2). 

Solution. Let the unknown term be k interest periods. 

8600 = 1000(0^ at .03) - 1000 '^(a* at .03), 



Case 1 
n k int. periods, 

p 2, i .03, 
12 - $1000, A - $8500. 



OCoff at .03) - 1000 '-^ 
*' Ja 

_8500(j a ot.Q3) 



(OB ol .03) - 8600(.Q2977 831) , 8 . 



PROBLEMS IN ANNUITIES 



71 



n 


(a^ at .03) 


9 


7.786 


k 


8.437 


10 


8.530 



The first and third entries at the left are from the column 
hi Table VIII for i - .03. k 



k 
periods. The term is - 



10 - - = 9.88 interest 
744 

4.94 years. Hence, for an 



annuity whose term is 4.75 years, the present value is 
less than $8500, while the present value would be greater than $8500 if the 
term were 5 years. 

NOTE. Values of k found by interpolation in Tables VII and VIII are in 
error by less than half of the interest rate per period. 1 In interpolating, use 
three decimal places of the table entries and compute the value of k to three 
decimal places. 

. EXERCISE XXX 2 
Find the terms of the annuities below. 



PBOB. 


PAYMENT 
INTERVAL 


INTEREST 
RATE 


PRESENT 
VALUE 


AMOUNT 


ANNUAL 
RBNT 


1. 


lyr. 


.05, m = 1 




$5000 


$ 500 


2. 


6 mo. 


.06, m = 2 




7500 


250 


3. 


1 yr. 


.03, m => 1 


$8000 




400 


4. 


3 mo. 


.08, m = 4 


9000 




1000 


5. 


6 mo. 


.05, m = I 


6500 




1300 


6. 


1 mo. 


.045, m =2 




3500 


600 


7. 


6 mo. 


.05, m = 2 


8500 




1000 


8. 


3 mo. 


.05, m = 2 




8600 


1000 


9. 


1 mo. 


.03, m = 1 


4600 




2500 


10. 


1 yr. 


.07, m = 1 




7450 


700 



11. For how many full years will it be necessary to deposit $250 at the 
end of each year to accumulate a fund of at least $3500, if the deposits 
earn 5%, compounded annually? 

^ 12. The cash value of a house is $15,000. In buying it on the install- 
ment plan a purchaser has agreed to pay $1000 at the end of each 6 months 
aa long as necessary. For how long must he pay if money is worth 6%, 
compounded semi-annually? 

32. Determination of the interest. rate. 

Example 1. Under what nominal rate, converted quarterly, is $7150 
the present value of an annuity whose annual rent is $880, payable quar- 
terly for 12 years and 6 months? 

1 For justification of this statement see Appendix. Note 6. 

s See supplementary Section 33 for other problems in whioh the term ie unknown. 



72 MATHEMATICS OF INVESTMENT 

Solution. Lot r be the unknown rate per period. 



Case 1 
n = 50 iut. periods, 

p = 1, i = r, 
R $220, A = $7150. 



7150 =220(a^ air), 
t nt r\ - 716 - *> finn 

a T l ~ l2(T ~ iz - ou U. 



.0176 

r 
.0200 



33.141 
32.500 
31.424 



The first and third entries at the left were obtained 
from the row in Table VIII for n == 50. Since 
.0200 - .0175 = .0025, 

r =.0175 +^- 7 (.0025) = .0175 + .00093 = .01843. 



The nominal rate iaj = 4r = 4(.01843) = .07372, converted quarterly. 

NOTEJ. A value of r obtained as above usually is in error by not more than * 
Jjth of the difference of the table rates used in the interpolation. Hence, 
r = .01843 prolmbly is in error by not more than J&(.0025) = .0001, and the 
nominal rate j = .07372 is in error by not more than .0004. We are justified 
only in saying that the rate is approximately .0737, with doubt as to the lost 
digit. 

Supplementary Example 2. Determine the nominal rate in Example 1 
accurately to hundredths of 1%. 

Solution. From Example 1, (0==-, at r) = 32.500, and r = .0184, approxi- 
mately. It is probable that ? is between .0184 and .0185, or else between 
.0184 and .0183. Since our tables do not use the rate .0184, we compute 

~ (1.0184)- 1 ' 50 log 1.0184 = 50(.0079184) 

.0184' ' - .39592. 



.(a^ai.0184) 
.0184) 



.40186 



.0184 



= 32.507. 



log (1.0184)- BO = 9.60408 - 10. 
(1.0184)- 60 = .40186. 



Since 32.507 is greater than 32.500, r must be greater than .0184, and probably 
is between .0184 arid .0185. By logarithms, (o at. .0186) = 32.438. 



i 


(ogjy| at i) 


.0184 


32.S07 


T 


32.500 


.0186 


32.438 



From interpolation in the table at the left, 
r = .0184 + 1 (.0001) - .018410. 

The nominal rate is j = 4 r = .073640, which is cer- 
tainly accurate to hundredths of 1%, and is probably 
accurate to thousandths of 1%. 

1 The author gives no theoretical justification for this statement. He has verified 
its truth for numerous examples scattered over the range of Tables VII and VIII. 



PROBLEMS IN ANNUITIES 73 

NOTE. We could obtain the solution of Example 1 with any desired degree 
of accuracy by successive computations as in Example 2. Our accuracy would 
be limited only by the extent of the logarithm tables at our disposal. 

In Example 1, the most simple formulas (Case 1, with p = 1) applied 
because the conversion period equaled the payment interval. In more 
complicated examples, the solution may be obtained by first considering a 
new problem of the simple type met in Example 1. 

Example 3. Under what nominal rate, converted semi-annually, is 
$7150 the present value of an annuity whose annual rent is $880, payable 
quarterly for 12 years and 6 months? 

Solution. Let the unknown nominal rate be j. We could use the formulas 
of Case 1, with p = 2, in the solution, but the work would be slightly compli- 
cated. Instead, we first solve the following new problem: "Determine the 
nominal rate, w, converted quarterly, under which the present value of the annuity 
will be $7150." We choose quarterly conversions here because the annuity is 
payable quarterly. This new problem is the Example 1 solved above, so that 
w = .0737. The rate j, compounded semi-annually, must be equivalent to 
the rate .0737, compounded quarterly, because the present value of the annuity 
is $7150 under both of these rates. Hence, the effective rates corresponding 
to these two rates must be the same. 1 From equation 17, if i represents the 
effective rate, 



= (l +i)', 1 +< - ( 



(1.0184).. 



= (1.0184)*; 1 + 3 - = (1.0184)" = 1.03714. 
a 

Table II was used in computing 1.03714. The desired nominal rate is 
j = 2(.03714) = .07428, or approximately .0743, with doubt as to the last 
digit. 



EXERCISE XXXI 2 

In the first ten problems find the nominal rates as closely 3 as is possible 
by interpolation in the tables. 

1 For a similar problem see Section 10, illustrative Example 3. 

s The Miscellaneous Problems at the end of the chapter may be taken up im- 
mediately after the completion of Exercise XXXI. 

If the instructor desires, the students may be requested to obtain accuracy to 
hundredths of 1%, as in Example 2 above. 



74 



MATHEMATICS OF INVESTMENT 



PKOB. 


ANNUAL 
RUNT 


PAYMENT 
INTHBVAL 


INTEREST 
PERIOD 


AMOUNT 


PBHSHNT 
VALUE 


THHM 


1. 


$1000 


1 year 


1 year 


$15,700 




12 yr. 


2. 


100 


1 year 


1 year 




* 1,785 


25 yr- 


3. 


600 


1 year 


1 year 




5,390 


15 yr. 


4, 


100 


6 mo. 


6 mo. 




1,110 


17 yr., 6 mo. 


6. 


400 


3 mo. 


3 mo. 


2,500 




5 yr., 3 mo. 


6. 


1000 


1 year 


lyear 


53,000 




26 yr. 


7. 1 


200 


3 mo. 


1 year 


2,750 




9yr. 


8. 1 


200 


3 mo. 


6 mo. 


2,750 




9yr. 


9. 


2400 


1 mo. 


1 year 




14,500 


Syr. 


10. 


500 


6 mo. 


3 mo. 


17,500 




24 yr., 6 mo. 



"^ 11. A man has paid $100 to a building and loan association at the end 
of each 3 months for the last 10 years. If he now has $5500 to his credit, 
at what nominal rate; converted quarterly, does the association compute 
interest? 

12. By use of the result of problem 11, find the effective rate of interest 
paid by the association of problem 11. 

18. It has been agreed to pay $1100 at the end of each 6 months for 
8 years. Under what nominal rate', compounded semi-annually, would 
this agreement be equivalent to a cash payment of $14,000? 

14. A fund of $12,000 has been deposited with a trust company in order 
to provide an income of $400 at the end of each 3 months, for 10 years, 
at which time the fund will be exhausted. At what effective rate does the 
trust company credit interest on the fund? 

HINT. First find the equivalent nominal rate, payable quarterly. 

SUPPLEMENTARY MATERIAL 

33. Difficult cases and .exact methods in finding the term. 
When the formulas of Case 2 apply to an annuity, it is necessary 
to use the explicit formulas III in finding the term if it is 
unknown, 

Example 1. The amount of an annuity is $8375, and the annual rent 
is $1700, payable semi-annually. What is the term if money is worth 
(.06, m = 4) ? 

1 See illustrative Example 3. The same preliminary work should be used for 
both of problems 7 and 8. First determine the nominal rate, converted quarterly, 
under which. $2750 is the amount. 



PROBLEMS IN ANNUITIES 



75 



Solution. Case 2 applies. Hence, let the unknown term be k years. 

8375 = 1700(8^ at .06, m = 4). 



Case 2 

n = k yr., p - 2, 

j = .06, m = 4, 

R = $1700, A = $8375. 



From Table V, the denominator is 2 (.030225). 

(1.015)" - 1 = 8375(2) (.030225) = 2g781 

1700 
(1.015)" - 1 + .29781 = 1.29781. (40) 

(a) To solve equation 40 by interpolation, we use entries from the column 
in Table V for i = .015. We obtain 



4 k = 17 + - . 17.507, or Jb = 4.377. 
1932 



(6) To obtain the exact value of k from equation 40, take the logarithm of 
both sides of the equation, using Table II for log 1.015. 

4 k log 1.015 = log 1.29781 ; 4 fc(.0064660) = .11321. 
11321 _ 11321 log 11321 = 4.05389 

log 2568.4 = 3.41270 
log k - 0.64119 




k = 



4(646.60) 2586.4 
k = 4.3771. 



The solutions of problems, treated by interpolation in Section 31, may 
be obtained by solving exponential equations, as in solution (6) above. 
In these exact solutions it is always necessary to use the explicit alge- 
braic expressions for the present values and the amounts of the annui- 
ties concerned. 

Example 2. If the rate is (.06, m = 2), find the term of an, annuity 
whose present value is $8500 and whose annual rent is $2000, payable 
quarterly. 

Solution. Let the unknown term be k interest periods. 



Case 1 
n = k int. periods, 

p = 2, i = .03, 
R - $1000, A = $8500. 



8500 = 1000(a^ at .03) = 1000 



1 - (1.03)-* 



8600(.02Q77831) 
1000 



,25312, 



2[(1.03)* - 1] 
(Table XI) 

(By Table I) 



(76) 



MATHEMATICS OF INVESTMENT 



(1.03)-* = 1 - .25312 = .74688. /. - k log 1.03 = log .74688. 
- fc(.0128372) = 9.87325 - 10 = - .12675. Gog 1-03 from Table II) 

fc = .12675 g 873g periodfi of 6 mon tha. 
1283.7 

The term is 5 = 4.9369 years. Compare this with the result by interpolation 
2 

in Example 2 of Section 31. 

EXERCISE XXXII 

1. At the end of each 6 months a man deposits $200 in a bank which 
credits interest quarterly at the rate 3%. For how many years must the 
deposits continue in order to create a fund of $3000? Use the exact 
method (&) of Example 1 above. 

2. If money is worth 6%, compounded monthly, for how long must 
payments of $2000 be made at the end of each 6 months in order to dis- 
charge a debt whose present liability is $30,000? Solve by both an inter- 
polation and an exact method. 

3. Solve illustrative Example 1 of Section 31 by the exact method. 

4. Solve problem 9 of Exercise XXX by the exact method. 

B. To create an educational fund for a daughter, a father decides to 
deposit $500 at the end of each 6 months in a bank which credits interest 
annually, from the date of deposit, at the rate 4 %. When will the fund 
amount to at least $6000? Solve by the exact method. 

MISCELLANEOUS PROBLEMS 

1. If money is worth 5%, effective, what equal payments should be 
made at the end of each year for 10 years in purchasing a house whose 
equivalent cash price is $5000? 

2. If a man saves $200 at the end of each month, when will he be able to 
buy an automobile, worth $3000, if his deposits accumulate at the rate 5 %, 
compounded semi-annually? 

3. A depreciation fund is being accumulated by equal deposits at the 
end of each month in a bank which credits 6 % interest monthly on deposits 
from date of deposit. What is the monthly deposit if the fund contains 
$7000 at the end of 5 years? 

' 4. In purchasing a house, worth $20,000 cash, a man has agreed to 
pay $5000 cash and $1000 semi-annually for 9 years. What interest rate, 
compounded sew-annually, is being used in the transaction? 



PROBLEMS IN ANNUITIES (77 J. 

6. An insurance policy, on maturing, gives the policy holder the option 
of an immediate endowment of $15,000 or an annuity, payable quarterly for 
10 years. Under the rate 3.5%, effective, what will be the quarterly pay- 
ment of the annuity? 

6. A fund of $50,000 has been deposited with a trust company which 
credits interest quarterly on all funds at the rate 5 %. For how long will 
this fund furnish a man payments of $1000 at the end of each 3 months? 
* 7. On the death of her husband, a widow deposited her inherited estate 
of $25,000 with -a trust company. If interest is credited semi-annually on 
the fund at the nominal rate 4 %, for how long will the widow be able to 
withdraw $800 at the end of each 6 months? 

8. A certain loan bureau lends money to heads of families on the fol- 
lowing plan : In return for a $100 loan, $9 must be paid at the end of each 
month for 1 year. What effective rate of interest is being charged? 

HINT. First find the nominal rate, compounded monthly. 

9. A certain homestead is worth $5000 cash. The government sold 
this to an ex-soldier under the agreement that he should pay $1000 at the 
end of each 6 months until the liability is discharged. If interest is at the 
rate (.04, m = 2), for how long must the payments continue? 

10. The annual rent of an annuity is $50, payable annually. The 
present value of the annuity is $400 and the amount is $600. Find the 
effective rate of interest by use of the relation 39 of Section 30. 

11. A certain farm has a cash value of $20,000. If money is worth 
(.05, m = 2), what equal payments, made at the beginning of each 6 
months for 6 years, would complete the purchase of the farm? 

12. A man borrowed $2000 under the agreement that interest should 
be at the rate (.06, m = 2) during the life of the transaction. He made no 
payments of either interest or principal for 4 years. At that time, he 
agreed to discharge all liability in connection with the debt by making equal 
payments at the end of each 3 months for 3 years. Find the quarterly 
payment. 



CHAPTER V 



THE PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 

34. Amortization of a debt. A debt, whose present value is A, 
is said to be amortized under a given rate of interest, if all lia- 
bilities as to principal and interest are discharged by a sequence of 
periodic payments. When the payments are equal, as is usually 
the case, they form an annuity whose present value must equal A, 
the original liability. Hence, most problems in the amortization 
of debts involve the present value formulas for annuities. Many 
amortization problems have been solved in previous chapters. 

Example 1. A man borrows $15,000, with interest payable annually 
at the rate 5%. The debt is to be paid, interest as due and original 
principal included, by equal installments at the end of each year for 5 
years, (a) Find the annual payment. (6) Form a schedule showing 
the progress of repayment (or amortization) of the principal. 

Solution. Let $x be the payment. The present 
value of the payment annuity, at the rate (.05, 
m = 1), must equal $16,000. 15000 = x(a at .05). 



Casel 
n = 5 int. periods, 



R 



1, i .05, 
, A = $15,030. 



x - 15000 



- $3464.622. 



AMORTIZATION SCHEDTJIJD 



YHAR 


OUTSTANDING 

PBINCIPAIi AT 

BEGINNING 
orYnAB 


INTEREST 

AT6 ?fc, 
DUB AT END 
OK YBAR 


ANNUAI, PAYMENT 
AT END OF YEAH 


FOB REPAYMENT 
OP PRINCIPAL 
AT END OF YEAR 


1 
2 
3 
4 
5 


$15,000.000 
12,285.378 
9,435,025 
6,442.154 
3,299.640 


$ 750.000 
614.269 
471.751 
322.108 
164,982 


$ 3,464.622 
3,464.622 
3,464.622 
3,464.622 
3,464.622 


$ 2,714.622 
2,850.353 
2,992.871 
3,142.514 
3,299.640 


Totals 


$46,462.197 


$2323.110 


$17,323.110 


$15,000.000 



78 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 79 



NOTH 1. The schedule shows that the payments satisfy the creditor's 
demands for interest and likewise return his principal in installments. If 
x = $3464.622 was computed correctly, we know, without forming the sched- 
ule, that these facts must be true because the present value of the five pay- 
ments is $15,000. The checks on the arithmetic done in the table are that the 
lust total should be $15,000, the sum of the second and the last should equal 
the third, and tho second should be interest on the first total for one year at 6%. 
Notice that tho repayments of principal increase from year to year, while tho 
interest, payments decrease. Amortization schedules are very useful in the 
bookkeeping of both debtor and creditor because the exact outstanding liability 
at every interest date is clearly shown. The outstanding principal, or liability 
at any date, is sometimes called the book value of the debt at that time. 

NOTE 2. Since money is worth 6%, in Example 1, we may assume that 
tho debtor invests tho $15,000 at 5% immediately after borrowing it. The 
accumulation of this fund should provide for all the annual payments, to be 
made to tho creditor, because then* present value is $15,000. A numerical veri- 
fication of this fact is obtained in the amortization table above if we merely alter 
the titles of tho columns, as below, leaving the rest of the table unchanged. 



YHAM 


IN FUND AT 
BHHINNINO 
op YSLMI 


iNTHnKBT 

RKGHIVHD AT 
END OF YBA.R 


PAYMENT TO 
CnBDiTon 
AT END OF YHAB 


TAKEN FROM FUND 
AT END OF YEAR 


1 


$15,000 


$750 


$3464.622 


$2714.622 



Thus, at the end of the first year, the debtor receives $760 from his invested 
fund and, in order to make the payment of $3464.622 to his creditor, he takes 
$2714.02 from tho principal. By the end of 6 years, the fund reduces to zero. 

Exampk 2. In Example 1, without using the amortization schedule, 
determine the principal outstanding at the beginning of the third year. 

Solution. The outstanding principal, or liability, is the present value 
of all payments remaining to bo made. These form an annuity whose term is 

three years. The outstanding principal is 

3464.62(0^ at .05) = $9435.03. 

This is the third entry of the first column 
of tho amortization schedule. 



P 



Case 1 

n 3 int. periods, 
1, i - .05, R - $3464.62. 



Exampk 3. A debt whose present value ia $30,000, bearing interest 
at the rate 4.6%, compounded semi-annually, is to be amortized in 10 
yearn by equal payments at the end of eaoh 3 months, (a) Find the quar- 
terly payment. (6) Find tho principal outstanding at the end of 6 years, 
after the payment due has been made. 



80 ' MATHEMATICS OF INVESTMENT 

Solution. (a) Let $x be the quarterly payment. The present value of the 
payment annuity must equal $30,000. 



Case 1 

n =*> 20 int. periods, 

p = 2, i = .0226, 

R = 2 x, A = $30,000. 



30000 = 2 x(a at .0225) - 2 x (o at .0225). 



15000 ji 



.0225 (ogyj at .0225) 
= 16006(.02237484) (OT294?q7) = $934.403. 



.0225 



(b) At the end of the 6th year, or the beginning of the 6th, the outstanding 
liability, L, is the present value of payments extending over 5 years. 

L = 1868.81 (ajjai. 0225). 



Casel 

n = 10 int. periods, 
.0225, ,- 2, R. $1868.81. L . $16]69L95 . 



L = 1868.81 ^5(0 oi .0225). 



EXERCISE XXXffi 

^ 1. A loan of $5000, with interest at 6%, payable semi-annually, is to be 
amortized by six semi-annual payments, the first due after 6 months, 
(a) Find the payment, to three decimal places. (&) Form the amor- 
tization schedule for the debt. 

2. In problem' 1, without using the amortization table, find the prin- 
cipal unpaid at the end of 1 year and 6 months, just after the payment 
due has been made. 

3. A man deposits $10,000 with a trust company which credits 5% 
interest annually. The fund is to provide equal payments at the end of 
each year for 5 years, at the end of which time the fund is to bo exhausted, 
(a) Find the annual payment to three decimal places. (&) Form a table 
showing the amortization of the fund. 

HINT. See Note 2, Section 34; think of the trust company as the debtor. 

4. In problem 3, without using the table, find the amount remaining 
in the fund at the end of 2 years, after the payment due has been made. 

6. A purchaser of a house owes $7500, and interest at 6% is payable 
semi-annually on all amounts remaining due. He wishes to discharge 
his debt, principal and interest included, by twelve equal semi-annual 
installments, the first due after 6 months. Find the necessary semi- 
annual payment. 

6. A street assessment of $500 against a certain piece of real estate is 
to be amortized, with interest at 6%, by six equal annual payments, the 
first due after \ year. What part of the assessment will remain unpaid 
8-t the beginning of the 4th year, after the payment due has been made? 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 81 

7. A house is worth $25,000 and the owner, on selling, desires the 
equivalent of interest at the rate 5%, payable semi-annually. (a) What 
quarterly installment, for 8 years, in addition to a cash payment of $5000, 
would satisfy the owner? (&) How much of the principal of the debt 
remains unpaid at the end of 3 years and 6 months, after the payment 
due has been made ? 

8. A debt of $12,000, with interest payable serai-annually at the rate 
5 %, is to be amortized in 10 years by equal semi-annual installments, the 
first due after 6 months. What part of the debt will remain unpaid at the 
beginning of the 6th year, after the payment due has been made? 

9. In problem S, what part of the llth payment is interest and what 
part is repayment of principal ? 

10. A debt will be discharged, principal and interest, at 6% effective, 
included, by payments of $1200 at the end of each year for 12 years, 
(a) What is the original principal of the debt? (b) What principal will 
remain outstanding at the beginning of the 5th year? (c) What part 
of the 5th payment will be interest and what part repayment of principal? 

11. A trust fund of $100,000 was created to provide a regular income 
at the end of each month for 20 years. If the trust company uses the 
interest rate 4%, converted semi-annually, what is the monthly payment, 
if the fund is to be exhausted by the end of 20 years? 

12. It was agreed to amortize a debt of $20,000 with interest at 5%, by 
12 equal annual payments, the first due in one year. The debtor failed 
to make the first four payments. What payment at the end of 5 years 
would bring the debtor up to date on liis contract? 

13. A debt of 538,000 is to bo amortized by payments of $2000 at the 
end of each 3 months for 6 years, (a) Find the nominal rate, compounded 
quarterly, at which interest is being paid, (fo) What is the effective rate 
of interest? 

14. A certain insurance policy on maturing gives the option of $10,000 
cash or $345 at the end of each months for 20 years. What rate of 
interest is being used by the insurance company? 

35. Amortization of a bonded debt. In amortizing a debt 
which is in the form of a bond issue, the periodic payments cannot 
be exactly equal. If the bonds are of $1000 denomination, for 
example, the principal repayments must be multiples of $1000, 
because any individual bond must be retired in one installment. 



82' 



MATHEMATICS OF INVESTMENT 



Example 1. Construct a schedule for the amortization, by 10 annual 
payments as nearly equal as possible, of a $10,000 debt which is outstand- 
ing in bonds of $100 denomination, and which bears 4% interest payable 
annually. The first payment is due at the end of 1 year. 

SoMion. Let &c be the annual payment, which would be made if the 
payments were to be equal. 

10000 - x(aat .04). 



Case 1 
n = 10 int. periods, 

p = 1, i = .04, 
R = $x, A = $10,000. 



x = 10000 



.04) 



$1232.91. 



The annual payments should be aa close as possible to $1232.91. Thus, at the 
ead of the 1st year, the interest due is $400, leaving 1232.91 400.00 
= $832.91 available for repayment of principal. Hence, retire 8 bonds, or 
$800 of the principal on this date, making a total payment of $1200. At the 
end of the next year 1232,91 - 368.00 = $864.91 is available for retiring bonds. 
Therefore, pay 9 bonds or $900 of the principal. 

AMORTIZATION SCHEDULE FOR A BONDED DEBT 



YBAB 


PniNOIPAL OUT- 
STANDING AT BEGIN- 
NING OF YBAB 


INTEREST Dun 
AT END OF YHAR 


BONDS RHTIBBD 
AT END OF YHAB 


TOTAL PAYMENT 
AT END OF YEAH 


1 


$10,000 


$400 


8 


$1,200 


2 


9,200 


368 


9 


1,268 


3 


8,300 


332 


9 


1,232 


4 


7,400 


296 


9 


1,196 


5 


6,500 


260 . 


10 


1,260 


6 


5,500 


220 


10 


1,220 


7 


4,500 


180 


11 


1,280 


8 


3,400 


136 


11 


1,236 


9 


2,300 


92 


11 


1,192 


10 


1,200 


48 


12 


1,248 


Totals 


$58,300 


$2,332 


100 


$12,332 



EXERCISE XXXTV 

^ 1. A $1,000,000 debt is outstanding in the form of $1000 bonds which 
pay 6% interest annually. Construct a schedule for the retirement of 
the debt, principal and interest included, by five annual payments as 
nearly equal as possible, the first payment due at the end of 1 year. 

2. A $1,000,000 issue of bonds, paying 5% interest annually, consists 
of 500 bonds of $100, 200 bonds of $500, 200 of $1000, and 130 of $5000 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 83 



denomination. Construct a schedule for the amortization of the debt 
by 10 annual payments as nearly equal as possible. 

HINT. In the schedule,, make a separate column for each class of bonds. 

36. Problems in which the periodic payment is known. If 
the present liability of a debt, the interest rate, and the size and 
frequency of the amortization payments are known, the term of 
the payment annuity can be found as in Section 31. 

Bxampk 1. A house is valued at $10,000 cash. It is agreed to pay 
$1200 cash and $1200 at the end of each 6 months as long as necessary 
to amortize the given cash value with interest at 5%, payable semi-an- 
nually. (a) For how long must the payments continue? (6) Construct 
an amortisation schedule. 

Solution. (a) After the cash payment of $1200, $8800 remains due. Let 
k be the time in interest periods necessary to 
amortize it with interest at the rate (.05, 
m - 2). 

8800 = 1200(0^ at .025) ; 
(fl at .025) = 7.333. 



Case 1 

n = k int. periods, 

p = 1, i = .025, 

R = $1200, A = $8800. 



By interpolation in Table VIII, k = 8.20. Hence, 8 full payments of $1200 
must be made in addition to the first cash payment. After the $1200 payment 
is made, at the end of 4 years, some principal is still outstanding because k is 
greater than 8. A partial payment will be necessary at the next payment 
date. These conclusions are verified in the schedule below. 

(6) AMORTIZATION SCHEDULE 



PAYMENT 
INTERVAL 


OtJTSTANDING PRIN- 
CIPAL AT BEGIN- 
NING OF INTERVAL 


INTEREST DUB AT 
END OP INTERVAL 


TOTAL PAYMENT 
AT END OF 
INTEBVAL 


PRINCIPAL RE- 
PAID AT END OP 
INTBBTAL 


1 


$8800.000 


$220.000 


$1200. 


$ 980.000 


2 


7820.000 


195.500 


. 1200. 


1004.500 


3 


6815.500 


170.388 


1200. 


1029.612 


4 


6785.888 


144.647 


1200. 


1055.353 


5 


4730.535 


118.263 


1200. 


1081.737 


6 


3648.798 


91.220 


1200. 


1108.780 


7 


2540.018 


63.500 


1200. 


1136.500 


8 


1403.518 


35.088 


1200. 


1164.912 


9 


238.606 


5.965 


244.571 


238.606 


Totals 


$41,782.863 


$1044.571 


$9844.571 


$8800.000 



84 ; MATHEMATICS OF INVESTMENT 

Exampk 2. Without using the amortization table, find the principal 
still unpaid in Example 1 at the end of 2 years, after the payment due 
has been made. 

Solution. Let $M be the amount remaining due. The payment of $M at 
the end of 1\ years, in addition to the payments already made, would complete 
the payment of the debt whose original principal was $8800. Hence, this 
"Old Obligation" must have the same value as the "New Obligations " 
Hated below. 



OLD OBLIGATION 


NHW OBLIGATIONS 


$8800 due at the beginning 
of the transaction. 


(a) $M due at the end of 2$ years. 

(&) Payments of $1200 due at the end of 
each 6 months for 2| years. 



To find M, write an equation of value, under the rate (.06, m = 2), with the 
end of 2J years as the comparison date. The sum of the values of obligations 
(&) is the amount of the annuity they form. 

8800(1. 025) B = M + 1200 (s^at .025). 
M = 8800(1.025) B - 1200(^0* .025) = 9956.39 - 6307.59 = $3648.80, (41) 

which checks with the proper entry in the table of Example 1. The debtor 
could close the transaction at the end of 2$ years by paying the regular in- 
stallment plus $3648.80 or (1200 + 3648.80) = $4848.80. 

NOTE 1. Recognize that 8800(1. 025) 5 is the amount the creditor should 
have at the end of 2i years if he invested $8800 at (.05, m = 2), whereas he 
actually has possession of only 1200 (s^ aL .025) as a consequence of the pay- 
ments received from the debtor. Hence, equation 41 shows that M is the 
difference between what the creditor should have and what he actually has. 

NOTE 2. By the method of Example 2 we can find the final installment in 
Example 1 without computing the amortization table. Let $ N be the amount 
remaining due just after the last full payment, at the end of 4 years. Then, 
N = 8800(1.025) 8 - 1200(85-, at .025) - 10721.946 - 10483.339 = $238.607. 

To close the transaction at the end of 4J years, the necessary payment is 
$238.607 plus interest for 6 months at 6%, or 238.607 + 5.966 $244.57. 

EXERCISE XXXV 

1. (a) How long will it take to amortize a debt whose present value 
is $10,000 if payments of $2000 are made ai the end of each year and 
if these payments include interest at the rate 5%, payable annually. 
(6) Form an amortization schedule for this debt. 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 85 

2. (a) Without using the table in problem 1, find the principal out- 
standing at the beginning of the 3d year. (&) Find the size of the final 
payment. 

3. A debt of $50,000, with interest payable quarterly at the rate 8%, 
is being amortized by payments of $1500 at the end of each 3 months. 
(a) What is the outstanding liability just after the 10th payment? 
(6) Find the final installment. 

4. A trust fund of $100,000 is invested at the rate 6%, compounded 
semi-annually. Principal and interest are to provide payments of $5000 
at the end of each 6 months until the fund is exhausted, (a) How many 
full payments of $5000 will be made? (6) What will be the size of the 
final partial payment? 

6. The purchaser of a farm has agreed to pay $1000 at the end of each 
3 months for 5 years, (a) If these payments include interest at the 
rate 6%, payable quarterly, what is the outstanding principal at the be- 
ginning of the transaction? (&) Find the outstanding liability at the 
beginning of the 3d year. Notice that, since the exact number of the re- 
maining payments is known, part (6) should be done like illustrative 
Example 2 of Section 34 ; it would be clumsy to use the method of illus- 
trative Example 2 of the present section. 

37. Sinking fund method. A sinking fund is a fund formed 
in order to pay an obligation falling due at some future date. In 
the following section, unless otherwise stated, it is assumed that 
the sinking funds involved are created by investing equal periodic 
payments. Then, the amount in a sinking fund at any time is the 
amount of the annuity formed by the payments, and examples 
involving sulking funds can be solved by use of the formulas for 
the amount of an annuity. Thus, to create a fund of $10,000 at the 
end of 10 years by investing $rc at the end of each 6 months for 
10 years, at the rate (.06, m = 2), x must satisfy 



10000 = .fa * - 03 > 5 - 1000 



Suppose that $A is borrowed under the agreement that interest 
shall be paid when due and that the principal shall be paid in one 
installment at the end of n years. If the debtor provides for the 
future payment of $1 at the end of n years by the creation of a 
sinking fund, invested under his own control, his debt is said to be 



86 



MATHEMATICS OF INVESTMENT 



retired by the sinking fund method. Under this method, the 
expense of the debt to the debtor is the sum of (a) and (6) below : 

(a) Interest on $A, paid periodically to the creditor when due. 

(6) Periodic deposits, necessary to create a sinking fund of $A, to pay the 
principal when due. 

NOTE 1. Recognize that the sinking fund is a private affair of the debtor. 
The rate of interest paid by the debtor on $A bears no relation to the rate of 
interest at which the debtor is able to invest his sinking fund. Usually, the 
desire for absolute safety for the fund would compel the debtor to invest it at 
a lower rate than he himself pays on his debt. 

Example 1. A debt of $10,000 is contracted under the agreement 
that interest shall be paid semi-annually at the rate 6%, and that the 
principal shall be paid in one installment at the end of 2^ years, 
(a) Under the sinking fund method, what is the semi-annual expense of 
the debt if the debtor invests his fund at (.04, m = 2) ? (6) Form a 
table showing the accumulation of the fund. 

Solution. (a) Let $z be the semi-annual deposit to the sinking fund, whose 
amount at the end of 24 years is $10,000. 

10000 = x 
1 



Case 1 
n = 5 int. periods, 

p = 1, i = .02, 
R = $3, S - $10,000. 



x = 10000- 



at .02) ; 

$1921.584. 



expense is 300 + 1921.58 



( Sg] ai .02) 

Interest due semi-annually on the principal of 
the debt is (.03) (10000) = $300. Semi-annual 
= $2221.58. 



(6) TABLE SHOWING GROWTH OP SINKING FUND 



PAYMENT 
INTERVAL 


IN POND AT 
BEGINNING OF 
INTERVAL 


INTEBBST AT 4% 
RECEIVED ON FUND 
AT END OF 
INTERVAL 


PAYMENT TO FUND 
AT END OF 
INTERVAL 


IN FUND AT END 
op INTERVAL 


1 








$192L584 


$1921.584 


2 


$1921.584 


$ 38.432 


1921.584 


3881.600 


3 


3881.600 


77.632 


1921.584 


6880.816 


4 


5880.816 


117.616 


1921.584 


7920.016 


5 


7920.016 


158.400 


1921.684 


10000.000 



Nora 2. The book value of the debtor's indebtedness, or his net indebted- 
ness, at any time may be defined as the difference between what he owes and 
what he has in his sinking fund. Thus, at the end of 2 years, the book value of 
the debt is 10000 - 7920.016 - $2079.984. 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS S7- 

NOTE 3. The amount in the sinking fund at any time is the amount of 
the payment annuity up to that date and can be found without forming the 
table (6). Thus, the amount in the fund at the end of 2 years is 
1921.584(8^ at .02) = $7920.02. 

EXERCISE XXXVI 

* J 1. A debt of $50,000, with interest payable semi-annually at the rate 
6%, is to be paid at the end of 3 years by the accumulation of a sinking 
fund, (a) If payments to the fund are made at the end of each 6 months 
and accumulate at the rate 3%, compounded semi-annually , what is the 
total semi-annual expense of the debt? (6) Form a table showing the 
accumulation of the fund. 

2. (a) In problem 1, without using the table, determine the amount 
in the sinking fund at the end of 2 years. (6) What is the book value 
of the debtor's indebtedness at this time? 

3. A loan of $10,000 bears 5% interest, payable serm-annually, and a 
sinking fund is created by payments at the end of each 6 months in order 
to repay the principal at the end of 4 years. Find the expense of the 
debt if the fund accumulates at the rate (.04, m = 2). 

4. A city issues $100,000 worth of bonds bearing 6% interest, payable 
annually, and is compelled by law to create a sinking fund to retire the 
bonds at the end of 10 years. If payments to the fund occur at the end 
of each year and are invested at 6%, effective, what is the annual ex- 
pense of the debt? 

6. How much is in the sinking fund in problem 4 at the beginning of 
the 7th year? 

6. A loan of $5000 is made under the agreement that interest shall 
be paid semi-annually at the rate 5.5% on all principal remaining due 
and that the principal shall be paid in full on or before the end of 6 years. 
(a\ Find the semi-annual expense if the debt is amortised by equal pay- 
ments at the end of each 6 months for 6 years. (&) Find the semi-an- 
nual expense to retire the debt by the sinking fund method at the end 
of 6 years, if payments to the fund are made at the end of each 6 months 
and are invested at (.04, m = 2). (c) Find the semi-annual expense 
under the sinking fund method if the fund earns (.06, m = 2) . (d) Which 
method is most advantageous to the debtor? 

7. A sinking fund is established by payments at the end of each 3 
months in order to accumulate a fund of $300,000 at the end of 15 years. 
Find the quarterly payment if interest is at the rate (.06, m = 2). 



. 

(OBI at $) 

Since - - r = i -J -- -, 



88 MATHEMATICS OF INVESTMENT 

8. To accumulate a fund of $100,000, payments of $5000 are invested 
at the end of each 6 -months at the rate 5%, compounded semi-annually. 
(a) How many full payments must be made? (6) How much must be 
paid on the last payment date? 

38. Comparison of the amortization and the sinking fund 
methods. To amortize a debt of $A, in n years, with interest 
payable annually at the rate i, by payments of R at the end of 
each year, we have A = R(ctn\ at i) or 

(42) 

R-Ai + Aj-' (43) 

(si at 

If the sinking fund method is used to retire this. debt, the amount 
in the fund at the end of n years is $A. If payments of $W are 
made to the fund at the end of each year and accumulate at the 
effective rate r, then A = W(sx\ at r) or 

W = A. l (44) 

(Sn\ at r) 

The annual interest on $A at the rate i is Ai, so that the total 
annual expense E under the sinking fund plan is 

E = Ai + W = Ai + A - ^-^ (45) 

fa\atr) 

When r = i in equation 45, E equals R, as given in equation 43. 
Thus, the amortization payment $R is sufficient to pay interest on 
$-4 at the rate i, and to create a sinking fund which amounts to 
&4 at the end of n years, if the fund also earns interest at the 
rate i. Hence, the amortization method may be considered as 
a special case of the sinking fund method, where the creditor is 
custodian of the sinking fund and invests it at the rate i. 

When r is less than i, (s^ at r} is less than (s^\ at i), so that 

A ,_ . ^ is greater than A -. - and E is greater than R. 
(Sn\atr) _ (%, at i) 

Similarly, when r is greater than i, the sinking fund expense is less 
than, the amortization expense. 



ft 



PAYMENT OF DEBTS BY PEEIODIC INSTALLMENTS 89 

NOTE. The conclusions of the last paragraph are obvious without the 
use of any formulas. If the debtor is able to invest his fund at the rate r, 
greater than i, his expense will be less than under the amortization method 
because, under the latter, he is investing a sinking fund with his creditor at the 
smaller rate i. 

NOTE. Equation 42 is sometimes called the amortization equation, and 

equation 44 is called the sulking fund equation. Table IX for may be called 

a *\ 
a table of amortization charges for a debt of $1 (A <= 1 in equation 42), and a 

table of the values of would be a table of sinking fund charges for a debt of $1. 



MISCELLANEOUS PROBLEMS 

In solving the more difficult problems of the set below, the student 
should recall that the writing of an equation of value, for a conveniently 
selected comparison, date, furnishes a systematic method of solution, 
as in illustrative Example 2 of Section 36. 

1. At the end of 5 years, a man will pay $15,000 cash for a house. 
(a) What equal amounts should he save at the end of each year to ac- 
cumulate the money if his savings earn 6%, effective? (6) What should 
he save at the beginning of each year in order to accumulate the money 
if the savings earn 6%, effective? 

2. A loan of $5000 is made, with interest at 6%, payable semi-annually. 
Is it better to amortize the debt in 6 years by equal semi-annual in- 
stallments, or to pay interest when due and to retire the principal in one 
installment at the end of 6 years by the accumulation of a sinking fund 
by semi-annual payments, invested at (.04, m = 2) ? 

3. A man, purchasing a farm worth $20,000 cash, agrees to pay $5000 
cash and $1500 at the end of each 6 months, (a) If the payments in- 
clude annual interest at the rate 5%, effective, how many full payments 
of $1500 will be necessary? (6) What is the purchaser's equity in the 
house at the beginning of the 3d year? 

4. A county has an assessed valuation of $50,000,000. The county 
borrows $500,000 at 5%, payable annually, and is to retire the principal 
at the end of 20 years through the accumulation of a sinking fund by 
annual payments invested at 4%, effective. By how much, per dollar 
of assessed valuation, will the annual taxes of the county be raised on 
account of the expense of the debt? 



V." 



MATHEMATICS OF INVESTMENT 



5. A debt of $25,000, with interest payable semi-annually at the rate 
6%, is to be amortized by equal payments, at the beginning of each 
6 months for 12 years, (a) Determine the payment. (6) At the be- 
ginning of the 4th year, after the payment due has been made, what prin- 
cipal remains outstanding? 

6. A debt of $12,000, with interest at 5%, compounded quarterly, is 
to be amortized by equal payments at the end of each 3 months for 8 years. 

At the end of 4 years, what payment, in addition to the one due, would 
cancel the remaining liability if the creditor should permit the future 
payments to be discounted, under the rate (.04, m = 4) ? 

7. A debt of $100,000, with interest at 5% payable annually, will be 
retired at the end of 10 years by the accumulation of a sinking fund by 
annual payments invested at 4%, effective. Considering the total an- 
nual expense of the debtor as an annuity, under what rate of interest is 
the present value of this annuity equal to $100,000 ? The answer obtained 
is the rate at which the debtor could afford to amortize his debt, instead 
of using the sinking fund method described in the problem. 

8. In order to accumulate a fund of $155,000, a corporation invests 
$20,000 at the end of each 3 months at the rate (.08, m = 4). (a) How 
many full payments of $20,000 must be made? (6) Three months after 
the last full payment of $20,000 is made, what partial payment will com- 
plete the fund? 

9. The original liability of a debt was $30,000, and interest is at the 
rate 5%, effective. Payments of $2000 were made at the end of each 
year for 7 years. At the end of that time it was decided to amortize the 
remaining indebtedness by equal payments at the end of each year for 
8 years. Find the annual payment. 

HINT. -Equations of value furnish a systematic method for solving prob- 
lems of this type. Let &c be the annual payment. Then, the value of the 
"Old Obligation" below must equal the sum of the values of the "New 
Obligations " on whatever comparison date is selected. 



OLD OBLIGATION 


NEW OBLIGATIONS 


$30,000 due at the begin- 
ning of the transaction. 


(a) $2000 due at the end of each year for 
7 years, 
(b) Eight annual payments of $s, the first 
. due at the end of 8 years. 



The end of 7 years is the most convenient comparison date. 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 91 

10. At the end of 5 years, $10,000 must be paid, (a) What equal 
amounts should the debtor deposit in a savings bank at the end of each 
6 months in order to provide for the payment, if his savings accumulate 
at the rate (.04, m = 2) ? (6) How much must he deposit semi-annually 
if his first deposit occurs immediately and the last at the end of 5 years? 

11. A debtor borrows $20,000, which is to be repaid, together with all 
accumulated interest at the rate (.05, m = 4), at the end of 6 years, 
(a) In order to pay the debt when due, what equal deposits must be 
made in a sinking fund at the end of each 6 months if the fund accumu- 
lates at the rate (.05, m = 2)? (6) At what rate, compounded semi- 
annually, could the debtor just as well have borrowed the $20,000, under 
the agreement that it be amortized in 6 years by equal payments at the 
end of each 6 months ? 

12. A certain state provided for the sale of farms to war veterans under 
the agreement that (a) interest should be computed at the rate (.04, m 
= 2), and (6) the total liability should be discharged by 10 equal semi- 
annual installments, the first due at the end of 3 years. Find the nec- 
essary installment if the farm is worth $6000 cash. 

13. Under what rate of interest will 25 semi-annual payments of 
$500, the first due immediately, amortize a debt of $9700? Determine 
both the nominal rate, compounded semi-annually, and the effective 
rate. 

14. A sinking fund is being accumulated by payments of $1000 at the 
end of each year. For the first 10 years the fund earns 6%, effective, 
and, for the next 6 years, 4%, effective. What is the size of the fund at 
the end of 16 years? It is advisable, first, to find the amount at the end 
of: 16 years due to the payments during the first 10 years. 

16. In order to create a fund of $60,000 by the end of 20 years, what 
equal payments should be made at the end of each 6 months if the fund 
accumulates at the rate (.04, m = 2) for the first 10 years and at 6%, 
effective, for the last 10 years? 

16. A fund of $250,000 is given to a university. The principal and 
interest of this fund are to provide payments of $2000 at the beginning 
of each month until the fund is exhausted. If the university succeeds 
in investing the fund at 5%, compounded semi-annually, how many full 
payments of $2000 will be made? 

17. Under what nominal rate, compounded semi-annually, would it 
be just as economical to amortize a debt in 10 years by equal payments 
at the end of each 6 months, as to pay interest semiTannually at the rate 



92 MATHEMATICS OF INVESTMENT 

6%, on the principal, and to repay the principal at the end of 10 years by 
the accumulation of a siring fund by equal payments at the end of each 
6 months, invested at the rate (.04, m = 2). 

18. A factory is worth $100,000 cash. At the time of purchase 
$25,000 was paid. Payments of $8000, including interest, were made 
at the end of each year for 6 years. The liability was completely 
discharged by six more annual payments of $9000, including interest, 
the first occurring at the end of the 7th year. What effective rate of 
interest did the debtor pay? 

HINT. Write an equation of value at the end of 6 years. Transpose all 
quantities in the equation to one side and solve by interpolation. Sec Appen- 
dix, Note 3, Example 2. 

SUPPLEMENTARY MATERIAL (} 

/ 39. Funds invested with building and loan associations. A 
building and loan association is a cooperative enterprise whose 
main purpose is to provide funds from which loans may be made 
to members of the association desiring to build homes. Some 
members are investors only, and do not borrow from the as- 
sociation. Others are simultaneously borrowers and investors. 
Shares of stock are sold to members generally in units of $100. 
Each share is paid for by equal periodic installments called dues, 
payable at the beginning of each month. Profits of the association 
arise from investing the money received as clues. Members share 
in the profits in proportion to the amount they have paid on their 
shares of stock, and then- profits are credited as payments on their 
stock. When the periodic payments on a $100 share, plus the 
credited earnings, have reached $100, the share is said to mature. 
The owner may then withdraw its value or may allow it to remain 
invested with the association. 

Over moderate periods of time, the interest rate received by an 
association on its invested funds is approximately constant. The 
amount to the credit of a member, who has been making periodic 
payments on a share, is the amount of the annuity formed by his 
payments, -with interest at the rate being earned by the association. 

Example 1. A member pays $2 at the beginning of each month on a 
share in an association whose funds earn 6%, compounded monthly. 
What is to the credit of the member just after the. 20th payment? 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 93 

Solution. The payments form an ordinary annuity of 20 payment inter- 
vals, if the term is considered to begin 1 month before the first payment is 
made. The amount on the 20th payment date is the amount of this ordinary 
annuity, or 2(a$Q\ at .005) = $41.96. 

Example 2. A member pays $1 at the beginning of each month on a 
$100 share. If the association is earning at the rate (.06, m = 12), when 
will the share mature? 

Solution. Let k be the number of installments necessary to bring the 
amount to the member's credit up to $100. - 

100 = (a^of .005). 

By interpolation in Table VII, k = 81.3. Just 
after the 81st payment, at the beginning of the 81st 
month, the member is credited with (sg^ at .005) 



Case 1 

n = k int. periods, 
p = 1, i - .005, 
R = $1, S = $100 

$99.558. By the beginning of the 82d month, 
this book value has earned (.005) (99.558) = $.498, and the new book value 
is 99.568 + .498 = $100.056. Hence, no payment is necessary at the begin- 
ning of the 82d month. Properly speaking, the share matured to the value 
$100 at a time during the 81st month, but the member ordinarily would be 
informed of the maturity at the beginning of the 82d month. 

Example 3. A $100 share matures just after the 83d monthly pay- 
ment of $1. (a) What rate, compounded monthly, is the association 
earning on its funds ? (6) Wnat is the effective rate? 

Solution. (a) Let r be the rate per period of 1 month. The amount of 
the payment annuity is $100, 



Case 1 
n s= 83 int. periods, 

p - 1, i = r, 
R = $1, S = $100. 



100 = 



By interpolation in Table VII, r = 
= .441%. The nominal rate is 12 (.00441) = .0529, 
compounded monthly. (6) The effective rate i is 
obtained from 
1 + i = (1.00441) 12 = 1.05422. (By Table n) 

Hence i = .0542, approximately. 

EXERCISE XXXVH 

1. A member pays $1 at the beginning of each month on a share in an 
association which earns 5%, compounded monthly. What is to the mem- 
ber's credit just after the 50th payment? 

2. When will the share of stock in problem I mature to the value 
$100, and what payment, if any, will be necessary on the maturity date? 

3. If an association earns 6%, compounded monthly, when will a $100 
share mature if the dues are $2 per month per share? 



94 MATHEMATICS OF INVESTMENT 

4. The monthly due on a $100 share is $.50, and the share will mature, 
approximately, just after the 131st monthly payment, (a) What rate, 
compounded monthly, does the association earn on its funds? (6) What 
is the effective rate? 

5. An association earns approximately 6%, compounded quarterly, 
on its funds, (a) Find the date, to the nearest month, on which a $100 
share will mature if the monthly due is $1, making your computation 
under the assumption that $3 is paid at the beginning of each 3 months. 
(&) Without any computation, tell why your result, as computed in (a), 
is smaller (or larger) than the actual result. 

6. The monthly due on each $100 share is $1, in an association where 
the shares mature just after the 80th payment. What is the effective 
rate earned on the shares? 

7. A man paid $30 per month for 80 months as dues on thirty $100 
shares in an association, and, to mature his stock at the beginning of the 
81st month, a partial payment of $6 was necessary. At what rate, com- 
pounded monthly, did his money increase during the 80 months? 

HINT. If he should pay $30 at the beginning of the 81st month, the book 
value of Ms shares would be $3024. 

40. Retirement of loans made by building and loan associa- 
tions. If a man borrows $A from a building and loan associa- 
tion, he is usually caused, at that time, to become a member of the 
association, by subscribing for $A worth of stock. He must pay 
monthly interest (usually in advance) on the principal of his loan 
and dues on his stock. When his stock matures, with the value 
$^4., the association appropriates it as repayment of the principal 
of the loan. Thus, if a man borrows $2000 from an association 
which charges borrowers 7% interest, payable monthly in advance, 
the interest due at the beginning of each month is ^(.07) (2000) 
= $11.67. The borrower subscribes for twenty $100 shares of 
stock on which he pays $20 as dues at the beginning of each month 
if the due per share is $1. Thus, the monthly expense of the debt 
is $31.67. Payments continue until the twenty shares mature 
with the value $2000, at which time the association takes them as 
repayment of the principal of the debt, and closes the transaction. 
This method of retiring a debt is essentially a sinking fund plan, 
where the debtor's fund is invested in stock of the association. 
The debtor is benefited by this method because the rate received 



PAYMENT OF DEBTS BY PERIODIC INSTALLMENTS 95 

on his stock investment is higher than could safely be obtained in 
the outside market. 

Example 1. A man borrows $2000 under the conditions of the pre- 
ceding paragraph. If shares in the association mature at the end of 
82 months, at what nominal rate, compounded monthly, may the bor- 
rower consider that he is amortizing his debt ? 

Solution. The debtor pays $31.67 at the beginning of each month for 82 
months. We wish the rate under which $2000 is the present value of this 
annuity due. Let r be the unknown rate per period of 1 month. The first 
$31.67 is paid cash and the remaining 81 payments form an ordinary annuity, 
.under Case 1, with p = 1. Hence, 

2000 = 31.67 + 31.67(0^ at r), 



By interpolation in Table VIII, r = &% + (%) = .681%. The nominal 
rateis 12(.00681) = .082, approximately, compounded monthly. The effective 
rate i, if desired, can be obtained, with the aid of Table II, from 
1 + i = (1.00681) 12 . 



EXERCISE 

1. An association charges borrowers 6% interest payable monthly in 
advance and issues $100 shares on which the monthly dues are $1 per 
share, (a) At what rate of interest, compounded monthly, may a bor- 
rower consider his loan to be amortized, if shares in the association mature 
at the end of the 84th month, without a payment at that time? (6) What 
is the effective rate of interest? 

2. Which would be more profitable, to borrow from the association 
of problem 1, or 'to pay 5% interest monthly in advance to some other 
lending source, and to repay the principal of the loan at the end of 84 
months by the accumulation of a sinking fund, at the rate 6%, com- 
pounded monthly, if payments to the fund are made at the beginning of 
each month for 84 months? 

3. An association charges borrowers 7% interest, payable monthly 
in advance, and issues $100 shares on which the monthly dues are $1 per 
share. If the shares mature at the end of 80 months, without a payment 
at that time, at what effective rate does a borrower amortize his debt? 

4. An association charges 6% interest payable monthly in advance, and 
issues $100 shares on which the monthly due per share is $.50. If the 
shares mature at the end of 130 months, without a payment at that 
time, what is the effective rate paid by a borrower? 



CHAPTER VI 
DEPRECIATION, PERPETUITIES, AND CAPITALIZED COST 

41. Depreciation ; .sinking fund plan. Fixed assets, such as 
buildings and machinery, diminish in value through use. Deprecia- 
tion is denned as that part of their loss in value which cannot be 
remedied by current repairs. In every business enterprise, the 
effects of depreciation should be foreseen and funds should be 
accumulated whose object is to supply the money needed for the 
replacement of assets when worn out. The deposits in these 
depreciation funds are called depreciation charges, 1 and are de- 
ducted periodically, under the heading of expense, from the cur- 
rent revenues of the business. 

NOTE 1. The replacement cost of an asset equals its cost when new minus 
its salvage or scrap value when worn out. Thus, if a machine costs $1000, 
and has a scrap value of $50 when worn out at the end of 10 years, its replace- 
ment cost is $950, the amount needed in addition to the scrap value in order 
to buy a new machine worth $1000. The replacement cost is also called the 
wearing value ; it is the value which is lost through wear during the life of the 
asset. 

A depreciation fund is essentially a sinking fund whose amount 
at the end of the life of the asset equals the replacement cost. 
Many different methods are in use for estimating the proper 
depreciation charge. Under the sinking fund method, the periodic 
depreciation charges are equal and are invested at compound inter- 
est at a specified rate. Under this plan, (a) the depreciation charges 
form an annuity whose amount equals the replacement cost, and 
(&) the depreciation fund is a sinking fund to which we may apply 
the methods for sinking funds used in Section 37. 

Exampk 1. A machine costs $1000 and it will wear out in 10 years. 
When worn out, its scrap value is $50. Under the sinking fund plan, 

i In this book, unless otherwise specified, we shall always assume that the charge 
for depreciation during each year is made at the end of the year, 

96 



DEPRECIATION AND CAPITALIZED COST 



97 



determine the depreciation charge which should be made at the end of 
each year for 10 years, if the fund is invested at 5%, effective. 

Solution. Let $2 be the annual charge. The replacement cost S = $950. 



Case 1 
n = 10 int. periods, 

p = 1, i .05, 
R=$x,S = $950. 



950 
x = 950- 



at .05), 



. 05) 



$75.529. 



DEPRECIATION TABLE 



YEAH 


[NT. Dun ON FUND 
AT END OF YBAB 


PAYMENT TO FUND 
AT END OF YBAB 


IN Fmn> AT 
END OF YEAR 


BOOK VALUE 
AT END car YBAB 




1 


$0 



$0 
75.529 


$ 
75.529 


$1000.00 
924.47 


2 


3.777 


75.529 


154.835 


845.16 


3 


7.742 


75.529 


238.106 


761.89 


4 


11.906 


75.529 


325.540 


674.46 


. 5 


16.277 


75.529 


417.346 


582.65 


6 


20.867 


75.529 


513.742 


486.26 


7 


25.687 


75.529 


614.958 


385.04 


8 


30.748 


75.529 


721.235 


278.76 


9 


36.062 


75.529 


832.826 


16747 


10 


41.641 


75.529 


949.996 


50.00 



In Figure 5 the growth of the fund 
and the decrease in the book value 
are represented graphically. 

NOTE 2. A good depreciation 
plan is in harmony with the funda- 
mental principle of economics that 
capital invested in an enterprise 
should be kept intact. Thus, at the 
end of 2 years in Example 1, the 
$154.84 in the fund should be con- 
sidered as capital, originally in- 
vested in the machine, which has 
been returned through the revenues 
of the business. The book value 
of the machine, 1000 - 154.84 = 
$845.16, is the amount of capital 
still invested in the machine. 

























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onn 




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7nn 








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S^ 


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r 


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inn 








c- 

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2 - 
.-Ye 


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ars 
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a. 


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) l( 



98 MATHEMATICS OF INVESTMENT 

The condition per cent of an asset at any time is the ratio of its 
remaining wearing value to its wearing value when new. 

Example 2. In Example 1, find the condition per cent at the end of 
6 years. 

Solution. Original wearing value is $950. Book value at end of 6 years is 
$486.26, and the remaining wearing value is 486.26 - 60 = $436.26. The 

condition per cent is 4 ^ 6 = .45923 or 45.923%. 
you 

EXERCISE XXXIX 

1. In illustrative Example 1 above, find the amount in the deprecia- 
tion fund at the end of 4 years, without using the depreciation table. 

2. A machine costs $3100 when new, it wears out in 12 years, and its 
final scrap value is $100. Under the sinking fund plan, determine the 
depreciation charge which should be made at the end of each 6 months 
if the fund accumulates at (.05, m = 2) . 

3. (a) In problem 2, without forming a depreciation table, find the 
amount in the depreciation fund, and the book value of the asset, at the 
end of 5 years. (6) What is the condition per cent at this time? 

/ 4. A building costs $100,000 and it will last 20 years, at the end of 
which time its salvage value is $5000. Under the pinking fund plan, 
when the fund earns 5%, effective, determine the size of the depreciation 
fund at the end of 6 years, if deposits in the fund are made at the end of 
each year. 

6. A motor truck has an original value of $2500, a probable life of 6 
years, and a final salvage value of $200. Its depreciation is to be covered 
by deposits in a fund at the end of each 3 months. Find the quarterly 
deposit if the fund earns (.055, m = 2). 

6. A manufacturing plant is composed of part (a) whose post is 
$90,000, life is 15 years, and salvage value is $6000, and part (&) whose 
cost is $50,000, life is 20 years, and salvage value is $5000. If deprecia- 
tion charges are made at the end of each year and accumulate at 4%, 
effective, what is the total annual charge for the plant? 

42. Straight line method. Consider an asset whose life is 
n years and whose replacement cost (wearing value) is $$, Sup- 
pose that annual depreciation charges are placed in a fund which 
does not earn interest. Then, in order to have $S at the end of 

cr 

n years, the annual charge must be -. The fund increases each 

n 



DEPRECIATION AND CAPITALIZED COST (99.) 

Sf 

year by -, and hence the book value decreases each year by - 

n n 

This method is called the straight line method, because we obtain 
straight lines as the graphs of the book value and of the amount 
in the fund (see problem 1 below). 

NOTE. An essential characteristic of the straight line method is that, 

under it, all of the annual decreases in book value are equal to -th of the total 

n 

wearing value S. It is usually stated, under this method, that is vrritten off 

n 

the book value each year for depreciation. Recall, from the table in Example 1 
of Section 41, that the annual decreases in book value are not equal under the 
sinking fund plan ; they increase as the asset grows old. 

NOTE. The straight line method is the special case of the sinking fund 
method, where the rate earned on the depreciation fund is 0%. Hereafter, any 
general reference to the use of the sinking fund method should be understood 
to include the straight line plan as one possibility. In supplementary Section 
48 below, another depreciation plan is considered which applies well to assets 
whose depreciation in early life is large compared to that in old age. The 
student is referred to textbooks on valuation and on accounting for many 
other special methods which are in use. Usually, each of these is particularly 
desirable for a certain type of assets. 

EXERCISE XL 

J 1. A building costs $50,000 and has a salvage value of $5000 when 
worn out at the end of 15 years, (a) Under the straight line method, 
form a table showing the amount in the depreciation fund and the book 
value at the end of each year. (6) Draw separate graphs of the book 
value and of the amount in the fund, using the years as abscissas. 

2. In problem 1, find the annual depreciation charge under the sink- 
ing fund method, where the fund earns 4%, effective, and compare with the 
charge in problem 1. 

43. Composite life. If the plant of an enterprise consists of 
several parts whose lives are of different lengths, it is useful to 
have a definition for the average or composite life of the plant as 
a whole. Let $S be the sum of the wearing values (replacement 
costs) and $D the sum of the annual depreciation charges for all 
parts. Under the sinking fund method let i be the effective rate 
earned on the depreciation fund. Then, the composite life is 
defined as the term of years, necessary for an annuity of $D, 




100 MATHEMATICS OF INVESTMENT 

paid annually, to have the amount $& If k is the composite life 

in years, then 

8 = D(SR at i), (46) 

which can be solved for k by interpolation. 

Under the straight line method of depreciation, fc is the number 
of times $D must be paid into a fund in order that the fund, with- 
out earning interest, should equal $S. Hence 

S = kD, k - |- (47) 

NOTE. In equation 46 place i = 0%. Then (s^ at 0) = k and equation 
46 becomes 5 = kD, as found above. This is an obvious consequence of the 
fact that the straight line method is the sinking fund method with i = 0%. 

Example 1. A plant consists of part A, with life 20 years, original 
cost $55,000, and scrap value $5000 ; part B with life 15 years, original 
cost $23,000, and scrap value $3000 ; part C, with life 10 years, origi- 
nal cost $16,000 and scrap value $1000. Determine the composite Me, 
(a) under the sinking fund method, with interest at 4%, effective, and 
(&) under the straight line method. 

Solution. (a) The total wearing value is 50,000 -f 20,000 + 15,000 
= $85,000. Under the sinking fund method, the annual charge for part A is 

50000 t = $1679.09. Similarly, the charges for B and C are $998.82 

(s^oi.04) 

and $1249.36, respectively. The total annual charge is $3927.27. Let k be 
the composite life. Then, the annuity of $3927.27, paid annually for k years, 
should have the amount $85,000, or 



Casel 

n = k int. periods, 
P = 1, * - .04, 
R - $3927.27, S = $85,000. 



85000 - 8027.27(1)0 at .04) ; 
4) =21.644. 



Table VII, k - 15.90 



.(b) Under the straight line method, the -annual depreciation charges for the 
parts A, B, and C are, respectively, $2500, $1333.33, and $1500. The total 
annual charge is $5333.33. The composite lif e k is 



NOTE. Compare the results above. When the rate on the sinking fund 
is 0% (the straight line method), the composite life differs from the life when 
i = 4%, by only (15.94 15.90) = .04 year. Thus, under the sinking fund 
method, regardless of the rate earned on the fund, the composite life may be 
obtained approximately by finding the Me under straight line depreciation. 



DEPRECIATION AND CAPITALIZED COST 



EXERCISE XLI 



v 1. Fhid the composite life for the plant with parts A, B, and C below, 
under the sinking fund method (a) at 3%, effective, and (ft at 6%, effec- 
tive, and under (c) the straight line method. 



PABT 


LIFE 


COST 


SCHAP VALUE 


A 


10 


$20,500 


$ 500 


B 


20 


35,760 


750 


C 


16 


19,000 


1000 



44. Valuation of a mine. A mine, or any similar property, is a 
depreciable asset which becomes valueless when all of the ore is 
removed. Part of the net revenue of the mine should be used to 
accumulate a depreciation, or redemption fund, which will return 
the original invested capital when the mine is exhausted. The 
revenue remaining, after the depreciation charge, is the owner's 
net return on his investment. 

NOTE. We shall assume, in this book, that the annual revenue, or royalty, 
from a mine, or similar enterprise, is payable in one installment at the end of 
the year. 

Example 1. A mine, whose life is 20 years, costs $200,000 cash. 
"What should be the net annual revenue in order to pay 6% interest, 
annually, on the invested capital, and to provide an annual deposit for 
a redemption fund which accumulates at 4%, effective? 

Solution. Let $x be the annual deposit in the sinking fund to provide 
$200,000 at the end of 20 years. 

200,000 - x(s at .04) ; x = $6,716.35. 



Case 1 
n = 20 int. periods, 

p = 1, i = .04, 
R - $z, 8 = $200,000. 



Annual interest at 6% on $200,000 is $12,000. 
Annual revenue required is 12,000 + 6,716.35 
= $18,716.35. 

Mining engineers furnish accurate estimates, for any given mine, 
of the life, k years, and of the annual revenue, $ JB. Suppose that a 
purchaser pays $P for a mine. If the annual revenue payments of 
$55 are exactly sufficient to amortize the original invested principal 
P at the effective rate i, then P is the present value of the annuity 
formed by the revenue installments, or 

P = R(an at i). (48) 



1102) MATHEMATICS OF INVESTMENT 

Recall x that the amortization payments are exactly sufficient to 
pay interest at the rate i on P and to accumulate a sinking (redemp- 
tion) fund at the rate i to repay P at the end of k years. Therefore, 
the price paid under the assumption that P is amortized is the price 
we should obtain on assuming that the investor receives the rate i 
on his investment and places the surplus revenue in a redemption 
fund which accumulates at the rate i. 

Consider determining the purchase price P if the buyer desires 
the effective rate i on his investment and is able to invest his 
redemption fund at the effective rate r. Let $D be the deprecia- 
tion charge deposited annually in the redemption fund, which 
accumulates to the amount P at the end of k years. Then 



Annual interest on $ P at the rate i is PL Since 

revenue = (interest on capital) + (depreciation charge), (49) 

fi-Fi + D-Fi + JV-V -?(*+ 1_Y (50) 

(sji atr) \ (sj] at r)/ ^ ' 

P = - -- (51) 



Example 2. The annual revenue from a mine will be $30,000 until 
it becomes exhausted at the end of 25 years. What should be paid for the 
mine if 8% is to be earned on the invested capital while a redemption 
fund accumulates at 5%? 

Solution. From formula 51 

P = 3000 . 3000 = $297,170. (Table IX) 

nft . 1 .10095246 ' V 

U8 + ( S2B1 oi.06) 

Nom In equation 61 place r = i, and use formula 39. One obtains 
p a jffi(ori at i), as obtained previously for this case in equation 48. 

EXERCISE XLII 

1. A purchaser paid $300,000 for a mine which will be exhausted at 
the end of 50 years. What annual revenue from the mine will be required 
to pay 7% on the investment and to provide an annual deposit in a re- 
demption fund which accumulates at 5%, effective? 

1 See Section 38. 



DEPRECIATION AND CAPITALIZED COST 103 

2. The annual revenue from a mine will be $50,000 until the ore is ex- 
hausted at the end of 50 years. A purchaser desires 7% on his investment. 
What should he pay for the mine if his redemption fund accumulates 
(a) at 7%; (6) at 5%; (c) at 4%? 

3. The privileges of a certain patent last for 10 years and the annual 
royalties from it will be $75,000. If a redemption fund can be accumu- 
lated at 5%, what should an investor pay for the patent rights if he de- 
mands 6% on his investment ? 

4. An investor in an oil property desires 10% on his investment and 
assumes that he can accumulate a redemption fund at 5%. What should 
he pay for an oil field whose net annual revenue for its 10 years of life is 
estimated at $100,000? 

5. A purchaser of a wooden ship estimates that the boat will be prac- 
tically valueless at the end of 6 years. If the net earnings for each of 
these years will be $50,000, what should the purchaser pay if he desires 
8% on his investment and accumulates a depreciation fund at 4%? 

45. Perpetuities. A perpetuity is an annuity whose payments 
continue forever. Present values l of perpetuities are useful in 
capitalization problems. 

Suppose that $1000 is invested at 6%, effective. Then, $60 
interest is received at the end of each year, forever. That is, at 6%, 
the present value of a perpetuity of $60, paid annually, is $1000. 
Similarly, if $A is invested at the rate i, payable annually, it will 
yield R = Ai interest annually, forever. Hence, the present value 
$.A of a perpetuity of $72 paid at the end of each year is obtained 
from Ai = R ; or, A = - (52) 

NOTE 1. If, in the paragraph above, we change the word year to interest 
period, it is seen that, when the interest rate per period is i, the present value 

R 

of a perpetuity of $12, paid at the end of each interest period, is 

% 

Example 1. At the end of each 6 months, $50 is required to clean a 
statue. If money is worth (.04, m = 2), what is the present value of 
all future renovation? 

Solution. The future renovation costs form a perpetuity whose present 

value, by formula 62, is -^ = $2500. 
02 

1 The notion of the amount of a perpetuity is meaningless and useless, since 
the end of the term of a perpetuity does not exist. 



104 MATHEMATICS OF INVESTMENT 

Consider a perpetuity of $1 paid at the end of each k years, and 
let (OK,, k at i) be its present value when money is worth the ef- 
fective rate i. In order to find a formula for a aa> t , first let us de- 
termine the installment $x which, if paid into .a fund at the end of 
each year for. k years, will accumulate to $1 at the end of k years. 
Then, $1 is the amount of the annuity of $x per annum and 

I = x(st\ at i) ; x ^- Therefore, a perpetuity of Ufcc per 

(St\ at i) 

annum will create a fund from which $1 can be paid at the end of 
each k years, forever. Hence, the present value of the perpetuity 
of $1 at the end of each k years is equal to the present value of the 
perpetuity of $#, per annum. Therefore, from formula 52 with 

R = x = 



i i i (sji at i) 

The present value $A of a perpetuity of $R paid at the end of 
each k years is 

:*") = ? 7^' (^) 



NOTE 2. Thus, if money is worth (.05, m = 1), the present value of a 
perpetuity of $90,000 paid at the end of each 20 years is 

A = PM- L_ . 9||p (.03024269) = $54,436.5. (Table IX) 

If i is not a table rate, formula 53 must be computed by inserting the explicit 
formula for (s^i ai i). 

NOTE 3. Recognize that formula 52, or formula 63, applies to a perpetuity 
whose first payment comes at the end of the first payment interval. The 
present value of a perpetuity due, or of a deferred perpetuity, can be obtained by 
the methods used for the corresponding type of annuity. 

EXERCISE XLHI 

1. (a) Find the present value of a perpetuity which pays $100 at the 
end of each 3 months, if money is worth (.08, m = 4). (6) -What is 
the present value if the payments occur at the beginning of each 3 months ? 
v/2. An enterprise will yield $5000 net profit at the end of each year. 
At 4%, find the capitalized value of the enterprise, where the "capitalized 
value at 4%" is the present value of all future earnings. 



DEPRECIATION AND CAPITALIZED COST 105- 

3. A bridge must be repainted each 5 years at a cost of $8000. If 
money is worth 5%, find the present value of all future repainting. 

4. A certain depreciable asset must be replaced at the end of each 25 
years at a cost of $50,000. At 6%, find the present value of all future 
replacements. 

6. Find the present value of an annuity of $1000 paid at the end of 
each year for 75 years, if money is worth (.04, m = 1). Compare the 
result with the present value of a perpetuity of $1000, paid annually. 

6. To repair a certain road, $1000 will be needed at the beginning of 
the 4th year and annually thereafter. Find the present value of all 
future repairs if money is worth 6%, effective. 

46. Capitalized cost. The capitalized cost of an asset is de- 
fined as the first cost plus the present value of all future replacements, 
which it is assumed will continue forever. Let $C be the first 
cost and $# the replacement cost of an asset which must be renewed 
at the end of each k years. Then, the capitalized cost $X equals 
$C plus the present value of a perpetuity of $R paid at the end of 
each k years. When money is worth the effective rate i, we obtain, 
on using formula 53, 

K-C + ZL-. (54) 

i (s^ati) 

If the replacement cost equals the first cost, R C. Then, on 
changing C to -r and on placing R = C in formula 54, 



i 



= i + 



at i) i\ (SB at i 



\ (See formula 39) 
)J 



i (a^ati) 

Example 1. A machine costs $3000 new and must be renewed at the 
end of each 15 years, (a) Find the capitalized cost when money is 
worth (.05, m = 1), if the final scrap value of the machine is $500; 
(6) if the scrap value is zero. 

Solution. (a) Use formula 64 with C - $3000 and R = $2500. 



K - 3000 + ; - - $5317.12. 
.05 



100 MATHEMATICS OF INVESTMENT 

(ft) When the scrap value is zero, C - R = $3000. From formula 65, 



NOTIS 1. If tho renewal coat of an asset is $1? and its life k years, the 
annual depreciation charge $D is given by R = D(s^ at i), or, 

(56) 

These future depreciation charges form a perpetuity of $D, paid annually, 
whoso present value is , or 



.1 D = l/fl 1A . 3 _J 

i i\ (s^jfrt i)/ * (a^oii) 



This renult is tlio same (see formula 54) as the present value of all future re- 
newal costs. Hence, the definition of capitalized cost may be restated to be 
" tho first cost plus tho present value of all future depreciation charges." 
NOTE 2. In formula 54, multiply both sides by i. Then 

Ki = Ci + R - - Ci + D. (See equation 56) 
(a ft at i) 

Thus, if an enterprise earns interest at the rate i on the capitalized cost K, 
tho revenue Ki provides for tho interest Ci at the rate i on the invested capital 
C and likewise for tho annual depreciation charge D. 

If two assets are available for serving the same purpose, that one 
should be used whoso capitalized cost is least. If their capitalized 
costs are the same, both assets are equally economical. 

Example 2. A certain type of pavement costs $12 per square yard, 
laid in place, and must be renewed at the same cost every 10 years. How 
much could a highway commission afford to pay to improve the pave- 
ment so that it would last 15 years, if money is worth 4%, effective? 

Solution. Let $* bo the cost per square yard of the improved type of 
pavement, whoso life is 15 years. If this type is just as economical na the old, 
its capitalized cost must bo tho same. The capitalized costs of the two types, 
as given by equation 66, are equated below : 

12 1 .,_. 1 

of .04) .04(0^0^.04) 

M) 12(11.1183874) $16 _ 450 . 
4) 8.1108958 

The commission could afford to pay anything less than (16.450 - 12) or $4.450 
to improve the old pavement. 



DEPRECIATION AND CAPITALIZED COST 

EXERCISE i XLIV 

1. Find the capitalized cost of a plant whose original cost is $200,000 
and whose life is 25 years, if its final salvage value is $15,000. Money 
is worth 4%. 

2. A section of pavement costing $50,000 has a lif e of 25 years. Find 
its capitalized cost if the renewal cost is $50,000, and if money is worth 3%. 

3. If it costs $2000 at the end of each year to maintain a section of 
railroad, how much would it pay to spend, immediately, to improve the 
section so that the annual maintenance would be reduced to $500? 
Money is worth 5J%. 

4. A bridge must be rebuilt every 50 years at a cost of $45,000. Find 
the capitalized cost if the first cost is $75,000, and if money is worth 3%. 

6. One machine costs $15,000, lasts 25 years, and has a final salvage 
value of $1000. Another machine for the same purpose costs $18,000, 
lasts 28 years, and has a salvage value of $2000. If money is worth 5%, 
which machine should be used? 

6. Would it be better to use tile costing $18 per thousand and lasting 
15 years, or to use other material costing $22 per thousand and lasting 
20 years, if money is worth 5% and if neither material has a scrap value? 

\f 7. A corporation is considering the use of motor trucks worth $5000 
each, whose life is 4 years and salvage value is zero. How much would 
it pay to spend, per truck, to obtain other trucks whose life would be 
6 years, and final salvage value zero ? Money is worth 5%. 

8. The interior of a room can be painted at a cost of $10 and the paint- 
ing must be repeated every 2 years. If money is worth 6%, how much 
could one afford to pay for papering the room if the paper would need 
renewal every 3 years? 

9. A certain manufacturing plant involves one part worth $100,000 
new and needing replacement every 10 years at a cost of $90,000, and a 
second part costing $52,000 new and needing renewal every 12 years 
at a cost of $50,000. What should be the net operating revenue in order 
to yield 7% on the capitalized cost? 

10. A certain dam will cost $100,000 and will need renewal at a cost of 
$50,000 every 10 years. If money is worth 4^%, how much could one 
afford to pay in addition to $100,000 to make the dam of permanent type? 

1 After this exercise the student may proceed immediately to the Miscellaneous 
Problems at the end of the chapter, 



108 



MATHEMATICS OF INVESTMENT 
SUPPLEMENTARY MATERIAL 



47. Difficult cases under perpetuities. Perpetuities are met 
to which formulas 52 and 53 do not apply. A systematic means 
for finding the present values of all perpetuities is furnished by 
infinite geometrical progressions. 

Example 1. If money is worth (.05, m = 2), find the present value 
of a perpetuity of $6 paid at the end of each 3 months. 

Solution. The present value $A of the perpetuity is the sum of the present 
values of all of the payments as listed below : 



Payment of $6 
due at end of 


3 mo. 


6 mo. 


9 mo. 


etc. 


to infinity . 


Present value 
of payment 


6(1.025)~* 


GCl.CESr 1 


6(1.025)"* 


etc. 


to infinitely 
many terms. 



A = 6[(1.025)-*+(1.026)^+ (1.025)-*+ . etc. to infinitely many terms}. 
The bracket contains an infinite geometrical progression whose first term a = 
(1.025)"*, and whose ratio w = (1.025)"*. The sum 1 of the series is - ; 
6(1.025)"* _ 6(1.025)"* (1.025)* 



1 - (1.025)-* 1 - (1.025)-* (1.025)* 



A 



6 



6 



.0124228 



$482.98. 



(Table X) 



(1.026)*- - 1 

NOTE. The formulas 52 and 53 of Section 45 can be obtained by the 
method of Example 1 (see problems 3 and 5 below). 

EXERCISE XLV 

Use geometrical progressions unless otherwise directed. 
1. . The annual rent of a perpetuity is $1000, payable in semi-annual 
installments. Find the present value when money is worth (.06, m = 1) . 

2. Find the present value of the perpetuity in problem 1 if money is 
worth (.06, m = 4). 

3. Derive the formula 52 for the present value of a perpetuity of $1 
paid annually, when money is worth the effective rate i. This present 

value is generally denoted by the symbol a* ; that is, (a w at i) = T- 
1 See Formula 22, Section, 91, 



DEPRECIATION AND CAPITALIZED COST 109/ 1 

4. (a) By use of a geometrical progression, find the present value of 
a perpetuity of $100 paid at the end of each 10 years, if money is worth 
5%, effective. (&) Compare with the result obtained by use of formula 53. 

6. Derive formula 63 for R (a a0i fc at i), the present value of a perpetu- 
ity of $E paid at the end of each k years, with money worth the effective 
rate i. 

6. At 6%, effective, find the capitalized value of an enterprise which 
yields a net revenue of $500 at the end of each month. 

7. Let (a>at i) represent the present value, when money is worth i, 
effective, of a perpetuity whose annual rent is $1, paid in p installments 

per year. Prove that (a$at i) = = 1. (a^ at i). 

Jp Jp 

8. If money is worth (.06, m = 4), find the present value of a perpetuity 
of $1000, paid semi-annually, by use of formula 53. 

9. If money is worth (.05, m = 2), find the present value of a perpetu- 
ity of $100 paid monthly, by use of the result of problem 7. 

^' 10. An irrigation system has just been completed. There will be no 
repair expense until the end of two years, after which $50 will be needed 
at the end of each 6 months. If money is worth 4%, effective, find the 
present value of the future upkeep. Solve by any method. 

48. Constant percentage method of depreciation. Under the 
constant percentage method, the book value decreases each year 
by a fixed percentage of the value at the beginning of the year. If 
the life of the asset is n years, the constant percentage r, expressed 
as a decimal, must be chosen so that the original cost $C is reduced 
to the residual book value $22 at the end of n years. The decrease 
in the first year is Cr, and the value at the end of 1 year is C Cr 
= C(l r). Similarly, the book value at the end of each year is 
(1 r) tunes the value at the beginning of the year. By the end 
of n years, the original value C has been multiplied n times by 
(1 r) , or by (1 r) n , and the residual scrap value R is C(l r) n . 
Therefore, we may obtain r from 

, (57) 



Each annual reduction in book value is the depreciation charge for 
that year, and, if we consider all of these reductions in book value 



110 



MATHEMATICS OF INVESTMENT 



placed in a depreciation fund which does not earn interest, the fund 
will contain the replacement cost at the end of n years. 

Example 1. For a certain asset, the original cost is $3000, the life is 
6 years, and the scrap value is $500. Find the annual percentage of de- 
preciatiou under the constant percentage method and form a table showing 
the changes in book value? 

Solution. From equation 67, 



log 5 = 9.22185 - 10. 
log (1 - r) = ilogi = 9.87031 - 10. 



V 3000 V o' 

1 - r = .74183. 
r = .25817. 

The book values in the table below were computed by 5-place logarithms. 
Thus, at the end of 3 years, the book value is B = 3000(1 r) 8 ; 
log (1 - r) s = 3 log (1 - r) = 3(9.87031 - 10) - 9.61093 - 10 

log 3000 = 3.47712 

log B = 3.08805; B = $1224.7. 

DEPRECIATION TABLE 



YBAB 


BOOK VALUE AT END 
OF YBAB 


DEPRECIATION 
DUBINQ YEAB 


IN DHPR. PTTND AT 
END OF "XEAB 


1 


$2225.5 


$774'.5 


$ 774.5 


2 


1651.0 


574.5 


1349.0 


3 


1224.7 


426.3 


1775.3 


4 


908.6 


316.1 


2091.4 


5 


674.0 


234.6 


2326.0 


6 


500.0 


174.0 


2500.0 



NOTE. When the scrap value R is relatively small, the method above 
gives ridiculously high depreciation charges in the early years. The method 
breaks down completely when R = 0. 

EXERCISE XLVI 

1. (a) Find the annual percentage of depreciation under the constant 
percentage method, for a machine whose original cost is $10,000, life IB 
5 years, and scrap value is $1000. (6) Form a depreciation table and 
draw a graph of the changes in book value. 

2. In problem 1, find the annual depreciation charge under the sinking 
fund plan, where the fund earns 4%, effective, and compare with the 
result of problem 1, 



DEPRECIATION AND CAPITALIZED COST (ill 

A machine, whose life is 20 years, costs $50,000 when new and has a 
scrap value of $5000 when worn out. Find the annual rate of deprecia- 
tion under the constant percentage method. 

4. For a certain asset, the depreciation in value during the early years 
of its life is known to be very great, as compared with the later years* 
Which of the three methods, straight line, sinking fund, or constant per- 
centage, would give a series of book values most in harmony with the 
actual values during the life of the asset? Justify your answer. 

MISCELLANEOUS PROBLEMS 

Depreciation, in problems below, is under the sinking fund method. 

1. An automobile costs $3500 when new, and its salvage value at the 
end of 6 years is $400. (a) If the depreciation fund earns 4%, by how 
much is the book value decreased during the 4th year? (6) By how 
much is the book value decreased during the 4th year, under the straight 
line method? 

2. A hotel has been built at a cost of $1,000,000 in an oil-boom city 
which will die at the end of 25 years. Assuming that the assets can be 
sold for $100,000 at that time, what must be the net annual revenue during 
the 25 years to earn 7% on the investment and to cover depreciation, 
where the depreciation fund earns 4%? 

3. A syndicate will build a theater in a boom city which will die at the 
end of 30 years. For each $10,000 unit of net annual profit expected, 
how much can the syndicate afford to spend on the theater if 8% is desired 
on the investment while a redemption fund to cover the initial investment 
is accumulated at 5%? Assume that the theater will be valueless at the 
end of 30 years. 

4. A machine, worth $100,000 new, will yield 12% net annual operating 
profit on its original cost, if no depreciation charges are made. If the 
life of the machine is 20 years, what annual profit will it yield if annual 
depreciation charges are made, where the depreciation fund accumulates 
at 4%, effective? Assume that the final salvage value of the machine 
is zero. 

6. The life of a mine is 30 years, and its net annual revenue is $50,000. 
Find the purchase price to yield an investor 7%, if the redemption fund 
accumulates at 4%. 

6. A mine will yield a net annual revenue of $25,000 for 20 years. 
It was purchased for $200,000. If, at this price, the investor considers 



112 MATHEMATICS OF INVESTMENT 

that he obtains 10% on his investment, at what rate does he accumulate 
his redemption fund? 

7. A certain railroad will cost $60,000 per mile to build. To maintain 
the roadbed in good condition will cost $500 per mile, payable at the be- 
ginning of each year. At the end of each 30 years, the tracks must be 
relaid at a cost of $30,000. What is the present value of the construction 
and of all future .maintenance and renewals, if money is worth 5%? 

8. Find the capitalized value at 6%, effective, of a farm whose net 
annual revenue is $3000. 

9. An automobile with a wearing value of $1200 has a life of 5 years. 
Upkeep and repairs cost the equivalent of $450 at the end of each year. 
What is the annual maintenance expense if the owner accumulates a de- 
preciation fund by annual charges invested at 5%? 

^0. A certain piece of forest land will yield a net annual revenue of 
$25,000 for 15 years, at the end of which time the cut-over land will be 
sold for $15,000. (a) If money is worth 6% to an investor, what should 
he pay for the property? (&) If the investor desires 9% on his in- 
vested capital, and assumes that he can accumulate a redemption fund 
at 5% to return his original capital at the end of 15 years, find the price 
he should pay for the land, by use of the method which was used in 
deriving formula 51 for the valuation of a mine. 



CHAPTER VII 
BONDS 

49. Terminology. A bond is a written contract to pay a 
definite redemption price $C on a specified redemption date and 
to pay equal dividends $D periodically until after the redemption 
date. The dividends are usually payable semi-annually, but may 
be paid annually or in any other regular fashion. The principal 
$F mentioned in the face of the bond is called the face value or 
par value. A bond is said to be redeemed at par if C = F (as is 
usually the case), and at a premium if C is greater than F. The 
interest rate named in a bond is called the dividend rate. The 
dividend $D is described in a bond by saying that it is the interest, 
semi-annual or otherwise, on the par value F at the dividend rate. 

Nona. The following is an extract from an ordinary bond : 

The Kansas Improvement Corporation acknowkdges itself to owe and, 
for value received, promises to pay to bearer FIVE HUNDRED DOLLARS 
on January 1st, 1926, with interest on said sum from and after January 
1st, 1920, at the rate 6% per annum, payable semi-annually , until the said 
principal sum is paid. Furthermore, an additional 10% of the said 
principal shatt be paid to bearer on the date of redemption. 

For this bond, F = $500, C = $550, and the semi-annual dividend D = $15 
is semi-annual interest at 6% on $500. A bond is named after its face F and 
dividend rate, so that the extract is from a $500, 6% bond. Corresponding to 
each dividend D there usually would be attached to the bond an individual 
coupon containing a written contract to pay $D on the proper date. 

50. When an investor purchases a bond, the interest rate i which 
he receives 'on his investment is computed assuming that he will 
hold the bond until it is redeemed. It is important to recognize 
that the investment rate i is not the same as the dividend rate of the 
bond, except in very special cases, because i depends on all of the 
following : $P, the price paid for the bond ; $C, the redemption 
price ; the time to elapse before the redemption date ; the number 
of times per year dividends are paid, and the size of $1), the 
periodic dividend. 

113 



114 MATHEMATICS OF INVESTMENT 

In this chapter we shall solve two principal problems. First, the 
determination of the price $P which should be paid for a specified 
bond if we know the investment rate demanded by the buyer. 
Second, the determination of the investment rate if we know the 
price which the investor had to pay. 

61. Purchase price to yield a given rate. The essential 
features of a bond contract are the promises (a) to pay $C on the 
redemption date and (6) to pay the annuity formed by the periodic 
dividends of SD. 1 If an investor desires a specified investment 
rate, the price $P he is willing to pay on purchasing the bond is 

p = (present value of $C due on the redemption date) (58) 
+ (present value of the annuity formed by the dividends), 

where present values are computed under the investor's rate. 

Example 1. A $1000, 6% bond, with dividends payable semi-annually, 
will be redeemed at 105% at the end of 15 years. Find the price to yield 
an investor (.05, m = 1). 

Solution. " At 105% " means at a premium of 5% over the par value. 
F = $1000, C = 1000 + 50 = $1050. The semi-annual dividend D = 
(.03)1000 = $30. The redemption price $1050 is due at the end of 15 
years. Hence, at the rate (.05, m = 1), 



Div. annuity, Case 1 

n = 15 int. periods, 

p - 2, i = .05, R - $60. 



P = 1050(1.05)-" + 60(0^0* .05). 

F = 506.07 + 60 5 s . (a at .05) = $1136.54. 
Ja 



EXERCISE XLVII 

1. A $1000, 5% bond, with dividends payable semi-annually, will be 
redeemed at 108% at the end of 7 years. Find the price to yield an in- 
vestor 6%, compounded semi-annually. 

The bonds in the table are redeemable at par. Find the purchase 
prices. The life is the time to the redemption date. 

1 When a bond is sold on a dividend date, the seller takes the dividend $D which 
is due. The purchaser will receive the future dividends, which form an ordinary 
annuity whose first payment is due at the end of one dividend interval, and whose 
last payment is due on the redemption date. 



BONDS 



PBOB. 


PAB VALUE 


Lira 


DIVIDEND 
RATH 


DIVIDENDS 
PAYABLE 


INVESTMENT 
RATH 


2. 


$ 1,000 


10 yr., 6 mo. 


5% 


semi-arm. 


(.06, m = 2) 


3. 


100 


17 yr. 


6% 


annually 


(.07, ro = 1) 


4. 


1,000 


14 yr. 


7% 


semi-ann. 


(.08, m = 1) 


6. 


500 


9yr. 


8% 


semi-ann. 


(.04, m = 2) 


6. 


2,000 


7yr. 


4% 


quarterly 


(.05, m = 4) 


7. 


1,000 


8 yr., 6 mo. 


3% 


semi-ann. 


(.06, m = 2) 


8. 


1,000,000 


13 yr., 6 mo. 


5J% 


semi-ann. 


(.04, m = 2) 


9. 


100,000 


19 yr. 


5% 


semi-ann. 


(.06, m = 1) 



10. A $10,000, 5% bond, whose dividends are payable annually, will be 
redeemed at par at the end of 30 years. Find the purchase prices to yield 
(a) 5%, effective ; (6) 7%, effective ; (c) 4%, effective. Compare your results. 

11. A $1000, 6% bond, whose dividends are payable semi-annually, is 
purchased to yield 5%, effective. Find the price if the bond is to be re- 
deemed at the end of (a) 5 years ; (6) 20 years ; (c) 75 years. Com- 
pare your results. 

s/ 12. A $100,000, 5% bond is redeemable at 110% at the end of 15 years, 
and dividends are payable annually. Find the price to yield (.06, m = 2). 

If a bond is redeemable at par (C = F) and if the investor's in- 
terest period equals the interval between successive dividends, 

it is easy to compute the premium (P F), the excess of the 
price P over par value F. Let k be the number of dividend periods 
to elapse before the bond matures, r the dividend rate per dividend 
interval, and i the investor's rate per interest period. Then, a 
dividend D = Fr is due at the end of each interest period and the 
redemption price F is due at the end of k periods. The equations 
below are easily verified. 



Div. annuity, Case 1 
7i = k int. periods, 
p = 1, i = i, R = Fr. 



P = 
P - F = 



[ at i) 



From formula 28, - Fi (a^ at i) = - Fi 
Therefore, P - F - Fr(a^ at i)'- Fi 






* 



- F. 



= F(l + i)-* - F. 



i) = (Fr - Fi)(a m at i). 



Premium = P - F = F(r - 0(45-, at i). 



(59) 



116 



MATHEMATICS OF INVESTMENT 



NOTE 1. Formula 59 shows that, when r is greater than i, P F is 
positive, or the bond is purchased at a positive premium over par value F. 
When r is less than i, P F is negative or the bond is purchased at a negative 
premium, that is, at a discount from the par value F. 

Example 2. A $1000, 6% bond, with dividends payable semi- 
annually, is redeemable at par at the end of 20 years, (a) Find the price 
to yield an investor (.05, m = 2). (6) To yield (.07, m = 2). 

Solution. (a) From formula 59 with F = $1000, the premium is P F 
= 1000C03 - .025) (a^ at .025) = 5(0^ at .025) = $125.51. P - F+ 125.51 
= $1125.51. (&) Premium = P - F = 1000(.03 - .035) (a^ at .035) - 
- 6(0^ at .035) = - $111.78. P = F - 111.78 = $882.22. In this case, 
we say the discount is $111.78. 

NOTE 2. Equation 59 could have been proved by direct reasoning. Sup- 
pose r is greater than i. Then, if an investor should pay $F for the bond, he 
would desire Fi as interest on each dividend date. Since each dividend is F r, 
he would be receiving (Fr Fi) = F(r i) excess income at the end of each 
interest period for k periods. Hence, he should pay, in addition to $F, a 
premium equal to the present value of the annuity formed by the excess income 
or F(r i)(a,j^at i). Similarly, when r is less than i, if the investor should 
pay $/*" for the bond, there would be a deficiency in income of Fi Fr = F(i r) 
at the end of each interest period. Hence, the present value of the defi- 
ciency or F (i r) (O F| at i) should be returned to the investor as a discount 
from the price F we supposed paid. 



VALUES TO THE NEAREST CENT OF A $100,000, 5% BOND WITH 
SEMI-ANNUAL DIVIDENDS 



INVEST. RATH 
WITH m 2 


TIMB TO REDEMPTION D/LTB 


101 YEARS 


11 YEABS 


Hi YEABS 


12 YEABS 


.0400 


108505.60 


108829.02 


109146.10 


109456.96 


.0405 


108060.01 


108365.61 


108665.14 


108958.72 


.0410 


107616.62 


107904.58 


108186.75 


108463.25 


.0415 


107175.43 


107445.93 


107710.93 


107970.54 


.0420 


106736.43 


106989.64 


107237.65 


107480.56 


.0425 


106299.59 


106535.71 


106766.91 


106993.30 


.0430 


105864.92 


106084.11 


106298.69 


106508.75 


.0435 


105432.40 


105634.84 


105832.97 


106026.89 


.0440 


105002.01 


105187.88 


105369.74 


105547.69 


.0445 


104573.75 


104743.21 


104908.99 


105071.16 


.0450 


104147.61 


104300.84 


104450.70 


104597.26 



BONDS _ 

NOTE 3. To facilitate practical work with bonds, extensive tables have 
been computed showing the purchase prices of bonds redeemable at par. 1 
The table on page 116 illustrates those found in bond tables. 

EXERCISE XLVm 

In the future use formula 59 to find P whenever F C and the in- 
vestor's interest period equals the dividend interval. Otherwise use the 
fundamental method involving formula 58. 

*! 1. Find the price to yield 4%, compounded serni-annually, of a $1000, 
5% bond, with dividends payable semi-annually, redeemable at par at the 
end of 15^ years. 

2. Verify all entries in the bond table on page 116, corresponding to 
the investment yields .04 and .045. 

3. A $5000 bond, paying a $100 dividend semi-annually, is redeemable 
at par at the end of 11 years. Find the price to yield (.06, m = 2). 

4. A man W signs a note promising to pay $2000 to M at the end 
of 5 years, and to pay interest semi-annually. on the $2000 at the rate 
6|%. (a) What will M receive on discounting this note immediately at a 
bank which uses the interest rate 7%, compounded semi-annually? 
(6) What will M receive if the bank uses the rate 7% effective? 

5. Find the price to yield 5%, effective, of a $10,000, 7% bond, with 
dividends payable annually, which is redeemable at par at the end of 
(a) 10 years ; (6) 15 years ; (c) 40 years, (d) Explain in a brief sen- 
tence how and why the price of a bond changes as the time to maturity 
increases, if the investor's rate is less than the dividend rate. 

6. Find the price to yield 6%, effective, of a $10,000, 4% bond with 
annual dividends, which is redeemable at par at the end of (a) 5 years ; 
(6) 10 years; (c) 80 years, (d) Explain in one brief sentence how 
and why the price of a bond changes as the time to maturity is increased, 
if the investor's rate is greater than the dividend rate. 

52. Changes in book value. On a dividend date, it is con- 
venient to use the term book value for the price $P at which a bond, 
would sell under a given investment rate i. Recall that this price 
$P, at which a purchaser could buy the bond, is the sum of the 
present values, under the rate i, of all payments promised in the 
bond. Hence, the dividends $D together with the redemption 
payment $(7 are sufficient to pay interest at rate i on the invested 

1 Sprague's Complete Bond Tables contain the purchase prices to the nearest cent 
for a bond of $1,000,000 par value, corresponding to a wide range of investment rates. 



118 



MATHEMATICS OF INVESTMENT 



principal $P, and to return the principal intact. If a bond is pur- 
chased at a premium over the redemption price $C, only $C of 
the original principal $P is returned at redemption. Therefore, 
the remaining principal, which equals the premium (P C) 
originally paid for the bond, is returned in installments, or is 
amortized, through the dividend payments. Thus, each dividend 
$D, in addition to paying interest due on principal, provides a 
partial payment of principal. These payments reduce the invested 
principal, or book value, from $P on the date of purchase to $C 
on the redemption date. 

Example 1. A $1000, 6% bond pays dividends semi-annually and 
will be redeemed at 110% on July 1, 1925. It is bought on July 1, 1922, 
to yield (.04, m = 2). Find the price paid and form a table showing the 
change in book value and the payment for amortization of the premium 
on each interest date. 

Solution. =$1100, D=$30. P = 1100(1.02) ~ a +30(or l ai. 02) =$1144.811. 

TABLE OP BOOK VALUES FOR A BOND BOUGHT AT A PREMIUM 



DATH 


INT. AT 4% Dun 
ON BOOK VALTJH 


DIVIDEND 
RBOBIVBD 


FOR AMOBTIZATIOJ* 
OF PREMIUM 


FINAL BOOK 
VALUE 


July 1, 1922 








$1144.811 


Jan. 1, 1923 


$22.896 


$30.000 


$7.104 


1137.707 


July 1, 1923 


22.754 


30.000 


7.246 


1130.461 


Jan. 1, 1924 


22.609 


30.000 


7.391 


1123.070 


July 1, 1924 


22.461 


30.000 


7.639 


1115.531 


Jan. 1, 1925 


22.311 


30.000 


7.689 


1107.842 


July 1, 1925 


22.157 


30.000 


7.843 


1099.999 



On Jan. 1, 1923, for example, interest due on book value is .02(1144.81) 
= $22.896. Hence, the $30 dividend pays the interest due and leaves (30 
22.896) =$7.104 for repayment, or amortization, of the premium; the 
new book value is 1144.811 - 7.104 = $1137.707. The check on the com- 
putation is that the fina.1 book value should be $1100, the redemption price. 

If a bond is purchased at a discount from the redemption price, 
that is, if P is less than C, the redemption payment C exceeds the 
original investment P by (C P). Hence, this excess must be 
the accumulated value on the redemption date of that part of the 
interest on the investment which the payments of D on the 
dividend dates .were insufficient to meet. Therefore, on each 
dividend date, the payment D is less than the interest due on 



BONDS 



119 



invested principal ; the interest which is not paid represents a new 
investment in the bond, whose book value is thereby increased. 
This writing up of the book value on dividend dates is called 
accumulating the discount because the book value increases from 
P on the date of purchase, to C on the redemption date, the total 
increase amounting to the original discount (C P). 

Example 2. A $1000, 4% bond pays dividends semi-annually and 
will be redeemed at 105% on January 1, 1924. It is purchased on Jan- 
uary 1, 1921, to yield (.06, m = 2). Find the price and form .a table 
showing the accumulation of the discount. 

Solution. C = $1050, and D = $20. 

P = 1060(1.03)-' + 20(ofi at .03) = $987.702. 

TABLE OP BOOK; VALUES FOB A BOND BOUGHT AT A DISCOUNT 



DATE 


INT. AT 8% Dura 
ON BOOK VA.LTJB 


DlVIDBND 

RBCBIVBD 


Fon AcouMuiiA.- 
TION OF DISCOUNT 


FINAL BOOK 
VALUE 


Jan. 1, 1921 








$ 987.702 


July 1, 1921 


$29.631 


$20.000 


$ 9.631 


997.333 


Jan. 1, 1922 


29.920 


20.000 


9.920 


1007.253 


July 1, 1922 


30.218 


20.000 


10.218 


1017.471 


Jon. 1, 1923 


30.524 


20.000 


10.524 


1027.995 


July 1, 1923 


30.840 


20.000 


10.840 


1038.835 


Jan. 1, 1924 


31.166 


20.000 


11.165 


1050.000 



In forming the row for July 1, 1921, for example, interest due at 6% 
is .03(987.702) - 29.631. Of this, only $20 is paid. The balance, 29.631 - 20 
= $9.631, is considered as a new investment, raising the book value of the 
bond to 987.702 + 9.631 = $997.333. In his bookkeeping on July 1, 1921, the 
investor records the receipt of $29.631 interest although only $20 actually came 
into his hands. Also, his books show a new investment of $9.631 in the bond. 

Recognize that, when a bond is purchased at a premium, the 
dividend D is the sum of the interest / on the investment plus a 
payment for amortization of the premium, or 

I = D (amortization payment). (60) 

Thus, in illustrative Example 1 above on Jan. 1, 1923, the interest is 
22.896 = 30 7.104. In accounting problems this fact is of importance. 
For instance, if a trust company purchases the bond of Example 1 for a 
trust fund, $1144.81 of the capital is invested. Suppose that the trust 
company considers all of each $30 dividend as interest and expends it for 
the beneficiary of the fund. Then, on July 1, 1925, the company faces 



(120* MATHEMATICS OF INVESTMENT? 

w' 

an illegal loss of $44.81 in the capital of the fund, because $1100 is re- 
ceived at redemption in place of $1144.81 invested. The company should 
consider only the entries in the 2d column of the table of Example 1 
as income for the beneficiary. 

Similarly, when a bond is bought at a discount, 

/ = D + (payment for accumulation of the discount) . (61) 

Thus, in illustrative Example 2 above on July 1, 1921, the interest is 
29.631 = 20 + 9.631. 

From equations 60 and 61 we obtain, respectively, 

(amortization payment} D I, 
(payt. for accumulation of discount} = I D. 

Let P and PI be the book values on two successive dividend dates, 
at the same yield. Then, if the bond is at a premium, P\ equals 
P minus the amortization payment, or PI = P (D T) ; 

P! = PO+ (7-D). (62) 

If the bond is at a discount, Pi equals P plug the payment for 
the accumulation of the discount or Pi = P + (/ D), the same 
as found in equation 62. Hence, equation 62 holds true for all 
bonds. 

EXERCISE XLIX 

s | 1. A $1000, 8% bond pays dividends semi-annually on February 1 
and August 1, and is redeemable at par on August 1, 1925. It is pur- 
chased on February 1, 1923, to yield (.06, m = 2) . Form a table showing 
the amortization of the premium. 

2. A $1000, 5% bond pays dividends annually on March 1, and is 
redeemable at 110% on March 1, 1931. It is purchased on March 1, 
1925, to yield (.07, m = 1). Form a table showing the accumulation of 
the discount. 

3. By use of formula 58, find the book value of the bond of problem 
2 on March 1, 1927, to yield (.07, m = 1) and thus verify the proper entry 
in the table of problem 2. Any book value in the tables of problems 
1 and 2 could be computed in this way without forming the tables. 

4. Under the investment rate (.04, m = 1), the book value of a $100, 
5% bond on January 1, 1921, is $113.55, and dividends are payable an- 
nually on January 1. Find the amount of the interest on the investment, 
and of the payment for amortization on January 1, 1922. 



4 1 



BONDS 121 

5. A $1000, 4% bond pays dividends annually on August 15 and is 
redeemable at par on August 15, 1935. An investor purchased it on 
August 15, 1923, to yield (.06, m = 1). (a) Without forming a table, 
find how much interest on invested capital should be recorded as re- 
ceived, in the accounts of the investor, on August 15, 1928. (6) How 
much new principal does the investor invest in the bond on August 15, 
1928? 

53. Price at a given yield between interest dates. The price 
of a bond on any date is the sum of the present values of all future 
payments promised in the bond. Let the investment rate be 

(i, m = 1), and suppose the last dividend was paid ~ th year ago. 

k 

At that time, the price P was the sum of the present values, 
at the rate (i, m = 1) of all future bond payments. To-day, the 
price P is the sum of the present values of these same payments 
because no more dividends have as yet been paid. Hence, 

P equals P accumulated at the rate (i, m = 1) for ^ years, or 

P = Po(l + i)*. (63) 

This price is on a strict compound interest yield basis. In practice, 
P is defined as PO, accumulated for - years at the rate i, simple 

K 

interest; P = Pof 1 + = i ), 

P = P -f Pofi (64) 

k 

That is, P equals PO plus simple interest on P from the last 
dividend date at the investment rate i. 

NOTB 1. The use of equation 64 favors the seller because it gives a 
slightly larger value of P than equation 63. The difference in price is neg* 
ligible except in large transactions. Use equation 64 in all problems in Ex- 
ercise L on page 123. 

Example 1. A $1000, 6% bond, with dividends payable July 1 and 
January 1, is redeemable at 110% on July 1, 1925. Find the price to yield 
(.05, m - 2) on August 18, 1922. 

Solution. July 1, 1922, was the lost dividend date. The price P then 
was Po = 1100(1.025)-" + 30(a u ~ja< .025) = $1113.77. Simple interest on 



122 MATHEMATICS OF INVESTMENT 

$1113.77 from July 1 to Aug. 16 at the investment rate 5%, is $6.96. The price 
on Aug. 16 is P = 1113.77 + 6.96 - $1120.73. 

It is proper to consider that the dividend on a bond accrues 
(or is earned) continuously during each dividend interval. Thus, 
d days after a dividend date, the 
(accrued dividend) = (simple int. for d days on the face F at dividend rate). (65) 

Example 2. In Example 1, find the accrued dividend on August 
16, 1922. 

Solution. From July 1 to Aug. 16 is 45 days. Accrued dividend is 
^fc (.06) (1000) =$7.50. 

NOTE 2. In using equation 65, take 360 days as 1 year, and find the 
approximate number of days between dates, as in expression 9, Chapter I. 

When a bond is purchased at a given yield between interest 
dates, part of the price P is a payment to the seller because of the 
dividend accrued since the last dividend date. The remainder of 
P is the present value of future dividend accruals and of the future 
redemption payment. This remainder of P corresponds to what 
was defined as the book value of a bond in Section 52. Hence, 
between dividend dates, the price is 

P = (Accrued Dividend to Date) + (Book Value) ; (66) 

(Book Value) = P - (Accrued Dividend). (57) 

Equation 67 is also true on dividend dates ; the accrued dividend 
is zero because the seller appropriates the dividend which is due, 
and hence the book value and the purchase price are the same, as 
they were previously defined to be in Section 52. 

Exampk 3. For the bond of Example 1 above, find the book value 
on August 16, 1922, to yield (.05, m = 2). 

Solution. On Aug. 16, P - $1120.73, from Example 1. The accrued 
dividend to Aug. 16 is $7.50, from Example 2. Book value on AUK 16 is 
1120.73 - 7.50 = $1113.23, from equation 67. 

NOTE 3. The accrued dividend, although earned, is not due-until the next 
chvidend date. Hence, theoretically, in equation 66 we should use, instead of 
the accrued dividend, its value discounted to date from the next dividend date. 
Thus, in Example 3 we should theoretically subtract the present value at 
COS, m - 2) on Aug. 16, 1922, of $7.50 due on Jan. 1, 1923, or 7.50(1.025)-$ 
- $7.35. The difference (in this case $.15) always is small unless a largo tranu- 
action is involved and, hence, it is the practice to use equation 67 as it standa. 



BONDS 



123.) 



/ EXERCISE L 

* 1. A $1000, 8% bond, with dividends payable January 16 and July 16, 
is redeemable at 110% on July 16, 1928. Find the purchase price and the 
book value on September 16, 1921, to yield (.04, m = 2) . 

Find the purchase prices and the book values of the bonds below on 
the specified dates. All bonds are redeemable at par. 



PHOB. 


PAH 

VALUE 


Div. 

RATH 


DIVIDEND 
DATES 


RBDBMP. 
DATE 


DATE OF 
PUBCHABH 


INVEST. 
RATE 


2. 


$1000 


5% 


June 1 


6/1/1932 


5/16/1922 


(.07, m = 1) 


3. 


100 


4% 


Jan. 1, July 1 


7/1/1940 


8/13/1931 


(.04, m = 2) 


4. 


5000 


44% 


May 1 


6/1/1952 


9/r /1924 


(.06, m = 1) 


5. 


1000 


6% 


June 1, Deo. 1 


6/1/1937 


8/16/1923 


(.05, m = 2) 


6. 


100 


64% 


May 1, Nov. 1 


5/1/1934 


3/1 /1927 


(.07, m = 2) 



The book value between dividend dates may be found very easily by 
interpolation between the book values at the last and at the next dividend 
dates. This method is especially easy if a bond table is available. 

Example 4. A $100, 6% bond pays dividends on July 1 and January 
1, and is redeemable at par on January 1, 1940. Find the book value and 
the purchase price on September 1, 1924, to yield (.04, m = 2). 

Solution. In the table below the book values 
to yield (.04, tn = 2) on 7/1/1924 and 1/1/1925 
were computed from equation 69. Let B be the 
book value on Sept. 1, which is of the way from 
July 1 to Jan. 1. Henoe, sinoe 122.938 - 122.396 
- .542, B = 122.938 - (.542) - $122.767. From 
equation 66, the purchase price P = 122.757 + 1 = 
$123.757. 



DATE 


BOOK 

VALUE 


7/1/1924 
9/1/1924 
1/1/1925 


$122.938 
B 
$122.396 



EXERCISE LI 

1. A $1000, 4% bond pays dividends annually on July 1, and is re- 
deemable at par on July 1, 1937. (a) By interpolation find the book 
value on November 1, 1928, to yield (.06, m = 1). (&) Find the purchase 
price on November 1, 1928. 

2. Find the book value in problem 1 by the method of illustrative 
example 3, Section 53, and compare with the result of problem 1. 

3. A $5000, 6% bond pays dividends semi-annually on May 1 and 
November 1, and is redeemable at par on November 1, 1947. By use of 
interpolation find the book value and the purchase price to yield (.04, 
?n - 2) on July 1, 1930, 



124 



MATHEMATICS OF INVESTMENT 



4. A $1000, 5% bond, with dividends payable March 1 and September 
1, is redeemable at par on March 1, 1935. By use of the bond table of 
Section 51, find by interpolation the book value to yield (.045, m = 2) 
on April 1, 1924. 

SUPPLEMENTARY NOTE. The interpolation method of illustrative Ex- 
ample 4, page 123, gives the same book value as is obtained by the method of 
Example 3, which uses equation 67. To prove this, lot the time to the present 

from the last dividend date be i th part of a dividend interval. Let Po be the 

/G 

book value on the last, and PI that on the next dividend date, P the purchase 
price to-day, D the periodic dividend, / the interest on P for a whole dividend 
interval at the investment rate, and B the book value of the bond to-day. 
First use the method of Example 3. Interest to date on Po at the investment 

rate is - (J), and the accrued dividend to date is - (D). From equation 64, 

K K 



Po -H rj and from equation 67, B 
/c 



l-D-p. 

7 T^ * 



I - D 



(68) 



By interpolation, as in Example 4, since the present is ^th interval from the 

K 

last dividend date, B is -th part of the way 

/C 



DATE 


BOOK 
VALUH 


Last div. date 


Po 


Present 


B 


Next div. date 


Pi 



from Po to PI, or B 
From equation 02, PI PO 

hence B 
tion 68. 



Po+|(Pi-Po). 



I D, and 
Po + ~ -, the same as in equa- 

rC 



Equation 68 shows that, when a bond is selling at a discount, the accumula- 
tion of the discount in - th interval is f- th of the total accumulation for the 
k k 

interval, for, in equation 61 it is seen that I D is the accumulation for the 



whole interval. Similarly, if we write equation 68 as B => Po 



D - I 

k ' 

it is seen that the amortization of the premium on a bond in -th interval is -th 
of the amortization for the whole interval. 

54. Professional practices in bond transactions. An investor 
buying a particular bond cannot usually demand a specified yield 
from his investment. He must pay whatever price is asked for 
that particular bond on the financial market. On bond exchanges, 



BONDS 125 

and in most private transactions, the purchase price of a bond is 
described to a purchaser as a certain quoted price plus the accrued 
dividend. 1 That is, the market quotation on a bond is what we 
have previously called the book value of the bond, in equation 67. 

NOTE 1. The quotation for a bond is given as a percentage of its par 
value. That is, a $10,000 bond, quoted at 93, has a book value of $9325. 

Example 1. A $10,000, 6% bond, with dividends payable June 1 
and December 1, is quoted at 93i on May 1. Find the purchase price. 

Solution. Quotation = book value = $9325.00. Accrued dividend since 
December 1 is $250. From equation 66, the price is 9325 + 250 = $9575. 

NOTE 2, Bond exchange methods are simplified by the quotation of book 
values instead of actual purchase a prices. If the yield at which a bond sells 
remains constant, the book value changes very slowly through the accumula- 
tion of the discount, or amortization of the premium, as the case may be. 
Hence, when the practice is to quote book values, the market quotations of 
bonds change very slowly and any violent fluctuation in them is due to a 
distinct change in the yields at which the bonds are selling. On the other 
hand, if the actual purchase price of a bond were the market quotation, the 
quotation would increase as the dividend -accrued and then, at each dividend 
date, a violent decrease would occur when the dividend was paid. Thus, even 
though the yield at which a bond were selling should remain constant, large 
fluctuations in its market quotation would occur . 

EXERCISE in 

1. (a) A $1000, 5% bond, with dividends payable February 1 and 
August 1, is quoted at 98.75 on May 1 ; find the purchase price. (&) If the 
bond is purchased for $993.30 on April 1, find the market quotation then. 

2. The interest dates for the 2d 4% Liberty Loan bonds are May 15 
and November 15. Take their closing quotation on the New York Stock 
Exchange from the morning newspaper and determine the purchase 
price for a $10,000 bond of this issue. 

3. A $1000, 6% bond whose dividend dates are January 1 and July 1 
is quoted at 103& on October 16, Find the total price paid by a purchaser 
if he pays a brokerage commission of I % of the par value. 

1 In bond market parlance, it is called accrued interest. The more proper word 
dividend has been consistently uaed in this book to avoid pitfalls which confront 
the beginner. As seen in Section 52, equations 00 and 61, the dividend is not 
the same as the interest on the investment. The terminology accrued interest in 
bond dealings must be learned by the student and appreciated to mean accrued 
dividend in the sonee of this chapter. 

1 The purchase price is called the flat price in bond parlance, as contrasted with 
the price, and accrued interest quotation customarily used. 



126 MATHEMATICS OF INVESTMENT 

56. Approximate bond yields. On a given date the book value 
of a bond is quoted on the market and the problem is met of de- 
termining the yield obtained by an investor on purchasing the 
bond and holding it to maturity. We first consider an approxi- 
mate method of solution, using mere arithmetic. 

NOTE. A bond salesman, in speaking of the yield on a bond, usually refers 
to an investment rate compounded the same number of times per year as divi- 
dends are paid. Thus, by the yield on a quarterly bond, he means the invest- 
ment rate, compounded quarterly. We shall follow this customary usage in 
the future. Moreover, in computing yields it is usual to neglect the accrued 
dividend and brokerage fee paid at the time of purchase in addition to the book 
value. A yield is computed with reference to the book value of the bond. 

The justification of the following rules is apparent on reading 
them. Let $5 be the quoted book value of a bond, t the time in 
years before its maturity, and $C its redemption price. The in- 
vested principal changes from $5 at purchase to $C at redemption, 
so that the average book value $B is given by B Q = %(B + C). 
Even though a bond pays dividends quarterly or semi-annually, 
in using the rules below proceed as if the dividends were payable 
annually at the dividend rate and let $D be this annual dividend. 

Rule 1. When the quoted book value B is at a premium over 

C. Compute $A, the average annual amortization of the pre- 
mium from A = remium . Compute $1, the average annual 
t 

interest on the investment from I = D A. 1 Then, the ap- 
proximate yield r equals the average annual interest divided by 

the average invested capital or r = -^ 

-Do 

Rule 2. When B is at a discount from C. Compute $T, the 

average annual accumulation of the discount from T = un . 

t 

Compute I from I = D + T* Then, the approximate yield r 
equals the average annual interest divided by the average invested 

capital, or r = -- 

BQ 

* See equation 60. See equation 61, 



BONDS 



fiak 



Example 1. A $1000, 5% bond pays dividends semi-annuaUy and is 
redeemable at 110%. Eleven years before its maturity, the book value 
is quoted on the market at 93. Estimate the yield. 

Solution. Considering its dividends annual, D = $50, C = $1100, and 
the book value B = $930. Using Rule 2, the average accumulation of the 
discount is 1 ^ = $15.5, and I = 50 + 15.5 = $65.5. The average invested 
capital is 4(930 + 1100) = $1015. The approximate yield r = ffcV = -065, 
or 6.5%. 

Example 2. A $1000, 5 % bond pays dividends on July 1 and January 
1 and is redeemable at par on January 1, 1961. Its quoted book value 
on May 1, 1922, is 113. Estimate the yield. 

Solution. Uae Rule 1 with B = $1130, t - 38f years, D = $50, and 
C = $1000. We find B = $1066. To find the average amortization of the 
premium we take t = 39, the nearest whole number, because the inaccuracy 
of our rule when t is large makes refinements in computation useless. A = -W 
= $3.3, I = 50 - 3.3 = $46.7, and therefore r = tffy = .044, or 4.4%. 

NOTE. -7- The author has experimentally verified that Rules 1 and 2 give 
estimated yields within .2% of the truth if : (a) the yield is between 4% and 8%, 
(b) the tune to maturity is less than 40 years, and (c) the difference between 
the dividend rate and the yield is less than 3%. Greater accuracy is obtained 
under favorable circumstances. For a bond whose term is more than 30 years, 
as in Example 2 above, take t as the whole number nearest to the time to 
maturity in years. In all other cases use the exact time to the nearest month. 

EXERCISE LIH 
Estimate the yields of the following bonds, by use of Rules 1 and 2. 



FBOB. 


PAGE 


To BB 


MARKET 


DrVlDHND 


TlMH TO 


i 








RATH 


PAID 




J * j 


$1000 


par 


107.24 


5% 


semi-ann. 


111 years 


2. a 


100 


par 


160.30 


7% 


semi-ann. 


25 years 


S.a 


1000 


par 


84.28 


4% 


semi-ann. 


40 years 


4. 


100 


110% 


96.50 


6% 


annually 


231 years 


5. 


100 


105% 


115.00 


5% 


annually 


8 years 


6. 


100 


116% 


98.75 


31% 


annually 


20 years 



1 Inspect the table of illustrative Example 2, Section 52. In that example, on 
computing the average semi-annual accumulation as in Rule 2, we obtain J(62.30) 
= $10.4, a result very close to all of the semi-annual accumulations. 

3 The yields in the first three problems, determined by accurate means, are am 
follows: (1) 4.2%; (2) 3.4%; (3) 4.9%. Compare your results as found from 
Boles 1 and 2 in order to form an opinion of their accuracy. 



/128 MATHEMATICS OF INVESTMENT 

7. A $10,000, 5% bond, -with dividends payable June 1 and December 
1, is redeemable at par on December 1, 1950. On May 23, 1925, it is 
quoted at 89. Estimate the yield. 

8. A Kingdom of Belgium 7|% bond, whose dividends are semi-annual, 
may be redeemed at 115% at the end of 8 years. Estimate its yield under 
the assumption that it will be redeemed then, if it is now quoted at 94. 

66. Yield on a dividend date by interpolation. When the 
quoted value of a bond is given on a dividend date, the yield may 
be determined by interpolation. When annuity tables, but no 
bond tables, are available, proceed as follows : 

(a) Find the estimated yield r as in Section 55. 

(&) Compute the book value of the bond at the rate TI nearest 
to r for which the annuity tables may be used. 

(c) Inspect the result of (&) and then compute the book value for 
another rate r z , chosen so that the true yield is probably between 
TI and r 2 . Select r 2 as near as possible to TV 

(d) Find the yield i by interpolation between the results in 
(&) and (c). 

Example 1. A $100, 6% bond, with semi-annual dividends, is redeem- 
able at par. The quoted book value, 10| years before maturity, ia 
$111.98. Find the yield. 



Solution. (a) Average n.Tnrma.1 interest 1 = 6 ifj^a = $4.9 ; estimated 
yield r = &fc = 4.6%. (b) Book value 10J years before maturity to yield 
(.045, m = 2) is $112.44 (by equation 59). (c) Since $112.44 is greater than 
$111.98, the yield is greater than .046, and is prob- 
ably between .045 and .05. The book value at 
(.05, m = 2) is $108.09. Let (i, m - 2) be the yield. 
In the table, 112.44 - 108.09 - 4.35,' 112.44 

- 111.98 = .46, and .06 - .045 = .005. Hence, 

* = -045 + jk (.005) - .0465, or the yield is ap- 



INVJEST. BATB 



.045, m = 2 

i, m =* 2 

.05, m = 2 



BOOK 
VALUE 



112.44 
111.98 
108.09 



proximately 4.55%, compounded semi-annually. 



NOTE 1. A' more exact solution J may be obtained as follows :. At the 
yield (.0455, m = 2), found above, compute the book value P, using logarithms 
in equation 59 because the annuity tables do not apply ; P = 100 + .725 

1 A solution as in Example 1 gives a result which is in error by not more than Ath 
of the difference between the table rates used in the interpolation. Wo ore, essen- 
tially, interpolating in Table VIII, and hence our result is subject only to the error 
we meet in using that table. 



BONDS 



(a^ } at .02275) = 111.998. Since $111.998 is greater than $111.980, the yield 
i is greater than .0455, and is probably between .0455 and .0456. By loga- 
rithms, the book value at (.0456, m = 2) is $111.910. From interpolation 
as in Example 1, i = .0455 + if (.0001) = .045520. The yield is 4.5520%, 
compounded semi-annually, with a possible error in the last decimal place. 

NOTE 2. The method of Example 1 is very easy if the desired book values 
can be read directly from a bond table (see problem 2 below) . If the bond table 
uses interest rates differing by $>&%, results obtained by interpolation in the 
table are in error by not more than a few .001%. Extension of the accuracy of 
a solution as in Note 1 is limited only by the extent of the logarithm tables at 
our disposal. 

NOTE 3. If the book value B is given on a day between dividend dates, 
the yield may be accurately obtained by the method of Section 68 below. An 
approximate result can be found by assuming B as the book value on the 
nearest dividend date and computing the corresponding yield. 



EXERCISE LIV 

Find the yield in each problem as in Example 1, page 128. If the in- 
structor so directs, extend the accuracy as in Note 1 above. 

1. A $100, 4% bond pays dividends on January 1 and July 1 and is 
redeemable at par on January 1, 1932. (a) Find the yield if the quoted 
value on July 1, 1919, is 89.32. (6) Find the effective rate of interest 
yielded by investing in the bond. 

v 2. A $100, 5% bond pays semi-annual dividends and is, redeemable 
at par. By use of the bond table of Section 51, find the yield if the quoted 
value 11 years before maturity is 107.56. 

For each bond in the table, par value is $100. Find the yields. 



PBOB. 


TO BE 

REDEEMED AT 


DIVIDEND 


TIME TO MATDiuTr 


BOOK 
VALTJH 










RATH 


PAYABLE 






3. 


110% 


6% 


annually 


30 years 


$ 78.50 


4. 


par 


41% 


semi-ann. 


15| years 


110.76 


6. 


par 


3% 


semi-ann. 


19 years 


83.30 


6, 


par 


6% 


annually 


12 years 


121.00 


7. 


105% 


6% 


semi-ann. 


24 years 


88.00 


8. 


par 


6% 


quarterly 


10 years 


107.00 



9. On January 1, 1923, a purchaser paid $87.22, exclusive of brokerage, 
for a $100, 4% bond whose dividends are payable July 1 and January 1 



130 MATHEMATICS OP INVESTMENT 

and which is redeemable at par on January 1, 1932. Find the yield ob- 
tained if the investor holds the bond to maturity. 

10. A $100, 5% bond pays semi-annual dividends and is redeemable 
at par at the end of 9 years. If it is quoted at 83.20, find the effective 
rate of interest yielded by the investment. 

67. Special types of bond issues. On issuing a set of bonds, a 
corporation, instead of desiring to redeem all bonds on one date, 
may prefer to redeem the issue in installments. The bonds are 
then said to form a serial issue. The price of the whole issue to 
net an investor a specified yield is the sum of the prices he should 
pay for the bonds entering in each redemption installment. 

Example 1. A $1,000,000 issue of 6% bonds was made on January 1, 
1920, with dividends payable semi-annually, and the issue is redeemable 
serially in 10 equal annual installments. Find the price at which all bonds 
outstanding on January 1, 1927, could be purchased to yield an investor 
(M,m = 2). 

Solution. There is $300,000 outstanding. The price of the bonds for 
$100,000, which are redeemable at the end of 1 year, is 1000 (an of .02) + 
100000 = $101,941.56; the prices of the bonds redeemable in the install- 
ments paid at the end of 2 years and of 3 years are $103,807.73 and $106,601.43, 
respectively. The total price of outstanding bonds is $311,350.72. 

An annuity bond, with face value $F, is a bond promising the 
payment of an annuity. The periodic payment $*S of the annuity 
is described as the installment which, if paid periodically during the 
life of the bond, is sufficient to redeem the face UPF in installments and 
to pay interest as due at the dividend rate on all of the face $F not 
yet redeemed. That is, the payments of $S amortize the face $F 
with interest at the dividend rate. When F and the dividend rate 
are known, S can be found by the methods of the amortization 
chapter. At a given investment yield, the price of an annuity 
bond is the present value of the annuity it promises. The annuity 
is always paid the same number of times per year as dividends 
are payable on the bond. 

Example 2. A certain ten-year, $10,000 annuity bond with the 
dividend rate 5% is redeemable in semi-annual installments of %8 each. 
(a) Find & (6) Find the purchase price of the bond, 5 years before 
maturity, to yield 6%, effective. 



BONDS .131 

Solution. (a) The payments of $S amortize $10,000 at (.05, m 2). 



Bond annuity, Case 1 

7i = 20 int. per., R = $5, 

A = $10,000, p - 1, i = .026. 



10,000 = S(a m at .025) ; 8 = $641.47. 



(6) The price A at the yield (.06, m = 1) is the present value of semi- 
annual payments of S made for 5 years. 



Case 1 

,(2) 



n = 5 int. per., 33 = 2, 
i = 06, # = $1282.94. 



A = 1282.94(ojf.ri .06) = $5484.09. 



EXERCISE LV 1 

1.- A $100,000 serial issue of 5% bonds, with dividends payable semi- 
annually, is redeemable in 5 equal annual installments. The issue was 
made July 1, 1927. On July 1, 1930, find the price of all outstanding 
bonds to net the investor (.06, m 2) . 

2. For the bonds purchased in problem 1, form a table showing, on each 
dividend date, the dividend received, the installment (if any) which is 
paid, the interest due on the book value, and the final book value. 

3. A house worth $12,000 cash is purchased under the following agree- 
ment : $2000 of the principal is to be paid at beginning of each year for six 
years ; interest at 6% is to be paid semi-annually on all principal outstand- 
ing. Two years later, the written contract embodying this agreement 
was sold to a banker, who purchased the remaining rights of the creditor 
!fco yield 7%, effective. What did the banker pay? 

J 4. A 5-year annuity bond for $20,000, with the dividend rate 6%, 
payable semi-annually, is issued on June 1, 1921. (a) Find the price on 
June 1, 1922, to yield (.03, m = 2). (&) Find the price on September 1, 
1922, to yield (.03, m -- 2). 

6. On June 1, 1924, find the price of the bond of problem 4 to yield 
(.06, m = 1). 

SUPPLEMENTARY MATERIAL 

58. Yield of a bond between dividend dates. If the quoted 
value of a bond is given on a day between dividend dates, the yield 
may be found by interpolation by essentially the same procedure, 
with steps (a) , (&) , (c) and (d) , as used in Section 56 on a dividend date . 

After the completion of Exercise LV, the student may immediately proceed to 
the consideration of the Miscellaneous Problems at the end of the chapter. 



( 132 ' 



MATHEMATICS OF INVESTMENT 



Example 1. A $100, 4% bond pays dividends annually on December 
1 and is redeemable at par on December 1, 1931. Find the yield on Feb- 
ruary 1, 1926, if the book value is quoted at 95.926. 

Solution. (a) As in Section. 55, the average annual interest on the in- 
vestment is Z = 4 + .71 = $4.71 ; the estimated yield is ^5 = .048. 
(b) The nearest table rate is 5%. To find the book value at 5% on Feb. 1, 1926, 
first compute the values at 5% on Dec. 1, 1925, and Dec. 1, 1926, the last and the 
next interest dates. The results, 94.924 and 96.671, are placed in the first 
row of the table below, and from them we find by interpolation the book value 

TABLE OF BOOK VALUES 



YIELD 


DEO. 1, 1925 


FEB. 1, 1020 


Dno. 1, 1020 


.05, m 1 
i, m = 1 
.045, m = 1 


$94.924 
97.421 


$95.048 
95.926 

97.485 


$95.671 
97.805 



on Feb. 1, 1926. Since 95.671 - 94.924 = .747, the book value on Feb. 1 at 
5% is 94.924 + 1(.747) = 95.048. (c) Since 95.048 is less than 95.926 (the 
given book value), the yield is less than .05 and is probably between .046 and 
.05. Prices at .045 on Dec. 1, 1925, and on Dec. 1, 1926, were computed and 
from them the book value on Feb. 1, 1926, at .045 was obtained by interpolation. 
(d) The yield i is obtained by interpolation in the column of the table for 
Feb. 1. 97.485 - 95.048 = 2.437 ; 97.485 - 95.926 = 1.559 ; .05 - .045 
= .005; hence i - .045 -f jjg .005 - .0482. The yield is approximately 
4.82%, compounded annually, with a possible small error in the last digit. 

NOTE. The method above is extremely simple if the desired book values 
can be read from a bond table. The accuracy of the solution can be extended 
by the method of Note 1, Section 56. 

EXERCISE LVI 

N^ 1. By use of the bond table of Section 51, find the yield of a $100, 5% 
bond, with dividends payable on September 1 and March 1, and redeem- 
able at par on September 1, 1928, if the quoted book value December 1, 
1917, is $106.78. 

2. The interest dates of a $100, 4% bond are July 1 and January 1, 
and it is redeemable at par on January 1, 1930. (a) Find the yield if it 
is quoted at 83.25 on September 1, 1923. (6) Find the effective rate of 
interest yielded by the bond. 



BONDS 133 

3. A man pays $87.22, exclusive of the brokerage commission, on Sep- 
tember 1, 1923, for a $100, 4% bond whose dividends are payable July 1 
and January 1, and which is redeemable at par on January 1, 1930. De- 
termine the yield, if the bond is held to maturity. 

4. A $1000, 5% bond pays dividends annually on June 16 and is re- 
deemable at 110% on. June 16, 1937. Find the yield if it is quoted at 
112.06 on November 16, 1932. 

6. The 3d Liberty Loan 4J% bonds are redeemable at par on Septem- 
ber 16, 1928. Interest dates are September 15 and March 15. If the 
bonds were quoted at 85 on May 15, 1921, what was the investment 
yield? 

MISCELLANEOUS PROBLEMS 

In the following problems, the word interest is used in the colloquial 
sense in connection, with bonds in place of the word dividend previously 
used. 

1. A certain $1000, 5% bond pays interest annually. It is stipulated 
that, at the option of the debtor corporation, it may be redeemed at par 
on any interest date after the end of 10 years. The bond certainly will be 
redeemed at par by the end of 20 years. At what purchase price would 
a purchaser be certain to obtain 6% or more on his investment? 

2. What is the proper price for the bond in problem 1 to yield 4%, or 
more? 

3 . In return for a loan of $5000, W gives his creditor H the following note : 



Norfolk, June 1, 1915. 

For value received, I promise to pay, to H or order, $5000 at the 
end of 6 years and to pay interest on this sum semi-annually at the 
rate 6%. Signed, W. 



On December 1, 1916, H sold this note to an investor desiring (.07, m = 2) 
on his investment. "What did H receive? 

4. What would H have received if he had sold the note to the same in- 
vestor as in problem 3 on February 1, 1917? 

5. Two $1000 bonds are redeemable at par and pay 4% interest semi- 
annually. Their quoted prices on a certain date to yield (.05, m = 2) 
are $973 and $941.11, respectively. Without using annuity tables, and 
without computation, state which bond has the longer term to run and 
justify your answer. 



134 MATHEMATICS OF INVESTMENT 

6. Determine the term of the bond in problem 5 quoted at $941.11., 

7. A $100, 4% bond, redeemable at par in 20 years, pays interest semi- 
annually. If it is quoted at $92.10, what is the effective rate of interest 
obtained by an investor? 

8. On June 1, 1921, a corporation has its surplus invested in bonds 
which are redeemable at par on June 1, 1928, and which pay interest 
semi-annually at the rate 5%. If the bonds are quoted at 102.74, would 
it pay the corporation to sell the bonds and reinvest the proceeds in 
Government bonds which net 4.65%, effective? 

9. A $1,000,000 issue of 5% bonds, paying interest annually, is to be 
redeemed at 110% in twenty annual installments. The first installmenl 
is to be paid at the end of 5 years and the last at the end of 24 years. II 
is desired that the annual payments (dividends on unpaid bonds and 
the redemption installment included)- at the end of, each year for the las- 
20 years shall be equal. Determine the payment. 

10. A house worth $12,000 is purchased under the following agreement 
$2,000 is to be paid cash and the balance of the principal is to be paid i\ 
four equal installments due at the ends of the 2d, 4th, 6th, and 8th years 
Interest at 6% ia to be paid semi-annually on all sums remaining due 
The note signed by the purchaser is sold after 3 years by the originf 
owner of the house. If the purchaser of the note demands 7%, compounde 
semi-annually, on his investment, what does he pay for the note? 

11. A trust fund of $20,000 is invested in bonds which yield 5% at 
nually. The trust agreement states that $ of the income shall be give 
to the beneficiary each year and that the balance shall be re-invested in 
savings bank which pays 5%, compounded annually. The whole fun 
shall be turned over to the beneficiary after 10 years. If money is wori 
6%, effective, to the beneficiary, what sum would he take now in place of h 
interest in the trust fund? Assume that he will live 10 years. 

12. A corporation can sell at par a $1,000,000 issue of 6j% bon( 
redeemable at par in 20 years and paying interest annually. To pay the 
at maturity the corporation would accumulate a sinking fund by annu 
deposits invested at 4%, effective. Would it be better for the corpor 
tion to sell at par a $1,000,000 issue of 5% bonds y if these are redeemat 
in such annual installments during the 20 years that the total annual pa 
ments, dividends and redemption payments included, will be equa 



REVIEW PROBLEMS ON PART I 135 

REVIEW PROBLEMS ON PART I 

1. In purchasing a farm, $5000 will be paid at the end of each year for 
10 years, (a) What is the equivalent cash price if money is worth 5%, 
effective? (6) What must be paid at the end of the 6th year to complete 
the purchase of the farm? 

2. A depreciation fund is being accumulated by semi-annual deposits 
of $250 in a bank paying interest semi-annually at the rate 5%. What 
is in the fund just after the 30th payment? 

3. A man wishes to donate to a university sufficient money to provide 
for the erection and the maintenance, for the next 50 years, of a building 
which will cost $500,000 to erect and will require $2000 at the end of each 
3 months to maintain. What should he donate if the university is able 
to invest its funds at 5%, compounded semi-annually? 

4. A debt of $100,000 bears interest at 6%, payable semi-annually. 
A sinking fund is being accumulated by payments at the end of each 6 
months to repay the principal in one installment at the end of 10 years. 
If the sinking fund earns 4% interest, compounded semi-annually, what is 
the total semi-annual expense of the debt? 

5. A debt of $100,000 is contracted under the agreement that interest 
at 6% shall be paid semi-annually on all sums remaining due. What 
payment at the end of each 6 months for 10 years will amortize this 
debt? 

6. By use of a geometrical progression derive the expression for the 
amount of an annuity whose annual rent is $2000, payable in semi-annual 
installments for 10 years, if money is worth 6%, compounded quarterly. 

7. Find the present value of an annuity whose annual rent is $3000, 
payable semi-annually for 20| years, if money is worth (.05, m = 4). 

8. A merchant owes $6000 due immediately. For what sum should 
he make out a 90-day, non-interest-bearing note, so that his creditor may 
realize $6000 on it if he discounts it immediately at a bank whose discount 
rate is 8%? 

9. If money is worth 5%, effective, find the equal payments which if 
made at the ends of the first and of the third years would discharge the 
liability of the following debts : (1) $1000 due without interest at the end 
of 3 years ; (2) $2000 due, with accumulated interest at the rate (06, m = 
2), at the end of 4 years. 

10. A trust fund of $100,000 is invested at 6%, effective. Payments of 
$10,000 will be made from the fund at the end of each year as long as pos- 



136 f MATHEMATICS OF INVESTMENT 

sible. (a) Find how many full payments of $10,000 will be made, 
(b) How much will be left hi the fund just after the last full payment of 
$10,000? 

11. Find the nominal rate of interest, compounded quarterly, under 
which payments of $1000 at the end of each 3 months for 20 years will be 
sufficient to accumulate a fund of $200,000. 

12. Find the purchase price, to yield 6%, effective, of a $100, 5% bond 
with interest payable semi-annually, which is to be redeemed at 110% 
at the end of 10 yeara. 

13. Estimate the yield on a bond which is quoted at 78, 10 years be- 
fore it is due, if it is to be redeemed at par and if its dividend rate is 
6%, payable annually. 

14. Find the capitalized cost of a machine, whose original cost is 
$200,000, which must be renewed at a cost of $150,000 every 20 years. 
Money is worth 5%, effective. 

'4 15. Find the yield of a $100, 6% bond, with semi-annual dividends, which 
is quoted at 93.70, 10 years before it is due, and is redeemable at par. 

16. A $100, 5% bond, quoted at 86.33 on September 1, 1926, yields 6% 
if held to maturity. The last coupon date was July 1. What is the pur- 
chase price on September 1 ? 

17. A man deposited $100 in a bank at the beginning of each 3 months 
for 10 years. What is to his credit at the end of 10 years if the bank pays 
8%, compounded quarterly? 

18. A man deposited $50 in a fund at the end of each month for 20 
years, at which time deposits ceased. What will be in the fund 10 years 
later if it accumulated for the first 20 years at the rate 6%, effective, 
and at the rate 4%, effective, for the remainder of the time? 

19. The cash price of a farm is $5000 and'money is worth (.06, m = 2) . 
What equal payments made quarterly will have an equivalent value, if 
the first payment is due at the end of 3 years and 9 months, and the last 
at the end of 134 years? 

20. A corporation issues $200,000 worth of 6% bonds, redeemable at 
par at the end of 15 years, with interest payable semi-annually. The 
corporation is compelled by the terms of the issue to accumulate a sulking 
fund, to pay the bonds at maturity, by payments at the end of each 6 
months, which are invested at (.04, m = 2). The bonds are sold by the 
corporation at 95 (95% of then* par value). Considering the total semi- 
annual expense as an annuity, under what rate of interest is the corpora- 
tion amortizing the loan it realizes from the bond issue? 



REVIEW PROBLEMS ON PART I 137 

21. The present liability of a debt is $100,000. It is agreed that pay- 
ments of $5000 shall be made at the end of each 6 months for 10 years, 
and that, during this time, the payments include interest at the rate 6%, 
payable semi-annually. Then, commencing with a first payment at the 
end of 10J years, semi-annual installments of $10,000 shall be paid as long 
as necessary to discharge the debt, (a) After the end of 10 years, if the 
payments include interest at the rate 5%, payable semi-annually, how many 
full payments of $10,000 must be made? (&) What part of the payment 
at the end of lOf years is interest on outstanding principal and what 
part is principal repayment? 

22. From whose standpoint, that of the debtor or of the creditor, is 
compound interest more desirable than simple interest? Tell why in 
one sentence. 

23. A 90-day note whose face value is $2000 bears interest at 6%. It 
is discounted at a bank 30 days before due. What are the proceeds if 
the banker's discount rate is 8%? 

24. A man borrows a sum of money for 72 days from a bank, charging 
5% interest payable in advance, (a) What interest rate is he paying? 
(&) What interest rate would he be paying if he borrowed money for 1 year 
from this bank? 

26. Estimate the yield of a bond whose redemption value is $135, 
whose dividends are each $10, and are paid annually, and whose purchase 
price 6 years before due is $147. 

26. (a) Find the yield J of a $100, 6% bond bought for $103.53 on October 
1, 1921. Coupons are payable semi-annually on February 1 and August 
1 and the bond will be redeemed at par on February 1, 1928. (&) Find 
the effective rate of interest yielded by the bond. 

27. The principal of a debt of $200,000 is to be paid after 20 years by 
the accumulation of a sinking fund into which 79 quarterly payments 
will be made, starting with the first payment in 6 months. Find the 
quarterly payment if the fund grows at 6%, compounded quarterly. 

28. To amortize a certain debt at 6%, effective, 40 semi-annual pay- 
ments of $587.50 must be made. Just after the 26th payment, what 
principal will be outstanding? 

29. (a) Find the capitalized worth at (.06, m = 12) of an enterprise 
which wnl yield a monthly income of $100, forever, first payment due 
now. (&) What is the present worth in (a) if the first monthly payment is 
due at the end of 6 months? 

1 Find the yield as in Section 68, or, if that section has not been studied, use the 
method of Section 55. 



138 MATHEMATICS OF INVESTMENT 

30. A bridge will need renewal at a cost of $100,000 every 25 years. 
Under 5% interest, what is the present equivalent of all future renewals? 

31. (a) By use of a geometrical progression determine a formula for 
the amount of an annuity whose annual rent is $20,000, which is paid 
quarterly for 30 years, if money is worth 7%, compounded annually. 
(6) Without a geometrical progression find the present value of the annuity. 

32. A house is worth $50,000. In purchasing it $20,000 is paid cash 
and the remainder is to be paid, principal and interest at (.05, m = 2) 
included, by semi-annual installments of $2000, first payment to be made 
at the end of 2 years, (a) Determine by interpolation how many whole 
payments of $2000 will be necessary. (6) What liability will be out- 
standing just before the last full payment of $2000? 

38. A debt of $50,000 is contracted and interest is at the rate 5%, 
compounded annually. The only payments (including interest) made 
were $5000 at the end of 2 years, and six. annual payments of $3000, 
starting with one at the end of 5 years. At the end of 10 years what ad- 
ditional payment would complete payment of the debt? 

34. If money is worth (.05, m = 2), find the equal payments which 
must be made at the ends of the 3d and 4th years in order to discharge 
the following liabilities : (1) $5000 due at the end of 6 years, without in- 
terest ; (2) $4000 due at the end of 5 years with all accumulated interest 
at (.06, m = 1). 

36. A debt of $100,000 is contracted and it is agreed that it shall be 
paid, principal and interest included, by equal payments at the end of 
each 6 months for 20 years. Interest is at the rate (4%, m = 2) for the 
first 10 years and at (5%, m = 2) for the next 10 years. What single 
rate of interest over the whole 20 years would have resulted in the same 
payments? 

36. A man invested $100,000 in a certain enterprise. At the ends of 
each of the next. 10 years he was paid $4000 and, in addition, he received 
a payment of $25,000 at the end of 6 years. At the end of 10 years he 
sold his investment holdings for $80,000. 'Considering the whole period 
of 10 years, what was the effective rate of interest yielded by the invest- 
ment? 

HINT. Write an equation of value; solve by interpolation as in Note 3 
in the Appendix. 

37. Mr. A borrows $5000 from B to finance his college course and gives 
B a note, promising to pay $5000 at the end of 10 years, together with all 
accumulations at 3%, compounded semi-annually. (a) What will A pay 



REVIEW PROBLEMS ON PART I 139 

at the end of 10 years? (&) At the end of 5 years, B sells A's promissory 
note to a bank, which discounts it, considering money as worth (.05, 
m = 1) . What does B realize from the sale ? 

38. Find the price at which a $100, 5% bond would be quoted on the 
market on September 1, 1922, to yield the investor (.06, m = 2). The 
bond is to be redeemed at par on August 1, 1928, and interest dates of the 
bond are August 1 and February 1. 

39. An industrial commission awards $10,000 damages to the wife 
of a workman killed in an accident, but suggests that this sum be paid out 
by a trust company in quarterly installments of $200, the first payment 
due immediately, (a) If the trust company pays (.04, m = 4) on money, 
for how long will payments continue ? (6) At the end of 10 years, the wife 
takes the balance of her fund. What amount does she receive? 

40. Determine the capitalized cost of a machine worth $5000 new, due 
to wear out in 20 years, and renewable with a scrap value of $1000. 
Money is worth .05, effective. 

41. Find the purchase price on December 1, 1920, of a $100, 6% 
bond with annual dividends, to yield at least 5%, if the bond may at the 
option of the issuing company be redeemed at 110% on any December 1 
from 1930 to 1935, inclusive, or at par on any December 1 from 1943 to 
1950. Justify your price. 

42. A father wills to his son, who is just 20 years old, $20,000 of stock 
which pays dividends annually at the rate 6%. The will directs that the 
earnings shall be held to his son's credit in a bank paying 3%, effective, 
and that all accumulations as well as the original property shall become 
the direct possession of his son on his 30th birthday. Assuming that the 
market value of the stock on the 30th birthday will be $20,000, what is the 
present value of the estate for the son on his 20th birthday, assuming that 
money is worth 4|% and that the son will certainly live to age 30? 

43. If money is worth (.06, m = 2), what equal installments paid at 
the ends of the 2d and 3d years will cancel the liability of the following 
obligations : (a) $1000 due without interest at the end of 5 years, and (6) 
$2000 due with accumulated interest at the rate 4%, compounded annually, 
at the end of 6 years? 

44. Two years and 9 months ago X borrowed $2000 from Y, and has 
paid nothing since then, (a) If interest is at the rate 6%, payable semi- 
annually, determine the theoretical compound amount which X should 
pay to settle his debt immediately. (6) Determine the amount by the. 
practical rule, 



140 MATHEMATICS OF INVESTMENT 

45. At the end of each 6 months, $200,000 is placed in a fund which 
accumulates at the rate (.06, m = 2). (a) How many full payments of 
$200,000 will be necessary to accumulate a fund of $1,000,000 ? (6) What 
smaller payment will be needed to complete the fund on the next date of 
deposit after the last $200,000 payment? 

46. Find the annual expense of a bond issue for $500,000 paying 5% 
annually, if it is to be retired at the end of 20 years by the accumulation 
of a sinking fund by annual payments invested at 4%, effective. 

47. In problem 46, at what effective rate of interest could the bor- 
rower just as well borrow $500,000 if it is agreed to amortize the debt by 
equal payments made at the ends of the next 20 years? 

48. How much is necessary for the endowment of a research fellowship 
paying $3000 annually, at the beginning of each year, to the fellow and 
supplying a research plant, whose original cost is $10,000, which requires 
$2000 at the beginning of each year for repairs and supplies? Money is 
worth 4%, effective. 

49. A banker employs his money in 90-day loans at 6% interest, pay- 
able in advance. At what effective rate is he investing his resources? 

50. Find the present value and the amount of an annuity of $50 per 
year for 20 years if money is worth 4%, payable annually. Use no tables 
and do entirely by arithmetic, knowing that (1.04) 20 = 2.191123. 

51. $100,000 falls due at the end of 10 years. The debtor put $8000 
into a sinking fund at the end of each of the first 3 years. He then decided 
to make equal a,nnua,1 deposits in his sinking fund for the remainder of the 
time in order to accumulate the necessary $100,000. If the fund earns 
(.04, m = T), what was the annual deposit? 

52. A corporation is to retire, by payments at the end of each of the 
next 10 years, a debt of $105,000 bearing 5% interest, payable annually. 
The tenth annual payment, including interest, is to be $15,000. The 
other nine are to be equal in amount and are to include interest. Deter- 
mine the size of these nine payments. 

53. Compute the purchase price to yield (,05, m = 4) of a $1000, 6% 
bond redeemable at 110% in 12J years, if it pays interest semi-annually. 

54. Compute the present value of an annuity whose annual rent is 
$3000, payable quarterly for 6 years, if interest is at the rate 5.2%, effective. 
- 66. The maximum sum insured under the War Risk Insurance Act pays 
$57.50 at the beginning of each month for 20 years certain after death 
or disability. What would be the equivalent cash sum payable at death, 
or disability, at 3$% interest? 



REVIEW PROBLEMS ON PART I 141 

66. A company issues $100,000 worth of 4%, 20-year bonds, which it 
wishes to pay at maturity by the accumulation of a sinking fund into 
which equal deposits will be made at the end of each year. The fund 
will earn 5% during the first five years, 4&% for the next 5 years, and 4% 
for the last 10 years. Determine the annual deposit. 

67. The amount of a certain annuity, whose term is 7 years, is $3595 
and the present value of the annuity is $2600. (a) Determine the effec- 
tive rate of interest. (&) Determine the nominal rate, if it is compounded 
quarterly. 

58. How long will it take to pay for a house worth $20,000 if interest 
is at 5%, effective, and if payments of $4000, including interest, are made 
at the beginning of each year? Find the last annual payment which will 
be made, assuming that the debtor never pays more than $4000 at one 
time. 

59. A sum of $1000 is due at the end of two years, (a) Discount it to 
the present time under the simple interest rate 6%. (6) Discount it under 
the simple discount rate 6%. (c) Discount it under the compound in- 
terest rate (.06, m = 1). 

60. A concern issues $200,000 worth of serial bonds, paying 5% in- 
terest annually. It is provided that $30,000 shall be used at the end of 
each year to retire bonds at par and to pay interest. How long will it 
take to retire the issue ? Disregard the denomination of the bonds. 

61. Find the value of a mine which will net $18,000 per year for 30 
years if the investment yield is to be 6% and if the redemption fund is 
to be accumulated at 3%, compounded annually. 

62. A man expects to go into business when he has saved $5000. He 
now has $2000 and can invest his savings at (5%, m 1) . How much 
must he save at the end of each year to obtain the necessary amount by 
the end of 5 years? 

63. Find by interpolation the composite life on a 4% basis of a plant 
consisting of : Part (A), with life 10 years, cost new, $13,000, scrap value, 
$2000; Part (B), with life 16 years, cost new, $20,000, and scrap value, 
$3000. 

64. How much could a telephone company afford to pay per $10 unit 
cost in improving the material in its poles in order to increase the length 
of life from 15 to 25 years? The poles have no scrap value when worn 
out, and money is worth (.05, m 1). 

65. What are the net proceeds if a. 9Q-day n.gt;Q for $1000, bearing 6% 
interest, is discounted at 8%? 



142 MATHEMATICS OF INVESTMENT 

66. X requests a 60-day loan of $1000 from a bank charging 6% in- 
terest in advance. How much money does the bank give him and what 
interest rate is X paying on the loan? 

67. A woman has funds on deposit in a bank paying (.04, m = 2). 
Should she reinvest in bonds yielding .0415, effective? 

68. How long will it take for a fund of $3500 to grow to $4750 if in- 
vested at the rate 6%, compounded quarterly? 

69. The sums $200, $500, and $1000 are due without interest in 1, 2, 
and 3 years respectively. When would the payment of $1700 equitably 
discharge these debts if money is worth (.06, m = 1) ? 

70. A father has 3 children aged 4, 7, and 9. He wishes to present 
each one with $1000 at age 21. In order to do so he decides to deposit 
equal sums in a bank at the end of each year for 10 years. If it is assumed 
that the children will certainly live and that the bank pays (5%, m = 1), 
how much must the father deposit annually? 

71. Which is worth more, if money is worth 6%, effective : (a) an in- 
come of 12 annual payments of $500, first payment to be made at the end 
of 2 years, or (&) 120 monthly payments of $50, first payment due at the 
end of 3 years and 1 month? 

72. A $100, 5% bond pays interest quarterly and is redeemable at 110% 
at the end of 10 years. Find its price to yield 6%, effective. 

73. Find the present value and the amount of an annuity of $3000 
payable at the end of each 3 years for 21 years if interest is at the rate 
(.05, m = 2). 

74. Find the nominal rate, converted quarterly, under which money 
will treble in 20 years. 

76. (a) What effective rate is yielded by purchasing at par a $100, 
4% bond, redeemable at par, which pays interest quarterly? (6) What 
rate, compounded semi-annually, does the investment yield ? 

76. (a) In order to retire a $10,000 debt at the end of 8 years a sinking 
fund will be accumulated by equal semi-annual deposits, the first due im- 
mediately and the last at the end of 7 years. Find the semi-annual 
payment if the fund is invested at the rate (.04, m = 2) . (6) Find the 
size of the payments, under the same rate, if the first is made immediately 
and the last at the end of 8 years. 

77. X lends $600 to B, who promises to repay it at the end of 6 years 
with all accumulated interest at (.06, m = 2). At the end of 3 years, 
B desires to pay in full. If X is now able to invest funds at only 4%, 
effective, what should the debtor pay? 



REVIEW PROBLEMS ON PART I 143 

78. Find the nominal rate, converted quarterly, which yields the 
effective rate .0635. 

79. (a) A house costs $23,000 cash. If interest is at the rate (.05, 
m = 1), what equal payments made at the beginning of each 6 months 
for 6| years will amortize the debt? (&) What liability is outstanding 
at the beginning of the 3d year before the payment due is made ? 

80. How much must a man provide to purchase and maintain forever 
an ambulance costing $6000 new, renewable every 4 years at a cost of 
$4500 and requiring annual upkeep of $1500 payable at the beginning of 
each year? Money is worth 4%, effective. 

81. A corporation was loaned $200,000 and, in return, made annual 
payments of $12,000 for 8 years in addition to making a final payment 
of $200,000 at the end of 9 years. What rate of interest did the corpora- 
tion pay? 

82. A loan of $100,000 is to be amortized by equal payments at the 
end of each year for 20 years. During the first 10 years the payments 
are to include interest at 5%, effective, and, during the last 10 years, in- 
terest at 6%, effective. Determine the annual payment. 

83. $10,000 is invested at 6%, effective. Principal and interest are to 
yield a fixed income at the end of each 6 months for 10 years, at the end of 
which time the principal is to be exhausted. Determine the semi-annual 
income. 

84. A house worth $10,000 cash is purchased by B. A cash payment 
of $2000 is made and it is agreed in the contract to pay $500 of principal 
at the end of each 6 months until the principal is repaid and, in addition, 
to pay interest at the rate 6% semi-annually on all unpaid principal. Just 
after the payments are made at the end of two years, an investor buys 
the contract to yield 7%, compounded semi-annually, on the investment. 
What does the investor pay? 

86. A farm worth $15,000 cash ia purchased by B, who contracts to 
pay $2000 at the beginning of each 6 months, these payments including 
semi-annual interest at 6%, until the liability is discharged. At the end 
of 4 years, just after the payments due are made, the contract signed by 
B is sold to an investor to yield him (.07, m = 2) on the investment. 
What does he pay? 

86. A state, in making farm loans to ex-soldiers, grants them the fol- 
lowing terms: (a) interest shall be computed at the rate (.04, m = 2) 
throughout the life of the loan ; (&) no interest shall be paid, but it shall 
accumulate as a liability, during the first 4 years ; (c) the total indebted- 



144 MATHEMATICS OF INVESTMENT 

ness shall be discharged by equal monthly payments, the first due at 
the end of 4 years and 1 month and the last at the end of 10 years. De- 
termine the monthly payment on a loan of $2000. 

87. (a) A boy aged 15 years will receive the accumulations at 5%, effec- 
tive, of an estate now worth $30,000, when he reaches the age 21. What is 
the present value of his inheritance at 3$%, effective, assuming that he 
will certainly live to age 21? (6) Suppose that the boy is to receive, an- 
nually, the income at 5% from the estate and to receive the principal at 
age 21. Find the present value of the inheritance at 3?%, effective. 

88. The quotation of a certain $100, 5% bond to-day (an interest date) 
is 88.37 and it yields 7% to an investor. Find the purchase price and 
market quotation 2 months later at the same yield. 

89. A note signed by Y promises to pay $1000 at the end of 90 days 
with interest at 5%. (a) What would the holder X obtain on selling the 
note 30 days later to a banker whose discount rate is 6%? (&) What 
would he obtain if the note were discounted under the simple interest 
rate 6%? 

90. A certain man invests $1500 at the rate (.04, m 1) on each of 
bis birthdays, starting at age 35 and ending at age 65. (a) At age 
65, what does he have on hand ? (6) Suppose that at age 65 he decides 
to save no more and to spend all of his savings by taking from them an 
equal amount at the end of each month for 15 years, and suppose that he 
will certainly live that long. What can he take per month if the savings 
remain invested at (4%, m = 1) ? (c) If he desires to have $5000 left at 
the end of the 15 years, what will be his monthly allowance? 

91. A depreciation fund is being formed by semi-annual deposits, to 
replace an article worth $10,000 new, when it becomes worn out after 6 
years, (a) If money is worth (5%, m = 1), what is the semi-annual 
charge if the scrap value of the article is $1000? (6) How much is in the 
depreciation fund just after the third deposit? (c) Find the condition 
per cent of the article at the end of 3 years. 

92. A debt of $50,000 is being amortized with interest at (.06, m = 2) 
by 24 equal Bemi-annual payments, the first payment cash. Find the 
payment and determine how much principal is outstanding just after 
the 12th payment. 

93. Find the present value of a perpetuity of $1000, payable semi- 
annually, if interest is at the rate 6%, effective. 

94. A man borrowed $10,000, which he agreed to amortize with interest 
at the rate 5%, payable annually, by equal payments, at the end of each. 



REVIEW PROBLEMS ON PART I 145 

year for 12 years. Immediately after borrowing the money he invested 
it at 7%, payable semi-annually. In balancing his books at the end of 
12 years, what is his accumulated profit on the transaction? 

95. A loan agency offers loans to salaried workers under the following 
plan. In return for a $100 loan, payments of $8.70 must be made at the 
end of each month for 1 year. Determine the nominal rate, compounded 
quarterly, under which the transaction is executed. 

96. A corporation can raise money by selling 6% bonds, with semi- 
annual dividends, at 95% of par value. To provide for their redemption 
at par at the end of 15 years, a sinking fund would be accumulated by in- 
vesting equal semi-annual deposits at (.04, m = 2). The corporation also 
can raise money by issuing, at par, 15-year, 7% annuity bonds redeemable 
in semi-annual installments, (a) Which method would entail the least 
semi-annual expense in raising $100,000 by a bond issue? (6) If money 
can be invested at (.04, m 2) by the corporation, what would be the 
equivalent profit, in values at the end of 15 years, from choosing the 
best method? 

97. A corporation will issue $1,000,000 worth of 5% bonds, paying 
interest semi-annually and redeemable at par in the following amounts : 
$200,000 at the end of 5 years ; $300,000 at the end of 10 years ; $500,000 
at the end of 15 years. A banking syndicate bids $945,000 for the issue. 
Under what interest rate is the corporation borrowing on the proceeds of 
the bond issue? 

98. An investor paid $300,000 for a mine and spent $30,000 additional 
at the beginning of each year for the first 3 years for running expenses. 
Equal annual operating profits were received beginning at the end of the 
3d year and ceasing with a profit at the end of 25 years, when the mine 
became exhausted. The investor reinvested all revenue from the mine 
at 5%, effective. What was the net operating profit for the last 23 years 
if, at the end of 25 years, he has as much as if he had received, and 
reinvested at 5%, effective, 8% interest annually on all capital invested 
in the mine and likewise had received back his capital intact at the end 
of 25 years? 

99. A man who borrowed $100,000 under the rate 6%, payable semi- 
annually, is to discharge all principal and interest obligations by equal 
payments at the end of each quarter for 8 years. At the end of 2 years, 
his creditor agrees to permit him to discharge his future obligations by 4 
equal semi-annual payments, the first due immediately, (a) What will 
be the semi-annual payment if the creditor, in computing it, uses the rate 



146 MATHEMATICS OF INVESTMENT 

5%, compounded semi-annually? (6) What will be the semi-annual 
payment if the rate (.07, m = 2) is used in the computation? 

100. A contract for deed is the name assigned to the following type of 
agreement in real estate transactions : In purchasing a piece of property 
worth $2000 cash B agrees to pay $500 cash and to pay $25 at the end of 
each month, these payments to include interest at the rate 6%, payable 
monthly, until the property is paid for. The owner A agrees on his part 
to deliver the deed for the property to B when payment is completed. 

Six months after the contract above was made, A sells it to an investor, 
who obtains the rate 7%, compounded monthly, on his investment. What 
does he pay, if A has already received the $25 due on the contract on 
this date? 



PART II LIFE INSURANCE 

CHAPTER VIII 
LIFE ANNUITIES 

69. Probability. The mathematical definition of probability 
makes precise the meaning customarily assigned to the words 
chance or probability as used, for example, in regard to the winning 
of a game. Thus, if a bag contains 7 black and 3 white balls and 
if a ball is drawn at random, the chance of a white ball being ob- 
tained is -^y because, out of 10 balls in the bag, 3 are white. 

Definition. If an event E can happen in h ways and fail in u 
ways, all of which are equally likely, the probability p of the event 
happening is 7, 

' - ~b w 

and the probability q the event failing is 



NOTE 1 . In the ball problem above, the event E was the drawing of a white 
hall ; h = 3, h + u = 10, p = A- The probability of failure q = &. The 
denominator (u + A) in the formulas should be remembered as the total number 
of ways in which E can happen or fail. 

From formulas 1 and 2, it is seen that p and q are both less than 
1. Moreover, 

, h , u _ u + h _ * 

P ^ q .h + u' r h + u u + h 

or the sum of the probabilities of failure and of success is 1. If 

an event is certain to happen, u = and p = - = 1. 

h 

j EXERCISE LVH 

/ 

1. An urn contains 10 white and 33 black balls. What is the probabil- 

ity that a ball drawn at random will be white? 

2. A deck of 52 cards contains 4 aces. On drawing a card at random 
from a deck, what is the probability, that it will be an ace? 



148 MATHEMATICS OF INVESTMENT 

;| 3. Out of a class of 50 containing 20 girls and 30 boys, one member ia 
chosen by lot. What is the probability that a girl will be picked? 

4. A cubical die with six faces, numbered from 1 to 6, is tossed. What 
is the probability that it will fall with the number 4 up ? 

t 6. A coin is tossed. What is the probability that it will fall head up ? 
-J 6. If the probability of a man living for at least 10 years is .8, find the 
probability of him dying within 10 years. 

7. If the probability of winning a game is , what is the probability 
of losing? 

NOTE 2. It is important to recognize that when we say, as in problem 7, 
above, "the probability of winning is ," we mean : (a) if a very large number 
of games are played, it is to be expected that approximately $ of them will be 
won, and (&) if the number of games played becomes larger and larger with- 
out bound, it is to be expected that the quotient, of the number of them which 
are won divided by the total played, will approach as a limiting value. We 
do not imply, for instance, that out of 45 games played exactly $ of 45, or 27 
games will be won. We must recognize that, if only a few games are played, 
it may happen that more, or equally well less, than of the total will be won. 

8. As a cooperative class exercise, toss a coin 400 times and record at 
each trial whether or not the coin falls head up. How many were heads 
out of (a) the first 10 trials; (&) 50 trials; (c) 400 trials? Compare in 
each case the number of heads with of the number of trials so as to 
appreciate Note 2, above. 

NOTE 3. The assumption in the definition of probability that all ways of 
happening or failing are equally likely, is a very important qualification. For 
example, we might reason as follows : A man selected at random will either 
live one day or else he will die before to-morrow. Hence, there are only two 
possibilities to consider, and the probability of dying before to-morrow is $. 
This ridiculous conclusion would neglect the fact that he is mare likely to live 
than to die, and hence our definition of probability should not be applied. 

60. Mortality Table. Table XIII was formed from the accu- 
mulated experience of many American life insurance companies. 
This table should be considered as showing the observed deaths 
among a group * of 100,000 people of the same age, all of whom 

1 The actual construction, of a mortality table is a very difficult matter and cannot 
be considered here. It is, of course, impossible to obtain for observation 100,000 
children of the same age, 10 years, and to keep a record of the deaths until all have 
died. However, data obtained by insurance companies, or census records of births 
and deaths, can be used to create a table equivalent to a death record of a repre- 
sentative group of 100,000 people of the same age, all of whom were alive at ago 10. 



LIFE ANNUITIES 149 

were alive at age 10. In the mortality table, l x represents the 
number of the group still alive at age x, and d a the number of the 
group dying between ages x and x + 1. Thus, IK = 89,032, and 
das = 718 (= 89032 - 88314). In general, d x = l x - Z^+i. Out 
of l x alive at age x, Ix+n remain alive at age x + n, and hence 
l a lx+n die between ages x and x -\- n. Thus, Z 2 6 l& r 2154 
die between ages 25 and 28. 

When the exact probability of the happening of an event is 
unknown, the probability may sometimes be determined by ob- 
servation and statistical analysis. Suppose that an event has 
been observed to happen h times out of m trials in the past. Then, 
as an approximation to the probability of. occurrence we may take 

v = This estimated value of p becomes increasingly reliable 
m 

as the number of observed cases increases. The statistical method 
is used in determining all probabilities in regard to the death or 
survival of an individual selected at random; our observed data 
is the tabulated record given in the mortality table. 

Exampk 1. A man is alive at age 25, (a) Find the probability that 
he will live at least 13 years. (6) Find the probability that he will die 
in the year after he is 42. 

Solution. (a) We observe Zst = 89,032 men alive at age 26. Of these, 

79,611 ( = Z 38 ) remain alive at age 38. The probability of living to age 38 is 

B 796U 



A* 785 
die in their 43d year. The probability of dying is p = ^i" 



EXERCISE LVm 

In the first nine problems find the probability : 
* 1. That a boy aged 10 will live to graduate from college at age 22. 
^ 2. That a man aged 33 will live to receive an inheritance payable at 
age 45. 

3. That a boy aged 15 will reach age 80. 

4. (a) That a man aged 56 will die within 5 years. (6) That he will 
die during the 5th year. 

6. That a man aged 24 will live to age 25. 
" 6. That a man aged 28 will die in his 38th year. 



150 MATHEMATICS OF INVESTMENT 

7. That a man aged 28 will die in the year after he is 38. 

8. That a man aged 40 will live at least 12 years. 

9. That a man aged 35 will live at least 20 years. 

''-' 10. If a man is alive at age 22, between what ages is he most likely to 
die and what is his probability of dying in that year? 

NOTE. The problems in probability solved in the future in the theory of 
life annuities and of life insurance, so far as it is presented in this book, will be 
like those of' Exercise LVIII. None of the well known theorems on probabil- 
ity are needed in solving such problems ; the mere definition of probability is 
sufficient. Hence, no further theorems on probability are discussed in this 
text. The student is referred for their consideration to books on college 
algebra. 

61. Formulas used with Table XTTT. In Table XIII, we verify 
that the number dying between ages 25 and 28 is Z 2 6 Ins = d%t 
+ d 2 e + ^27, the sum of those dying in their 25th, 26th, or 27th 
years. Similarly, those dying between ages x and x + n are 



From Table XIII, Zg 8 = 0, and hence Z 87; !&, etc., are zero because 
all are dead before reaching age 96. The group of l x alive at age a 
are those who die in the future years, so that 

Z = d x + ds+i + + d w . 

It is convenient to use " (x} " to abbreviate a man aged x. Le' 
n p a represent the probability that (x) will live at least n years, or 
that (x) will still be alive at age x + n. Since Za; +n remain aliv< 
at age (x -\- n), out of la, alive at age x, 

* 



When n = 1, we omit the n = 1 on n p x and write p a for the prob 
ability that (jc) will live 1 year ; 



P. = -f 1 ' (6 

la 

The values of p^ are tabulated in Table XIII. Let q x represen 
the probability that (x} will die before age (x + 1). Since a 
of the group of l x die in the first year, 

2. = 



150 MATHEMATICS OF INVESTMENT 

7. That a man aged 2S will die in the year after he is 38. 

8. That a man aged 40 will live at toast 12 years. 

9. That a man aged 35 will live at least 20 years. 

1 ' 10. If a man is alive at ago 22, between what ages is he most likely to 
die and what is his probability of dying in that year? 

NOTH. The problems hi probability solved in the future in the theory of 
life annuities and of life inmiranee, HO fur us it, in presented in this book, will bo 
like those of 'Exercise LV.III. None of the wnll known theorems on probabil- 
ity are needed in solving Huoh problems; the more definition of probability is 
sufficient. Honee, no further theorems on probability arc disuusHed in this 
text. The student is referred for their consideration to books on college 
algebra. 

61. Formulas used with Table Xm. In Table XIII, we verify 
that the number dying botwoon ages 25 and 28 is l^ l& = rZ 2 r> 
+ dQ -\- dvj, the sum of those dying in their 25th, 2Gth, or 27th 
years. Similarly, those dying between ages x and x -f- n are 

Z* L+ n = d x + d x+ i + + d+i-i. (3) 

From Table XIII, Zoo = 0, and hence 1&, Z 08 , etc., are zero because 
all are dead before reaching age 90. The group of l x alive at age x 
are those who die in the future years, so that 

lx = d x + d x+i + + <ZgG. (4) 

It is convenient to use " (x) " to abbreviate a man aged x. Let 
n ps represent the probability that (x} will live at least- n years, or, 
that (x) will still be alive at age x + n. Since l x+n remain alive 
at age (x + n), out of l x alive at ago x, 



When 7i = l,wa omit then = 1 on n p a and write p a far the prob- 
ability that (x) will live 1 year j 

p. = *-"- (6) 

Lfo 

The values of y) K are tabulated in Table XIII. Let q a represent 
tho probability that (x) will die before ago (x + 1). Since d x 
of tho group of l a die in tho first year, 

<1 - ( ' (7) 



LIFE ANNUITIES 151 

The values of q a are tabulated in Table XIII. Let n \q a represent 
the probability that x will die in the year after reaching age (x + ri), 
between the ages (re + ri) and (x + n + 1). Of the original. 
group of l a alive at age x, d^+n die in the year after reaching age 
(x + ri). Hence 



Let n ([x represent the probability that x will die before reaching 
age x + n. Of the group of l e alive at age x, (l x Ix+n) will die 
before reaching age x + n, and hence the probability of dying is 

I n *+" ~ 8 - fQ~\ 

|n(Z* -- 7 -- (V) 

IB 

Example 1. State in words the probabilities denoted by the following 
symbols and find their values by the formulas above : 

(a) npas', (&) isl&a; (c) lug* 
Solution. (a) npas is the probability that a man aged 25 will be alive at 



age 42 ; npa 6 = ^ = L^r^n' W I 6 !? 22 ^ ^ e probability that a man aged 22 
Its 89032 

will die in the year after he reaches age 37; i&\q& == y =, j~ (c) lugaa 

ijj 91192 

is the probability that a man aged 22 will die before he is 15 years older (before 
reaching age 22 + 16 - 37) ; | 18 <? M - lsL=JlL. 



EXERCISE LIX 

1. Find the probability that a man aged 25 will live at least (a) 30 
years ; (6) 40 years ; (c) 70 years. 

'' 2. Find the probability that a man aged 30 will die in the year after 
reaching age 40. 

3. From formula 7 find the probability of a man aged 23 dying within 
1 year, and verify the entry in Table XIII. 

4. From formula 6 find the probability of a man aged 37 being alive 
at age 38, and verify the table entry. 

sj 5. Find the probability that a man aged 33 will die in the year after 
reaching age 55. 

6. State in words the probabilities represented by the following sym- 
bols and express them as quotients by the formulas above: lap^J 15)242; 
wl?a; Mas; 270j* loPsaJ Pa', M I (Zsa- 



152 MATHEMATICS OF INVESTMENT 

7. From formulas 5 and 9 prove that nPx = 1 |n2 

NOTE. If we consider the event of (x) living for at least n years, the 
failure of the event means that (x} dies within n years. Hence, the result of 
problem 7 should be true because, from Section 59 the sum of the probabilities 
of the success and of the failure of an event is 1, and p = 1 q. 

8. What is the probability of a man aged 26 dying some time after he 
reaches age 45? 

9. Verify formula 4 for x 90. 

10. Verify formula 3 for x = 53 and n = 5. 

62. Mathematical expectation; present value of an expecta- 
tion. If a man gambles in a game where the stake is $100, and 
where his probability of winning is .6, his chances are worth 
.6(100) = $60. Such a statement is made precise in the 
following 

Definition. If p is the probability of a person receiving a sum 
$S, the mathematical expectation of the person is pS. 

If the sum $S is due at the end of n years, the mathematical 
expectation at the end of n years is pS. If money is worth the ef- 
fective rate i, the present value $4 of the expectation is given by 

A = PS(1 + if". (10) 

NOTE. In the future, the arithmetical work in all examples will be per- 
formed by 5-place logarithms. 

Example 1. If money is worth 3$%, find the present value of the ex- 
pectation of a man aged 25 who is promised a payment of. $5000 at the end 
of 12 years if he is still alive. 

Solution. The probability p of receiving the payment is the probability of 
the man living to age 37, or p = upao- From equation 10, the present value 
of the expectation is 

A = ^55000(1.035)-" = fiOM(1.085)-*V ( Fo rmula 5) 

A = $2986.4. B (Tables VI and XIII) 

NOTE. In Example 1 it would be said that the payment of $6000 at the 

end of 12 years is contingent (or dependent) on the survival of the man. For 

brevity, in using formula 10, we shall speak of the present value of a contingent 

payment instead of, more completely, the present value of the expectation of 

this payment. 

EXERCISE LX 

1. In playing a game for a stake of. $50, what is the mathematical 
expectation of a player whose probability of winning is .3 ? 



LIFE ANNUITIES 153' 

Norm. Suppose that a professional gambler should operate the game of 
problem 1 and charge each player the value of his mathematical expectation 
as a fee for entering the game. Then, if a very large number of players enter 
the game, the gambler may expect to win, or lose, approximately nothing. 
This follows from the facts pointed out in Note 2, Section 59, because, if a 
large number play, approximately .3 of them may be expected to win the 
stake, and the money won would, in this case, be approximately equal to the 
total fees collected by the gambler. If, however, the gambler should admit 
only a few players to the game, he might happen to win, or equally well lose, 
a large sum, because out of a few games he has no right to expect that exactly 
3 of them will be won. The principle involved in this note is fundamental 
in the theory of insurance, and finds immediate application in problem 4, 
below. In any financial operation which is essentially similar to that of the 
professional gambler above, the safety of the operator depends on his obtain- 
ing a large number of players for his game. 

2. At the end of 10 years a man will receive $10,000 if he is alive. At 
5% interest, find the present worth of his expectation if his probability 
of living is .8. 

1 3. A young man, aged 20, on entering college is promised $1000 at the 
end of 4 years, if he graduates with honors. At 5% interest, find the pres- 
ent value of his expectation. 

4. Out of 1,000,000 buildings of a certain type, assume that the equiva- 
lent of 2500 total losses, payable at the end of the year, will be suffered 
through fire in the course of one year, (a) If an owner insures his build- 
ing for $20,000 for one year, what is the present value of his expectation, 
at 3% interest? (&) What is the least price that a fire insurance company 
could be expected to charge for insuring his building? 

5. A certain estate will be turned over to the heir on his 23d birthday. 
If the estate will then be worth $60,000, what is the present worth of the 
inheritance if money is worth 4$% and if the heir is now 14 years old ? 

6. A boy aged 15 has been willed an estate worth $10,000 now. The 
will directs that the estate shall be allowed to accumulate at the rate 
(.04, m = 2} until the heir is 21. If money is worth 3J% to the boy, find 
the present value of his expectation. 

63. Present value of a pure endowment. If $1 ia to be paid 
to (x) when he reaches age (v + ri), we shall say he has (or is prom- 
ised) an n-year pure endowment of $1. Let n E x be the present 
value of this endowment when money is worth the effective rate i. 
The probability p of the endowment being paid equals the prob- 



154 MATHEMATICS OF INVESTMENT 

ability of (x) living to age x + n, or p = n p x . Hence, from for- 
mula 10, with S 1, 

n E a = np*(l + i)~ n = Za+n(1 7 + *)"". (Formula 5) 

LX 

In the future we shall use u as an abbreviation for the discount 
factor (1 + O" 1 - Thus > v = (1 + 0~S w 2 = (1 + i)" 2 , etc., v n = 
(l+fl-. Hence, ^ _ ^ (u) 

*X 

The present value $A of an ?i-year pure endowment of $.R to a 
man aged x is given by 

(12) 



*x 

Norm 1. Remember the subscript (a; '+ n) on Z a+n in formula 12 as the 
age at which (x) receives the endowment. 

Example 1. A man aged 35 is promised a $3000 payment at age 39. 
Find the present value of this promise if money is worth 6%, effective. 

Solution. The man aged 35 has a 4r-year pure endowment of $3000. 
From formula 12, its present value is 

A = 3000(^6) = S 00 ?" 4 * 39 = SOQOft; 06 )"^ - $2290.3. (Tables VI, XIII) 

*36 ItS 

NOTE 2. Formula 11 may be derived by the following method. The 
present value JS^, is the sum which, if contributed now by a man aged x, will 
mate possible the payment of $1 to him at the end of n years, if money can be 
invested at the rate i, effective. Suppose that l a men of age x make equal 
contributions to a common fund with the object of providing all survivors of 
the group with $1 payments at the end of n years. Since l a + men will survive, 
the necessary payments at the end of n years total $i a+n . The present value 
of this amount at the rate i is l a+n (l + i)~" Z ffl+ u n , which is the sum needed 
in the common fund. Hence, the share which each of the Z people must con- 
tribute is v ni^ 

I. ' 
the same as obtained in formula 11. 

EXERCISE LXI 

1. A man aged 31 is promised a gift of $10,000 when he reaches age 41. 
Find the present value of the promise at 3^% interest. 

2. State in words what is represented by $2000 (lyJB'as) and find its 
value at 5% interest. 



f jf- 

LIFE ANNUITIES ('155.' 

v ' 3. A will specifies that the estate shall be turned over to the heir, now 
aged 23, when he reaches 30 years of age. If the estate will then amount 
to $150,000 find the present value of the inheritance at 4% interest. 

4. A man aged 25 has $1000 cash. What pure endowment, payable 
at the end of 20 years, could he purchase from an insurance company which 
will compute the endowment at a 4% rate? Make use of equation 12, to 
determine the unknown quantity R. 

6. If money is worth 3%, what endowment payable at age 45 could a 
man aged 30 purchase for $7500? / ..__. 

/ i - . - - ., f\ t \ 
tf~ '*"' " ' f - -t-'jt'- - 

64. Whole life annuity. A whole life annuity is an annuity 
whose periodic payments continue as long as a certain individual 
(or individuals) survives. We shall deal only with the case 
where one individual is concerned. In speaking of a life annuity 
we shall always mean a whole life annuity unless otherwise specified. 

NOTE 1. The periodic payments of all annuities will be supposed equal 
and will be due at the ends of the payment intervals unless otherwise stated. 
When no rate of interest is specified, it will be understood as the rate i, effectiye. 

Exampk 1. If money is worth 3%, find the present value of a life 
annuity of $1000 payable annually to a man aged 92. 

Solution. He is promised a payment, or endowment, of $1000 at age 93, 
another at 94, and a third at 95, which he will receive if lie is alive when they 
are due. No payment is possible after he is of age 95 because he is certainly 
dead at age 96. The present value A of his expectation from the annuity is 
the sum of the present values of the equivalent three endowments, due in 1, 
2, and 3 years, foom formula 12, the present values of these endowments are 
1000i#o2, lOOO^oa, and lOOOa^sa. Hence, by use of formula 12 and tables 
VI and XIII, we obtain 

A = 1000(1.002 



wt / 
(1.036)^ 8 \ 

Let a x be the present value of a ife annuity of $1 payable at the 
end of each year to a man now aged x. This annuity is equivalent 
to pure endowments of $1 payable at ages (x -f 1), (x + 2), , 
to age 95. The present values of these endowments are tabulated 
below, and a x equals their sum. 



156; 



MATHEMATICS OF INVESTMENT 



AQB AT WHICH SI ENDOW- 
MENT IB PAYABLE 


TIME FROM Now UNTIL ENDOW- 
MENT IB PAYABLB 


PRHSBNT VALUE OF THH 
ENDOWMBNT 


3+2 


1 yr. 
2yr. 


ku 


A " t 


95 


95 x 


95-3! EX 



On adding the last column we obtain 

a x = 



'95 



(13) 



The present value $A of a life annuity of %R paid at the end of 
each year is given by A = Ra x . 

NOTE 2. IE contrast to the annuities certain considered in Part I, life 
annuities are called contingent annuities, because their payments are con- 
tingent (or dependent) on the survival of (x). The life annuity was inter- 
preted as heing paid to (x). Recognize that a, is the present value of pay- 
ments made at the end of each year during the life of (x), regardless of who 
receives the payments. 

EXERCISE LXH 

1. By the process of Example 1 above, find the present value of a life 
annuity of $2000 payable at the end of each year to a man now aged 91, 
if money is worth 5%. 

4 2. If money is. worth 6%, find the present value of a whole life pension 
of $1000 -payable at the end of each year to a man now aged 92. Use 
formula 13. 

3. (a) By use of formula 13, write the explicit expression which 
would be computed in finding the present value of a life annuity of $500 
paid at the end of each year to a man aged 65, if money is worth 6%. 
(6) How many multiplications would be necessary in computing the nu- 
merator? 

4. By the method used in deriving the formula for , find the present 
value of a life annuity of $1000 payable at the end of each 3 years to a 
man aged 35, if money is worth (.04, m = 1). Do not compute the ex- 
pression obtained. 



LIFE ANNUITIES 157 

66. Commutation symbols. Auxiliary symbols (such as D h 
and Nb below), called commutation symbols, are used in life an 
nuity and insurance formulas. From formula 11 for nE X) we obtain 

n E a = vn ^ n = 

l a V l x 

Let Dh be an abbreviation for y^*, or 

D k - irt*. ' (14) 

Thus, DBO = v 60 l5Q. Hence, v*l x = D x , v^^lg+n = Dx+n, and 

A = %* (15) 

UK 

The present value $A of an Ti-year pure endowment of $J2 is 

A - R(JBJ - ^5=*=- (16) 

Z/B 

Example 1. Compute AB if money is worth 3%. 

Solution. D S8 - ii = (1.035)- se (89032) = 37674. 

NOTE. It is very customary for insurance companies to use 3i% as the 
rate in annuity computations. The values of DI O , DH, to DOB at 3J% are 
tabulated in Table XIV, and the result of Example 1 above is seen to check the 
proper table entry. Formulas 15 and 16 may be used, in connection with 
Table XTV, only when the rate is 3J%. Tables of the values of D* at, other 
rates 1 are found in collections of actuarial tables. In problems in this book, 
when the rate is not 3J%, formulas 11 and 12 must be used. 

To simplify formula 13 for a a , .multiply numerator and de- 
nominator by V. We obtain, 



Since v*l 9 = D a , V+H^i = >+!, etc., v 96 ^ = D 



9 B, 



Introduce Nk as an abbreviation for the sum of all JD's from Db 

to D 96 : Nk = Dk + Dk+i + + DM. (18) 

Thus, JVoo = DM 4- D n + D 92 + D 93 + D M + D 9B . Since the 
numerator in formula 17 is Na+i, 

a ^*H. - ^IQ'i 

Q x ^ i \-*-"/ 

1 See Tables of Applied Mathematics, by Glover. 



158 ' MATHEMATICS OF INVESTMENT 

The present value $A of a life annuity of $R per year to (x) is 

A = Ra x = *^i- (20) 

MX 

NOTE. The values of Nk are tabulated in Table XIV for the rate 34%. 
For this rate, formulas 19 and 20 may be used in connection with Table XIV. 
For all other rates the previous formula 13 must be used. The values of Nk 
for a few other interest rates are found in actuarial tables. 

Exampk 2. If money is worth 3%, what life annuity, payable at 
the end of each year, can a man aged 50 purchase for $10,000? 
Solution. Let R be the payment of the annuity. From formula 20, 

10000 = .8(050) = R^, 
DM 

p 10000D60 10000(12498.6) 9.70000 
R = -Jfc 169166 * 738 ' 83 - 

EXERCISE LXTTT 

1. Compute the value of D& for i = .035 and verify the entry in 
Table XIV. 

2. By use of formula 18 and Table XIV for the D's, find the value of 
(a)N 66 -, (&) tf M ; (c) N m . 

3. Find the present value of a life annuity of $1000 at the end of each 
year for a man aged 24, at 3J%. 

4. (o) Find the present value of a pure endowment of $3500 at the 
end of 12 years for a man aged 33, at 3|%. (6) Find the present value of 
the endowment at the rate 4%. 

5. A man aged 65 is promised a pension of $2000 at the end of each 
year as long as he lives, (a) If money is worth 3|%, find the present value 
of his pension. (&) What is the present value if $2000 is to be paid at 
the beginning of each year? 

4 6. An estate is worth $100,000 and is invested at 5%, effective. The 
annual income is willed to a woman, aged 30, for the rest of her life. Find 
the present value of her inheritance if money is worth 3J%. 

7. A man aged 45 has agreed to pay a $75 insurance premium at the 
end of each year as long as he lives. At 3$% interest, what is the present 
value of his premiums from the standpoint of the insurance company? 

8. A man aged 26 has agreed to pay $50 insurance premiums at the 
end of each year for the rest of his life. At 3J%, what is the present value 
of his premiums? 



LIFE ANNUITIES 



159 



9. A man aged 60 gives $10,000 to an insurance company in return for 
an annuity contract promising him payments at the end of each year as 
long as he lives. If money is worth 3|% to the company, what annual 
payment does he receive ? 

10. From the formulas previously developed, prove that 
a x = vp x (l 



NOTE. Recognize that this formula would make the computation of a 
table of the values of a, very simple. First, we should- compute at&, which 
is zero ; then, 094 = vpu(l + ctt&) gives the value of OM, etc., for a ra , <ZM, , 
down to au>. 

66. Temporary and deferred life annuities. A temporary life 
annuity of $R per year for n years to (x) furnishes payments of 
$jffi at the end of 1 year, 2 years, etc., to the end of n years, if (x) 
continues to live. The payments cease at the end of n years, 
even though (x) remains alive. Let a^\ represent the present 
value of a temporary Me annuity of $1 paid annually for n years 
to (Jt). This annuity promises n pure endowments whose present 
values are tabulated below. 



AQB AT WHIOH SI 
ENDOWMENT is PAYABLE 


TlMB FHOM NOW UNTIL 

ENDOWMENT is PAYABLE 


PBESENT VALUE 
OF THE ENDOWMENT 


X+ 1 

'z + 2 
x + n 


1 yr. 
2 yr. 

nyr. 


77 _ W ^B+I 
lJ2/ a - 

va 

E v*U z 


ZJCia - 7 
L-e 

IP _ n k+n 


n jG/ai ~ - 

LX 



The sum of the present values is 

a an\ ^ 



or 



vx 



(21) 



Formula 21 applies for all interest rates. To obtain a formula 
in terms of N and D (which, with our tables, will be useful only 



160 MATHEMATICS OF INVESTMENT 

when the rate is &%), multiply numerator and denominator in 
equation 21 by v x . 



From formula 18, 



Hence, #3+1 - 2V I+n+1 = >*+! + D^+a + + D x+n , and there- 
fore 

a*n = N ** -">** (22) 

t'x 

If the annual payment of the temporary annuity is $E, the present 
value $A is given by 

A = R(a x *\) = R ( N ** ~ ^ + " +l) (23) 

"* 

The definition of a deferred life annuity is similar to that for" a 
deferred annuity certain (see Section 26, Part I). A life .annuity 
of $1 per year, whose term is deferred 10 years, to a man aged 30, 
promises the first $1 payment at the end of (10 + 1) or 11 years, 
and $1 annually thereafter. Let n \Q>x be the present value of a 
life annuity of $1 per year, whose term is deferred n years, to a 
man aged x. The first payment of the deferred annuity is due 
at the end of (n + 1) years. It is clear that a whole life annuity 
of $1 per year to a man aged x pays him $1 at the end of each year 
for the first n years, and also at the end of each year after that, 
provided that he lives. The payments during the first n years 
form a temporary life annuity whose present value is a^. The 
payments after the nth. year are those of the deferred annuity, 
whose present value we are representing by nja,. Hence, the pres- 
ent value a x of the whole life annuity is the sum of the other two 
present values or 

a x Jo, + a*i; (24) 

n|a* - o, - a^\. (25) 



LIFE ANNUITIES / 161 , 

On using formulas 19 and 22 in formula 25, 




(27) 

The present value $A of a life annuity of $72 per year, deferred 
n years, for a man aged x, is 

- A = *(!<**) = *%5i- (28) 

l) x 

EXERCISE LS3V 

1. At 3?%, find the present value of a life annuity of $1000 per annum, 
deferred 20 years, to a man aged 23. 

2. At 3%, find the present value of a life annuity of $2000 paid an- 
nually for 25 years to a man aged 45. 

3. If money is worth 5%, find the present value of a life annuity of 
$1000 paid annually for 3 years to a man aged 27. 

4. A man aged 25 will pay 20 annual premiums of $50 each on a life 
insurance policy, if the man remains alive. If the first premium is cash, 
find their present value, at 3%. ' 

5. A man aged 50 gives an insurance company $10,000 in return for a 
contract to pay him a fixed income at the end of each year for 20 years, 
if he lives. If money is worth 3i% to the company, what is the annual 
income? Use formula 23. 

v/ 6. A man aged 40 pays an insurance company $20,000 in return for 
a contract to pay him a life annuity whose first annual payment will be 
made when his age is 65. Find the annual payment, if money is worth 
3&% to the insurance company, by use of formula 28. 

7. A corporation has promised to pay an employee, now aged 48, a 
pension of $1000 at the end of each year, starting with a payment on his 
61st birthday. At 3%%, what is the present value of this obligation? 

NOTE. Any pension system instituted by a company constitutes a definite 
present obligation whose value can be determined by finding, as in the prob- 
lem above, the present value of the pension promised to each employee. 

\/ 8. A man aged 43 estimates his future earnings at $5000 at the end of 
each year for the next 25 years. At 3f%, find the capitalized (present) 
value of his earning power. 



162 MATHEMATICS OF INVESTMENT 

67. Annuities due. A life annuity due is one whose payments 
occur at the beginnings of the payment intervals, so that the first 
payment is cash. Let a z be the present value of a life annuity 
due of $1 paid annually to (x). A cash payment of $1 is due and 
the remaining payments of $1 at the end of each year form an 
ordinary life annuity whose present value is a x . Hence, the pres- 
ent value of the annuity due is given by 

a x = 1 + a x . (29) 

From formula 19, 

_ i _i_ N x+ i _ D x +N x+ i _ P.+ (IWi+ At+H ----- hflro) /cm 
a a -l+ -- - (30) 

.-* (3D 

MX 

The present value $A of a life annuity due of $.R paid annually is 
A = *(a) = $& (32) 

L>x 

Let a.\ be the present value of a temporary life annuity due, 

whose term is n years,' paying $1 annually, to a man aged x. The 

first $1 is paid cash and the remaining payments form an ordi- 

nary temporary life annuity whose term is (n 1) years. Hence, 

asSi = 1 -h Oxt=i\> (33) 

From formula 22, a x 

Hence, a^ = 1 + 

1 D a D a 

Since D x + N x+ i = N x , 

* X n\= N *~ N * +n ' (34) 



= Nx + l ~ 



-n 
AWi- 


D x 

N- x+n Dx + 


D x 

N .-, N , 

1.1 (B+l JV ffi+n. 



The present value $A of a temporary annuity due of $# payable 
annually for n years to (x) is 



A = *(a^|) = * ~ *+ . (35) 

"X 

Example 1. In a certain insurance policy, the present value of the 
benefits promised to the policyholder is $3500. If the polioyholcler is 
of age 27, what equal premiums should he pay to the insurance company 
at the beginning of each year for 10 years, in payment of the policy, if 
money is worth 3J% to the company? 



LIFE ANNTJITIES < 163 

Solution. Let $R be the annual premium. The premiums form a tem- 
xary life amiuity due whose present value equals $3600. Prom formula 
, with A = $3600, x = 27, and n = 10, 



3500 = V- ^) R = 3600 

' 



R = 3500(3^01) = 
287510 

NOTE 1, In discussing premiums on life insurance policies, formulas 32 
d 35 are of great use. Formulas 16, 28, 32, and 35 of this chapter are the 
as we shall use moat frequently in the future. 

Summary of present value formulas 
Pure endowment : A = #(*) = R ^ - (16) 

DX 

Whole life annuity: A = R(a x )=^^- (20) 

DX 

Temporary life annuity : A = R(a^ = R Nx+1 ~ Nx ^^- (23) 

D x 

Deferred life annuity : A = R( n \ a*) = R ^i . (28) 

D x 

Whole life annuity due : A = R(eL x ) = R^' (32) 

* 

Temp, life annuity due : A = /?(a^) = R N * ~ N ^ n - (36) 



MISCELLANEOUS PROBLEMS 

L. A man aged 40 pays $10,000 to an insurance company in return for 

ontract to pay him a fixed annual income for life, starting with a pay- 

nt on his 60th birthday. Find the annual income if money is worth 

& to the company. 

I. At age 65 a man considers whether he should (a) pay his total 

ings of $20,000 to an insurance company for a life annuity whose first 

mal payment would occur in 1 year, or (6) invest his savings at 6%, 

ictive. Find the difference in his annual income under the two methods, 

uming that money is worth 3|% to the insurance company. 

1. In problem 2, what will be received by the heirs of the man at his 

,th if he adopts plan (a) ? What will they receive under plan (6) ? 



164 MATHEMATICS OF INVESTMENT 

4. A certain insurance policy taken out by a man aged 28 calls for 
premiums of $200 at the beginning of each year as long as he lives. Find 
the present value of these premiums at 3%. 

5. A certain insurance policy matures when the policyholder is of age 
35 and gives Him $2000 cash or the optipn of equal payments at the be- 
ginning of each year for 10 years as long as he lives. If money is worth 
3&%, find, the annual payment under the optional plan. 

6. A boy of 16 has been left an estate of $100,000, which is invested 
at 5%, effective. If he lives, he will receive the income annually for the 
next 10 years and the principal of the estate when he reaches age 26. If 
money is worth 3|%, find the present value of his inheritance. 

7. Derive formula 13 for a f by the mutual benefit fund reasoning used 
in Note 2, Section 63. Thus, at the end of 1 year, $k+! will be needed for 
payments ; $Z I+a at the end of 2 years, etc., $Ze 6 at the end of (95 x) 
years. Discount all of these payments and divide by l a , 

8. Derive formula 21 for a^ by the mutual fund method of reason- 
ing. 

'( 9. A man aged 22 agrees to pay $50 as the premium on an insurance 
policy at the beginning of each year for 10 years if he lives. Find the 
present value of his premiums at 3% interest. 

10. The present value of the benefits promised in a certain insurance 
policy is $8000. If the policyholder is aged 30, what equal premiums 
should he agree to pay at the beginning of each year for 15 years, provided 
he lives, if money is worth 3$% to the insurance company? 

11. A man is to receive a life annuity of $2000 per year, the first pay- 
ment occurring on his 55th birthday. If he postpones the annuity so 
that the first annual payment will occur on his 65th birthday, what will 
be the annual income, if the new annuity has the same present value as 
the former one, under 3|% interest? 

12. A certain professor at age 66 enters upon a pension of the Carnegie 
Foundation which will pay $2000 at the end of each year for life. In 
order to have, at age 65, an amount equal to the present value at 3J% of 
the pension he is to receive, what equal sums would the professor have 
had to have invested annually at 5%, assuming that his first investment 
would have occurred at age 41 and his last at age 65? 



CHAPTER DC 
LIFE INSURANCE 

68. Terminology. Insurance is an indemnity or protection 
against loss. The business of insuring people against any variety 
of disaster is on a scientific basis only when a large number of in- 
dividuals are insured under one organization, so that individual 
losses may be distributed over the whole group according to some 
scientific principle of mutuality. That is, each of the insured 
should pay in proportion to what he is promised as an insurance 
benefit. In this chapter we shall discuss the principles and most 
simple aspects of the scientific type of life insurance furnished 
by old line, or legal reserve companies. 

When an individual is insured by a company, he and the com- 
pany sign a written contract, called a policy. The individual is 
called a policyholder, or the insured. In the contract the com- 
pany promises to pay certain sums of money, called benefits, 
if certain events occur. The person to whom the benefits are to 
be paid is called the beneficiary. The insured agrees to pay cer- 
tain sums called gross or office premiums in return for the con- 
tracted benefits. The policy date is the day the contract was 
entered into. The successive years after this date are called 
policy years. 

The fundamental problem of a company is to determine the pre- 
miums which should be charged a policyholder in return for speci- 
fied benefits. Every insurance company adopts a certain mor- 
tality table and an assumed rate of earnings on invested funds 
as the basis for its computations. We shall use the American 
Experience Table and 3%, as is the custom among many com- 
panies. The net premiums for a policy are those whose present 
value is equal to the present value of the policy benefits under the 
following assumptions : (a) the benefits from the policy mil be paid 
at the ends of the years in which they fall due; (&) the company's 
funds will earn interest at exactly the specified rate (3% in our case) ; 

166 



166 



MATHEMATICS OP INVESTMENT 



(c) the deaths among the policyholders will occur at exactly the rate 
given by the mortality table (Table XIII in our case). Under these 
assumptions, if a company were run without profit or administra- 
tive expense, it could afford to issue policies in return for these net 
premiums. The actual gross premiums for a policy are the net 
premiums plus certain amounts which provide for the adminis- 
trative expense of the company and for added expense due to vio- 
lations of the theoretical conditions (a), (6), and (c) assumed above. 
In computing gross premiums, insurance companies use their own 
individual methods. Our discussion is concerned entirely with 
net premiums and related questions. 

Nona. In the future, if the interest rate in a problem is not given, it is 
understood to be 3%. 

69. Net single premium ; whole life insurance. If a policy- 
holder agrees to pay all premium obligations in one installment, it 
is payable immediately on the policy date and is called the single 
premium for the policy. The net single premium is the present 
value on the policy date of all benefits of the policy. 

A whole life insurance of $R on the life of (x) is an agreement by 
the company to pay $R to the beneficiary at the end of the year in 
which (x) dies. A policy containing this contract is called a 
whole life policy. 

Example 1. Find the net single premium for a whole life policy for 
$1000 for a man aged 91. 

Solution. Suppose that the company issues whole life policies for $1000 
to ZDI, or 462 men of age 91. During the first year, d 9 i - 246 men will die; 
$246,000 in death claims will be payable to beneficiaries at the end of 1 year. 
The present value of this payment is 246,000(1. 035) ~ l = 246,000 v. The 
other entries below are easily verified. 



Pouor YEAB 


DEATHS DURING YBAB 


BENEFITS DBB AT END 
OP YHAB 


PIIBBTIINT VALOT op 

BmNHfflTH 


1 


d 9 i = 246 


$246,000 


246,000 v 


2 


d 9 a - 137 


137,000 


137,000 w 


3 


dgs = 58 


58,000 


58,000 it 


4 


dw-lS 


18,000 


18,000 v* 


5 


d 8B =3 


3,000 


3,000 V 6 - 



LITE INSURANCE 



167 



Hence, on the policy date, the insurance company should obtain through the 
net single premiums from the lai men, a fund equal to the sum of the last col- 
umn. Thia sum, divided by 462, is the share or net single premium paid by 
each of the ZBI men. By use of Table VI, we find that each pays 



246000 v + 137000 + 58000 t> + 18000 v* + 3000 v 5 
462 



$943.93. 



Let lAs be the net single premium for a whole life insurance of 
$1 on the life of (x). To obtain A a by the method of Example 1 
above, suppose that a company issues whole life policies for $1 
insurance to each of l a men of age x. During the first policy 
year, d x will die ; $d ffl in death benefits is payable to beneficiaries 
at the end of 1 year. The present value of these benefits at the 
rate i is d x (1 + i}~ 1 = vd a . The other entries below are easily 
verified. 



POLICY YHAB 


DEATHS DUBINQ YBAB 


BENEFITS Dim AT END 
op YEAR 


___^^_ 

PBUBBNT VALtra os 
BEND PITS 


1 


d, 


^ 


wd, 


2 


dx+i 


9d x +i 


t^dz+i 


3 


d : + * 


V.* 


"^ 


96-s 


,;. 





^., 



In the last row, notice that when the group reaches age 95, the 
policies have been in force (95 re) years, or the (96 :c)th year 
is just entered on. Honce, dw is due at the end of (96 x) 
years. From the net single premiums paid on the policy date, the 
company must obtain a fund equal to the sum of the values in 
the last column. The share of each of the l a men, or his net single 
premium A x > is 

A =1 v( ** + y2rfg+1 + y3 ^ g+2 "^ ---- *" &*~* d u . (36) 



On multiplying numerator and denominator above by v*, 
+ v*+*d f+l + + 



168 MATHEMATICS OF INVESTMENT 

Introduce a new symbol C* = v k+1 dk. Thus, C& = 

v 94 cZ B a, etc., v^^dj, = C X ) V^dx+i == CWij and ir^d^ = C B 5. Hence 

ri \ fi I ... ' ^ 

4 _ ^a "t" v^g+l ~T T ^96. 

A ~ D x 

Introduce a new symbol 

M k = C k + Cft+i H h C 9B - (37) 

Thus, M 92 = C B2 4- Cgg + C B4 + C B5 ; M s - C a + + C^. 

Hence 4 X = ^- (38) 

X 

The net single premium A for a whole life policy of $72 for (x) is 

(39) 



NOTE. The values of Mb for the rate 3%, v = (1.035) ~ l , ore given in 
Table XIV. 

EXERCISE LXV 

Use formulas 38 and 39 unless otherwise directed. 

1. Compute the values of (7 9 4 and of Ceo at 3$% ; verify the entries 
for M 94 and for M 9B in Table XIV. 

4 2. By the method of illustrative Example 1, page 166, find the net 
single premium for a whole life insurance for $1000 for a man aged 93, if 
interest is at the rate 5%. 

3. Find the net single premium for a whole life insurance of $1000 on 
the life of a man, (a) aged 90 ; (6) aged 50 ; (c) aged 30 ; (d) aged 10. 

g 4. How much whole life insurance can a man aged 50 purchase from 
a company for $1500 cash? 

6. How much whole lif e insurance can a man aged 35 purchase from a 
company for $2000 cash? 

70. Term insurance. An n-year term insurance for $R on 
the life of (x) promises the payment of $R at the end of the year' 
in which (x) dies, only on condition that his death occurs within 
n years. Thus, a 5-year term insurance gives no benefit unless 
(jc) dies' within 5 years. Let A 1 ^ represent the present value of 
an Ti-year term insurance for $1 on the life of (x}. To obtain the 
value of A, assume that the company issues n-year term in- 



LIFE 



169 



surance policies for $1 to each of Z a men aged x. The present values 
of the benefits which will be paid are tabulated below ; the policy 
has no force after n years. 



POLICY YEAR 


DEATHS DURING YEAH 


BENEFITS DUB AT END 
OP YHAH 


PBBBENT VALUE 
OF BENEFITS 


1 

2 


d, 
d a +i 


94. 

$da+I 


vd x 
&d a+ i 


n 


dj;+n-l 


$dx+n-l 


f^s+n-l 



The net single premium paid by each man is the sum of the last 
column, divided by l a , or 

[ .i2J I ... I .nJ 

(40) 



in t 

3711 z a 

On multiplying numerator and denominator above by if and on 
using the symbol C k = 



-i + 



-\ 



\- C 



9B , 



Since M x = C x + C I+i + + 

and M x+n = 

it is seen that the numerator in equation 41 is M x M^ n ; hence 

^ ^ ~ M ^+". (42) 

*n\ D; ^ 

The net single premium $A for T^year term insurance of $B on 
the lifo of (x) is 

A = J R4* = g(^ ~ M x+ n). (43) 

*n| ^ 

The not fiinglo premium for a 1-year term insurance for (x) is 
called the natural premium at age x. The natural premium for 
$1 insurance is obtained from bquation 42, with n = 1 : 

Natural Premium - 4*-, *** ~ M * +1 - ^-, (44) 

**x *^< 



where M - 



C s because of formula 37. 



170- MATHEMATICS OF INVESTMENT 

EXERCISE LXVI 

Use formulas 42 and 43 unless otherwise specified. 

1. By use of the method used in deriving formula 40, find the expres- 
sion for the net single premium for a 3-year term insurance for $1000 on 
the life of a man aged 25 and compute its value at 5% interest. 
'- r 2. Find the net single premium for a 10-year insurance for $2000 on 
the life of a man* aged 31. 

3. Find the natural premium for $1 insurance at age 22 ; at age 90. 

4. (a) Find the net single premium for a whole life insurance of $1000 
at age 50. (6) Find the net single premium at age 50 for a 10-year term 
insurance for $1000. 

J 6. How much term insurance for 10 years can be purchased for $2000 
cash by a man aged 35? 

6. How much term insurance for 10 years can be purchased for $2000 
cash by a man aged 55? 

71. Endowment insurance. An n-year endowment insur- 
ance of $R on the life of a man aged x furnishes 

(a) a payment of $R at the end of the year in which (x) dies, if he 

dies within n years, and (b) a pure endowment of $J5 to (x) at 

the end of n years if (x) is alive at that time. 

Thus, a 20-year endowment insurance of $1000 pays $1000 at 
death, if it occurs within 20 years ; or, if (x) is alive at the end of 
20 years, he receives the endowment of $1000. Let A X n\ repre- 
sent the net single premium (or present value) of an n-year en- 
dowment insurance of $1 on the life of (x}. The present valuo 
Axft is the sum of the present values of (a) the n-year term in- 
surance for $1 on the life of (x}, and of (6' the n-yoar pure endow- 
ment of $1 to (x}. Hence, on using formulas 16 and 42, 




(45) 

If the endowment insurance is for $A, the net single premium $A 
is given by Af* - Af*+n + *+n) 



LIFE INSURANCE , 171 

EXERCISE LXVH 

1. (a) Compute the net single premium for a $1000, 20-year endow- 
ment insurance on the life of a man aged 23. (&) Compute the present 
value of ti pure endowment of $1000 payable to the man at age 43. 
(c) Find the net single premium for a 20-year term insurance for $1000 
on the life of the man aged 23, by using (a) and (&). 

'2. Find the net single premium for a 10-year endowment insurance for 
$5000 on the life of a inun aged 30. 

3. (a) Find the net single premium for a 10-year endowment insurance 
for $3000 on the life of a man aged 26. (6) Find the net single premium 
for a 10-year term insurance for $3000 on his life, (c) From the results 
of (a) and (&), find the present value of a 10-year pure endowment of $3000 
for the man. 

f /4. How much 20-year endowment insurance can a man aged 33 pur- 
chase for $3000 cash? 

6. How much 10-year endowment insurance can a man aged 45 pur- 
chase for $2500 cash? 

72, Annual premiums. If the net premiums for a policy are 
payable annually, instead of in one installment (the net single 
premium), they must satisfy the condition that the 

(pr. val. of annual premiums) = (net single premium), (47) 
because the net single premium is the present value of the policy 
benefits. When paid annually, the premiums for a policy are 
always equal and are payable at the beginnings of the years, as 
long as the policyholdor lives. 

Example 1. (a) Find the net single premium for a 10-year term in- 
surance for $10,000 on the life of a man aged 46. (&) Find the equivalent 
annual premium which the man might agree to pay for 10 years, if he lives. 

Solution, (a) From formula 43, the not single premium is 

10000(4*^) = 1QQ 00(M4 fl " MM) = $1119.30. (Table XIV) 

4010 1 , JJlft 

That is, the present value of the insurance benefits is $1119.30. (b) Let P be 
the annual premium. The 10 premiums form a 10-year life annuity due 
paid by a man aged 46. Their present value ie P(a 46 iji), and it must equal 
$1119.30. Hence, 

1119.30 - P(a) - pN * Ntt ' (Formula 35) 



1119.30 Dq m $137.33, (Table XIV) 



172 



MATHEMATICS OF INVESTMENT 



The annual payments of $137.33 have a value equivalent to $1119.30 paid 
cash. 

NOTE 1. The solution of (a) was not necessary in order to solve (6) above. 
Thus, we may write, immediately, from equation 47, 



10000 W6- 



- N 



In insurance practice the most simple forms of insurance policies 
are those tabulated below. Their names, policy benefits, and 
manner of premium payment should be memorized. All premiums 
are payable in advance, at the beginning of the year. .The num- 
bers assigned are merely for later convenience in this book. 



NUMBBB 


NAME OF POLICY 


POLICY BHNBFITS 


PBBMIUMB PAID 


I 


Ordinary life 


Whole life insurance 


Annually for life 


II 


w-payment lif e 


Whole life insurance 


Annually for n years 


III 


?i-year term 


7i-year term insurance 


Annually for n years 


rv 


Tfc-year endowment 


(a) Ti-year pure endowment 
(6) 7i-year term insurance 


Annually for n years 



To determine the net annual premiums for these policies, we 
use the fundamental equation 47, and the method x of Note 1 
above. Consider an ordinary life policy for $1 for a man aged x. 
Let P a be the net annual premium. The premiums paid by the 
man aged x form a whole life annuity due whose present value is 
P a ,(a a ). The net single premium for the policy is A& Hence, 
from equation 47, 

JP.(a,} -A.; Px~~ - 7T 2 - (Formulas 32, 38) 



1 The student ia advised to solve problejn 1 of 
reading the rest of the section. 



(48) 
we &XVIH below before 



LIFE INSURANCE 173, 

Let n Px be the net annual premium for an n-payment life policy 
for $1 for a man aged x. The premium payments by (x) form an 
?>-year Me annuity due whose present value is nPsCa^). Hence, 
from equation 47, 

n P*(a.*nd - A*', nP* Nx ~ Nx + n = ^=- (Formulas 35, 38) 

Uy U x 

nPx = M * (49) 



It is left as an exercise (problem 3, below) for the student to 
prove that the net annual premium P^ for an n-year term insur- 
ance policy for $1 for (x) is given by 



M x+n _ 

-- 



,.. 

(60) 



Let P.^ be the net annual premium for an n-year endowment 
policy for $1 for (x). The premiums paid by the man aged x form 
an w-year life annuity due whose present value is P a -ni(a a ^ 1 ). The 
net single premium for the policy is A^. From equation 47, 



- M,+ n + D^ t (Formulas 35, 45) 



, -*1* ^. , (51) 

P*x JVjc+n 

If the policies I to IV are for $J? instead of $1, the annual pre- 
miums are found from formulas 48 to 51 by multiplying by R. 

EXERCISE LXVIE 

1. By the method of Note 1, page 172, find the net annual premium 
for a 5-payment life policy for $1000 for a man aged 45. 
J 2. A man aged 29 has agreed to pay 15 annual premiums of $100 for a 
certain policy. Find the net single premium for the policy, 

3. Establish formula 50 for the net annual premium for an n-year term 
insurance policy for $1 for a man aged &. 

4, The net single premium for a certain policy for a man aged 26 is 
$3500, (a) Wbftt is the net annual premium if hQ agress to pay premiums 



174/ MATHEMATICS OF INVESTMENT 

annually for life? (&) What is the net annual premium if he agrees to 
pay annually for 12 years? 

6. For a man aged 25-, find the net annual premium (a) for an ordinary 
life policy for $2000 ; (&) for a 20-payment life policy for $2000. 

6. Find the net annual premium for a 10-year term policy for $1000 
for a man (a) aged 32 ; (&) aged 42 ; (c) aged 62. 

7. Find the net annual premium for a 20-year endowment policy for 
$1000 for a person of your own age. 

8. (a) Find the net annual premium for a 5-year term policy for $1000 
for a man aged 40. (&) Find the natural premiums for a $1000 insurance 
at each of the ages 40, 41, 42, 43, and 44. (c) Compare the result of (a) 
with the five results in (6) . 

9. A whole life insurance policy for $1000 taken at age 30 states that 
the annual premiums were computed as if : (a) term insurance of $1000 
were given for the first year, and (&) an ordinary whole life policy were 
then written when the man reaches age 31. Find the net premium (a) 
for the first year, and (&) for the subsequent years. 

NOTE. Such a policy is very common and is said to be written on tho 
1-year term plan. The advantages from an insurance company's standpoint 
are apparent after reading the next chapter. 

10. A certain endowment policy for $1000 taken at age 23 provides 
that the net premium for the 1st year is that for 1-year torm insurance 
and that the net premiums for the remaining 19 years are those for a 19- 
year endowment policy for $1000, taken at age 24. Find the net annual 
premium (a) for the first year ; (&) for subsequent years. This policy is 
.another example of the 1-year term plan. 

1 11. How much insurance on the 20-payment life plan can a man aged 
32 purchase for a net annual premium of $75 ? It is advisable, first, to 
find the equivalent net single premium. 

12. How large a 20-year endowment policy can a man aged 23 pur- 
chase for net annual premiums of $100? 

13. (a) For a boy aged 16, find the not annual premium for a $1000 
endowment policy, which matures at age 85. (/;) Find the- not annual 
premium for an ordinary life policy for $1000 tulam at ago l(i. (e) Ex- 
plain the small difference between the results. 

14. A man aged 30 takes out a policy which provides him with $10,000 
insurance for the first 10 years, $8000 for the next 10 yours, and $5000 
for the remainder of his life. He is to pay premiums annually for 10 
years. Find the net annual premium. 



LIFE INSURANCE 175 

15. A certain policy on maturing at age 55 offers the option of a pure 
endowment of $2000, or an equivalent amount of paid up whole life in- 
surance, that is, as much insurance as the $2000, considered as a net single 
premium, will buy. Find the amount of paid up insurance. 

NOTE ON GHOSH PHRMIUMB. Premiums previously discussed were net 
premiums, or present values of the benefits to be paid under the policy. In 
conducting on insurance company there is expense due to the salaries paid 
to administrative officials, the commissions paid to agents for obtaining new 
policyholders, the expense of the medical examination of policyholders, book- 
keeping expense, etc. To provide for these items and for unforeseen con- 
tingencies, it is necessary for the company to add to the net premium an 
amount called the loading. The not premium plus the loading is the gross or 
actual premium paid by the policyholdor. Sometimes the loading is de- 
termined as a certain percentage of the not premium plus a constant charge 
independent of the nature of the policy. Sometimes the loading may be 
determined simply as n percentage of the net premium, the percentage either 
being independent of the policyholder's age, or varying with it. Each com- 
pany uses its own method for loading, but the resulting gross premiums of all 
large, well-managed companies are essentially the same. 

SUPPLEMENTARY MATERIAL 

73. Net single premiums as present values of expectations. 

A whole life policy on a life aged x promises only one payment, 
due at the end of the year in which (x) dies. However, we may 
think of the policy as promising payments at the end of each 
year up to the man's 96th birthday, the payment at each date 
being contingent on his death during the preceding year. Then, 
the method used in deriving formulas for life annuities may be 
used to obtain the present value, or net single premium, for the 
policy. 

Consider obtaining the net single premium A a for a whole life 
insurance of $1 on the life of (*). At the end of 1 year, $1 will be 
paid if (*) dies during the preceding year. The probability of 

(x) dying in this year is ~^j from formula 10 with S 1, the 

IK 

present value of the expectation of this payment is ~ (1 + i)~ l 

l a 

or ^2. The other present values below are verified similarly, 

la 



176 



MATHEMATICS OF INVESTMENT 



PAYMENT OF SI 
WILL BE MADE 
AT END OF 


IF MAN DEBS BETWEEN AGES 


PROBABILITY 
OF PAYMENT 
BEING MADE 


PRESENT VAT/CH 
OB- THE PAYMENT 


lyr. 


x and (x + 1) 


d, 
I, 


vd x 

I, 


2yr. 
96 - x yr. 


(a; + 1) and (x + 2) 
95 and 06 


"Ir 


I, 


i x 


I, 



The expression obtained for A x on adding the present values in 
the last column is the same as previously obtained in formula 36. 

NOTE. From the present point of view, an insurance company may be 
likened to a gambler who plays against all of the beneficiaries of the policies. 
The net single premiums are the present values of the expectations of the 
beneficiaries. So many players are involved as opponents of the company 
that the probabilities of winning and of losing as given by the mortality table 
will be practically certain to operate. Hence, the company will neither lose 
nor win in the long run. 

EXERCISE L33X 

1. By the method which was used above to obtain A x , find the expres- 
sion for the present value of a 10-year term insurance policy for $1000 
on a life aged 20. 

2. By the method above, derive the formula 40 for A 1 ^. 

74. Policies of irregular type. Equation 47 enables us to 
find the premiums for any policy for which the present value of 
the benefits is known. 

Example 1. A policy written for a man aged 32 promises the follow- 
ing benefits : (a) Term insurance for $5000 for 28 years ; (ft) a life annuity 
of $1000 paid annually, first payment due at age 60. It is agreed that 
premiums shall be paid annually for 28 years. Find P, the net annual 
premium. 

Solution. The present value of benefit (a) is 6000(A3 33ffl ) ; benefit 
(b) is a life annuity, term deferred JJ7 years, whose present value is 1000(jff|aa). 



LIFE INSURANCE 



177 



The annual premiums form a 28-year temporary annuity due, whose present 
value is P-,). Hence, from equation 47, 



Aa 

1000 N 60 + 500Q(M a - 



^) = $234.43. 



(Formulas 35, 42, 28) 
(Table XIV) 



EXERCISE LXX 

Find the periodic premium payment for each policy described. The 
age of policyJwlder is the age at the time the policy was written. 



Pnon. 


BBXEFITS OP POLICY 


A an OF 
POLIOY- 

HOLDBB 


METHOD OP 
PATINO 
PREMIUMS 


1. 


(a) 10-year term insurance for $1000. 
(6) A pure endowment of $2000 at the end 
of 10 years. 


27 


10 annual 
premiums 


2. 


(a) Term insurance of $10,000 for 20 years. 
(6) Life annuity of $1000 paid annually, 
first payment at age 06. 


46 


20 annual 
premiums 


3. 


Life annuity of $1000 paid annually, first pay- 
ment at age 65. 


30 


25 annual 
premiums 


4. 


Life annuity of $1000 paid annually, first pay- 
ment at age 70. 


45 


10 annual 
premiums 


6. 


(a) Term insurance of $10,000 for first 10 
years, 
(b) Term insurance for $6000 for next 20 years, 
(c) Life annuity of $2000, paid annually with 
first payment at age 65. 


45 


20 annual 
premiums 



6. In what way does the policy in problem 1 differ from a 10-year 
endowment policy? 

NOTE. The policy of problem 4 is called an annuity policy and is a familiar 
form for those wishing protection in thoir old age. This same feature of 
protection is present in the policy of problem 2. 



CHAPTER X 
POLICY RESERVES 

75. Policy reserve. At age 30, the natural premium for $1000 
insurance, that is, the net single premium for 1-year term insur- 
ance for $1000, is found to be $8.14. This is the sum which each 
of Zao men of age 30 should pay in order to provide benefits of 
$1000 in the case of all of the group who will die within 1 year. 
The $8.14 premium is the actual expense of an insurance company 
in insuring a man aged 30 for $1000 for 1 year. The expense of 
insurance for 1 year increases continually during life, after an early 
age, being $17.94 per $1000 insurance at age 55 and $139.58 at 
age 80. 

Consider a man aged 30 who takes out a $1000, ordinary life 
policy. Throughout life he pays a net level premium (that is, a 
constant premium) of $17.19, as obtained from formula 48, and is 
insured for $1000 all during life. The expense of the company in 
insuring him during the first year is the natural premium, $8.14. 
Hence, in the first year the man pays (17.19 8.14) = $9.05 
more than the expense. The insurance company may be consid- 
ered to place this unused $9.05 in a reserve fund which will accu- 
mulate at interest for future needs. Up to age 54, each annual 
premium of $17.19 is more than the expense of insurance and the 
company places the excess over expense in the reserve fund. At 
age 55 the $17.19 premium is less than the insurance expense, 
which is $17.94. The deficiency, (17.94 - 17.19) = $.75, IB taken 
from the reserve fund. From then on until the end of life, the 
expense of insurance is met more and more largely from tho re- 
serve fund. Thus, at age 80, the expense is $130,58 (tho natural 
premium, as given above) so that (139.58 - 17.19) - $122,39 
comes from the reserve. 

For every insurance policy (except a 1-year term policy) where 
a net level premium P is paid, tho annual expense of insurance 

178 



POLICY RESERVES 



179 



during the early policy years is less than the premium P. Hence, 
the insurance company should place the unused parts of the pre- 
miums in a reserve fund and accumulate it at interest to answer 
the future needs of the policy. When the expense of insurance, 
in later years, becomes greater than the level premium, the defi- 
ciency is made up by contributions from the reserve fund. The 
reserve funds should be regarded as a possession, of the policy- 
holders, merely held and invested by the insurance company. 

The reserve on a policy at the end of any policy year, before the 
next premium due is paid, is called the terminal reserve for that 
year. In this chapter we consider the determination of the ter- 
minal reserve for a given year. 

Example 1. Form a, table showing the terminal reserves for the first 
6 years for a 5-payment life policy for $1000 written at age 40. 

Solution. From formula 49, the not annual premium is $89.4674. Assume 
that the company issues the same policy to each of Z*o = 78,106 men. The 
following table shows the disposition of the funds received as premiums. 



POUOT 

YHAB 


PREMIUMS PAID 
AT START off 
YEAH 


RBsHirvm FUND 
AT STAHT OF 
YBAB 


ACCOM. FUND 
AT 34%, AT END 
on 1 YEAH 


RBBIJIIVII FUND 
AVTEB DEATH 
RMNHPITS AT 
END ov YEAH 


RBSHRVH 
ran SUB- 
vivon AT 
END aw 
YBAII 


1 


$6,987,100 


$ 6,987,160 


$ 7,231,710 


S 6,466,710 


$ 84 


2 


0,918,725 


18,385,435 


13,848,925 


13,074,925 


171 


3 


6,849,485 


19,924,410 


20,021,764 


19,836,704 


262 


4 


6,779,201 


20,01(5,025 


27,547,586 


26,750,586 


357 


5 


6,707,903 


33,458,549 


34,029,508 


33,817,598 


456 








33,817,608 


35,001,224 


34,173,224 


466 



For example, at tho beginning of the first year, (78, 106) (89.4674) is received in 
premiums. At tho and of the your, interest at 3J% added to the premium 
fund gives $7,231,710. During tho your, (k 70S deaths ooeurred so that 
$706,000 is payable to beneficiaries, leaving $6,460,710. There are Zi = 77,341 
survivors; the total fund $0,400,710, divided by 77,341, gives $84 as tho 
share, or reserve, per policy. At the beginning of the 2d year, 77,341 men pay 
premiums, etc. After tho 6th year, no more premiums will be received, HO that 
all death benefits in the future come from the fund, $33,817,698, on hand at 
tho end of 5 years, and it future accumulations at interest, 



180 MATHEMATICS OF INVESTMENT 

EXERCISE LXXI 

1. Assume that an insurance company issues $1000 ordinary life poli- 
cies to each, of ^ men of age 92. Compute a table showing the disposi- 
tion of funds received as premiums and the total reserve per policy at the 
end of each year. 

76. Remaining benefits of a policy; computation of the re- 
serve. At any time after a policy is written, the remaining 
benefits of a policy are the promised payments of the policy as 
they affect the policyholder at his attained age. 

Example 1. A certain insurance policy, written for a man aged 32, 
promises Kim (a) temporary life insurance for $1000 for 25 years ; (&) a 
pure endowment of $1000 payable at the end of 25 years ; (c) a lif e annuity 
of $1000 payable annually, first payment at age 60. (1) Describe the re- 
maining benefits, 8 years later. (2) Find the present value of the re- 
maining benefits, 8 years later. 

Solution. (1) Eight years later, the attained age of the man is 40 years. 
The policy promises (a) term insurance for $1000 for 17 years to a man aged 
40 ; (&) a pure endowment of $1000 payable at the end of 17 years to a man now 
aged 40 ; (c) a deferred life annuity, for a man aged 40, of $1000 paid annually. 
Since the first payment of the annuity is due at age 60, which is 20 years later, 
the annuity is deferred 19 years. 

(2) The present value of the remaining benefits at age 40 is the sum of the 
present values or net single premiums for the three individual benefits or 



1000 A + lOOOiT^o + 1000i.]o4o, 
which can be computed by use of the proper formulas. 

Consider the conditions in regard to a policy, written for a man 
aged x, n years after the policy date. The attained age of the 
policyholder is x + n, and the reserve fund for the policy contains 
a certain amount $"7, the terminal reserve at the end of n years. 
The company is liable for the remaining benefits of the policy, and 
the policyholder is liable for the future premiums, Since all 
future benefits must be. paid from the reserve and from the future 
Dremiums, the following equation is satisfied : 

-ingle premium for \ /Pr.val.atagex + n\ /Terminal \ 
. val. of) remaining ) = (' of .net premiums J-H reserve at ) (52) 
is at age x + n / \due in the future/ \age x + n' 



POLICY RESERVES 181 

To find the terminal reserve on a policy, first find the net annual 
premium and then use equation 52. 

Example 2. Find the terminal reserve at the end of 6 years on a 20- 
year endowment policy for $1000 written at age 24. 

Solution. (a) The net annual premium is 1000 (P z ^\) = $39.085, from 
formula 51. (&) The remaining benefits at the attained age of 30 years are 
a pure endowment of $1000 payable at the end of 14 years to a man now aged 
30 and term insurance for $1000 for 14 years on a life aged 30 ; in other words, 
the benefits form a 14-year endowment insurance for $1000 for a man aged 30. 
The remaining premiums form a 14-year temporary annuity due. Let V be the 
reserve at the end of 6 years. From equation 62, 



1000(Jfao -M + Z>) _ 39.086(^0 - AT). (Formulas 46, 35) 

Dao D$o 

v 1000(Jf 30 - M u + Pit) - 39.Q85(J\r 30 - Jfa) 

Pn 
V = 66 - $217.9. (Table XIV) 



The method used in Example 2 may be applied in the case of 
any standard poli cy to obtain a general formula for the reserve at 
the end of a given number of years. For example, consider an 
ordinary life policy for $1 written for a man aged x. Let n V a 
represent the terminal reserve at the end of n years. The net 

annual premium is P a = ^- ? - The remaining benefit at the at- 

rf x 

tained age (x + n) is whole life insurance for $1 for a man aged 
(x + n). The remaining premiums form a whole life annuity 
due of P m payable annually by a man now aged (x + ri). From 
equation 52, n y a = Axn 




NOTB 1. For the advantage of the insurance actuary, who has occasion to 
compute the reserves on numerous policies, it is advisable to develop general 
formulas and convenient numerical methods for the computation of reserves. 
In the case of a student meeting the subject for the first time, it is more im- 
portant to appreciate thoroughly the truth of equation 62. Such appreciation 
is attained only by direct application of the equation. The problems of 
Exorcise LXXII below should be solved by direct application of equation 52, 
as was done in illustrative Example 2 above, 



MATHEMATICS OF INVESTMENT 

EXERCISE LXXn 1 

J l. If the net single premium for the remaining benefits of a policy is 
$745, and if the present value of the future premiums is $530, what is the 

reserve? 

i 

~i 2. At an attained age of 42, the net single premium for the remaining 
benefits of a policy is $750. There are six annual premiums of $50 remain- 
ing to be paid, the first due immediately. Find the policy reserve. 

3. At the attained age of 44, the reserve on a certain policy is $500. 
Annual premiums of $25, the first due immediately, must be paid for the 
remainder of life. Find the present value of the remaining policy benefits. 
" 4. A $1000, 10-payment life policy is written at age 34. (a) Find the 
reserve on the policy at the end of 6 years. (6) Find the reserve at the 
end of 10 years. 

6. A $1000, 5-payment life policy is written at age 40. (a) Take the 
premium, as computed in illustrative Example 1, Section 75; compute 
the reserve at the end of 3 years and compare with the result given in the 
table of that example. (6) Find the terminal reserve at the end of 5 years 
and compare with the table. 

6. A $2000, 20-year endowment policy is written at age 33. (a) Find 
the terminal reserve at the end of 15 years. (&) What is the terminal 
reserve at the end of 20 years, before the endowment is paid? 

7. In the case of a 1-year term policy, why is the reserve zero at the 
end of the year? 

i 8. An ordinary life policy for $5000 is written at age 25. Find the 
terminal reserve at the end of 15 years. 

9. Derive a formula for the terminal reserve at the end of n years for 
an w-payment life policy written at age x. (6) Derive a formula for the 
reserve at the end of m years, where m is greater than n. 

10. Find the reserve at the end of 5 years for a 10-year term policy for 
$10,000 written at age 35. 

11. (a) Find the reserve at the, end of 5 years for an ordinary life policy 
for $10,000 written at age 35. (&) Compare your answer with that in 
problem 10 and give a brief explanation of the difference. 

12. A man aged 25 pays the net single premium, for a 10-year term in- 

rance for $1000. What is the policy reserve, 5 years later? 
L3. A man aged 30 pays the net single premium for a whole life insur- 
ance for $1000. Ten years later, what is the policy reserve? 

1 After the completion of Exorcise LXXII, tho student may proceed immediately 
to the Miscellaneous Problems at tho end of tho chapter. 



POLICY RESERVES 183 

14. Derive a general formula, as in equation 53, for the reserve at the 
end of m years for an n-year endowment policy for $1 written at age x. 

NOTE 2. The method for computing reserves, furnished by equation 52, 
is called the prospective method because the future history of the policy is the 
basis for the equation. Retrospective methods also are used. 

NOTE 3. Insurance companies are subject to legal regulation. It is 
usually specified by state law that, at periodic times, an insurance company 
must show net assets equal to the sum of the reserves for all of its outstanding 
policies. The law specifies a standard mortality table and interest rate to be 
used. A company is insolvent if it cannot show net assets equal to the neces- 
sary reserve. It is likewise recognized by law that a company's reserve be- 
longs to its policyholders as a whole. Hence, the reserve on a policy is the basis 
for its cash surrender value, the amount which the company must pay to a 
policyholder if he decides to withdraw from the company and surrender his 
policy. The cash surrender value equals the reserve, minus a surrender charge. 
The surrender charge in most states is specified by law and may be considered 
as a charge by the insurance company on account of the expense entailed in 
finding a new policyholder to take the place of the one withdrawing. This 
charge is legitimate because the theoretical reserve was computed by the 
company on the assumption that it had so many policyholders that the laws of 
averages, as dealt with in using the mortality table, would hold. Hence, the 
number of policyholders must be maintained and any one withdrawing should 
pay for the expense of obtaining a new policyholder in his place. 

NOTE 4. It should be recognized that the discussion in the preceding three 
chapters is merely an introduction to the subject of life annuities and of life 
insurance. We have not considered joint life, or survivorship annuities and 
insurance. Moreover, the subject of reserves requires a thorough treatment, 
beyond what we have given, from both the theoretical and the computational 
standpoint. The surplus of a company, its manner of declaring dividends to 
policyholders, and many other practical questions connected with the account- 
ing and business methods of insurance companies have not even been men- 
tioned. The student who wishes to pursue the subject farther is referred to the 
Text Book of the Institute of Actuaries, and to the courses of study described 
by the Educational Committees of the Actuarial Society of America and of the 
American Institute of Actuaries, 

SUPPLEMENTARY EXERCISE LXXni 

Students working the problems below should have previously completed 
Supplementary Section 74 of Chapter IX. 

1. A policy written for a person aged 38 promises whole life insur- 
ance for $10,000, and a life annuity of $1000 payable annually, with the 



184 MATHEMATICS OF INVESTMENT 

first payment at age 65. Premiums are payable annually for 20 years, 
(a) Find the reserve at the end of 10 years. (6) Find the reserve at the 
end of 20 years. 

2. A policy written at ago 27 promises $1000 term insurance for 20 
years and a pure endowment of $5000 at the end of 20 years. Premiums 
are payable annually for 10 years. Find the reserve at the end of G 
years. 

3. A policy written at age 15 promises 20-year endowment insurance 
for $1000, and the premiums are payable annually for 10 years. () De- 
termine the reserve at the end of 10 years. (6) Determine the reserve 
at the end of 9 years. 

. 4. A certain pure annuity policy written at age 40 promises a life 
annuity of $1000 with the first payment at age 61, The premiums are 
payable annually for 21 years. Find the reserve (a) at the ond of 5 years ; 
(&) at the end of 20 years. 

NOTE, When a corporation or association promises a pension to a person, 
its act is equivalent to writing a pure annuity policy for the person involved. 
Hence, a pension association should be considered solvent only whon its reserve 
fund is equal to the sum of the reserves on each of its pension contracts. As 
judged by this standard, there are an unfortunately largo numbor of insolvent 
pension associations in operation. Their insolvency does not become apparent 
until after they have been operating long enough so that the theoretical reserve 
(which they do not possess) becomes necessary in order to moot liabilities 
falling due. 

5. A group of workers of the same age entered a pension association 
which promised $500 annual payments for life, starting with payments at 
age 61. At ago 55, 10,000 workers remain alivo. They aro required to 
pay $50 at the beginning of each year up to and including their 00th 
birthdays. How much should tho asHociation have on hand as a reserve 
before the $50 payments due at ago 55 have boon made? 

MISCELLANEOUS PROBLEMS ON INSURANCE* 

1. "Write a sample of each of tho following tyjxis of insurance policies, 
stating the age of the policyholder, tho bencfitH ho will reooivo, and how 
he is required to pay premiums : (a) 20-payment life ; (b) 10-year endow- 
ment; (c) ordinary whole life ; (d) 10-year term, 

1 Insurance companies mentioned in those problems aro aeaumed to operate under 
assumptions (a), (b), and (o) of Section, 68, 



POLICY RESERVES 185 

2. (a) A man aged 47 desires to set aside a sufficient sum which he can 
invest at 5%, effective, to pay him an annual income of $1000 for 10 years, 
starting with a payment on his 61st birthday. Find the amount set aside, 
assuming that ke will certainly live to age 70. (6) At age 47 what would he 
have to pay to an insurance company for a contract to, pay him $1000 at 
the end of each year for life, with the first payment at age 61, with the 
understanding that the company would compute the charge in accordance 
with the principles of scientific life insurance, at 3^%? 

3. A woman offers $3000 to a benevolent organization on condition 
that the organization pay her 5% interest thereon at the end of each year 
for life. If the organization can purchase the required annuity for her 
from an insurance company, which uses the rate 3|%, will it pay to accept 
her offer if she is 55 years old ? 

4. According to a will, a trust fund of $200,000 will go to a charity at 
the death of a girl who is now aged 19, and she is to receive the income at 
4% for the remainder of her life. On a 3|% basis, find the present value of 
(a) her inheritance and of (6) the bequest to the charity. 

5. A man borrows $200,000, on which he pays 5% interest annually. 
The principal is due at the end of 8 years. To protect his creditor he is 
compelled to take out an 8-year term insurance policy for $200,000. 
Assume that the man will certainly live to the end of 8 years, and find 
the present value at 6%, effective, of all of his payments on account of the 
debt, assuming that he pays merely the net premiums for his insurance 
as computed by a company which uses the rate 3i%. His age is 40 years. 

6. A man aged 35 pays the net single premium on a whole life insurance 
for $1000. What is the policy reserve 10 years later? 

7. A man aged 30 took out a 10-payment life policy. At the end of 
10 years he desires to convert it into a 20-year endowment insurance as 
of that date. How much paid up endowment insurance will he obtain 
if the company permits all of his reserve to be used for that purpose? 
Notice that his reserve is the net single premium for the new insurance. 

8. (a) Find the net annual premium at age 43 for an ordinary life policy 
for $2000. (6) Suppose that the man is alive at the end of 25 years. 
Find the reserve on his policy and compare it with the sum he would have 
on hand if he had invested all of his annual premiums at 5%, effective, 

9. A man aged 42 borrows $100,000 and agrees to pay 4% interest 
annually. He agrees to provide for the payment of the principal at his 
death, or at the end of 10 years if he lives, by taking out a 10-year endow- 
ment policy for $100,000, with the creditor as beneficiary. The debtor 



186 MATHEMATICS OF INVESTMENT 

considers his future payments, assuming (1) that he will pay merely 
the net premiums at 3% for his policy, (2) that he will certainly live to 
the end of 10 years, and (3) that he is able to invest his money at 7%, 
effective. He asks if it would pay to borrow $100,000 elsewhere at 6%, 
payable annually, with the agreement that the principal may be re- 
paid at the end of 10 years through the accumulation of a sinking fund, 
(a) Which method is best? (6) In terms of present values, how much 
could the debtor save by selecting the best method? 

10. Compare the net single premiums for whole life insurance for 
$1000 (a) at ages 25 and 26; (6) at ages 75 and 76. (c) For which pair 
is the change in cost greatest? 



PART III AUXILIARY SUBJECTS 
CHAPTER XI 
LOGARITHMS 

77. Definition of logarithms. Logarithms are exponents. 
The logarithm of a number N with respect to a base a, where a 
is > 0, 7* 1, is the exponent of the power to which a must bo 
raised to obtain N. That is, by definition, if 

#*-*, (1) 

then, the logarithm of N with respect to the base a is re ; or, in 
abbreviated form, l oga N = x. (2) 

Thus, since 49 = 7 2 / then Iog 7 49 = 2 ; since 1000 = 10 3 , then 
logio 1000 = 3. Also, if logs 'N = 2, then, from equation 1, 
N = 5 2 ; if Iog 6 N = 4, then N = 6 4 = 1296. 

In the future, whenever we talk of the logarithm of a number 
we shall be referring to a positive number N. This is necessary 
because, in the definition of a logarithm, the base a is positive, 
and hence only positive numbers N have logarithms as long as 
the x in equation 1 is a real number. 

EXERCISE LXXIV 

1. Since 3 a - 9, what is logs 9? 

2. Since 5 4 = 625, what is logG 625? 

3. Since 100 10 2 , what is logm 100? 

4. Since 2 3 8, what is log* 8? 

6. Since 10 = 1, what is Iog 10 l? Since 17 1, what is logn 1? 
Since every number to the power zero is 1, what is the logarithm of 1 
with respect to every base ; that is, since a = 1, what is loga 1 ? 

6. What is Ipg 8 36? 8. What is log a 16? 10. .What is Iog 6 25? 

7. What is logio 10,000? 9. What is Iog 7 7? 11. What is log a? 

187 



188 MATHEMATICS OF INVESTMENT 



/12. If Iog4# = 2, findiV. 19. Find log 10 1. 

13. If logs N = 4, find N. 20. Find log lfl 4. 

14. H logic N = 5, find N. 21. Find logioo 10. 

15. If lo&W = i, find N. 22. If 10 1 - 6 = 31 .62, find Iog 10 31.62. 

16. If lo&N = i find N. 23. If 10- 009 = 5, find logio 5. 

17. If loga# = 6.5, find N. 24. If 10 2 - 4814 = 303, find Iog 10 303. 

18. Find Iog 9 81. 25. If 10- 4771 = 3, find log w 3. 

Express in another way the fact that : 

26. Iogio86.6 = 1.9370. 29. Iog 10 4730 = 3.6749. 

27. logio 684 = 2.8351. 30. 343 = 7 3 . 

28. logio 6.6 = .8195. 8 1. V = 1.732. 

32. If N = i, find log* N. HINT. i = (4)" 1 . 

33. Iftf = 4,findlog 2 tf. HINT. ^ - 3 2" 1 . 

lo lo A 

34. If N = .1, find logio JV. HINT. .1 = A- 

36. Find logio .001. Find logio .00001/: Find logio .0000001. 

78. Properties of logarithms. Logarithms have properties 
which make them valuable tools for simplifying arithmetical 
computation. 

Property I. [The logarithm of., the product of two numbers M 
and N is equal to the sum of the logarithms of M and N : 

log tt AflV = log* Jf + log, tf. (3) 

Proof. Let logo M = x and logo N ** y. Then, 

since logo-W = x, then M = a*, (Def. of logarithms) 

and since logo AT = y, then N = a". (Def. of logarithms) 

Hence, MN = o a a = a a+v . (Law of oxpozionte) 

Since . MN=a a+v , thenlog a MN^x+y. (Def . of logarithms) 

Hence, logo MN log a M + logo N. (Subst. x = log M',y** logo ^V) 

Property n. The logarithm of the quotient of two numbers, M 
divided by N, is equal to the logarithm of the numerator minus the 
logarithm of the denominator : 

lOga j - lOga M- loga N. (4) 

Proof. Let logo M = x and logo N = y. * Then, 
since log a M = x, then M = a", (Def. of logarithms) 

and since log a N = y, then N = a". (Def. of logarithms) 



LOGARITHMS 189 

TUf n* 

Hence, 41 = <L = a*-//. (La W O f exponents) 

Since = a""", then logo ^ = x y. (Def. of logarithms) 

Hence, logo ^ = Iog M logo N. (Subst. x = logo M;y^ Iog 2V) 

Property HE. The logarithm of a number N, raised to a .power 
Jc, is k times the logarithm of N: 

lOga-JV* = felOgaW. (5) 

Proof. Let logo N = x; then N = a*, by the definition of logarithms. 

Hence, . N k = (a") 6 = a* x . (Law of exponents) 

Since N h a* 1 , then logo N h = kx. (Def. of logarithms) 

Hence, logo N h = k logo N. (Subst. x = logo N) 

NOTE. In the future we shall deal entirely with logarithms to the base 10. 
Hence; for convenience, instead of writing logio N we shall write merely log N, 
understanding that the base always is 10. Logarithms to the base 10 are called 
Common Logarithms ; the name Briggs' logarithms is also used, in honor of an 
Englishman named Henry Briggs (1556-1630), who computed the first table 
of Common Logarithms. 

Example 1. Given that: log 2 = .3010, log 5 = .6990, log 17 = 
1.2305,_fuid the logarithms of each of the following numbers: 34, 85, 
, Vl7, 25. 

Solution. log 34 - log 2(17) - log 2 + log 17 - .3010 + 1.2305= 1.5315. 
log 85 = log 5(17) - log 5 + log 17= .6990 + 1.2306 = 2.2295. (Prop. I) 
log jr. = i og 17 _ fog 5 =* 1.2305 - .6990 = .5315. (Prop. II) 

log Vl7 - log 17* = i log 17 = J(1.2305) = .61525. (Prop. Ill with k = *) 
log 25 - log 6 = 2 log 5 = 2(.6990) - 1.3980. (Prop. Ill with k - 2) 

EXERCISE LXXV 

In the problems below find the logarithms of the given numbers, given 
that : 

log 2 = .3010 log 3 = .4771 ' log 5 = .6990 

log 7 - .8451 log 11 - 1.0414 log 13 = 1.1139 

log 17 - 1.2305 log 23 1.3617 log 29 = 1.4624 

1. 6 2, 9 3. 46 4. 61 B. -^ 6. A 

7. 20 r 8. V 9. -^5 10. V7 11. 49 12. 16 

13. 5 14. ^17 IB. Jf 16. 50 17. 55 18. 154 

19. 20. U 21. 10 22. 100 23, 1000 24. 10,000 



190 



'MATHEMATICS OF INVESTMENT 



26. 230 26. 2300 27. 23,000 28. 230,000 29. .1 = A 30. .01 = ^ 
31. .001 32. .0001 33. .5 = A 34. .05 36. .005 36. .0005 

79. Common logarithms. If one number N = 10* is larger 
than another number M = 10", then x must be larger than y. 
Since x = log N and y = log M, it follows that, if N is larger 
than M, then log N is larger than log M . Thus, since 9 is larger 
than 7, log 9 must be larger than log 7. 

The table below gives the logarithms of certain powers of 10. 



SINOB: 


THHN: 


10000 


= 10* 


log 10000 = 4 


1000 


= 10" 


log 1000 = 3 


100 


= 10* 


log 100 = 2 


10 


= 10 1 


log 10 = 1 


.1 


= A = lo- 1 


log .1 = - 1 


.01 


- T*IF = 10-* 


log .01 = - 2 


.001 


- T*W = lo- 3 


log .001 = - 3 



Consider the number 7, or any other number between 1 and 10. 
Since 7 is greater than 1 and less than 10, log 7 is greater than log 
1, which is 0, and is less than log 10, which is 1. That is, since 7 
is between 1 and 10, log 7 lies between and 1. Hence, log 7 = 
+ (a proper fraction). From a table of logarithms, as described 
later, log 7 = .84510, approximately, so that the fraction men- 
tioned above is .84510. Similarly, since 750 is between 100 and 
1000; log 750 lies between 2 and 3 ; therefore, log 750 = 2 + (a 
proper fraction) ; since 5473 is between 1000 and 10,000, log 5473 
= 3 + (a proper fraction). In the same manner, since .15 lies 
between .1 and 1, log .15 lies between 1 and 0, and hence 1 
log .15 = 1 + (a proper fraction). In general, the logarithm 
of every positive number can be expressed as an integer) either positive 
or negative, plus a positive proper fraction. 

The integral part of a logarithm is called its characteristic. 
When a number N is greater than 1, the characteristic of log N 
is positive ; when N is less than 1, the characteristic of log N is 
negative, 

1 Any number between 1 and oan be expressed as 1 + (a proper fraction). 
Thus, - .67 - -* 1 + .43 ; - .88 - - 1 + .12, etc, 



LOGAEITHMS 191 

The fractional part of a logarithm is called its mantissa. 

Thus, given that log 700 = 2.84510, the characteristic of log 700 is 2, and 
the mantissa is .84510 ; given that log .27 = 1 -|- .43136, the characteristic 
of log ,27 is 1 and the mantissa is .43136. 

80. Properties of the mantissa and the characteristic. Given 
that log 3.8137 = .58134, then, by use of Properties I and II of 
Section 78, and from the logarithms of powers of 10 given in Sec- 
tion 79, we prove the following results : 

log 3813.7 = log 1000(3.8137) = log 1000+log 3.8137=3+.58134=3.68134. 
log 381.37 = log 100(3.8137) = log 100 +log 3.8137 =2 +.58134 =2.58134. 
log 38.137 = log 10(3.8137) = log 10 +log 3.8137 = 1 +.58134 = 1.58134. 
log 3.8137 = 0.58134. 



log .38137 = log = log 3.8137- log 10 - .68134- 1 = - 1+. 58134. 

log .038137 = log ^j|? = log 3.8137- log 100 = .58134- 2 = - 2 +.58134. 
log .0038137 = log = log 3.8137- log 1000 = .58134- 3 = - 3+.58134. 



NOTE. The characteristics of the logarithms above could have been 
obtained as in Section 79. Thus, since 3813.7 lies between 1000 and 10,000, 
log 3813.7 lies between 3 and 4 ; log 3813.7 = 3 + (a proper fraction). There- 
fore, the characteristic of log 3813.7 is 3, as found above. 

From inspection above, we see that .58134 is the mantissa of all 
of the logarithms. This result, which obviously would hold for 
any succession of digits as well as it does for the digits 3, 8, 1, 3, 7, 
may be summarized as follows : 

Rule 1. The mantissa of the logarithm of a number 2V de- 
pends only on the succession of digits in N. If two numbers have 
the same succession of digits, that is, if they differ only in the posi- 
tion of the decimal point, their logarithms have the same mantissa. 

The logarithms above also illustrate facts about the character- 
istic. 

Rule 2, The characteristic of the logarithm of a number 
greater than 1 is positive and is 1 less than the number of digits 
in the number to the left of the decimal point. 

NOTE. Thus, in accordance with Rule 2, 3 is the characteristic of log 
3813.7 j 2 is the characteristic of log 381.37, etc, Eule 2 is justified in general 



192 MATHEMATICS OF INVESTMENT 

by recognizing that, if a number N has (k + 1) digits to the left of the decimal 
point, then N is between 10* and lO* 41 ; hence log N is between k and (k + 1) 
and log N = k + (a proper fraction). That is, k is the characteristic of log N. 

Rule 3. If a number N is less than 1, the characteristic of 
log AT is a negative integer ; if the first significant figure of N 
appears in the fcth decimal place, then the characteristic of log N 
is ft. 

Thus, the first significant figure of .38137 is 3 and appears in the first decimal 
place, and, in accordance with Rule 3, the characteristic of log .38137 is 1. 
The first significant figure of .038137 appears in the 2d decimal place, while 
the characteristic of log .038137 is - 2, etc. 

NOTE. It may appear strange to the student that we write, for example, 
log .0038137 = 3 + .58134, instead of performing the subtraction. For 
every number N which is less than 1, log N is a negative number ; thus, log 
.0038137 = 3 + .58134 = - 2.41866. Written in this way, the mantissa 
.58134 and the characteristic 3 are lost sight of. We write the logarithm in 
the form 3 + .58134 to keep the characteristic and the mantissa in a 
prominent position. 

Since the mantissa depends merely on the succession of digits 
in the number, it is customary. to speak of a mantissa as corre- 
sponding to a given succession of digits without thinking of any 
decimal point being associated with the digits. Thus, above, we 
would say that the mantissa for the digits 38137 is .58134. 

Example 1. Given that the mantissa for the digits 5843 is .76664, 
find log 5843 ; log 584.3 ; log 58,430,000; log 5.843 ; log .0005843. 

Solution. The characteristic of log 6843 is 3 ; hence, log 5843 = 3.76664. 
Similarly, log 584.3 =2.76664; log 58,430,000=7.76664; log 6.843 - 0.76664 ; 
log .0005843 = - 4 + .76664. 

EXERCISE LXXVI 

1. Given that .75101 is the mantissa for. the digits 56365, find log 
5636.5; log 56365; log 563.65; log 56,365,000 ; log .0056365; log .56365. 

2. Given that .93046 is the mantissa for 85204, find log 85.204 ; log 
852,040,000; log 8.5204; log "85204; log .085204; log .0000085204, ' 

3. Given that .39863 is the mantissa for 2504, find log 2504 ; ' log 2.504 ; 
log 25,040; log .2504; log .00000000002504. 

4. Given that log 273.7 = 2.43727, find log 2.737; log 27.37; 
log 27,370; log .02737; log .002737. Moke use of Rule 1. 



LOGARITHMS 



193 



5. Given that log 68,025 = 4.83267, find log 68.025; log 6.8025; 
log .68025 ; log 6802.5 ; log .00068025. 

6. What is the mantissa of log 1 ; of log 10; of log 10,000; of log .1 ; 
of log .00001? 

81. Tables of mantissas. The mantissa for a given succession 
of digits can be computed by the methods of advanced mathematics. 
The computed mantissas are then gathered in tables of logarithms 
which, more correctly, should be called tables of mantissas. Ex- 
cept in special cases, mantissas are infinite decimal fractions. 
Thus the mantissa for 10705. is .02958667163045713486 to 20 deci- 
mal places. In a 5-place table of logarithms, this mantissa would 
be recorded correct to 5 decimal places, giving .02959. In an 8- 
place table, it would be recorded as .02958667, correct to 8 deci- 
mal places. 

NOTB. Table I in this book is a 5-place table of logarithms. A decimal 
point is understood hi front of each tabulated mantissa. To find the mantissa 
for N = 3553, for example, go to the sixth page of Table I. Find the digits 
355 in column headed N ; the mantissa for 3553 is entered in the corresponding 
row under the column headed 3. The entry is " 060," but the first two digits 
of tlie mantissa are understood to be " 65," the same as for the first entry in 
the row. Thus, the mantissa for 3553 is .55060. From Table I the student 
should now verify that : 



FOR THE DIGITS BHLOW 


THE MANTISSA is 


3630 


.55991 


3947 


.59627 


4589 


.66172 


9331 


.96993 


9332 


.96997 


9333 1 


.97002 



Example 1. Find log 38570 ; log .008432. 

Solution. By inspection, the characteristic of log 38570 is 4 ; the mantissa 
as found in Table I is .58625. Hence, log 38570 = 4.58626. The character- 
istic of log .008432 is - 3; log .008432 = - 3 + .92593. 

1 In Table I, for 9333, wo find the entry " *002. ' ' The asterisk (*) on the " 002 " 
means that tho first two digits are to be changed from 96, as at the beginning of 
the 1 row, to 97. 



194 



MATHEMATICS OF INVESTMENT' 



In order to obtain greater convenience in computation, it is 
customary to write negative characteristics in a different manner 
than heretofore. Thus, in log .008432 = - 3 + .92593, change 
the - 3 to (7 - 10). Then log .008432 = - 3 + .92593 = 
7 - 10 + .92593 = 7.92593 - 10. Recognize cle 4 arly that log 
.008432 = - 3 + .92593 = - 2.07407. We verify that 7.92593 - 10 
= 2.07407. The two ways introduced for writing log .008432 
are merely two different ways of writing the negative number 
2.07407, which is the actual logarithm involved. Similarly, log 
.8432 = - 1 + .92593 = 9.92593 - 10; log .000'000'000'ob8432 
= '- 12 + .92593 = 8 - 20 + .92593 = 8.92593 - 20, etc. 

NOTE. The change from the new form to the old or vice versa is oasy. 
Thus, given that log .05383 = 8.73102 - 10, we see that the characteristic is 
(8 - 10) or - 2; given that log .006849 = - 3 + .76708, then log .006849 
= 7.76708 - 10. 

EXERCISE LXXVH 

1. What are the characteristics of the following logarithms : 9.8542 10 ; 
7.7325 - 10; 6.5839 - 10; 4.3786 - 10? 

2. Write the following logarithms in the other form : 3 + .5678 ; 
- 5 + 7654; - 7 + .8724; - 1 + .9675. 

3. Write the following logarithms as pure negative numbers : 3 + 
.5674; - 1 + .7235; 9.7536 - 10; 7.2539 - 10. 

4. By use of Table I verify the logarithms given below : 



N 


Loo N 


N 


Loo# 


3616. 


3.54593 


35.88 


1.65486 


.01832 


8.26293 - 10 


1.170 


0.06819 


889,900 


5.94934 


.0008141 


0.91008 - 10 


.6761 


9.83001 - 10 


. 27,770 


4.44358 


621.8 


2.79365 


.00004788 


5.08015 - 10 



NOTE. When the characteristic of log 2V is 0, log N is equal to Us man- 
tissa. Thus, log 1.578 = 0.19811. Hence, a table of mantissas is a tiiblo of 
the actual logarithms of all numbers between 1 and 10. 

82. Logarithms of numbers with five significant figures. If a 

number N has five significant digits, log N cannot be read di- 
rectly from the table. We must use the process of interpolation 
as described in the following examples. 



LOGARITHMS 



195 



Exampk 1. Find log 25.637. 

Solution. The characteristic is 1. To find the mantissa, recognize that 
25.637 is between 25.630 and 25.640 ; the mantissas for 2563 and for 2564 were 
read from Table I and the logarithms of 25.630 and 25.640 are given in the 
table below. Since 25.637 is .7 of the way from 
25.630 toward 25.640, we assume l that log 25.637 
is .7 of the way from 1.40875 toward 1.40892. The 
total way, or difference, is .40892 - .40875 = 
.00017; .7 of the way is .7(.00017) = .000119. 
We reduce this to .00012, the nearest number of 
five decimal places. Hence, 



NUMB an 


LOGABrrHM 


25.630 
25.637 
25.640 


1.40875 

? ? 

1.40892 



log 25.637 = 1.40875 + .00012 = 1.40887. 

NOTE. At first, the student should do all interpolation in detail as in 
Example 1 above. Afterward, he should aim to gain speed by doing the 
arithmetic mentally. The small tables in the column in Table I headed PP, 
an abbreviation for proportional parts, are given to reduce the arithmetical 
work. 

Exampk 2. Find log .0017797. 

Solution. The characteristic is 3 or (7 10). The digits 17797 form 
a number between 17790 and 17800. The tabular difference between the corre- 
sponding "mantissas is (.25042 .25018) = .00024, 
or 24 units in the 6th decimal place. Since 17797 
is .7 of the way from 17790 to 17800, we wish .7(24). 
By multiplication, ,7(24) = 16.8. This should be 
found without multiplication from the small table 
headed 24 under the column PP. From this table 
we read .1(24) = 2.4, .2(24) - 4.8, etc., .7(24) = 



NUMBER 


MANTISSA 


17790 


.25018 


17797' 


? ? 


17800 


.25042 



16.8. Hence, the mantissa for 17797 is .25018 + .17 = .25035, and 
log .0017797 - 7.25035 - 10. 

NOTE. The following situation is sometimes met in interpolating. Sup- 
pose that .6(15) 7.5 is the part of the tabular difference which-we must add. 
Wo may, with equal justification, call 7.5 cither 7 or 8. As a definite rule in 
this book, whenever such' an ambiguity is met, we agree to choose the even 
number. Hence, we choose 8 above. Similarly, in using .7(15), or 10.5, we 
should call it 10, because we have a choice between 10 and 11. 

1 This assumption is justified by the first paragraph of Section 70, Since 25,037 
ia between 26.030 and 25.640, log 25,637 must be between log 25.030 and log 25.040. 
In interpolating as in Example 1, we merely go ono step farther than, tliis admitted 
fact when we assume that the change in the logarithm is proportional to the change 
in the number. This assumption, although not exactly true, is sufficiently accurate 
for all practical purposes. 



196 



MATHEMATICS OF INVESTMENT 



EXERCISE LXXVIH 

1. Verify the following logarithms : 

log 256.32 = 2.40878 log 8966.1 = 3.95211 

log 13.798 = 1.13982 log 931.42 = 2.96915 

log .073563 = 8,86666 - 10 log 33.581 = 1.52609 

log .59834 - 9.77695 - 10 log .00047178 = 6.67374 - 10 

log 1.1675 = 0.06725 log 676.93 = 2.83064 

2. Find the logarithms of the following numbers : 

18.156 .31463 .061931 151.11 

5321.7 83196 48.568 6319.1 

67.589 113.42 384.22 9.3393 

.031562 .92156 .52793 .000031579 

.009567 5.6319 1.1678 83.462 

83. To find the number when the logarithm is given. 

Example 1. Find N if log# = 7.67062 - 10. 

Solution. Since the characteristic is (7 10) = 3, the first significant 
figure of 2V will appear in the 3d decimal place ; N = .00 . . . . To find the 
digits of N, we must obtain the number whose mantissa is .67062. We 
search for this mantissa, or those nearest to it, in Table I ; we find .67062 as 
the mantissa of 4684. Hence, N = .004684. 

Example 2. Find N if log N = 5.41152. 

Solution. We wish the 6-figure number whose mantissa is .41152. On 
inspecting Table I we find the tabular mantissas .41145 and .41162 between 
which .41162 lies. The total way, or tabular difference, between .41145 and 
.41162 is .00017, or 17 units in the 5th decimal place. 
The partial difference .41152 - .41146 > .00007, or 
7 units hi the 6th decimal place. Hence, .41152 is 
fr of the way from .41145 to .41162. Wo then assume 
that the number x, whose mantissa is .41162, is -^ of 
the way from 25790 to 25800. The total way, or dif- 
ference, is 10 units in the 6th place; ^(10) 4.1; 
the nearest unit is 4. Hence, .41152 is the mantissa 
of 25790 + 4 - 25794. Since the characteristic of log N is 5, N = 267,940. 

Nora. The arithmetic in Example 2 above is simplified by use of the table 

headed 17 under the column of proportional parts. In Example 2 wo desire 

' />1 .0), which we can easily obtain if we know -fr oorroot to tho nearest tenth. 

m the table headed 17, we read .4(17) - 6.8, or ^ - .4; .6(17) - 8.6, 

P? - .5. Since 7 is between 6.8 and 8,5, ^ is between .4 and .5, but is 

rest to .4. Thus, ^(10) - 4, to the nearest unit. With practice, this 



NUMBEH 


MANTISSA 


25790, 

X 

25800 


.41145 
.41152 
.41162 



LOGARITHMS 107 

result should be obtained almost instantaneously. Thus, look under the table 
headed 17 for the number nearest to 7 ; we find 6.8 ; at the left it is shown tfiat 
this is A of 17 ; hence ^(10) = 4. 

EXERCISE LXXIX 

1. Find the numbers corresponding to the given logarithms and 
verify the answers given : 

log N = 3.21388; N = 1636.4. log N - 3.75097; N - 5636. 

log N = 8.40415 - 10; N = .02536. log N = 0.46839; AT = 2.9403. 

log N - 2.16931; N - 144.31. log N = 3.33590; N - 2167.2. 

log N = 9.52163 - 10; N - .33238. log N - 8.66267 - 10; N = .044944. 

log N =0.89651; N - 7.8797. log N - 0.36217; N - 2.2499. 

2. Find the numbers corresponding to the following logarithms : 

log N = 5.21631 log N - 3.19008 log N - 9.64397 - 10 

log N = 1.39876 log N - 7.56642 - 10 log N - 2.57938 - 10 

log N = 8.95321 - 10 . log N = 0.89577 log N = 1.77871 

log N = 4.32111 - 10 log N - L21352 log JV - 7.77853 

log N = 2.15678 log N - 8,45673 - 10 log AT = 3.15698 

84. Computation of products and of quotients. 

Exampk 1. Compute P - 787.97 X .0033238 X 14.431. 

Solution. From Property I of Section 78, log P is the sum of the logarithms 
of the factors. From Table I, 

log 787.97 =2.89661 
log .0033238 = 7.52163 - 10 
log 14.431 -1.15031 
(add) log P -11.57746 -10 = 1.67746 
. From Table I, P - 37.796. 



Solution. From Property II of Section 78, log Q equals the logarithm of 
the numerator minus the logarithm of the denominator. Both numerator 
and denominator are products whose logarithms are determined by Property I. 

log 4.8031 - 0.68152. log 78797 - 4.89651 

log 269.97 - 2.43131 log 253.6 =2.40415 

log 1.6364 0.21389 (add) log Denom, 7.30066 

(add) log Numer. - 3.32672 
log Denom. 7.30066 
(subtract) log Q = ? 



198 MATHEMATICS OF INVESTMENT 

We recognize that the result on subtracting will be negative. To obtain log Q 
in standard form, we add and also subtract 10 from the log numerator. 

log Numer. - 8.32672 = 13.32672 - 10 

log Denotn. = 7.30066 

(subtract) logQ = 0.02606 - 10; Q = .00010618. 

NOTE. Before computing any expression by logarithms, a computing form 
should be made. Thus, the first operation in solving Example 2 above was to 
write down the following form : 

log 4.8031 - log 78797 = 

log 269.97 = log 253.6 = 

log 1.6364 = (add) logDenom. = 

(add) log Numer. = 

log Denom. = 

(subtract) log Q => 

A systematic form prevents errors and makes it easy to repeat the work if it is 
desired to check the computation. 

EXERCISE LXXX 

Compute by logarithms : 

1. 563.7 X 8.2156 X .00565. 2. 4.321 X 21,98 X .99315. 

675.31 4 66.854. 

13.215' ' 2356.7 

.008315 6 783.12 X 11.325 

.0003156' ' 8932 

86 X 73 X 139.68 fi 9.325X631.75. 

3215.7 X .4563 ' ' .8319 X .5686 

ft .42173 X .21667 1Q 5.3172 X .4266 

.3852 X. 956 ' ' 18.11X31.681 

85. Computation of powers and of roots. 

Exampk 1. Find (.3156) 4 . 

Solution. From Property III of Section 78 with fc 4, 
log (.3166)* - 4 log .3156 - 4(9.49914 - 10) - 37.99600 - 40 - 7.99656 - 10. 
From Table I, (.3156)* - .009921. 

Example 2. Find ^856.31. 

SoMion. -^856.31 - (856.31)*. From Property III with k -*, log ^ 

- * log 866.31 2 ' 93 263 - 0.97754 j hence, ^856\3T - 9.4960. 
3 



LOGARITHMS 



199 



Exampk 3. Find ^08361; ^.08351. 

Solution. Since #.08351 = (.08351)*, we obtain from Property III, 
log ^08351 - Jlog .08351 - 8 ' 9217 f ~ 1Q . If we divide this as it stands, 

D 

we obtain 1.48696 J& a most inconvenient form. Hence, we add and, at 
the same time, subtract 50 from log .08351 in order that, after the division by 6, 
the result will be in the standard f orm for logarithms with negative character- 
istics. Hence, 
log v'.'oWl = 8 -92174 - 10 _ 50 + 8.92174 - 10 - 50 _ 58.92174 - 60 



log ^ 

From Property III, log 



6 6 

9.82029 - 10 ; hence, from Table I, 



6 
.66113. 



log .08361 



8.92174 - 10 



28.92174 - 30 



9.64058 - 10; hence, ^.08361 - .43710. 



EXERCISE LXXXI 

Compute by logarithms : 
1. (175) 2 . 2. (66.73) 8 . 

4. V53T2. 6. ^.079677. 

7. (353.3 X 1.6888) 2 . 8. ^199^62. 

10. (1.06)*. 11. (1.03)". 



13. 



1 



16 



85.75 

56.35 X 4.3167 



14. 



(45.6) 2 



8. (.013821) 4 . 

6. (.38956)*. 

9. (1.05) 7 . 

12. (1.06) 29 . 

16. (1.03)" 8 - 1 



(1.03)' 



V 



'21.36 X V52L9 
HINT. For this problem, the computing form is : 



17. 
19. 


log 56.36 = 
log 4.3157 - 


log 621.9 - 
f i log 521.9 = 
1 log 21 .36 - 


(add) log Numer. = 
log Denom. 


(add) log Denom. = 


(subtract) log fract. = 
4 log frftct. 

535 X 831,6 X (1.03) 8 


Result = 
18. (189.5)*. ' 

20. V896.33. 
M .03166 X 75.31 


475 X 938 
^00356. 
(163.2) a X 257.3 


1893.2 X 35830 


221.38 X (.3561) a 



28. 



24, (1.035)"", 





200 MATHEMATICS OF INVESTMENT 

86. Problems in computation. It is very important to realize 
tbat the properties I, II, and III of logarithms may bo used in com- 
puting products, quotients, and powers, but that they may not 
bs used in computing differences or sums except in the auxiliary 
Banner illustrated below, _ 

v Tin 4 n V896+ (.567) (35.3) 
Example 1. Compute Q = 532 - (15 31)* -- 

Solution. By logarithms, ve perform each of the three computations below. 

log V896 - * tag 896 - 2 ' 9 ^ 231 = 1.47616; V896 = 29.934. 
2 

log .567 - 9,76358 - 10 
log 35.3 = 1,84777 

log prod. - 11,30135 - 10; (.567) (35.3) = 20.016. 
leg (15.31)' = 2 log 16.31 = 3(1.18498) = 2.36996; hence, (15.31) 3 234.40. 

e *. above, 



log 49.949 = 1.69853 = 11.69853 - 10 
log 297.60 = 2.47363 - 2.47363 

(subtract) log Q = 9.22490 - 10; hence, Q = .16784. 

NOTE. A computation cUne with a 5-place table of logarithms will give 
results which are accurate to 4 significant figures, but the 5th figure always 
tvill be open to question. Each mantissa in the table, and each of those we 
determine by interpolation is subject to an error of part of 1 unit hi the 5th 
(fecimal place, even though all of our interpolation is done correctly. During a 
Jong computation, these accumulated errors hi the logarithms, together with 
tie allowable error due to oil* final interpolation, cause an unavoidable error 
in the 5th significant figure <vf our final result. Therefore, if a number with 
BUore than 5 significant figures, such as 2,986,633, is mot in a computation with 
A,5-place table, we should reduce this number to 2,986,500, tho nearest number 
having 5 significant figures, before finding its logarithm. To retain more than 
& significant digits is fictitious accuracy, since our final results will bo accurate 
fa only 4 digits. For the sanw reason, in looking up tho number corresponding 
fa a given logarithm, the interpolation should not be carried beyond the near- 
est unit in the 5th significant place. 

NOTE. Logarithmic computation of products, quotients, and powers must 
d&al entirely with positive ntwnbers, according to the statements of Section 77. 
Hence, if negative numbew a?e involved, we first compute the expression by 
logarithms as if all numbers vere positive, and then by inspection determine 
tie proper sign to be assigned to the result. Thus, to compute (- 75.3) X 
( - 8.392) X (- 32.15) we firsb find 75.3 X 8.392 X 32.15 - 20316 ; then, we 
ft&te that a negative sign rnngt be attached, giving 20316 as the result* 



LOGARITHMS 201 



EXERCISE LXXXH 
Compute by logarithms : 

(35.6) 2 + 89.53 . 1.931 X 5.622 - 




V11L39 - 2.513 ' 5.923 

3 dOS) 5 ~ 1- 4 1 ~ (1.Q4)- 4 . 

(1.03)* -1 ' -04 

6 (1-07) 8 ~ 1. 6 251 + 63.95 X 41.27 

.07 ' 787 

7. 395X856. 8. ||||. 

9. (Iog395)(log856). 10. |^|^|- That is, compute 

That is, compute 
(2.59660) (2. 93247). 

11 log 88-2 - 3 log (1.04) 

654 log 2 ' 

13 Io 6 6 - 532 14. log 8.957 

' log 1.04' log 1.06' 

16. 153.5(1.025) 10 . 16. 35.285(1.04)- B . 

17. (1.05)*. 18. (1.035)*. 
19. 12[(1.02)A - 1]. 20. log <85 + 3 . 

, 87. Exponential equations. An equation in which the unknown 
is involved in an exponent is called an exponential equation. 
Thus, 3* 7 = 27 is an exponential equation for 2. In this sec- 
tion we shall treat exponential equations of the type that can be 
solved by use of the following rule : 

Rule, To solve a simple exponential equation, take the log- 
arithm of both sides of the equation and solve the resulting 
equation. <- 

Exampk 1. Solve the equation IS 2 ** 2 = (356)5*. 

Solution. Toko tho logarithm of both sides of the equation, making use 
of Property I. Then (2 x -f 2) log 13 - log 366 + * log 6, or 



202 MATHEMATICS OF INVESTMENT 

(2 x + 2) (1.11394) - 2.66146 + s(.69897). (Table I) 

2.22788 x + 2.22788 - 2.66146 + .69897 x. 

1.62891 x = .32367; 

.32357 011*4 log -32367 = 9.60997 - 10 

1.62891 ' log 1.5289 = 0.18438 

(subtract) log x = 9.32659 - 10 

The exponential equations met in applications to the mathe- 
matics of investment are of the form 

A' = B, (6) 

where A and B are constants, and where v is a function of the un- 
known quantity. 

Example 2. Solve (1.07) 2n = 4.57. 

Solution. Taking the logarithm of both sides, we obtain 

2 n.]og 1.07 = log 4.57; n = lo ? 4 ' 57 . = ' 659 ' 92 = *&&. 

2 log 1.07 2(.02938) .05876 

log -.85082 = 9.81949 - 10 
log .05876 = 8.76908 - 10 
(subtract) logn = 1.05041; n =11.231. 

EXERCISE LXXXm 

Solve the following equations : 

1. (1.05)" = 6.325. 2. 15* = 95. 

3. 12 a+1 = 38. 4. (1.025) 2 " = 3.8261. 

6. S3* = 569. 6. 2 n = 31. 

7. 5* = 27(2-). 8. 25(6*) - 282. 



HINT. Clear the equation of fractions and reduce to the form of equation 
6, obtaining (1.035) n = 1.06875. 

10. (1.045)"" = .753. 

HINT. The equation becomes - n log 1.045 = 9.87679 - 10 = - .12321. 
11. (1.03)-* - .8321. 12. 850(1.05)" = 1638. 

13. 65.30(1.025)-" - 52.67, 14. 750 CLOg)*-l = 3500> 

-02 



LOGARITHMS 203 

SUPPLEMENTARY MATERIAL 

88. Logarithms to bases different from 10. To avoid confu- 
sion we shall explicitly denote the bases for all logarithms met in 
this section. From Section 77, x = log JV satisfies the equation 
a* = N. By solving this exponential equation, we can find x 
when N and a ane given. Thus, taking the logarithm to the base 
10 of both sides of a x = N, we obtain 

logio N 




or x = log, N = - . logiotf. (7) 



NOTE. Equation 7 enables us to find the logarithm of any number with 
respect to a given base a, provided that we have a table of logarithms to the 
base 10. The quantity logio a is called the modulus of the system of logarithms 
to the base 10 with respect to the system to the base a. 

The natural system of logarithms is that system where the base 
is the number e = 2.718281828 ---- The number e is a very im- 
portant mathematical constant and logarithms to the base e are 
useful in advanced mathematics.- From an 8-place table, we find 

logio e = 0.43429448; log .43429448 = 9.63778431 -10. 
Exampk 1. Find log, 35. 

Solution, Let x log. 35. Then, e a = 36 ; taking the logarithm of both 
sides to the base 10, x logio - logio 35 ; 

x a iQRio 35 = 1.54407 log 1.5441 - 10.18868 - 10 

logio e 0.43429* log .43429 9.63778 - 10 

x 3.5555. (subtract) log x = 0.55090 

EXERCISE LXXXIV 

1. Find log. 76; logs 10; log, 830; log, 657. 

2. Find the natural logarithm of 4368. 

3. Find logo 353; logs 10; Iogg895; Iogi 5 33. 

4. If a and 6 arc any two positive numbers, prove that 

logs N <=> log a N - logs a. 

HINT. Let x Iog N and y - log& N. Then N = a* = b. Take the 
logarithm with respect to the base & of both siclee of the equation ft" = a*, 



CHAPTER XII 
PROGRESSIONS 

89. Arithmetical progressions. A progression is a sequence 
of numbers formed according to some law. An arithmetical pro- 
gression is a progression in which each term is obtained from the 
next preceding term by the addition of a fixed constant called 
the common difference. Thus, 3, 6, 9, 12, , etc., is an arith- 
metical progression in which the common difference is 3. Simi- 
larly, 3, f, 2, f , , etc., is an arithmetical progression in which 
the common difference is ( ^-). 

Let a represent the first term of an arithmetical progression, d 
the common difference, and n the number of terms in the progres- 
sion. Then, in the progression, 



a = 1st term, 

a + d = 2d term, 

a + 2d = 3d term, 



a + 3d = 4th term, 

- etc. (8) 

a + (n l)d = nth term. 



If we let I represent the last, or the nth, term, we have proved that 
I = a + (n -1 ) d. (9) 

If we start with the last term, the next to the last term is formed by 
subtracting d, the second from the last by subtracting 2 d, etc. 
That is, in going backward, we meet an arithmetical progression 
with the common difference ( d). Thus, 



Z = last term, 
Z d = 1st from last term, 
Z 2 d = 2d from last term, 



i! 3 d = 3d from last term, 
etc. - (10) 

a = Z - (n l)d = (n l)st from 
last. 



Let s represent the sum of the terms of the progression. Then, 
we obtain equation 11 below by using the terms aj3 given in equa- 
tions 8, and equation 12 by using equations 10. 

s = a +a da 2d ..-etc. . + [a + (n - l)fl. (11) 

etc,,-.-+[Z-(n-l)4 (12) 
304 



PROGRESSIONS 205 

On adding equations 11 and 12, we obtain 

2 a - (a + I) + (a + I) + ( + J) + ' ' ' etc. + (a + I). (13) 
There are n terms in equation 13, one corresponding to each term 
of the progression. Hence, 2 s = n(a + I), or 

s=2( a + Z). (14) 

2 

If any three of the quantities (a, d, n, I, s) are given, the equa- 
tions 9 and 14 enable us to find the other two. We call (a, d, n, I, s) 
the elements of the progression. 

Example 1. In an arithmetical progression with the first term 3, 
the 6th term is 28. Find the common difference and the intermediate 
terms. 

Solution. We have a = 3, n = 6, and I = 28. Hence, from equation 9, 
28 = 3 + 5 d ; 5 d = 25 ; d = 5. The terms of the progression are 3, 8, 13, 
18, 23, 28. 

EXERCISE LXXXV 

1. Find the last term and the sum of the progression 

3, 5, 7, 9, ... to twelve terms. 

2. Find the sum of the progression 5, 4, 3, 2, . . . , to eighteen 'terms. 

3. Find the last term and the sum of the progression 
1000(.05), 950(.05), 900(.05), . . . etc., to twenty terms. 

4. If 10 is the first term and 33 is the 20th term of an arithmetical 
progression, find the common difference and the sum of the progression. 

5. If 15 is the 4th term and 32 is the 10th term of an arithmetical 
progression, find the intermediate terms. 

90. Geometrical progressions. A geometrical progression 

is a progression in which each term is formed by multiplying the 
preceding term by a fixed constant r. The number r is called 
the common ratio of the progression because the ratio of any term 
to the preceding term is equal to r. Thus, 4, 12, 36, 108, - 
etc., is a geometrical progression with the common ratio r => 3, 
The sequence 

(1.05), (1.05) 2 , (1.05) 8 , (1.05)*, etc., 
is a geometrical progression with the ratio r * (1.05). 



206 MATHEMATICS OF INVESTMENT 

Let a represent the first term, r the common ratio, and n the 
number of terms in a geometrical progression. Then, 



ar 6 = 5th term, 
. . '. etc., 

= (73, __ i) B t term, 



a = 1st term, 
ar = 2d term, 
ar 2 = 3d term, 
ar 8 = 4th term, 

If we let I represent the last, or nth, term, we have proved that 



Let s represent the sum of the terms of the progression. Then 
s = a + ar + ar 2 + etc. + ar n ~ 2 + ar"" 1 , (16) 
rs = ar 4- ar 2 + ar 8 + etc. + ar"" 1 + * (17) 
On subtracting equation 17 from equation 16, all terms will cancel 
except a from equation 16 and ar n from equation 17. Thus, 
s T-S = 5(1 r) = a ar n . 

Hence, s = a -f^ = 5^1 ( 18 ) 

Since I ar n-1 , then rl = ar n ; on substituting this in the first 

fraction of equation 18, rl a 

s = _ T (19) 

Example 1. Find the sum of 1 + -J- + i + etc ---- to six terms. 
Solution. Use formula 18 with a = 1, r ~ 4, and n = 6. 

S= l-* = } = 24' 
Exampk 2. Find an expression for the sum of 

1 + (1.05) " + (1-05) + (1.05) 1 - 6 + etc. . . . + (1.05) 38 - 5 . , 
Solution. The terms form a geometrical progression for which o 1, 
r = (1.05)- B , and I - (1.05) W - B . From formula 19, 

' - 1 ra (l.Q5) M - 1 . 



(1.05) - B - 1 (1.05) - 1 

EXERCISE LXXXVI 

1. Find the last term and the sum of 25, 5, 1, t, A, etc. to seven 
terms. 

2. Find the last term and the sum of 2, 4, 8, . . . etc, to eighteen terms. 

3. Find the ratio, the number of terms, and the sum for the progres- 
sion 3, 9, 27, ... t? t ; to 729. 



PROGRESSIONS 207 

4. Find the sum of 2, 1, , etc. to eight terms. 

5. Find the sum of 1 + * + H ----- j-rk- 
Find expressions for the following sums : 

6. (1.05) + (1.Q5) 2 + (1.05) 8 + etc. + (1.05) 28 . 

7. (1.04) 2 + (1.04) 4 + (1.04) 8 + etc. + (1.04) 38 . 

8. (1.06) - 215 + (LOG)" 24 + (1.06)' 28 + . . . etc. + (1.06) "\ 

9. (1.03) - 1 + (1.03)' 2 + (LOS)' 8 + - - etc. + (1.03)-". 
10. (1.02) + (1.02) J + (1.02) 8 + etc. + (1.02) 80 . 

91. Infinite geometrical progressions. Consider the follow- 
ing hypothetical example. A certain jar contains two quarts of 
water. One quart is poured out; then, ^ of the remainder, or 
^ quart, is poured out ; then, % of the remainder, or quart, is 
poured out, etc., without ceasing. The amounts poured out are 

1, , i, -g-, etc. to infinitely many terms. 
The sum of the amounts poured out up to and including the nth 
pouring is , , 1 , 1 , , 1 

Sn = 1+ 2 + 4 + ' ' + 2^i' 

Since the amount originally in the jar was 2 quarts, s n can never 
exceed 2. Also, it is clear intuitionally that, as n increases with- 
out bound, s n must approach the value 2 because the amount of 
water left in the jar approaches as the process continues. We 
can prove this fact mathematically ; from formula 19, 

2) = 2 - (20) 



1 
As n grows large without bound, continually decreases and 

approaches zero. Thus, from equation 20 we prove that, as n 
increases without bound, s n approaches the limit 2, as was seen 
intuitionally above. Hence, we may agree, by definition, to call 
this value 2 the sum of the infinite geometrical progression, or to say 

2 = l + -f-+''- etc. to infinitely many terms. 
This example shows that a sensible definition, in accordance with 
our intuitions, may be given for the sum of an infinite geometrical 
progression, 



208 MATHEMATICS OF INVESTMENT 

In general, consider any infinite geometrical progression for 
which the ratio r is numerically less than 1, that is, for which r 
lies between 1 and + 1. The terms of the progression are 

a, ar, ar 2 , ar 3 , etc. to infinitely many terms. 
Let s n represent the sum of the first n terms of the progression : 

s n = a + ar + ar 2 + + ar n ~ l . 

The statement as n approaches infinity will be used as an abbre- 
viation for the statement as n increases wthout bound. 

The sum S of an infinite geometrical progression is defined as 
the limiting value, if any exists, approached by s n as n approaches 
infinity. 

From formula 18, 

a ar n a ar n 





As n approaches infinity, it is evident that r n approaches zero l 
because r is numerically less than 1. Hence, from equation 21 

it is seen that, as n approaches infinity, s n approaches _ as a 

limiting value, because the other term in equation 21 approaches 
zero. Since, by definition, this limiting value of s n is the value 
we assign to the sum 

S = a + ar + ar 2 + etc. to infinitely many terms, 
we have proved that a 

1 r 

Example 1. Find the sum of the progression 

(1.04) "* + (1.04) " + (1.04)" 8 H ---- etc. to infinitely many term. 

Solution. The ratio of the infinite geometrical progression is r = (1.04)~ a ; 
a = (1.04) ~*. From formula 22, the sum is 

S 



(1.04)-* 



1 - (1.04)-' 

Exampk 2. Express the infinite repeating decimal .08333 as 
a fraction. 

Solution. We verify that .08333 equals .08 plus 

.003 + .0003 + .00003 + etc. to infinitely many terms. 

1 For a rigorous proof of this intuitional fact the student is referred to the theory 
pf limits as presented, for example, in books on the Calculus. 



PROGRESSIONS 209 

These terms form an infinite geometrical progression with a = .003, and r = .1. 
Their sum is ~^- = :92. Hence, 

.08333 = .08 + = JL + JL = .?A = -L. 
.9 100 900 300 12 

NOTE. By the method of Example 2 above, any infinite repeating decimal 
can be shown to represent a fraction whose numerator and denominator are 
integers. 

EXERCISE LXXXVH 

Find the sums of the following progressions : 

1. 2 + 1 + i + to infinitely many terms. 

2. 5 + l+i + -jfr+---to infinitely many terms. 

8 - (W + (Ti5y' + (i^ + ^ 

4. (1.04)- 1 + (1.04)" 8 + (1.04) " 3 H to infinitely many terms.- 

6. (1.03)~* + (1.03)" 4 + (1.03)~ H to infinitely many terms. 

6. (l.Oir 1 + (1.01)~ 2 + (l.Oir 8 + ... to infinitely many terms. 
Express the following infinite decimals as fractions : 

7. .333333 -. 8. .66666 

9. .11111 . 10. .41111 -. 

11. .5636363 -. 12. .24222222 . 



APPENDIX 

Note 1 

Proof of Rule l; Section 16, Part I. Consider the equation 
2 = (1 + r). 

The solution of this equation for n is the time required for money to 
double itself if r is the rate per period. On taking the logarithms, 
with respect to the base e = 2.71828 . . . , of both sides of the equa- 
tion, we obtain l og 2 

71 " log (1 + r) f 
where " log " means " log fl ." From textbooks on the Calculus, we 

find that log (1 + r) =r-^+^ ----- r(l - | + ~ -), 

and from a table of natural logarithms we obtain log 2 = .693. Hence 
.693 .693. ,r r 2 , * 



On 1 neglecting the powers of r in the parenthesis from r 2 on, we obtain 
as an approximate solution 

.693 , .693 .693 , , 
n = -- = -- r '5O- 

r 2 r 

Note 2 

Proof of Rule 1, Section 17, Part I. Consider three obligations 
whose maturity values are Si, 89, and 83, which are due, respectively, 
at the ends of n\, n a , and nj years. We shall prove Rule 1 for this 
special case. The reasoning and the details of proof are the same for 
the case of any number of obligations. Let i be the effective rate of 
interest, and let n bo the equated time. By definition, n satisfi.es 
the equation 

iT" 1 + &(1 + ff"" 1 + A(l + i)-" 3 . (1) 
211 



212 MATHEMATICS OF INVESTMENT 

By use of the binomial theorem, we obtain the following infinite series 



"(I 



. 

2 

Since i is small, we make only a slight error if we use only the first 
two terms of each infinite series as an approximate value for the 
corresponding power of (1 + i). On using these approximations in 
equation 1, we obtain 

(1 - ni)GSi + S + &) = Si(l - ni*) + S 2 (l - n*i) + &(1 - *) 
On expanding both sides and on solving for n, we obtain 



ni + n a + na 
which establishes Rule 1 for the present case. 

Note 3 

Solution, of equations by interpolation. The method of inter- 
polation which the student has used in connection with logarithm 
and compound interest tables can be used in solving equations whose 
solution would otherwise present very great difficulties. 

Example 1. Solve for n in the equation 

5(1.06)" = 7.5 + 7.6(.00)n. (1) 

Solution. On rewriting the equation and on using the abbreviation F(n) 
fov the left member, wo obtain 

F(n) - 5(1.06)" - 7.6 - .45 n - 0. 

We desire a value n fc such that F( - 0. If we find a value n - n t such 
that F&I) is negative, and another value n - ni such that F<j) is positive, 
then it will follow that there is a value n k, between n\ and nt t such that 
/?(*) - 0. That is, there must be a solution n h between Wi and n*. From 
a rough inspection of Table V we guess that the solution is greater than n - 19. 
With the aid of Table V wo compute Fw for n - 19, 20, and 21. F(iw 
_ .922; F(ao) - - .464; J^OD - -H .048. Hence, there is a solution 
n k of the equation between n * 20 and n - 21, We find fc by interpola- 



APPENDIX 



213 



0. The total dif- 
The partial differ- 




tion in the table below where we use the fact that Fw = 
ference in the tabular entries is .048 ( .464) = .512. 
ence is ( .464) = .464. Hence, since is $ff 
= .91 of the way from .464 to +.048, we assume 
that the solution k is .91 of the way from 20 to 21, or 
that k = 20 + .91 = 20.91. Of course, this is only an 
approximate solution of the equation, but such a one 
is extremely useful in practical applications. An in- 
spection of the equation shows that there cannot be 
any other solution because 5(1.06)" increases much more rapidly than .45 n, 
and hence F(n) will be positive for all values of n greater than 21. 

Example 2. A man invests $6000 in the stock of a corporation. 
He receives a $400 dividend at the end of each year for 10 years. At 
the end of that time lie sells his holdings for $5000. Considering the 
whole 10-year period, at what effective rate may the man consider 
his investment to have been made ? 

Solution. Let r be the effective rate. With the end of 10 years as a 
comparison date, we write the following equation of value : 

6000(1 + r) 10 - 5000 + 400^ at r}, 

Fw - 6000(1 + r) 10 - 5000 - 400( aini at r) = 0. (1) 

We shall solve equation 1 by interpolation. 

If the $1000 loss in capital had been uniformly distributed over the 10 years, 
the loss per year would have been $100. Hence, under this false (but ap- 
proximately true) condition, the net annual income would have been $300. 
The average invested capital would have been $(6000 + 5000) = $5500. 
Hence, since -^nnr = -055, we guess 1 .055 as an approximation to the solution 
of the equation. When r .055, F(.osS) <=> + 98.73. Since this is positive, 
the solution must be less than .055. We find ^(.oe) 257.80. Hence, the 
solution r = k of equation 1, for which F(K) = is between r => .056 and 
r = .05. We interpolate in the table below. 98.7 - (- 267.8) - 356.5; 
- (- 257.8) 267.8; .055 - .06 - .005. Hence, 

- .05 + .0036, 



or. 



approxi- 
The solution could be obtained ac- 



r 


*V) 


.05 


- 257.S 


r = k 





.055 


98.7 



' 366.5 

mately, k <= .054. 

curatcly to hundredths (or to thousandths, or less) of 
1%, if desired, by the method used in Example 2, Sec- 
tion 32, Part I. 

Note 4 

Abridged multiplication. Consider forming the product (11.132157) 
X (893.214) . We decide in advance that wo desire the result accurately 

1 Notice the uimilaiity between this reasoning and that employed in Section 55 of 
Parti, 



214 



MATHEMATICS OF INVESTMENT 



to the nearest digit in the second decimal place. The ordinary mul- 
tiplication would proceed as at the left below, while the abridged 
method proceeds as at the right. 



OHDINABY METHOD 


ABIUDOBD METHOD 


11.132157 
893.214 


xxxxx 
11.132157 
893.214 


Multiply 
by 


44528628 
11132157 
22264314 
33396471 
100189413 
89057256 


8905.7256 
1001.8935 
33.3963 
2.2264 
.1113 
.0444 


800 
90 
3 
.2 
.01 
.004 


9943.398481598 


9943.3975 


Add 


ResuU = 9943.40 


Result = 9943.40 



In multiplying by the abridged method we proceed as follows : 

Since we desire the result to be accurate in the 2d decimal place, we carry 
two extra places, or four decimal places, for safety. To multiply by 893.214 
we multiply in succession by 800, 90, 3, .2, .01, and .004 and then add the 
results (this is the same as is done in the ordinary method of multiplication, 
except that the multiplications are performed in the reverse order). We 
first multiply by 800, that is, we multiply by 8 and then move the decimal 
point. All digits of 11.132157 are used in this operation in order to obtain 
four significant decimal places in the result. To obtain four decimal places 
when multiplying by 90 we need one less digit of 11.132157 ; we put X over 
the "7" to indicate that we multiply 11.13215 at this time. Wo put X over 
the "5" and then multiply 11.1321 by 3; we put X over the last " 1 " and 
then multiply 11.132 by .2 ; etc. The advantages of this method are obvious. 
Less labor is involved, the decimal point is accurately located, and fewer mis- 
takes will occur in the final addition, 

Note 6 

Accuracy of the interpolation method in solving for the time in the 
compound interest equation. Consider the equation 

A = (1 + r) n , ' (1) 

where A and r are known. To determine the value of n by interpola- 
tion, we first find from our interest table (Table V if A > 1, Table VI 
if A < 1) two integers ni and n a , n a n\ 1, such that the oorre- 



APPENDIX 



215 



spending values A\ = (1 + r) n i and A z = (1 + r)" 1 include A between 
them. That is, AI< A< A*. Then, as obtained by interpolation, 
the solution of equation 1 is 

A A , 

"AT I **- * 1 

J\ =s n\ -\ 

The exact solution of equation 1 is obtained by taking the logarithm 
of both sides ; log A = n log (1 + r), where log means log,. 



_ log A 
log (1 + r) 



From equation 2 we obtain 

dn _ 1 

dA Alo g (l+r: 

Hence since dn/dA is positive, 
n is an increasing function of 
A. Moreover, since dtn/dA 2 is 
negative, the graph of n as a 
function of A, with the A-&XJB 
horizontal, will be concave 
downward, as in Figure 6, dis- 
torted for illustration. It is 
seen graphically that the dif- 
ference between n, as given in 
equation 2, and N is given 
by the line EF in the figure. 
This error is less than DH, 
where H is the point in which 
the tangent (drawn at P) in- 
tersects the ordinate at J.a. 
Since CD - 1, 



tfn 
dA* 



- 1 



4 2 log(l 



(2) 
(3) 




TlAi A 

Fia. 6 



, -. , 
Ailog(l+r) 



- 1. 



Since A 2 = (1 + r)" 1 = (1 + r)(l + r)" 1 ** Ai(l + r}, A t - Aj. = 
rAi, Hence, on inserting the infinite series for log (1 + r), as obtained 
from any textbook on the Calculus, 

if _rf '_, 

r 



DH 



ft* M 



r 2 , r 3 



- 1 = 



216 MATHEMATICS OF INVESTMENT 




If r < .10. as is the case in the tables of this book. DH <-(}, 

-2\.95/ 

which is approximately r, if computed to only two decimal places. 
Hence, if we are computing results to only two decimal places, a solu- 
tion of equation 1, obtained by interpolation, is in error by at most 
Jr. 

Note 6 

Accuracy of the interpolation method in solving for the time in the 
annuity equations. Consider the equation 

0^ at r) = S, (1) 

where S and r are known. From equation 1, on inserting the explicit 
algebraic expression for (s| at r), we obtain 

t 1 + r)n - 1 = 8; (1 + 70" - Sr + 1. 

T 

If we solve equation 1 for n by interpolation in Table VII, our solution 
is the same as. we should obtain in solving the equivalent equation 

(1 + r) n = A (2) 

(where A = Sr + 1) for n, by interpolation in Table V. For, the 
solution of equation 1 by interpolation would be 



while that for equation 2 would be 

-- _ L oV + 1 - Sir - 



which is the same as the result for equation 1. Hence, it follows 
from Note 6 of the Appendix that the error in the solution of equation 
1 obtained by interpolation in Table VII is at most i of the interest 
rate r. Similarly, it follows that, if we should solve for n in the equa- 
tion (ctn\ at r) = A, by interpolating in Table VIII, the error of the 
result would be at most r. 



INDEX 



Numbers refer to pages 



Abridged multiplication, 213 

Accrued dividend, on a bond, 126 

Accumulation factor, 16 

Accumulation of diacount, on a bond, 
119 

Accumulation problem, 16 

American Experience Table of Mor- 
tality, 148 

Amortization, of a debt, 78; of the 
premium on a bond, 118 

Amortization equation, 89 

Amortization plan, 78; bonded debt 
retired by a, 81 ; comparison of 
sinking fund method with the, 88 ; 
final payment under the, 84 

Amortization schedule, for a debt, 78; 
for the premium on a bond, 118 

Amount, at compound interest, 15; 
at simple interest, 1; in a sinking 
fund, 87 ; of an annuity certain, 39 

Annual premium; see net annual 
premium 

Annual rent of an annuity, 39; de- 
termination of the, 68 

Annuities certain, 39; formulas for, 
43, 47, 60; interpolation methods 
for, 70, 72; summary of formulas 
for, 60 

Annuity; see annuity certain, and 
life annuity 

Annuity bond, 130 

Annuity certain, 39; amount of an, 
39; annual rent of an, -39; con- 
tinuous, 02; deferred, 69; determi- 
nation of the annual rent of an, 68 ; 
interest rate borne by an, 71 ; term 
of an, 70; due, 56; payment in- 
terval of an, 39 ; present value of an, 
39; term of an, 39 

Annuity due, certain, 56 ; life, 162 

Annuity policy, 177 



Approximate bond yield, 126 
Arithmetical progression, 204 
Asset, scrap value of an, 96; wearing 

value of an, 96; condition per cent 

of an, 98 
Average date, 34 
Averaging an account, 34 

Bank discount, 9 , 

Base of system of logarithms, 187 

Beneficiary, 165 

Benefit of a policy, 166 

Binomial theorem, 63 

Bond, 113 ; accumulation of discount 
on a, 119; accrued dividend on a, 
126; amortization of premium on 
a, 118; approximate yield on a, 
126; book value of a, 117, 122; 
changes m book value of a, 117; 
dividend on a, 113; face value of 
a, 113; flat price of a, 125; pur- 
chase price of a, 114, 121; quoted 
price of a, 125 ; redemption price of 
a, 113; the yield of a, 126; yield 
on a, by interpolation, 128, 131 

Bond table, 116 

Book value, of a debt, 86; of a de- 
preciable asset, 97 

Book value of a bond, on an interest 
date, 117; between interest dates, 
122 

Briggs' system of logarithms, 189 

Building and loan associations, 92; 
dues of, 92; interest rates earned 
by, 93; loans made by, 94; profits 
in, 92; shares in, 92; time for 
stock to mature in, 62 

Capitalized cost, 105 
Cash surrender value, 183 
Characteristic of a logarithm, 190 
Common logarithms, 189 



217 



218 



INDEX 



Numbers refer to pages 



Commutation symbols, 157 

Comparison date, for comparing 
values, 26; for writing an equa- 
tion of value, 27 

Composite life, 99 

Compound amount, 14; for a frac- 
tional period, 20 

Compound interest, 14; accumula- 
tion problem under, 15; amount 
under, 14; continuous conversion 
under, 35 ; conversion period under, 
14; discount problem under, 15; 
effective rate under, 18; nominal 
rate under, 18 

Concluding payment, under amortiza- 
tion process, 84 

Condition per cent, 98 

Contingent annuity, 39 

Contingent payment, present value 
of a, 152 

Continuous annuity, 62 

Continuously converted interest, 35 

Conversion period, 14 

Deferred annuity, certain, 59 

Deferred life annuity, 159 

Depreciation, 96; constant percen- 
tage method for, 109; valuation 
of mining property under, 101; 
sinking fund plan for, 96; straight 
line method for, 98 

Discount, banking use of, 9 ; problem 
of, under compound interest, 15; 
bond purchased at a, 119; rate of, 
7; simple, 7 

Discount factor, 16 

Discounting of notesj under simple 
discount, 10; under compound 
interest, 24 

Dividend, on a bond, 113 

Dues of a building and loan associa- 
tion, 92 

. Effective rate of interest, 18 
Endowment insurance, 170; also see 

pure endowment 
Equated date, 33 
Equated time, 33 ; equation for, 33, 

212 



Equation of value, 27; comparison 

date for, 27 

Exact simple interest, 2 
Exponential equation, 201; use of, 

in annuity problems, 75 

Force of interest, 36 

Geometrical progressions, 205; use 
of, under annuities certain, 41, 43, 
46 ; use of, under perpetuities, 108 

Glover's tables, 167 

Graphical representation, of accu- 
mulation under interest, 23; of a 
deferred annuity, 59 ; of an annuity 
due, 57 ; of depreciation, 97 

Gross premium, 165 

Infinite geometrical progressions, 207 ; 
use of, under perpetuities, 108 

Insurance, 165 ; endowment, 170 ; gross 
premium for, 165, 175 ; net annual 
premium for, 171; net premiums 
for, 165; net single premium for, 
166; policy of, 165; term, 168; 
whole lif e, 166 

Insurance policy, 165; beneficiary of 
an, 165; benefits of an, 165; policy 
date of an, 165; endowment, 172; 
w-paymcnt life, 172; n-yoar term, 
172 ; ordinary life, 172 ; reserve on 
an, 178; whole life, 166 

Insurance premium ; seo premium 

Interest, 1 ; compoxind, 14 ; converted 
continuously, 35 ; effective rate of, 
18 ; exact, 2 ; force of, 36 ; graphi- 
cal representation of accumulation 
under, 23 ; in advance, 9 ; nominal 
rate of, 18; ordinary, 2; rate of, 
1; simple, 1 

Interest period, 14 

Interpolation, annuity problems 
solved by, 70, 72, 213 ; book valuo 
of bond between interest dates by, 
123; compound interest problems 
solved by, 29; use of, in logarith- 
mic computation, 194; yield of 
bond by, 128, 131 



INDEX 



219 



Numbers refer to pages 



Investment yield of a bond, 113, 126; 
by approximate method, 126; by 
interpolation, 128, 131 

Legal reserve insurance company, 166 

Level premium, 178 

Life annuity, 155 ; deferred, 159 ; due, 
162; present value of a, 165, 159, 
163; temporary, 159; whole, 155 

Life insurance ; see insurance 

Loading, 175 

Logarithms, 187; base of a system 
of, 187; Briggs 1 system of, 189; 
change of base of, 203 ; . characteris- 
tics of, 190; common, 189; man- 
tissas of, 190; Napierian, 203; 
natural, 203; properties of, 188; 
use of tables of, 194, 196 

Mathematical expectation, 152; net 
single premium as a, 175; of a con- 
tingent payment, 152 

Mantissa, 190 

Mining property, valuation of, 101 

Modulus, of a system of logarithms, 
203 

Mortality, American Experience Table 
of, 148 

n-payment endowment policy, 172 
Tirpaymont life policy, 172 
7i-year term policy, 172 
Natural premium, 169 
Net annual premium, 171; 
endowment policy, 173 ; 



for 
for 



an 
an 

irregular policy, 176; for an w- 
payment life policy, 173 ; for an n- 
year term policy, 173 ; for ordinary 
life policy, 172 

Net premiums, 165 

Net single premium, 166, 175; for 
endowment insurance, 170; for 
irregular benefits, 176; for term 
insurance, 168; for whole, life in- 
surance, 166 

Nominal rate of interest, 18 

Notes ; sec discounting of notes 

Office premium, 165 

Old lino insurance company, 165 



Drdinary life policy, 172 
Ordinary simple interest, 2 

Par value of a bond, 113 

Payment of a debt, amortization 
process for, 78 ; amortization sched- 
ule for, 78; building and loan 
association arrangement for, 94; 
comparison of amortization and 
sinking fund methods for, 88; 
sinking fund method for, 85 

Pensions, present value of, 184 

Perpetuities, 103; infinite geometri- 
cal progressions applied to, 108; 
present values of, 103, 104; use of, 
in capitalization problems, 105 

Policy ; see insurance policy 

Policy date, 165 

Policyholder, 165 

Policy year, 165 

Premium, annual, 171; gross, 165; 
level, 178; natural, 169; net, 165; 
net annual, 171 ; net single, 166 

Premium on a bond, formula for the, 
115 ; amortization schedule for the, 
118 

Present value, of a contingent pay- 
ment, 152; of a pure endowment, 
153; of a life annuity, 155, 159, 
162 ; of an annuity certain, 39 ; of 
life insurance, 165; under com- 
pound interest, 15; under simple 
discount, 7 ; under simple interest, 2 

Principal, 1 ; amortization of, 78 

Probabilities of life, 150 

Probability, 147 

Progressions, 204; arithmetical, 204; 
geometrical, 205 ; infinite geometri- 
cal, 207 

Proportional parts, 195 

Prospective method of valuation, 183 

Pure endowment, 153 ; present value 
of a, 153, 157 

Rate of discount, 7 

Rate of interest, 1 ; borne by an an- 
nuity, 71; effective, 18; nominal, 
18 ; paid by a borrower of a build- 



220 



INDEX 



Numbers rofer to pages 



ing and loan association, 95; 
yielded by a bond, 126, 128, 
131 

Redemption fund, for a mine, 101 
Reserve, terminal, 178; table show- 
ing growth of a, 179; formula for 
the, 180 

Scrap value, 96 

Serial bonds, 130 

Simple discount, 6 

Simple interest, 1; exact, 2; ordi- 
nary, 2 ; six per cent rule for, 3 

Sinking fund, 85; amount in a, 87; 
table showing growth of a, 86 

Sinking fund equation, 89 

Sinking fund plan, for depreciation, 
96 ; for retiring a debt, 85 

Six per cent rule, 3 

Straight line method for depreciation, 
98 



Temporary life annuity, 159 

Term, of an annuity certain, 39 ; de- 
termination of the, 70 

Terminal reserve, 178; for an ordi- 
nary life policy, 181; prospective 
method for obtaining the, 183 

Term insurance, 168 

Time, to double money, 30, 211 

Valuation, of a mine, .101 ; of an 

insurance reserve, 180 
Value, cash surrender, 183; of an 

obligation, 24 
Values, comparison of, 26 

Wearing value, 96 
Whole life annuity, 155 
Whole life insurance policy, 166 

Yield of a bond, by approximate 
method, 126; by interpolation, 
128, 131 



TABLE I 

COMMON LOGARITHMS OF NUMBERS 
TO FIVE DECIMAL PLACES 

Pages 2 to 19 

TABLE II 
COMMON LOGARITHMS OF NUMBERS 

FROM 1.00000 to 1.10000 
TO SEVEN DECIMAL PLACES 

Pages 20 to 21 



N 





1 


9 


3 


4 


5 


' 6 


7 


8 


9 




PP 


100 


00000 


043 


087 


130 


173 


217 


260 


303 


340 


88'J 






01. 
02 
03 


432 
860 
01284 


475 
903 
326 


518 
945 
368 


561 
988 
410 


604 
+030 
452 


647 
+072 
494 


089 
+116 
536 


732 
+167 
678 


775 
+199 
020 


817 
+242 
002 




44 43 42 


04 
05 
06 

07 
08 
09 


703 
02119 
531 

938 
03342 
743 


745 
160 
672 

979 
383 
782 


787 
202 
612 

*019 
423 
822 


828 
243 
653 

*060 
463 
862 


870 
284 
. 694 

+100 
503 
902 


912 
325 
736 

+141 
543 
941 


953 
306 
776 

+181 
583 
981 


005 
407 
816 

+222 
023 
+021 


+036 
449 
857 

+202 
003 
+000 


+078 
490 
898 

+302 
703 
+100 


1 
2 
3 
4 
5 
G 
7 
8 



4.4 4.3 4.2 
8,8 8.0 8.4 
13.2 12.0 12.0 
17.0 17.2 10.8 
22.0 21.5 21,0 
20.4 2C.8 2fl.2 
30.8 30.1 20.4 
3G.2 34.4 33.0 
30.0 38.7 37.8 


110 


04139 


179 


218 


258 


297 


336 


376 


415 


454 


493 






11 
12 
13 


532 
922 
05308 


671 
961 
346 


610 
999 
385 


650 
*038 
423 


689 
+077 
461 


727 
+115 
600 


760 
+154 
638 


805 
+102 
676 


844 
+231 . 
014 


883 
+200 
652 




41 40 30 


14 
15 
16 

17 
18 
19 


690 
06070 
446 

819 
07188 
555 


729 
108 
483 

856 
225 
591 


767 
145 
521 

893 
262 
628 


805 
183 
568 

930 
298 
664 


843 
221 
505 

987 
335 
700 


881 
258 
633 

+004 
372 
737 


918 
206 
670 

+041 
408 
773 


950 
333 
707 

*078 
445 
809 


094 
371 
744 

+116 
482 
840 


+032 
408 
781 

+151 
518 
882 


1 
3 

4 

5 

7 
8 



4.1 4.0 3.1) 
8.2 8.0 7.H 
12.3 12.0 11.7 
10.4 10.0 15.0 
20.S 20.0 10.0 
24.0 24.0 23.4 
28.7 28.0 27.3 
32.8 32.0 31.2 
36,9 30.0 36.1 


120 


918 


954 


990 


+027 


+063 


*099 


+135 


+171 


+207 


+243 






21 
22 
23 


08279 
636 
991 


314 
072 
*026 


360 
707 
*061 


386 
743 
+006 


422 
778 
+132 


458 
814 
+167 


493 
849 
+202 


529 
884 
*237 


505 
020 
+272 


600 
066 
+307 




38 87 8ft 


24 

25 
26 

.27 
28 
29 


09342 
691 
10037 

380 
721 
11059 


377 
726 
072 

415 
766 
093 


412 
760 
106 

449 
789 
126 


447 
795 
140 

483 
823 
160 


482 
830 
176 

517 
857 
193 


617 
864 
209 

561 
890 
227 


552 
899 
243 

585 
924 
201 


587 
934 
278 

010 
058 
204 


621 
908 
312 

053 
002 
327 


656 
+003 
340 

087 
+02. 1 ) 
361 


1 
3 

6 
6 
7 
B 



3.8 3,7 3.0 
7.0 7.4 7.2 
11.4 11.1 10.8 
15.2 14.8 14.4 
10.0 18.5 18.0 
22.8 22.2 21.0 
20.0 20.0 20.2 
30.4 20.6 28.8 
34.2 33.3 32.4 


130 


394 


428 


461 


494 


528 


561 


504 


628 


601 


094 






31 
32 
33 


727- 
12067 
385 


760 
090 
418 


793 
123 
450 


826 
156 
483 


860 
189 
516 


893 
222 
648 


926 
254 
581 


959 
287 
613 


992 
320 
046 


*024 
352 
678 




35 84 33 


34 
35 
36 

37 
38 
39 


710 
13033 
354 

672 
988 
14301 


743 
066 
386 

704 
+019 
333 


776 
098 
418 

735 
*051 
364 


808 
130 
460 

767 
*082 
395 


840 
162 
481 

799 
+114 
426 


872 
194 
513 

830 
+145 
457 


905 
226 
545 

862 
+176 
489 


937 
258 
577 

803 
+208 
520 


969 
290 
609 

025 
+239 
551 


*001 
322 
640 

056 
+270 
582 


2 

I 
6 

7 

1 


3.0 3.4 3.3 
7.0 0.8 0.6 
10,5 10.2 0.0 
14.0 13,6 13.2 
17.5 17.0 10.0 
21.0 20.4 10.8 
24.5 23.8 23.1 
28.0 27.2 20.4 
31.0 30.0 20.7 


110 


613 


644 


676 


706 


737 


768 


799 


820 


860 


801 






41 
42 
43 

44 
45 
46 

47 
48 
40 


922 
15229 
534 

836 
16137 
435 

732 
17026 
319 


953 
259 
664 

866 
167 
465 

761 
056 
348 


983 
290 
694 

897 
197 
495 

791 
085 
377 


+014 
320 
025 

927 
227 
524 

820 
114 
406 


+045 
351 
655 

967 
256 
554 

850 
143 
435 


+076 
381 
685 

987 
286 
584 

870 
173 
464 


+106 
412 
715 

+017 
316 
613 

909 
202 
493 


+137 
442 
746 

+047 
346 
643 

938 
231 
522 


*168 
473 
770 

+077 
376 
673 

967 
260 
851 


*198 
503 
806 

+107 
406 
702 

997 
289 
580 


2 

I 
5 

a 

7 

! 


32 31 30 

8.2 3Tl S70~ 
0.4 0.2 0.0 
0.0 0.3 0.0 
12.8 12,4 12,0 
10.0 16.0 10,0 
19.2 18.0 18.0 
22.4 21.7 21.0 
20.6 24.8 24.0 
28.8 27.0 27.0 


150 


609 


638 


667 


696 


725 


754 


782 


811 


840 


sto 






IT ' 





1 


2 


8 


4 


, 5 


6 


7 


8 


ft 




PP 



N 





1 


2 


3 


4 


5 


6 


1 


8 


9 




] 


pp 


150 


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260 


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299 


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320 


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336 


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43 


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182 


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220 


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250 


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100 


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115 


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173 


181 


188 


195 


203 


210 


217 


225 




92 


232 


240 


247 


254 


202 


200 


270 


283 


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93 


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313 


320 


327 


335 


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97 


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721 


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743 


750 


757 


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779 


780 


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081 


087 


094 


*000 


*007 


*014 




01 


82020 


027 


033 


040 


040 


053 


000 


000 


073 


079 




02 


080 


002 


000 


105 


112 


119 


125 


132 


138 


145 




03 


151 


158 


164 


171 


178 


184 


101 


107 


204 


210 


7 


04 


217 


223 


230 


236 


243 


240 


250 


263 


269 


276 


1 0.7 
2 1.4 


05 


282 


280 


205 


302 


308 


315 


321 


328 


334 


341 


3 2.1 


06 


347 


354 


360 


367 


373 


380 


387 


303 


400 


406 


4 2.8 
5 3.5 


67 


413 


410 


428 


432 


430 


445 


452 


458 


465 


471 


4.2 
7 4.0 


US 


478 


484 


491 


407 


504 


510 


517 


523 


530 


536 


8 6.6 


60 


543 


540 


556 


562 


669 


576 


582 


588 


595 


601 


6.3 


670 


607 


614 


020 


627 


633 


640 


646 


653 


659 


666 




71 


672 


670 


685 


692 


608 


705 


711 


718 


724 


730 




72 


737 


743 


750 


756 


763 


760 


776 


782 


789 


705 




73 


802 


808 


814 


821 


827 


834 


840 


847 


853 


800 




74 


800 


872 


870 


885 


802 


898 


005 


911 


018 


024 




76 


030 


937 


043 


050 


950 


063 


909 


976 


982 


088 




76 


005 


*001 


*008 


*014 


*020 


HQ27 


*033 


*040 


*046 


*052 




77 


83050 


065 


072 


078 


085 


001 


097 


104 


110 


117 




78 


123 


120 


130 


142 


149 


156 


101 


168 


174 


181 




79 


187 


193 


200 


206 


213 


210 


225 


232 


238 


245 




680 


251 


257 


264 


270 


270 


283 


280 


296 


302 


308 




81 


315 


321 


327 


334 


340 


347 


353 


360 


366 


372 




82 


378 


385 


391 


308 


404 


410 


417 


423 


429 


436 


9 


83 


442 


448 


455 


461 


467 


474 


480 


487 


403 


490 


1 0.6 
























a 1.2 


84 


506 


512 


518 


525 


531 


637 


544 


550 


556 


503 


3 1.8 


85 


500 


676 


682 


588 


594 


601 


007 


613 


020 


626 


4 2.4 


86 


632 


630 


645 


651 


058 


664 


070 


677 


083 


689 


5 3.0 
6 3.6 
























7 4.2 


87 


606 


702 


708 


715 


721 


727 


734 


740 


740 


763 


S 4.8 


88 


750 


765 


771 


778 


784 


700 


707 


803 


809 


816 


9 0.4 


80 


822 


828 


835 


841 


847 


853 


800 


860 


872 


879 




600 


886 


801 


807 


004 


910 


016 


023 


920 


935 


042 




01 


048 


054 


000 


067 


973 


970 


085 


092 


008 


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02 


84011 


017 


023 


029 


030 


042 


048 


065 


061 


067 




03 


073 


080 


086 


002 


008 


105 


111 


117 


123 


130 




04 


136 


142 


148 


165 


161 


167 


173 


180 


186 


192 




05 


108 


205 


211 


217 


223 


230 


236 


242 


248 


255 




06 


201 


267 


273 


280 


280 


202 


298 


305 


311 


317 




97 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 




08 


380 


302 


398 


404 


410 


417 


423 


429 


436 


442 




00 


448 


464 


400 


406 


473 


479 


485 


491 


497 


604 




700 


510 


516 


522 


528 


535 


541 


647 


553 


559 


566 




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1 


3 


3 


4 


5 


6 


7 


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6 


7 


8 


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700 


84510 


516 


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528 


635 


541 


547 


663 


650 


500 




01 


572 


678 


584 


590 


597 


603 


609 


615 


621 


028 




02 


634 


640 


646 


652 


058 


005 


671 


877 


683 


089 




03 


606 


702 


708 


714 


720 


720 


733 


739 


746 


751 




04 


767 


763 


770 


770 


782 


788 


704 


800 


807 


813 




05 


819 


825 


831 


837 


844 


850 


856 


802 


808 


874 




08 


880 


887 


893 


899 


905 


911 


917 


924 


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942 


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954 


960 


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973 


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034 


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040 


052 


058 




7 


09 


065 


071 


077 


083 


080 


095 


101 


107 


114 


120 


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0.7 


710 


126 


132 


138 


144 


150 


150 


103 


160 


17fi 


181 


2 
3 


1.4 
2.1 


11 


187 


103 


190 


205 


211 


217 


224 


230 


230 


242 


4 


2.8 
8.5 


12 


248 


264 


260 


266 


272 


278 


285 


201 


207 


303 


Q 




13 


309 


315 


321 


327 


333 


330 


345 


362 


368 


304 


7 


4!o 
























8 


5.0 


14 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 





0.3 


15 


431 


437 


443 


440 


455 


461 


407 


473 


470 


485 




16 


401 


407 


503 


600 


516 


622 


628 


634 


510 


540 




17 


562 


568 


564 


570 


576 


582 


588 


504 


000 


OOfl 




18 


612 


018 


625 


631 


637 


643 


640 


655 


001 


007 




19 


673 


670 


685 


691 


097 


703 


700 


715 


721 


727 




720 


733~ 


730 


745 


751 


75T 


763 


769 


775 


781 


788 




21 


704 


800 


806 


812 


818 


824 


830 


836 


842 


848 






22 


854 


860 


866 


872 


878 


884 


800 


890 


002 


908 






23 


014 


920 


920 


932 


038 


944 


950 


066 


002 


908 


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24 


974 


980 


986 


992 


098 


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25 


80034 


040 


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052 


058 


064 


070 


076 


082 


088 


4 


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26 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 





3lO 
3.6 


27 


153 


159 


165 


171 


177 


183 


180 


195 


201 


207 


7 


4.2 


28 


213 


219 


225 


231 


237 


243 


240 


255 


201 


207 


8 
g 


t)A 


20 


273 


279 


285 


291 


297 


303 


308 


314 


320 


320 






730 


332 


338 


344 


350 


356 


362 


308 


374 


380 


386 




31 


302 


308 


404 


410 


415 


421 


427 


433 


430 


445 




32 


451 


467 


463 


469 


475 


481 


487 


403 


400 


504 




33 


610 


516 


522 


628 


534 


640 


546 


562 


568 


504 




34 


570 


576 


581 


687 


603 


500 


605 


611 


017 


623 




35 


620 


635 


641 


646 


652 


668 


664 


670 


676 


682 




36 


688 


604 


700 


706 


711 


717 


723 


,720 


735 


741 






37 


747 


763 


750 


764 


770 


776 


782 


788 


704 


800 





HL 


38 


806 


812 


817 


823 


829 


835 


841 


847 


863 


860 


1 
2 


i"n 


30 


804 


870 


876 


882 


888 


804 


000 


900 


Oil 


017 


3 


1:8 


740 


923~ 


929 


935 


941 


947 


953 


058 


964 


070 


970 


fi 


2.0 
2.15 


41 


iili" 


988 


904 


090 


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7 


3.0 
3.6 


42 


87040 


046 


052 


058 


064 


070 


076 


081 


087 


003 


8 


4.0 


43 


009 


105 


111 


116 


122 


128 


134 


140 


146 


151 





4.5 


44 


157 


163 


160 


175 


181 


186 


102 


198 


204 


210 




45 


216 


221 


227 


233 


239 


245 


251 


260 


202 


208 




46 


274 


280 


286 


291 


297 


303 


309 


316 


320 


320 




47 


332 


338 


344 


349 


365 


361 


307 


373 


370 


384 




48 


300 


396 


402 


408 


413 


419 


425 


431 


437 


442 




40 


448 


454 


460 


466 


471 


477 


483 


489 


406 


600 




750 


06 


512 


518 


523 


529 


635 


641 


647 


552 


568 




- N 





1 


a 


3 


4 


5 


6 


7 


8 


9 


PP 



14 



N 





1 


8 


3 


4 


5 


6 


7 


8 





PP 


750 

51 
52 
53 

54 
65 
56 

67 
58 
59 

760 

61 
62 
03 

64' 
05 
66 

67 
08 
69 

770 

71 
72 
73 

74 
75 
76 

77 
78 
79 

780 

81 
82 
83 

84 
85 
86 

87 
88 
89 

790 

01 
02 
03 

04 
05 
00 

97 
98 
00 

800 


87506 


512 


518 


523 


629 


535 


541 


547 


552 


558 


G 

l" 0.6 
2 1.2 
3 1.8 
4 2.4 
5 3.0 
6 3.6 
7 4.2 
8 4.8 
9 5.4 

5 

1 0.5 
1.0 
1.5 
4 2.0 
5 2.5 
3.0 
7 3.5 
8 4.0 
4.5 


564 
022 
679 

737 
795 
852 

010 
067 
88024 

081 

138 
105 
252 

300 
366 
423 

480 
536 
503 


570 
028 
685 

743 
800 
868 

015 
073 
030 


576 
633 
601 

749 
806 
864 

921 
078 
036 


581 
630 
607 

754 
812 
860 

027 
984 
041 


687 
645 
703 

700 
818 
875 

033 
090 
047 


593 
651 
708 

766 
823 
881 

038 
900 
053 


599 
056 
714 

772 
829 
887 

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604 
602 
720 

777 
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574 
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104 


110 


110 


121 


127 


133 


144 
201 
258 

315 
372 
420 

485 
542 
598 


150 
207 
204 

321 
377 
434 

401 
547 
604 


156 
213 
270 

320 
383 
440 

407 
553 
610 


101 
218 
275 

332 
380 
440 

502 
559 
615 


107 
224 
281 

338 
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451 

508 
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173 
230 
287 

343 
400 
467 

513 
570 
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178 
235 
202 

340 
400 
403 

510 
670 
032 


184 
241 
208 

355 
412 
468 

525 
581 
638 


190 
247 
304 

360 
417 
474 

530 
587 
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649 


065 


600 


660 


072 


077 


683 


689 


004 


700 


705 
762 
818 

874 
030 
986 

80042 
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154 

200" 

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321 
370 

432 
487 
542 

507 
053 
708 

703 


711 
707 
824 

880 
030 
002 

048 
104 
159 


717 
773 
820 

885 
941 
907 

053 
109 
165 


722 
770 
835 

801 
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050 
115 
170 


728 
784 
840 

897 
953 
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120 
170 


734 
700 
846 

902 
058 
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070 
120 
182 


730 
705 
852 

908 
064 
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070 
131 
187 


746 
801 
857 

913 
069 
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081 
137 
103 


750 
807 
803 

019 
976 
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087 
143 
108 


756 
812 
808 

925 
981 
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092 
148 
204 


215 


221 


226 


232 


237 


243 


248 


264 


260 


271 
320 
382 

437 
492 
548 

003 
058 
713 


270 
332 
387 

443 
498 
553 

000 
004 
710 


282 
337 
303 

448 
604 
550 

014 
000 
724 


287 
343 
308 

454 
509 
504 

620 
075 
730 


203 
348 
404 

450 
515 
570 

025 
080 
736 


208 
364 
400 

405 
520 
575 

631 
080 

741 


304 
300 
416. 

470 
520 
581 

030 
691 
740 


310 
365 
421 

476 
531 
586 

042 
697 
762 


315 
371 
420 

481 
537 
502 

647 
702 
757 


708 


774 


779 


785 


700 


700 


801 


807 


812 


818 
873 
027 

982 
00037 
001 

140 
200 
255 

300 


823 
878 
033 

988 
042 
097 

161 
206 
200 


820 
883 
038 

003 
048 
102 

157 
211 
260 


834 
880 
044 

098 
053 
108 

162 

217 
271 


840 
894 
949 

*004 
069 
113 

168 
222 
270 


845 
900 
066 

*009 
064 
119 

173 
227 
282 


851 
905 
060 

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009 
124 

170 
233 
287 


850 
Oil 
066 

*020 
075 
129 

184 
238 
293 


862 
916 
971 

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080 
135 

189 
244 
298 


867 
022 
077 

+031 
086 
140 

196 
249 
304 


314 


320 


325 


331 


336 


342 


347 


352 


358 


N 





1 


2 


3 


4 


S 





7 


8 





PP 



N 





1 


2 


3 


4 


S 


6 


7 


8 


9 


PP 


800 


90300 


314 


320 


325 


331 


336 


342 


347 


352 


358 




01 


363 


360 


374 


380 


385 


390 


396 


401 


407 


412 




02 


417 


423 


428 


434 


430 


445 


450 


455 


461 


466 




03 


472 


477 


482 


488 


493 


400 


504 


609 


516 


520 




04 


526 


531 


636 


542 


547 


553 


558 


563 


569 


574 




05 


580 


685 


590 


596 


601 


607 


612 


617 


623 


628 




06 


634 


630 


644 


650 


655 


660 


666 


671 


677 


682 




07 


687 


603 


698 


703 


700 


714 


720 


725 


730 


736 




08 


741 


747 


762 


757 


763 


768 


773 


779 


784 


789 




09 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




810 


840 


854 


850 


865 


870 


875 


881 


886 


801 


807 




11 


002 


907 


013 


018 


024 


020 


034 


040 


945 


950 




12 


056 


061 


066 


972 


977 


082 


088 


003 


098 


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13 


01000 


014 


020 


025 


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036 


041 


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052 


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6 
























1 oT" 


14 


062 


068 


073 


078 


084 


080 


094 


100 


105 


110 


2 1.2 


15 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 


3 1.8 


16 


160 


174 


180 


185 


190 


106 


201 


206 


212 


217 


t P 


17 


222 


228 


233 


238 


243 


240 


254 


259 


265 


270 


? ii 


18 


276 


281 


286 


201 


297 


302 


307 


312 


318 


323 


8 4i8 


18. 


328 


334 


339 


344 


350 


355 


360 


365 


371 


376 


0.4 


830 


381 


387 


392 


307 


403 


408 


413 


418 


424 


420 




21 


434 


440 


445 


450 


455 


461 


466 


471 


477 


482 




22 


487 


492 


498 


503 


508 


514 


510 


524 


529 


535 




23 


540 


645 


551 


556 


561 


566 


572 


577 


582 


587 




24 


593 


598 


603 


609 


614 


610 


624 


630 


635 


640 




25' 


646 


651 


656 


661 


666 


672 


677 


682 


687 


693 




26 


698 


703 


709 


714 


710 


724 


730 


736 


740 


745 




27 


761 


756 


761 


766 


772 


777 


782 


787 


793 


708 




28 


803 


808 


814 


819 


824 


820 


834 


840 


845 


850 




29 


856 


861 


866 


871 


876 


882 


887 


892 


897 


003 




830 


908 


913 


918 


024 


020 


034 


939 


044 


050 


955 




31 


960 


065 


971 


076 


081 


986 


991 


007 


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32 


92012 


018 


023 


028 


033 


038 


044 


040 


054 


059 


5 


33 


065 


070 


076 


080 


085 


091 


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101 


106 


111 


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2 1.0 


34 


117 


122 


127 


132 


137 


143 


148 


153 


168 


163 


3 1.5 


35 


169 


174 


179 


184 


180 


106 


200 


205 


210 


215 


4 2.0 


36 


221 


226 


231 


236 


241 


247 


252 


257 


262 


207 


5 2.5 
6 3.0 


37 


273 


278 


283 


288 


203 


208 


304 


300 


314 


319 


7 .5 
8 4.0 


38 


324 


330 


335 


340 


345 


350 


355 


361 


366 


371 


4.5 


39 


376 


381 


387 


302 


307 


402 


407 


412 


418 


423 




840 


428 


433 


438 


443 


440 


454 


459 


464 


469 


474 




41 


480~ 


485 


490 


495 


500 


605 


611 


516 


521 


526 




42 


531 


536 


542 


547 


652 


657 


562 


567 


572 


578 




43 


583 


588 


593 


508 


603 


600 


614 


619 


624 


629 




44 


634 


630 


646 


650 


655 


660 


665 


670 


675 


681 




45 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 




46 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 




47 


788 


703 


799 


804 


800 


814 


819 


824 


829 


834 




48 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 




40 


891 


896 


901 


906 


911 


016 


921 


927 


932 


937 




850 


942 


947 


952 


067 


062 


967 


973 


978 


983 


088 




MT 





1 


2 


3 


4 


5 


8 


7 


8 


9 


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16 



N 





1 


a 


3 


4 


5 


G 


7 


8 





V 


P 


850 


02942 


947 


052 


957 


962 


967 


073 


078 


083 


088 






51 
52 
53 

54 
55 
56 

57 
58 


993 
93044 
095 

146 
107 
247 

298 
340 


008 
040 
100 

151 
202 
252 

303 
354 


*003 
054 
105 

156 
207 
258 

308 
350 


*008 
050 
110 

161 
212 
263 

313 
364 


*013 
064 

115 

166 
217 
268 

318 
369 


*018 
060 
120 

171 
222 
273 

323 
374 


*024 
075 
125 

176 
227 
278 

328 
370 


*029 
080 
131 

181 
232 
283 

334 
384 


*034 
085 
136 

186 
237 
288 

330 
380 


*039 
000 
141 

102 
242 
293 

344 
304 




6 


59 


309 


404 


400 


414 


420 


425 


430 


435 


440 


445 


1 


0.6 


860 


450 


455 


460 


465 


470 


476 


480 


485 


400 


495 


3 


1.8 


61 
62 
63 

64 
65 
66 

67 
68 
60 


600 
551 
601 

651 
702 
752 

802 
852 
002 


505 
556 
606 

056 
707 
757 

807 
857 
907 


510 
661 
611 

661 
712 
762 

812 
862 
912 


515 
506 
616 

666 
717 
767 

817 
807 
917 


520 
571 
621 

671 
722 
772 

822 
872 
922 


526 
576 
626 

676 
727 
777 

827 
877 
027 


531 
681 
631 

682 
732 
782 

832 
882 
932 


536 
586 
636 

687 
737 
787 

837 
887 
937 


641 
501 
641 

692 
742 
702 

842 
802 
942 


646 
696 
646 

697 
747 
707 

847 
897 
047 


5 

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3.0 
3.0 
4.2 
4.8 
6.4 


870 


952 


057 


002 


967 


072 


077 


982 


987 


092 


007 






71 
72 
73 

74 
75 
76 

77 
78 
70 


94002 
002 
101 

151 
201 
250 

300 
349 
390 


007 
057 
106 

156 
206 
255 

305 
354 
401 


012 
062 
111 

161 
211 
260 

310 
359 
409 


017 
067 
116 

166 
216 
265 

315 
364 
414 


022 
072 
121 

171 
221 
270 

320 
369 
419 


027 
077 
126 

176 
226 
275 

325 
374 
424 


032 
082 
131 

181 
231 
280 

330 
379 
420 


037 
086 
136 

186 
236 
285 

335 
384 
433 


042 
091 
141 

191 
240 
290 

340 
380 
438 


047 
090 
146 

100 
245 
205 

345 
394 
443 


2 
3 

5 


a 7 

9 


5 

0.6 
1.0 
1.5 
2.0 
2.5 
3.0 
3.6 
4.0 
4.6 


880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


498 






81 
82 
83 

84 
85 
86 

87 
88 
89 


408 
547 
596 

645 
694 
743 

702 
841 
800 


503 
552 
601 

650 
000 
748 

707 
846 
806 


607 
657 
606 

655 
704 
753 

802 
851 
900 


512 
562 
611 

660 
709 
758 

807 
856 
905 


617 
667 
616 

665 
714 
763 

812 
861 
010 


522 

571 
621 

670 
719 
708 

817 
866 
916 


527 
576 
626 

675 
724 
773 

822 
871 
910 


532 
581 
630 

680 
729 
778 

827 
876 
024 


537 
586 
635 

685 
734 
783 

832 
880 
020 


542 
591 
640 

689 
738 
787 

836 
885 
934 


2 
3 


4 

0,4 
0.8 
1.2 


800 


039 


914 


949 


954 


050 


963 


968 


073 


978 


983 


5 


1.0 
2.0 


91 
92 
93 

94 
95 
96 

97 
98 
90 


088 
05036 
085 

134 
182 
231 

270 
328 
376 


003 
041 
000 

130 
187 
236 

284 
332 
381 


008 
046 
005 

143 
102 
240 

280 
337 
386 


*002 
051 
100 

148 
197 
245 

294 
342 
300 


*007 
056 
105 

153 
202 
250 

209 
347 
305 


*012 
061 
109 

158 
207 
255 

303 
352 
400 


*017 
066 
114 

163 
211 
260 

308 
357 
405 


*022 
071 

no 

168 
216 
265 

313 
361 
410 


*027 
076 
124 

173 
221 
270. 

318 
366 
416 


*032 
080 
120 

177- 
226 
274 

323 

371 
419 



7 
8 
B 


2.4 
3.8 
3.2 
3.0 


000 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 






N 





1 


% 


3 


4 


S 


6 


7 


8 


9 


1 


P 



17 



N 





1 


2 


3 


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5 


6 


7 


8 


9 


PP 


900 


95424 


429 


434 


439 


444 


448 


453 


458 


403 


408 




01 


472 


477 


482 


487 


492 


497 


501 


500 


611 


516 




02 


521 


525 


530 


535 


540 


545 


650 


554 


550 


564 




03 


569 


574 


678 


,583 


588 


593 


598 


002 


607 


612 




04 


617 


622 


626 


631 


636 


641 


646 


650 


055 


060 




05 


665 


670 


674 


679 


684 


689 


694 


098 


703 


708 




06 


713 


718 


722 


727 


732 


737- 


742 


740 


751 


756 




07 


761 


766 


770 


775 


780 


785 


789 


794 


790 


804 




08 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 




09 


856 


861 


866 


871 


876 


880 


885 


890 


805 


899 




910 


904 


909 


914 


918 


923 


028 


933 


938 


942 


947 




11 


952 


967 


901 


960 


971 


076 


980 


985 


000 


005 




12 


999 


+004 


+009 


+014 


+019 


*023 


*028 


+033 


+038 


+042 




13 


96047 


052 


057 


061 


066 


071 


076 


080 


085 


090 


5 
























1 00 


14 


005 


099 


104 


100 


114 


118 


123 


128 


133 


137 


2 1.0 


15 


142 


147 


152 


150 


161 


166 


171 


176 


180 


185 


3 1.5 


16 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


4 2.0 
5 2.0 


17 


237 


242 


246 


251 


250 


261 


265 


270 


275 


280 


3.0 
7 3.5 


'18 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 




19 


332 


336 


341 


340 


360 


355 


360 


305 


369 


374 


4lfl 


920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 




21 


426 


431 


435 


440 


445 


450 


454 


459 


464 


468 




22 


473 


478 


483 


487 


492 


497 


501 


500 


511 


516 




23 


520 


525 


530 


534 


539 


544 


548 


553 


558 


502 




24 


567 


572 


577 


581 


586 


591 


695 


000 


605 


600 




25 


614 


619 


624 


628 


633 


G38 


642 


047 


652 


050 




26 


661 


666 


670 


675 


680 


685 


689 


094 


609 


703 




27 


708 


713 


717 


722 


727 


731 


730 


741 


745 


750 




28 


755 


759 


704 


709 


774 


778 


783 


788 


792 


797 




29 


802 


806 


811 


810 


820 


825 


830 


834 


830 


844 




930 


848~ 


853 


868 


862 


867 


872 


870 


881 


880 


800 




31 


895 


900 


904 


909 


914 


918 


923 


928 


032 


937 




32 


942 


946 


951 


956 


960 


965 


970 


974 


979 


984 


4 


. 33 


988 


993 


997 


+002 


+007 


+011 


+016 


+021 


+025 


+030 




34 


97035 


039 


044 


049 


053 


058 


063 


067 


072 


077 


2 Oil 
3 1.2 


35 


081 


086 


090 


095 


100 


104 


109 


114 


118 


123 


4 1.0 


36 


128 


132 


137 


142 


140 


151 


155 


160 


165 


169 


5 2.0 
6 2.4 


37 


174 


179 


183 


188 


102 


197 


202 


206 


211 


216 


7 2.8 
8 32 


38 


220 


225 


230 


234 


239 


243 


248 


253 


257 


202 


o a!o 


39 


267 


271 


276 


280 


285 


290 


294 


209 


304 


308 




910 


313 


317 


322 


327 


331 


330 


340 


346 


350 


354 




41 


359 


364 


368 


373 


377 


382 


387 


391 


390 


400 




42 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




43 


451 


466 


460 


465 


470 


474 


470 


483 


488 


493 




44 


497 


502 


506 


611 


510 


620 


525 


629 


534 


530 




45 


543 


548 


552 


667 


562 


666 


571 


575 


580 


585 




46 


589 


694 


598 


603 


607 


612 


017 


621 


626 


630 




47 


635 


640 


644 


649 


663 


658 


663 


667 


072 


676 




48 
49 


681 
727 


685 
731 


690 
736 


695 
740 


699 
745 


704 
749 


708 
754 


713 
759 


717 
763 


722 
768 




950 


772 


777 


782 


786 


791 


7.95 


800 


804 


809 


813 




N 





1 


* 


8 


A 


5 


6 


7 


8 


9 


PP 



18 



N 





1 


2 


3 


4 


A 


6 


7 


8 





PP 


050 

51 
52 
C3J 

54 
55 
50 

57 
58 
50 

060 

61 
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G3 

04 
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00 

07 
08 
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970 

71 
72 
73 

74 
75 
70 

77 
78 
70 

080 

81 
82 
83 

84 
85 
80 

87 
88 
80 

000 

01 
02 
03 

04( 
05 
00 

07 
08 
00 

1000 


97772 


777 


782 


780 


701 


705 


800 


804 


809 


813 




1 0.5 
2 1.0 
3 1.5 
4 2.0 
B 2.6 
3.0 
7 8.5 
8 4.0 
4.6 

4 
T-5T" 

2 0.8 
3 1.2 
4 1.6 
6 2.0 
6 2.4 
7 2.8 
8 3.2 
3.0 


818 
864 
000 

055 
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040 

001 
137 
182 


823 
868 
914 

050 
005 
050 

000 
141 
180 


827 
873 
018 

904 
000 
055 

100 
140 
101 


832 
877 
023 

008 
014 
050 

105 
150 
105 


830 
882 
028 

073 
010 
064 

100 
165 
200 


841 
886 
032 

078 
023 
068 

114 
159 
204 


845 
801 
037 

082 
028 
073 

118 
164 
209 


850 
896 
041 

987 
032 
078 

123 
108 
214 


855 
900 
946 

901 
037 
082 

127 
173 
218 


859 
905 
960 

996 
041 
087 

132 
177 
223 


227 


232 


230 


241 


246 


260 


254 


250 


263 


268 


272 
318 
363 

408 
463 
498 

543 
CSS 
032 

677 


277 
J22 
307 

412 
457 
502 

547 
502 
037 


281 
327 
372 

417 
402 
507 

552 
597 
041 


280 
331 
376 

421 
466 
511 

550 
001 
046 


200 
330 
381 

426 
471 
510 

561 
605 
050 


295 
340 
385 

430 
476 
520 

505 
610 
655 


200 
345 
300 

435 
480 
525 

670 
614 
050 


304 
349 
304 

439 
484 
529 

674 
610 
064 


308 
354 
300 

444 
480 
534 

579 
623 
668 


313 
368 
403 

448 
403 
538 

583 
628 
673 


082 


080 


001 


605 


700 


704 


700 


713 


717 


722 
707 
811 

856 
900 
046 

080 
00034 
078 

iiif 


720 
771 
810 

860 
006 
040 

004 
038 
083 


731 
770 
820 

865 
000 
054 

008 
043 
087 


736 
780 
825 

800 
014 
058 

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047 
002 


740 
784 
820 

874 
018 
063 

*007 
052 
006 


744 
789 
834 

878 
023 
007 

012 
056 
100 


740 
703 
838 

883 
027 
972 

*016 
061 
105 


753 
798 
843 

887 
032 
076 

+021 
065 
100 


758 
802 
847 

802 
036 
981 

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069 
114 


762 
807 
851 

896 
941 
985 

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074 
118 


127 


131 


136 


140 


145 


140 


154 


158 


162 


107 
211 
255 

300 
344 
388 

432 
470 
520 

eoT 


171 
210 
200 

304 
348 
302 

430 
480 
524 


176 
220 
204 

308 
352 
300 

441 
484 
528 


180 
224 
209 

313 
357 
401 

445 
480 
533 


185 
220 
273 

317 
361 
406 

440 
403 
637 


189 
233 
277 

322 
360 
410 

464 
408 
542 


103 
238 
282 

326 
370 
414 

458 
602 
640 


108 
242 
286 

330 
374 
419 

403 
606 
650 


202 
247 
201 

336 
379 
423 

407 
511 
555 


207 
251 
295 

339 
383 
427 

471 
615 
550 


508 


572 


577 


581 


585 


600 


594 


509 


603 


007 
051 
095 

730 
782 
820 

870 
013 
957 


612 
050 
000 

743 
787 
830 

874 
917 
901 


016 
060 
704 

747 
701 
835 

878 
922 
065 


621 
004 
708 

752 
795 
830 

883 
026 
070 


625 
660 

712 

750 
800 
843 

887 
930 
074 


620 
673 
717 

700 
804 
848 

801 
935 
078 


034 
677 
721 

705 
808 
852 

800 
030 
083 


638 
082 
726 

769 
813 
866 

900 
044 
087 


642 
686 
730 

774 
817 
861 

904 
948 
901 


647 
691 
734 

778 
822 
806 

909 
052 
096 


00000 


004 


009 


013 


017 


022 


026 


030 


036 


030 


N 





1 


a 


8 


4 


5 


6 


7 


8 





PP 



19 



N 





i 


2 


3 


4 


5 


6 


7 


8 


9 


1000 

1001 
1002 
1003 


0000000 


0434 


0869 


1303 


1737 


2171 


2605 


3030 


3473 


3007 


4341 
8677 
001 3009 


4775 
9111 
3442 


5208 
9544 
3875 


5642 
9977 
4308 


6076 
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4741 


6610 
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6174 


6943 
+1277 
5607 


7377 
*1710 
6039 


7810 
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6472 


8244 
+2670 
6005 


1004 
1005 
1006 


7337 
002 1661 
5980 


7770 
2093 
6411 


8202 
2625 
6843 


8636 
2957 
7275 


9067 
3389 
7706 


9499 
3821 
8138 


9932 
4253 
8560 


*0384 
4685 
0001 


*0796 
5116 
0432 


+1228 
5S48 
0803 


1007 
1008 
1000 

1010 

1011 
1012 
1013 


003 0295 
. 4605 
8912 


0726 
5036 
9342 


1157 
5467 
9772 


1588 
6898 
*0203 


2019 
6328 
*0033 


2451 
6760 
*1063 


2882 
7190 
*1403 


3313 
7620 
*1024 


3744 
8051 
*2364 


4174 
8481 
+2784 


0043214 


3644 


4074 


4504 


4033 


5363 


5703 


6223 


6052 


7082 


7612 
0051805 
6094 


7941 
2234 
6523 


8371 
2663 
6952 


8800 
3092 
7380 


9229 
3521 
7809 


9659 
3960 
8238 


*0088 
4370 
8666 


*0617 
4808 
0004 


*0047 
5237 
9523 


*137fl 
6000 
0(151 


1014 
1016 
1016 


006 0380 
4660 
8937 


0808 
5088 
9365 


1236 
5516 
9792 


1664 
6944 
+0219 


2092 
6372 
*0647 


2521 
6799 
*1074 


2940 
7227 
*1601 


3377 
7656 
*1928 


3806 
8082 
*2355 


4233 
8. r >lQ 
*2782 


1017 
1018 
1019 

1020 

1021 
1022 
1023 


007 3210 
7478 
0081742 


3637 
7904 
2168 


4064 
8331 
2594 


4490 
8767 
3020 


4017 
9184 
3440 


5344 
9010 
3872 


5771 
*0037 
4208 


0198 
+0403 
4724 


0024 
+0880 
5160 


7061 
*131fl 
6,170 


6002 


6427 


6853 


7279 


7704 


8130 


8550 


8981 


0407 

~3669~ 
7007 
*2151 


0832 


009 0257 
4509 
8758 


0683 
4934 
9181 


1108 
6359 
9605 


1533 
5784 
*0030 


1959 
6208 
+0454 


2384 
6633 
+0878 


2800 
7058 
*1303 


3234 
7483 
+1727 


4084 
81192 
+2675 


1024 
1026 
1026 


010 3000 
7239 
Oil 1474 


3424 
7662 
1897 


3848 
8086 
2320 


4272 
8510 
2743 


4696 
8933 
3166 


5120 
9357 
3590 


5544 
9780 
4013 


6967 
*0204 
4436 


0301 
+0027 
4859 


681/5 
+1050 
5282 


1027 
1028 
1029 

1080 

1031 
1032 
1033 


5704 
9931 
012 4154 


6127 
*0364 
4576 


6550 
*0776 
4998 


6973 
*1198 
5420 


7396 
+1621 
5842 


7818 
*2043 
0264 


8241 
*2465 
0685 


8064 
+2887 
7107 


0086 
+3310 
7520 


0600 
+3732 
7051 


8372 


8794 


9215 


9637 


+0059 


*0480 


*0901 


+1323 


+1744 


+2106 


013 2587 
6797 
014 1003 


3008 
7218 
1424 


3429 
7639 
1844 


3850 
8059 
2264 


4271 
8480 
2685 


4692 
8901 
3105 


5113 
9321 
3625 


5534 
9742 
3945 


5055 
*0102 
4305 


0370 
+0683 
4785 


1034 
1035 
1036 


5205 
9403 
015 3598 


5625 
9823 
4017 


6046 
*0243 
4436 


6465 
*0662 
4866 


6885 
*1082 
5274 


7305 
*1501 
5693 


7725 
+1920 
6112 


8144 
+2340 
0531 


8564 
*2760 
0950 


8084 
*3178 
7300 


1037 
1038 
1039 

1040 

1041 
1042 
1043 


7788 
016 1974 
6155 


8206 
2392 
6573 


8625 
2810 
6991 


9044 
3229 
7409 


9462 
3647 
7827 


9881 
4065 
8245 


*0300 
4483 
8663 


*0718 
4001 
9080 


+1137 
5319 
0408 


*1556 
5737 
0010 


017 0333 


0751 


1168 


1586 


2003 


2421 


2838 


3256 


3673 


4000 


4507 
8677 
0182843 


4924 
9094 
3259 


5342 
9511 
3676 


5759 
9927 
4092 


6176 
*0344 
4508 


6593 
+0761 
4926 


7010 
+1177 
5341 


7427 
*1594 
6757 


7844 
*2010 
0173 


8200 
+2427 
6580 


1044 
1045 
1046 


7005 
019 1163 
5317 


7421 
1578 
5732 


7837 
1994 
6147 


8253 
2410 
6562 


8669 
2825 
6977 


9084 
3240 
7392 


9500 
3656 
7807 


9916 
4071 
8222 


+0332 
4480 
8037 


+0747 
4002 
0062 


1047 
1048 
1049 

1000 


9467 
020 3613 
7765 


9882 
4027 
8169 


*0296 
4442 
8583 


*0711 
4856 
8997 


*1126 
5270 
9411 


*1540 
6684 
9824 


+1955 
6099 
*0238 


+2309 
6513 
+0652 


*2784 
0927 
+1066 


*3108 
7341 
+1470 


021 1893 


2307 


2720 


3134 


3547 


3961 


4374 


4787 


5201 


6614 


N 


e 


1 


2 


3 


A 


5 


6 


V 


8 






20 



N 





1 


a 


8 


4 


5 


6 


7 


8 





1050 

1061 
1052 
1053 


021 1893 


2307 


2720 


3134 


3547 


3001 


4374 


4787 


5201 


5614 


0027 
022 0157 
4284 


0440 
0570 
4606 


0854 
0983 
6109 


7207 
1390 
5621 


7680 
1808 
5933 


8093 
2221 
0345 


8600 
2034 
6758 


8919 
3046 
7170 


9332 
3459 
7582 


9745 
3871 
7994 


1064 
1055 
1056 


8406 
023 2525 
0030 


8818 
2036 
7060 


9230 
3348 
7402 


9042 
3759 
7873 


+0054 
4171 
8284 


*0406 
4582 
8095 


+0878 
4994 
9106 


*1289 
5405 
9517 


+1701 
5817 
9928 


*2113 
6228 
+0339 


1057 
1058 
1059 

1000 

1001 
1002 
1003 


024 0750 
4857 
8960 


1101 
5207 
9370 


1672 
6678 
9780 


1982 
6088 
+0190 


2393 
0498 
+0600 


2804 
0900 
+1010 


3214 
7310 
*1419 


3625 
7720 
*1829 


4030 
8139 
+2230 


4446 
8540 
*2649 


026 3059 


3408 


3878 


4288 


4097 


5107 


5510 


5020 


0336 


6744 


7164 
026 1245 
6333 


7603 
1054 
5741 


7972 
2003 
6150 


8382 
2472 
6558 


8701 
2881 
0967 


9200 
3289 
7375 


9009 
3098 
7783 


*0018 
4107 
8192 


+0427 
4515 
8600 


+0836 
4924 
9008 


1004 
1005 
1000 


9416 
027 3496 
7572 


9824 
3904 
7979 


*0233 
4312 
8387 


*0641 
4719 
8794 


*1049 
6127 
9201 


+1457 
5535 
9009 


+1865 
5942 
+0010 


1-2273 
6350 
*0423 


+2680 
0757 
+0830 


*3088 
7166 
+1237 


1007 
1008 
1009 

1070 

1071 
1072 
1073 


0281044 
5713 
9777 


2051 
0119 
+0183 


2458 
0520 
+0500 


2865 
0932 
*0990 


3272 
7339 
*1402 


3679 
7746 
*1808 


4086 
8162 
"2214 


4492 
8558 
*2020 


4899 
8904 
+3020 


6306 
9371 
+3432 


029 3838 


4244 


4049 


5065 


5401 


5867 


0272 


6078 


7084 


7489 


7895 
030 1948 
6997 


8300 
2353 
0402 


8706 
276S 
6807 


9111 
,3163 
7211 


9516 
3568 
7610 


9922 
3973 
8020 


*0327 
4378 
8425 


+0732 
4783 
8830 


+1138 
51S8 
9234 


+1543 
5592 
9638 


1074 
1075 
1070 


0310043 
4085 
8123 


0447 
4489 
8520 


0851 
4893 
8930 


1256 
5290 
9333 


1000 
5700 
9737 


2064 
0104 
*0140 


2468 
6508 
+0544 


2872 
0912 
+0947 


3277 
7315 
*1360 


3681 
7719 
+1754 


1077 
1078 
1079 

1080 

1081 
1082 
1083 


032 2157 
0188 
033 0214 


2560 
0590 
0017 


2903 
0993 
1010 


3367 
7396 
1422 


3770 
7799 
1824 


4173 
8201 
2220 


4676 
8004 
2029 


4979 
9007 
3031 


5382 
9400 
3433 


5785 
9812 
3835 


4238 


4640 


5042 


5444 


5846 


6248 


6050 


7052 


7453 


7855 


8267 
0342273 
0285 


8659 
2074 
0680 


9000 
3075 
7087 


9402 
3477 
7487 


9864 
3878 
7888 


+0266 
4279 
8289 


+0607 
4080 
8690 


+1008 
5081 
9091 


+1470 
5482 
0491 


*1871 
6884 
9892 


1084 
1085 
1080 


035 0293 
4297 
8298 


0003 
4608 
8008 


1004 
6098 
9098 


1495 
5498 
9498 


1895 
5898 
9898 


2290 
0208 
+0297 


2696 
6098 
"0007 


3096 
7098 
+1097 


3407 
7498 
+1490 


3807 
7808 
*1806 


1087 
1088 
1089 

1000 

1001 
1092 
1093 


030 2295 
0289 
037 0279 


2606 
0688 
0078 


3004 
7087 
1070 


3494 
7480 

1476 


3893 
7885 
1874 


4203 
8284 
2272 


4602 
8083 
2071 


5091 
9082 
3070 


5491 
9481 
3408 


6800 
9880 
3867 


4205 

8248 
038 2220 
6202 


4003 


5062 


5400 


5868 


0257 


0655 


7053 


7451 


7849 


8040 
2024 
0500 


9044 
3022 
0090 


9442 
3419 
7393 


0839 
3817 
7791 


+0237 
4214 
8188 


+0035 
4612 
8585 


+1033 
6000 
8982 


+1431 
5407 
9370 


+1829 
6804 
9770 


1094 
1095 
1096 


039 0173 
4141 
8100 


0570 
4538 
8502 


0967 
4934 
8898 


1304 
6331 
9294 


1701 
6727 
0690 


2158 
0124 
*008Q 


2554 
,6520 
10482 


2051 
0917 
+0878 


3348 
7313 
+1274 


3746 
7709 
+1070 


1097 
1098 
1090 

1100 


040 2060 
0023 
9977 


2462 
0410 
*0372 


2858 
0814 
"0707 


3254 
7210 
+1102 


3660 
7005 
+1G57 


4045 
8001 
+1952 


4441 
8306 
+2347 


4837 
8701 
+2742 


5232 
9187 
*3137 


5028 
9682 
+3B32 


041 3927 


4322 


4710 


51U 


5506 


6900 


6295 


0690 


7084 


7479 


N 





1 


9 


3 


4 


5 


6 


7 


8 


9 



21 



TABLE HI The Number of Each Day of the Year 



fa hi 
Q n 

Ss 


d 

4 


i 


| 


1 


I 


1 


! 


1 


i 


i 


1 


1 


E 

31 


I 


1 


32 


00 


91 


121 


152 


182 


213 


244 


274 


305 


335 


i 


2 


2 


33 


61 


92 


122 


153 


183 


214 


245 


275 


306 


336 


2 


3 


3 


34 


62 


93 


123 


154 


184 


215 


246 


276 


307 


337 


3 


4 


4 


35 


63 


94 


124 


155 


185 


216 


247 


277 


308 


338 


4 


ft 


6 


36 


64 


95 


126 


156 


180 


217 


248 


278 


309 


339 


& 


6 


6 


37 


65 


9G 


126 


157 


187 


218 


240 


279 


310 


340 





7 


7 


38 


66 


97 


127 


158 


188 


219 


250 


280 


311 


341 


7 


8 


8 


39 


67 


98 


128 


150 


189 


220 


261 


281 


312 


342 


8 


9 


S 


40 


68 


99 


120 


160 


190 


221 


252 


282 


313 


343 


9 


10 


10 


41 


69 


100 


130 


161 


191 


222 


253 


283 


314 


344 


10 


11 


11 


42 


70 


101 


131 


162 


192 


223 


254 


284 


315 


345 


11 


12 


12 


43 


71 


102 


132 


163 


193 


224 


255 


285 


316 


340 


12 


18 


13 


44 


72 


103 


133 


164 


194 


225 


256 


286 


317 


347 


13 


14 


14 


45 


73 


104 


134 


165 


195 


226 


257 


287 


318 


348 


14 


15 


15 


46 


74 


105 


135 


166 


196 


227 


258 


288 


310 


340 


15 


16 


16 


47 


75 


106 


136 


167 


197 


228 


260 


280 


320 


350 


10 


17 


17 


48 


76 


107 


137 


" 168 


198 . 


229 


260 


200 


321 


351 


17 


18 


18 


49 


77 


108 


138 


169 


199 


230 


261 


201 


322 


352 


18 


19 


19 


50 


78 


109 


139 


170 


200 


231 


202 


202 


323 


353 


19 


20 


20 


51 


79 


110 


140 


171 


201 


232 


203 


203 


324 


354 


20 


21 


21 


52 


80 


111 


141 


172 


202 


233 


264 


204 


325 


355 


21 


22 


22 


53 


'81 


112 


142 


173 


203 


234 


265 


205 


320 


356 


22 


23 


23 


54 


82 


113 


143 


174 


204 


235 


260 


206 


327 


357 


23 


24 


24 


55 


83 


114 


144 


175 


205 


236 


267 


207 


328 


358 


24 


25 


25 


56 


84 


115 


145 


176 


206 


237 


268 


208 


320 


350 


25 


2G 


20 


57 


85 


116 


146 


177 


207 


238 


269 


200 


330 


360 


26 


7 


27 


58 


86 


117 


147 


178 


208 


239 


270 


300 


331 


361 


27 


28 


28 


50 


87 


118 


148 


179 


209 


240 


271 


301 


332 


362 


28 


29 


29 




88 


119 


149 


180 


210 


241 


272 


302 


333 


363 


29 


30 


30 




89 


120 


150 


181 


211 


242 


273 


SOU 


334 


364 


30 


31 


31 




90 




151 




212 


243 




304 




365 


31 



NOTB. la leap years, alter February 28, add 1 to the tabulated number. 



TABLE IV Ordinary and Exact Interest at 1% on $10,000 



Ezact Interest for 1 to 865 Days 


Ordinary Interest for 1 to 360 Days 


Days 


Interest 


For days below add to 
interest column 


Days 


Interest 


For days below add to 
interest column 


980 


40 


60 


$80 


120 


940 


160 


980 


1 


$ .2739726 


74 


147 


220 


293 


1 


$ .2777778 


73 


146 


217 


280 


2 


.5479452 


75 


148 


221 


294 


2 


.5555556 


74 


140 


218 


290 


3 


.8219178 


76 


140 


222 


295 


3 


.8333333 


76 


147 


219 


291 


4 


1.0958904 


77 


150 


223 


296 


4 


1.1111111 


76 


148 


220 


292 


5 


1.3098030 


78 


161 


224 


297 


5 


1.3888880 


77 


140 


221 


293 





1.0438356 


79 


152 


225 


298 





1.0006607 


78 


150 


222 


294 


7 


1.9178082 


80 


163 


220 


299 


7 


1.0444444 


70 


151 


223 


296 


8 


2.1917808 


81 


154 


227 


300 


8 


2.2222222 


SO 


152 


224 


296 


e 


2.4057534 


82 


155 


228 


301 


9 


2.5000000 


81 


153 


225 


297 


10 


2.7397200 


83 


150 


229 


302 


10 


2.7777778 


82 


154 


226 


298 


11 


3.0130980 


84 


157 


230 


303 


11 


3.0566656 


83 


155 


227 


299 


12 


3.2870712 


85 


158 


231 


304 


12 


3.3333333 


84 


150 


228 


300 


13 


3.5010438 


86 


159 


232 


305 


13 


3.0111111 


85 


157 


229 


301 


14 


3.8350104 


87 


100 


233 


300 


14 


3.8888880 


80 


158 


230 


302 


15 


4.10Q5890 


88 


101 


234 


307 


15 


4.1060007 


87 


150 


231 


303 


10 


4.383501G 


89 


102 


235 


308 


10 


4.4444444 


88 


160 


232 


304 


17 


4.0575342 


90 


103 


236 


300 


17 


4,7222222 


89' 


101 


233 


305 


IS 


4.9315008 


91 


104 


237 


310 


18 


5.0000000 


90 


102 


234 


306 


19 


5.2054795 


92 


105 


238 


311 


19 


5.2777778 


91 


163 


235 


307 


20 


6.4794521 


93 


100 


230 


312 


20 


5.5555556 


92 


164 


236 


308 


21 


5.7534247 


94 


107 


240 


313 


21 


5.8333333 


03 


105 


237 


309 


22 


0.0273973 


95 


108 


241 


314 


22 


6.1111111 


94 


106 


238 


310 


23 


0.3013099 


90 


109 


242 


315 


23 


6.3888889 


05 


167 


239 


311 


24 


0.5753425 


97 


170 


243 


310 


24 


0.6000007 


00 


108 


240 


312 


25 


0.8493151 


98 


171 


244 


317 


25 


0.0444444 


07 


109 


241 


313 


20 


7.1232877 


99 


172 


245 


318 


26 


7.2222222 


98 


170 


242 


314 


27 


7.3972003 


100 


173 


246 


319 


27 


7.5000000 


99 


171 


243 


315 


28 


7.0712320 


101 


174 


2-17 


320 


28 


7.7777778 


100 


172 


244 


310 


29 


7.9452055 


102 


175 


248 


321 


20 


8.0555566 


101 


173 


246 


317 


30 


8.2191781 


103 


170 


249 


322 


30 


8.3333333 


102 


174 


246 


318 


31 


8.4931507 


104 


177 


250 


323 


31 


8.6111111 


103 


175 


247 


319 


32 


8.7071233 


105 


178 


251 


324 


32 


8.8888889 


104 


176 


248 


320 


33 


9.0410959 


106 


179 


252 


325 


33 


9.1006007 


105 


177 


249 


321 


34 


9.3150085 


107 


180 


253 


326 


34 


9.4444444 


100 


178 


260 


322 


3C 


9.5890411 


108 


181 


254 


327 


35 


9.7222222 


107 


179 


251 


323 


30 


9.8630137 


109 


182 


255 


328 


30 


10.0000000 


108 


ISO 


252 


324 


37 


10.1309803 


110 


183 


260 


329 


37 


10.2777778 


109 


181 


253 


325 


38 


10.4109580 


111 


184 


257 


330 


38 


10.5565650 


110 


182 


254 


326 


39 


10.0849315 


112 


185 


268 


331 


30 


10.8333333 


111 


183 


255 


327 


40 


10.9589041 


113 


186 


259 


332 


40 


11.1111111 


112 


184 


250 


328 


41 


11.2328707 


114 


187 


200 


333 


41 


11.3888889 


113 


185 


257 


329 


42 


11.5008493 


115 


188 


201 


334 


42 


11.0060607 


114 


186 


258 


330 


43 


11.7808210 


11C 


189 


202 


335 


43 


11.0444444 


115 


187 


269 


331 


44 


12.0547945 


117 


190 


203 


330 


44 


12.2222222 


110 


188 


260 


332 


45 


12.3287071 


118 


191 


204 


337 


45 


12.5000000 


117 


189 


261 


333 


40 


12.0027397 


119 


192 


205 


338 


40 


12.7777778 


118 


190 


262 


334 


47 


12.8707123 


120 


193 


206 


339 


47 


13.0555650 


119 


191 


263 


335 


48 


13.1500849 


121 


194 


207 


340 


48 


13.3333333 


120 


192 


264 


330 


49 


13.4240575 


122 


195 


208 


341 


40 


13.0111111 


121 


193 


205 


337 


60 


13.0980301 


123 


190 


269 


342 


50 


13.8888889 


122 


194 


266 


338 


51 


13.9720027 


124 


197 


270 


343 


51 


14.1006607 


123 


195 


267 


339 


52 


14.2405753 


125 


198 


271 


344 


52 


14.4444444 


124 


196 


268 


340 


53 


14.5205470 


120 


199 


272 


345 


53 


14.7222222 


126 


197 


269 


341 


54 


14.7945206 


127 


200 


273 


340 


64 


15.0000000 


126 


198 


270 


342 


56 


15.0084932 


128 


201 


274 


347 


55 


15.2777776 


127 


190 


271 


343 


50 


15.3424058 


120 


202 


275 


. 348 


60 


15.5555550 


128 


200 


272 


344 


57 


15.0104384 


130 


203 


270 


349 


57 


15.8333333 


129 


201 


273 


345 


58 


15.8904111 


131 


204 


277 


350 


58 


16.1111111 


130 


202 


274 


340 


59 


16.1043830 


132 


205 


278 


351 


69 


10.3888889 


131 


203 


275 


347 


00 


10.4383602 


133 


200 


279 


352 


00 


10.0660007 


132 


204 


276 


348 


01 


10.7123288 


134 


207 


280 


363 


01 


10.0444444 


133 


205 


277 


349 


02 


10.9803014 


136 


208 


281 


354 


02 


17.2222222 


134 


300 


278 


350 


03 


17.2002740 


136 


209 


282 


355 


03 


17.5000000 


135 


207 


279 


351 


04 


17.5342406 


137 


210 


283 


350 


64 


17.7777778 


136 


208 


280 


352 


05 


17.8082192 


138 


211 


284 


367 


05 


18.0665566 


137 


209 


281 


353 


60 


18.0821918 


139 


212 


285 


368 


06 


18.3333333 


138 


210 


282 


351 


07 


18.3501044 


140 


213 


280 


369 


67 


18.6111111 


139 


211 


283 


355 


08 


18.6301370 


141 


214 


287 


300 


08 


18.8888889 


140 


212 


284 


350 


09 


18,9041090 


142 


215 


288 


301 


00 


10.1060067 


141 


213 


286 


357 


70 


19.1780822 


143 


210 


280 


302 


70 


10.4444444 


142 


214 


286 


358 


71 


19.4520548 


144 


217 


200 


363 


71 


10.7222222 


143 


215 


287 


350 


72 


19.7260274 


145 


218 


291 


364 


72 


20.0000000 


144 


216 


288 


360 


73 


20.0000000 


140 


219 


292 


306 















TABLE V COMPOUND AMOUNT OF 1 

(1 + 0" . 



n 


5% 


1% 


% 


!% 


1% 


1 
2 
3 
4 
5 


1.0041 6667 
1.0083 5069 
1.0125 5210 
1.0167 7112 
1.0210 0767 


1.0050 0000 
1.0100 2500 
1.0160 7513 
1.0201 5050 
1.0252 6125 


1.0058 3333 
1.0117 0069 
1.0170 0228 
1.0235 3830 
1.0296 0894 


1.0075 0000 
1.0150 6025 
1.0220 0017 
1.0303 3910 
1.0380 6073 


1.0100 0000 
1.0201 0000 
1.0303 0100 
1.0400 0401 
1.0510 1005 


6 

7 
8 
9 
10 


1.0252 6187 
1.0205 3370 
1.0338 2352 
1.0381 3111 
1.0424 5666 


1.0303 7761 
1.0355 2940 
1.0407 0704 
1.0460 1058 
1.0511 4013 


1.0355 1440 
1.0415 6400 
1.0470 3064 
1.0537 4182 
1.0598 8805 


1.0458 5224 
1.0530 0613 
1.0015 9885 
1.0606 6084 
1.0775 8255 


1.0015 2015 
1.0721 3635 
1.0828 5671 
1.0936 8527 
1.1046 2213 


11 
12 
13 
14 
15 


1.0468 0023 
1.0C11 0100 
1.0555 4174 
1.0690 3983 
1.0643 5025 


1.0663 9683 
1.0616 7781 
1.0060 8620 
1.0723 2113 
1.0770 8274 


1.0660 7133 
1.0722 0008 
1.0785 4511 
1.0848 3002 
1.0011 6483 


1.0866 0441 
1.0038 0000 
1.1020 1045 
1.1102 7553 
1.1186 0259 


1.1166 0835 
1.1268 2503 
1.1380 9328 
1.1494 7421 
1.1600 0806 


16 
17 
18 
10 
20 


1.0087 9100 
1.0732 4430 
1.0777 1621 
1.0822 0070 
1.0867 1589 


1.0830 7115 
1.0884 8051 
1.0030 2894 
1.0003 9868 
1.1048 9658 


1.0075 2000 
1.1030 3222 
1.1103 7182 
1.1168 4890 
1.1233 6395 


1.1260 0211 
1.1354 4455 
1.1430 6039 
1.1525 4000 
1.1611 8414 


1.1726 7864 
1,1843 0443 
1.1061 4748 
1.2081 0895 
1.2201 9004 


21 
22 
23 
24 
25 


1.0912 4387 
1.0967 9072 
1.1003 5662 
1.1049 4134 
1.1095 4520 


1.1104 2006 
1.1169 7216 
1.1215 5202 
1.1271 6078 
1.1327 9558 


1.1290 1600 
1.1365 0808 
1.1431 3771 
1.1408 0002 
1.1505 1322 


1.1008 9302 
. 1.1780 6722 
1.1875 0723 
1.1964 1353 
1.2053 8663 


1.2323 9104 
1,2447 1680 
1.2571 6302 
1.2607 3465 
1.2824 3200 


26 
27 
28 
20 
30 


1.1141 6836 
1.1188 1073 
1.1234 7244 
1.1281 5358 
1.1328 5422 


1.1384 5055 
1.1441 5185 
1.1498 7261 
1.1566 2197 
1.1614 0008 


1.1032 5055 
1.1700 4523 
1.1708 7040 
1.1837 3657 
' 1.1006 4000 


1.2144 2703 
1.2235 3523 
1.2327 1175 
1.2410 5700 
1.2612 7176 


1.2052 5631 
1.3082 0888 
1.3212 9007 
1.3345 0388 
1.3478 4802 


81 
32 
33 
34 
35 


1.1375 7444 
1.1423 1434 
1.1470 7308 
1.1518 5346 
1.1566 5284 


1.1672 0708 
1.1730 4312 
1.1780 0833 
1.1848 0288 
1.1907 2G89 


1.1076 8610 
1.2045 7202 
1.2115 0860 
1.2186 6034 
1.2267 7523 


1.2006 5G30 
1.2701 1122 
1.2796 3706 
1.2802 3434 
1.2980 0350 


1.3613 2740 
1.3749 4068 
1.3880 0000 
1.4025 7600 
1.4160 0270 : 


36 
37 
38 
30 
40 


1.1014 7223 
1.1063 1170 
1.1711 7133 
1.1700 5121 
1.1809 5142 


1.1966 8052 
1.2026 6303 
1.2080 7725 
1.2147 2063 
1.2207 0424 


1.2320 2550 
1.2401 1705 
1,2473 5107 
1.2546 2780 
1.2610 4055 


. 1.3086 4537 
1.3184 6021 
1.3283 4866 
1.3383 1128 
1.3483 4801 


1.4307 6878 
1.4450 7647 
1.4505 2724 
1.4741 2251 
1.4888 6373 


41 
42 
43 
44 
45 


1.1858 7206 
1.1008 1319 
1.1067 7491 
1.2007 5731 
1.2057 0046 


1.2268 0821 
1.2330 3270 
1.2301 978(5 
1.2453 0385 
1.2510 2082 


1.2003 0701 
1.2707 1220 
1.2841 6960 
1.2010 5062 
1.2001 8525 


1.3584 0123 
1,3080 4060 
1,3780 1460 
1.3802 5642 
1.3906 7684 


1.B037 5237 
1.5187 8089 
1.5330 7779 
1.5403 1757 
1.5048 1075 


46 
47 
48 
4 

50 


1.2107 844Q 
1.2158 2940 
.1.2208 9530 
1.2259 8242 
1.2310 0068 


1.2578 7802 
1.2641 6832 
1.2704 8016 
1.2768 4161 
1.2832 2681 


1.3067 6383 
1.3143 8662 
1.3220 5383 
1.3207 6686 
1.3375 2283 


1.4101 7341 
1.4207 4071 
1.4314 0533 
1.4421 4087 
1.4520 5603 


1.5804 5885 
1.5902 6344 
1.0122 2608 
1.0283 4834 
1.6446 3182 



TABLE V COMPOUND AMOUNT OF 1 



n 


5 , 
12% 


|% 


i% 


!% 


1% 


51 
62 
3 

1 
55 


1.2302 2002 
1.2413 7114 
1.2405 4352 
1.2517 3745 
1.2509 5302 


1.2890 4194 
1.2900 0015 
1.3025 70(50 
1.3090 8340 
1.3150 2887 


1.3453 2504 
1.3531 7277 
1.3010 6028 
1.3090 0583 
1.3700 9170 


1.4038 5411 
1.4748 3301 
1.4858 9420 
1,4970 3847 
1.6082 002G 


1.6010 7814 
,1.0770 8892 
1.0944 0581 
1.7114 1047 
1.7285 2457 


56 
57 
58 
59 
00 


1.2021 0033 
1.2074 4040 
1.2727 3050 
1.2780 3354 
1.2833 5868 


1.3222 0702 
1.3288 1805 
1.3354 0214 
1.3421 3940 
1.3488 5016 


1.3850 2415 
1.3931 0340 
1.4012 2000 
1.4004 0374 
1.4170 2520 


1.5106 7826 
1.5300 7009 
1.5424 6740 
1.5540 2583 
1.5660 8103 


1.7458 0082 
1.7032 (5702 
1.7800 0000 
1.7087 0001) 
1.8160 0670 


61 
02 
63 
G4 
65 


1.2887 0001 
1.2940 7561 
1.2994 0700 
1.3048 8204 
1.3103 1905 


1.3556 9440 
1.3023 7238 
1.3001 8424 
1.3700 3010 
1.3S29 1031 


1.4258 0474 
1.4342 1240 
1.4425 7870 
1.4500 9374 
1.4694 6787 


1.5774 2303 
1.6892 5431 
1.0011 7372 
l.Giai 8252 
1.0252 8130 


1.8348 0307 
1.8532 1230 
1.8717 4443 
1.8904 0187 
1.0003 0040 


66 
07 
68 
69 
70 


1.3167 7872 
1.3212 0113 
1.3267 6038 
1.3322 0458 
1.3378 4580 


1.3808 2480 
1.3007 7399 
1.4037 5785 
1.4107 7604 
1.4178 3053 


1.4679 7138 
1.4705 3454 
1.4851 4700 
1.4938 1102 
1.5025 2492 


1.0374 7100 
1.0407 6203 
1.0021 2517 
1.0745 9111 
1.0871 5055 


1.0284 0015 
1.9477 4475 
1.0072 2220 
1.9808 9442 
2.0Q07 0337 


71 
78 
73 
74 
75 


1.3434 2010 
1.3490 1774 
1.3546 3805 
1.3002 8298 
1.3659 5082 


1.4249 1908 
1.4320 4428 
1.4392 0450 
1.4464 0052 
1.4536 3252 


1.5112 8906 
1.5201 0550 
1.6280 7279 
1.5378 0170 
1.5408 6283 


1.0098 0418 
1.7125 5271 
1.7253 9085 
1.7383 3733 
1.7513 7480 


2.0208 3100 
2.0470 0931 
2.0075 7031 
2.0882 4001 
2,1001 2847 


70 
77 

78 
79 
80 


1.3710 4220 
1.3773 5746 
1.3830 0045 
1.3888 5035 
1.3046 4627 


1.4009 0069 
1.4082 0519 
1.4755 4022 
1.4829 2395 
1.4903 3857 


1.5558 8620 
1.5040 0220 
1.5740 0115 
1.5832 7334 
1.5925 0910 


1.7045 1017 
1.7777 4400 
1.7010 7708 
1.8045 1015 
1.8180 4308 


2.1302 1076 
2.1515 2195 
2.1730 3717 
2.1947 0754 
2.2107 1522 


'81 
89 
83 
81 
85 


1.4004 6729 
1.4062 9253 
1.4121 6209 
1.4180 3005 
1.4230 4454 


1.4977 9026 
1.5052 7921 
1.5128 0561 
1.5203 6964 
1.6279 7148 


1.6017 0874 
1.0111 4257 
1.0205 4090 
1.6290 0405 
1.0396 0236 


1.8310 7031 
1.8454 1001 
1.8502 5763 
1.8732 0100 
1.8872 5008 


2.2388 8237 
2.2012 7119 
2.2838 SHOO 
2.3007 2274 
2.3297 8007 


86 
87 
88 
89 
90 


1.4298 7704 
1.4358 3540 
1.4418 1811 
1.4478 2568 
1.4638 5820 


1.5350 1134 
1.6432 8040 
1.5510 0585 
1.5587 6087 
1.5665 6408 


1.0490 0012 
1.8586 8667 
1.6683 6134 
1.6780 0344 
1,6878 8232 


1.9014 0536 
1.0150 0500 
1.0300 3330 
1.9446 0865 
1.9590 9240 


2.3630 8787 
2.3700 1875 
2.4003 8404 
2.4248 8879 
2.4480 3207 


91 
92 
93 
M 
95 


1.4590 1003 
1.4050 0002 
1.4721 0735 
1.4782 4113 
1.4844 0047 


1.6743 8745 
1.5822 6930 
1.5901 7069 
1.5981 2154 
1.6001 1216 


1.0977 2830 
1.7070 3172 
1.7175 9290 
1.7270 1210 
1.7376 8993 


1.0737 8505 
1.0886 8905 
2.0035 0340 
2.0186 2974 
2.0330 6871 


2.4731 1000 
2.4978 6019 
2.5228 2801) 
2.5480 5098 
2,6735 3765 


96 
97 
98 
99 
100 


1.4905 8547 
1.4967 9024 
1.5030 8289 
1.6092 0653 
1.5155 8420 


1.6141 4271 
1.6222 1342 
1.6303 2449 
1.6384 7011 
1.6406 6849 


1.7478 2040 
1.7580 2211 
1,7082 7724 
1.7786 0219 
1.7880 6731 


2.0489 2123 
2.0642 8814 
2.0797 7030 
2.0953 0858 
2.1110 8384 


2.5002 7203 
2.0252 0506 
2.0515 1831 
2.0780 3340 
2.7048 1383 



26 



TABLE V COMPODTTD AMOUNT OF 1 

(1 + 0" 



n 


i% 


1% 


s% 


! 


1% 


101 
102 
103 
104 
105 


1.0218 9019 
1.5262 4044 
1.6346 0811 
1.6410 0231 
1.6474 2315 


1.G549 0183 
1.0631 7634 
1.6714 9223 
1.0708 4000 
1.0882 4894 


1.7094 0295 
1.8098 0047 
1.8204 5722 
1.8310 7065 
1.8417 5783 


2.1269 1097 
2.1428 6885 
2.1589 4036 
2.1751 3242 
2.1014 4591 


2.7318 6197 
2.7591 8059 
2.7807 7239 
2.8146 4012 
2.8427 8052 


106 
107 
108 
109 
110 


1.5538 7075 
1.5003 4521 
1.5608 4006 
1.6733 7518 
1.5700 3001 


1.6060 9018 
1.7051 7303 
1.7136 0950 
1.7222 6800 
1.7308 7034 


1.8525 0142 
1.8633 0768 
1.8741 7097 
1.8851 0067 
1.8061 0014 


2.2078 8175 
2.2244 4087 
2.2411 2417 
2.2579 3200 
2.2748 0710 


2.8712 1438 
2.8999 2653 
2.9289 2579 
2.9582 1605 
2.9877 9720 


111 
112 
113 
114 
115 


1.5805 1396 
1.5031 2443 
1.5007 0246 
1.0004 2812 
1.0131 2167 


1.7395 3373 
1.7482 3140 
1.7500 7266 
1.7667 5742 
1.7746 8621 


1.9071 0070 
1.9182 9100 
1.0204 8104 
1.9407 3726 
1.9620 5822 


2.2019 2860 
2.3001 1807 
2.3204 3645 
2.3438 8472 
2.3614 6386 


3.0170 7517 
3.0478 5192 
3.0783 3044 
3.1001 1375 
3.1402 0489 


110 
117 
118 
119 
120 


1.0108 4201 
1.6205 0220 
1.0333 0973 
1.0401 7543 
1.0470 09GO 


1.7834 6014 
1.7023 7644 
1.8013 3832 
1.8103 4501 
1.8103 9073 


1.0034 4522 
1.0748 0866 
1.9804 1800 
1.9080 0634 
2.0006 0138 


2.3791 7484 
2.3070 1865 
2.4149 0620 
2.4331 0876 
2.4513 5708 


3.1716 0093 
3.2033 2300 
3.2353 5623 
3.2677 0980 
3.3003 8689 


121 
122 
123 
124 
125 


1.0538 7204 
1.6007 0317 
1.0070 8302 
1.0746 3170 
1.6816 0933 


1.8284 0372 
1.8376 3619 
1.8468 2437 
1.8560 5849 
1,8653 3878 


2.0213 8440 
2.0331 7581 
2.0450 3600 
2.0500 6538 
2.0080 6434 


2.4607 4226 
2.4882 6532 
2.5009 2731 
2.6257 2927 
2.6446 7224 


3.3333 9076 
3.3607 2407 
3.4003 9192 
3.4343 0584 
3.4087 3980 


120 
127 
128 
129 
130 


1.0886 1603 
1.6056 5193 
1.7027 1715 
1.7008 1181 
1.7100 3002 


1.8746 6548 
1.8840 3880 
1(8034 5000 
1.9020 2629 
1.9124 4002 


2.0810 3330 
2.0931 7260 
2.1053 8284 
2.1170 6424 
2.1300 1728 


2.6637 6728 
2.5829 8540 
2.6023 5785 
2.0218 7653 
2.0415 3060 


3.5034 2719 
3.5384 6147 
3.5738 4008 
3.6005 8454 
3.6456 8030 


131 
132 
138 
134 
135 


1.7240 8002 
1.7312 7303 
1.7384 8727 
1.7467 3097 
1.7530 0485 


1.9220 0313 
1.0316 1314 . 
1.9412 7121 
1.9500 7757 
1.0607 3245 


2.1424 4238 
2.1640 3900 
2.1076 1044 
2.1801 5425 
2.1028 7182 


2.0613 6115 
2.6813 1128 
2.7014 2112 
2.7216 8177 
2.7420 0430 


3.0821 3719 
3.7180 6856 
3.7501 4815 
3.7037 0903 
3.8310 4673 


130 
137 
138 
139 
140 


1,7003 0903 
1.7670 4305 
1.7760 0884 
1.7824 0471 
1.7808 3130 


1.9705 3012 
1.9803 8880 
1.0002 9074 
2.0002 4210 
2.0102 4340 


2.2056 0357 
2.2185 2904 
2.2314 7137 
2.2444 8828 
2.2576 8113 


2.7626 0000 
2.7833 8005 
2.8042 6640 
2.8252 8731 
2.8404 7007 


3.8009 6319 
3.0086 6282 
3.0477 4045 
3,9872 2695 
4.0270 9922 


141 
142 
143 
144 
145 


1.7072 8902 
1.8047 7773 
1.8122 0763 
1.8108 4837 
1.8274 3158 


2.0202 0402 
2.0303 9609 
2.0405 4808 
2.0507 5082 
2.0010 0457 


2.2707 6036 
2.2830 0640 
2.2073 1071 
2.3107 2074 
2.3241 0905 


2.8678 2554 
2.8803 3424 
2.9110 0424 
2.9328 3077 
2.0548 3305 


4.0073 7021 
'4.1080.4391 
4.1401 2435 
4.1000 1660 
4.232G 2175 


146 
147 
148 
14 
150 


1.8350 4688 
1,8426 0100 
1.8503 6978 
1.8580 7906 
1.8668 2106 


2.0713 0950 
2.0816 0614 
2.0020 7447 
2.1026 3484 
2.1130 4762 


2.3377 6778 
2.3513 9470 
2.30B1 1117 
2.3780 0705 
2.3927 8461 


2,0709 0430 
2.0003 2176 
3.0218 1067 
3.0444 8020 
3.0673 1389 


4.2748 4697 
4.3175 9544 
4.3607 7139 
4.4043 7910 
4.4484 2290 



27 



TABLE V COMPOUND AMOUNT OP 1 
(1 + i)" 



n 


l|% . 


l\% 


1|% 


l|% 


2% 


l 

2 
3 

4 
5 


1.0112 6000 
1.0226 2666 
1.0341 3111 
1.0467 6600 
1.0676 2094 


1.0126 0000 
1.0261 6626 
1.0370 7070 
1.0609 4634 
1.0640 8216 


1.0160 0000 
1.0302 2600 
1.0460 7838 
1.0013 0355 
1.0772 8400 


1.0176 0000 
1.0353 0026 
1.0534 2411 
1.0718 5003 
1.0000 1050 


1.0200 0000 
1.0404 0000 
1.0012 0800 
1.0824 3210 
1.1040 8080 


6 

7 
8 
9 
10 


1.0694 2716 
1.0814 6821 
1.0036 2462 
1.1060 2780 
1.1183 6068 


1,0773 8318 
1.0908 6047 
1.1044 8610 
1.1182 0218 
1.1322 7083 


1.0934 4320 
1.1098 4401 
1.1204 9250 . 
1.1433 8098 
1.1006 4083 


1.1097 0235 
1.1291 2215 
1.1488 8178 
1.1089 8721 
1.1894 4440 


1.1201 0242 
1.1480 8607 
1.1718 5938 
MOW) 0207 
1.2189 0442 


11 

is 

13 

14 

15 


1.1300 6124 
1.1436 7444 
1.1666 4078 
1.1606 6186 
1.1827 0032 


1.1464 2422 
1.1607 6462 
1.1762 6306 
1.1800 6476 
1.2048 2018 


1.1779 4804 
1.1960 1817 
1.2135 5244 
1.2317 5573 
1.2602 3207 


1.2102 6077 
1.2314 3031 
1.2529 8050 
1.2740 1082 
1.2972 2780 


1.2433 7431 
1.2082 4170 
1.2080 0003 
1.3104 787(1 
1.3458 081)4 


16 
17 
18 

IB 
90 


1.1060 1480 
1.2004 6007 
1.2230 7660 
1.2368 3611 
1.2607 6062 


1.2198 8066 
1.2361 3817 
1,2605 7739 
1.2662 0961 
1.2820 3723 


1.2080 8555 
1.2880 2033 
1.3073 4004 
1.3200 6075 
1.3408 5601 


1.3100 2035 
1.3430 2811 
1.3005 3111 
1.3004 4540. 
1.4147 7820; 


1.3727 8571 
1.4002 4142 
1.4282 4025 
1.4508 1117 
1.4859 4740 


21 
22 
28 
34 
85 


1.2648 2146 
1.2700 6071 
1.2034 4003 
1.3070 0123 
1.3227 0613 


1.2980 6270 
1.3142 8848 
1.3307 1709 
1.3473 5106 
1.3641 9294 


1.3070 6783 
1.3876 0370 
1.4083 7715 
1.4206 0281 
1.4509 4635 


1.4305 3081 
1.4647 2871 
1.4003 0140 
1.5104 4,279 
1.5420 8054 


1.5150 0634 
1.5450 7007 
1.57(18 0920 
1.0084 3725 
1.0400 0500 


J86 
27 
28 
29 
80 


1.3376 8667 
1.3626 3442 
1.3678 6166 
1.3832 3080 
1.3088 0134 


1.3812 4636 
1.3986 1002 
1.4160 0230 
1.4336 0221 
1.4616 1336 


1.4727 0953 
1.4948 0018 
1.5172 2218 
1.5300 8051 
1.6630 8022 


1.5009 8200 
1.5974 5739 
1.0254 1200 
1.0538 5702 
1.6828 0013 


1.0734 1811 
1.7068 8048 
1.7410 2421 
1.7758 4400 
1.8113 0168 


31 
32 
33 
84 
; 35 


1.4146 3786 
1.4304 6140 
1.4466 4308 
1.4628 1760 
1.4792 7430 


1.4607 6863 
1.4881 3051 
1.6067 3214 
1.5265 6020 
1.6446 3687 


1.6806 2642 
1.6103 2432 
1.0344 7918 
1.6580 0637 
1.6838 8132 


1,7122 4913 
1.7422 1349 
1.7727 0223 
1.8037 2452 
1.8362 8070 


1.8475 8882 
1.8846 4069 
1.9222 3140 
1.9000 7003 
1.0908 8055 


36 
87 
88 
39 
40 


1.4060 1613 
1.6127 4610 
1.6297 6367 
1.6460 7341 
1.6643 7687 


1.6630 4382 
1.6834 0312 
1.6032 8678 
1.6233 2787 
1.6436 1046 


1.7001 3954 
1.7347 7603 
1.7007 0828 
1.7872 1025 
1.8140 1841 


1.8074 0727 
1.0000 8689 
1.0333 3841 
1,9071 7184 
2.0015 0734 


2.0308 8734 
2.0800 8500 
2.1222 0879 
2.1047 4477 
2.2080 3000 


41 
42 
48 
44 
45 


1.6810 7611 
1.5097 7334 
1.6177 7079 
1.6360 7071 
1.6643 7638 


1.6641 6471 
1.6840 6677 
1.7060 2885 
1.7273 6421 
1.7480 4614 


1.8412 2808 
1.8088 4712 
1.8968 7082 
1.9263 3302 
1.0542 1301 


2.0300 2530 
2,0722 0024 
2.1086 3090 
2.1454 3010 
2.1829 7522 


2.2522 0040 
2.2972 4447 
2.3431 8936 
2.3900 5314 
2.4378 5421 


46 
47 

48 
49 
50 


1.6720 8710 
1.6018 0821 
1.7108 4106 
1.7300 8801 
1.7406 6160 


1.7708 0707 
1.7929 4306 
1.8153 6485 
1.8380 4679 
1.8610 2237 ' 


1.0835 2021 
2.0132 7010 
2.0434 7820 
2.0741 3046 
2.1052 4242 


2.2211 7728 
2.2600 4780 
2.2005 0872 
2.3308 4170 
2.3807 8803 


2.4800 1120 
2.5303 4361 
2.15870 7039 
2.0388 1170 
2.0016 8803 



TABLE V COMPOUND AMOUNT OF 1 

(1 + *)" 



n 


*I% 


i;% 


1|% 


1|% 


2% 


51 
53 
58 
54 
55 


1.7602 3395 
1.7891 3784 
1.8092 0504 
1.8206 1988 
1.8602 0310 


1.8842 S615 
1.0078 3872 
1.0310 8670 
1.0558 3279 
1.0802 8070 


2.1308 2100 
2.1088 7337 
2.2014 0047 
2.2344 2757 
2.2679 4398 


2.4224 5274 
2.4048 4006 
2.5070 8046 
2.5518 7012 
2.5065 2785 


2.7454 1079 
2.8003 2810 
2.8503 3475 
2.0134 0144 
2.9717 3007 


6 
57 
58 
5 ' 
60 


1.8710 1788 
1.8020 6684 
1.0133 5260 
1.0348 7780 
1.0506 4518 


2.0050 3420 
2.0300 9713 
2.0554 7335 
2.0811 6070 
2.1071 8135 


2.3010 6314 
2.3304 0269 
2.3715 3008 
2.4071 1308 
2.4432 1078 


2.0410 0708 
2.0882 0161 
2.7362 4503 
2.7831 1182 
2.8318 1028 


3.0311 0529 
3.0917 8859 
3.1530 2430 
3.2166 0085 
3.2810 3070 


61 
62 
63 
04 
65 


1.9780 6744 
2.0000 1733 
2.0234 2765 
2.0461 9121 
2.0002 1087 


2.1335 2111 
2.1001 0013 
2.1871 0250 
2.2145 3241 
2.2422 1407 


2.4708 0807 
2.5170 0009 
2.5548 2208 
2.5031 4442 
2.0320 4168 


2.8813 7306 
2.0317 9709 
2.9831 0354 
3,0343 0785 
3.0884 2574 


3.3466 5140 
3.4135 8443 
3.4818 5612 
3.5514 9324 
3.6225 2311 


66 
67 
68 
. 69 
70 


2.0024 8940 
2.1160 2990 
2.1308 3533 
2.1030 0848 
2.1882 6245 


2,2702 4174 
2.2980 1976 
2.3273 5251 
2.3504 4442 
2.3858 9097 


2.6715 2221 
2.7115 9504 
2.7522 0890 
2.7035 6300 
2.8354 5629 


3.1424 7319 
3.1974 0647 
3.2534 2213 
3.3103 5702 
3.3682 8827 


3.6040 7357 
3.7088 7304 
3.8442 5050 
3.9211 3551 
3.9995 5822 


71 

72 
78 

74 

75 


2.2128 7020 
2.2377 0508 
2.2020 3904 
2.2833 0801 
2.3141 4249 


2.4167 2372 
2.4450 2027 
2.4704 0427 
2.6074 6045 
2.5387 0358 


2.8770 8814 
2.9211 5706 
2.0049 7533 
3.0004 4906 
3.0546 9171 


3.4272 3331 
3.4872 0900 
3.6482 3607 
3.6103 3020 
3.6736 1008 


4.0795 4930 
4.1011 4038 
4.2443 6318 
4.3292 5045 
4.4158 3540 


76 

77 
78 
79 
80 


2.3401 7650 
2.3666 0358 
2.3031 2070 
2,4200 4042 
2.4472 7498 


2.5705 2850 
2.6020 6011 
2.6351 0336 
2.6681 3327 
2.7014 8404 


3.1004 1060 
3.1469 1674 
3.1041 2050 
3.2420 3230 
3.2006 6279 


3.7377 0742 
3.8032 0888 
3.8697 6503 
3.0374 8502 
4.0063 9102 


4.5041 5216 
4.5042 3521 
4.0801 1001 
4.7798 4231 
4.8754 3016 


81 

82 
83 
84 
85 


2.4748 0082 
2.5020 4840 
2.5308 0310 
2.5502 7473 
2.5880 0657 


2.7352 6350 
2.7004 4417 
2.8040 6222 
2.8301 1300 
2.8746 0101 


3.3400 2273 
3.3901 2307 
3.4409 7402 
.3.4925 8954 
3.5449 7838 


4.0765 0378 
4.1478 4260 
4.2204 2984 
4.2942 8737 
4.3694 3740 


4.0729 4704 
5,0724 0600 
5.1738 5504 
5.2773 3214 
5.3828 7878 


86 

87 
88 
89 
90 


2.0171 8232 
2.0466 2502 
2.6764 0016 
2.7066 0966 
2.7300 5780 


2.0105 3444 
2.0409 1612 
2.0837,6257 
, 3.0210 4048 
3.0688 1260 


3.5981 5306 
3.6521 2535 
3,7069 0723 
3.7025 1084 
3.8189 4851 


4.4450 0255 
4.5237 0584 
4.6028 7070 
4.6834 2003 
4.7653 8080 


5.4005 3636 
5.6003 4708 
5.7123 5402 
5.8266 0110 
5.9431 3313 


91 
92 
93 

8 


2.7677 4867 
2.7988 8584 
2.8303 7331 
2.8022 1501 
2.8044 1402 


3.0070 4776 
3.1367 0085 
3.1740 6786 
3.2146 4483 
3.2548 2789 


3.8702 3273 
3.9343 7022 
3.9933 0187 
4.0532 0275 
4.1140 9214 


4.8487 7406 
4.0336 2853 
5.0199 0703 
6.1078 1646 
5.1972 0324 


6.0619 0579 
6.1832 3570 
6.3069 0042 
0.4330 3843 
6.6616 9920 


96 
97 
98 
99 
100 


2.9269 7700 
2.0500 QB50 
2.0932 0462 
3.0268 7807 
3.0600 3045 


3.2065 1324 
3.3367 0710 
3.3784 1600 
3.4206 4020 
3.4634 0427 


4.1758 0362 
4.2384 4067 
4.3020 1718 
4.3066 4744 
4.4320 4565 


5.2881 5429 
5.3800 9009 
5.4748 5919 
5.5706 6923 
5,6681 5504 


6.6929 3318 
6.8267 9184 
6.0633 2708 
7.1026 9423 
7.2440 4612 



TABLE V COMPOUND AMOUNT OP 1 

(1 + 0" 



n 


2 -Of 
t'O 


2 -or 
2/0 


2|% 


3% 


3|% 


1 

2 
3 

4 
5 


1.0225 0000 
1.0465 0625 
1.0600 3014 
1.0930 8332 
1.1176 7769 


1.0250 0000 
1.0506 2500 
1.0768 9063 
1.1038 1280 
1.1314 0821 


1.0276 0000 
1.0557 5625 
1.0847 8955 
1.1146 2126 
1.1452 7334 


1.0300 0000 
1.0609 0000 
1.0027 2700 
1.1255 0881 
1.1592 7407 


1.0350 0000 
1.0712 2500 
1,1087 1788 
1.1476 2300 
1,1876 8631 


e 

7 

8 
9 
10 


1.1428 2644 
1.1085 3001 
1.1048 3114 
1.2217 1484 
1.2492 0343 


1.1506 9342 
1.1886 8575 
1.2184 0290 
1.2488 6297 
1.2800 8454 


1.1767 6830 
1.2091 2049 
1.2423 8055 
1.2765 4602 
1.3110 6103 


1.1040 6230 
1.2208 7387 
1.2067 7008 
1.3047 7318 
1.3430 1638 


1.2202 5533 
1.2722 7926 
1.3168 0004 
1.3028 0735 
1.4105 9870 


11 
12 
13 
14 
15 


1.2773 1050 
1.3060 4900 
1.3354 3611 
1.3654 8343 
1.3962 0680 


1.3120 8666 
1.3448 8882 
1.3785 1104 
1.4129 7382 
1.4482 9817 


1.3477 2144 
1.3847 8378 
1.4228 6533 
1.4019 9413 
1.5021 0890 


1.3842 3387 . 
1.4257 6080 
1.4685 3371 
1.5125 8972 
1,5579 6742 


1.4590 6072 
1.5110 0866 
1.5639 5606 
1.6186 9452 
1.6753 4883 


16 
17 

18 
19 
20 


1.4276 2146 
1.4597 4294 
1.4925 8716 
1.5261 7037 
1.5605 0920 


1.4845 0562 
1.5216 1826 
1.5506 5872 
1.6986 5010 
1.6386 1644 


1.5435 0944 
1.6859 5595 
1.6296 6973 
1.6743 8200 
1.7204 2843 


1.0047 0644 
1.6528 4763 
1.7024 3300 
1.7535 0005 
1.8061 1123 


1.7339 8604 
1.7046 7555 
1.8574 8020 
1.9225 0132 
1.0897 8886 


31 
99 
23 
24 
25 


1.5956 2066 
1.6316 2212 
1.8682 3137 
1.7057 6658 
1.7441 4632 


1.6795 8185 
1.7215 7140 
1.7646 1068 
1.8087 2695 
1.8539 4410 


1.7677 4021 
1.8163 6307 
1.8663 0278 
1.9176 2610 
1.9703 6082 


- 1.8002 9467 
1.9161 0341 
1.0735 8651 
2.0327 0411 
2.0037 7703 


2.0694 3147 
2,1316 1168 
2.2061 1448 
2.2833 2849 
2.3632 4408 


26 
27 

28 
29 
30 


1.7833 8962 
1.8235 1588 
1.8645 4499 
1.9064 9725 
1.0493 9344 


1.9002 9270 
1.9478 0002 
1.0964 9502 
2.0464 0739 
2.0976 6758 


2.0245 4575 
2.0802 2075 
2.1374 2682 
2.1962 0600 
2.2566 0173 


2.1506 9127 
2.2212 8901 
2.2879 2708 
2.3565 6551 
2.4272 6247 


2.4459 6856 
2.6315 6711 
2.6201 7100 
2.7118 7708 
2.8067 9370 


31 
32 
33 
34 
35 


1.9D32 5479 
2,0381 0303 
2.0839 6034 
2.1308 4945 
2.1787 9356 


2.1500 0677 
2.2037 5604 
2.2588 5086 
2.3153 2213 
2.3732 0619 


2.3186 5828 
2.3824 2138 
2.4479 3797 
2.5152 5620 
2.5844 2581 


2.5000 8035 
2.5760 8270 
2.6523 3524 
2.7319 0530 
2.8138 6246 


2.0060 3148 
3.0067 0750 
3.1119 4235 
3.2208 6033 
3.3336 9045 


36 
37 
38 
39 
40 


2.2278 1642 
2.2779 4229 
2.3291 9599 
2.3810 0290 
2.4351 8897 


2.4325 3532 
2.4033 4870 
2.5556 8242 
2.6195 7448 
2.6850 6384 


2.6554 0752 
2.7286 2370 
2.8035 5810 
2.8806 5595 
2.0508 7399 


2.8982 7833 
2.0852 2608 
3.0747 8348 
3.1670 2608 
3.2620 3779 


3.4502 6611 
3.8710 2543 
3,6900 1132 
3.8263 7171 
3.9592 5072 


41 
42 
43 
44 
45 


2.4809 8072 
2.5460 0528 
2.6032 0040 
2.6618 6444 
2.7217 6639 


2.7521 9043 
2.8209 0520 
2.8015 2008 
2.9638 0808 
3.0379 0328 


3.0412 7052 
3.1249 0546 
3.2108 4036 
3.2991 3847 
3.3898 6478 


3.3598 9893 
3.4606 9689 
3.6645 1077 
3.8714 6227 
3.7815 9584 


4.0978 3381 
4.2412 5799 
4.3807 0202 
4.5433 4160 
4.7023 5865 


46 
47 
48 
4 
50 


2.7829 0590 
2.8456 1331 
2.9096 3061 
2.9751 0650 
3.0420 4640 


3.1138 6086 
3.1916 9713 
3.2714 8966 
3.3532 7680 
3.4371 0872 


3.4830 8606 
3,5788 7093 
3.6772 8088 
3.7784 1535 
3.8823 2177 


3.8960 4372 
4.0118 0503 
4.1322 5188 
4.2562 1044 
4.3839 0602 


4.8669 4110 
5.0372 8404 
5.2136 8898 
5.3960 6459 
5.5849 2680 



30 



TABLE V COMPOUND AMOUNT OP 1 



n 


2l% 


2% 


2|% 


3% 


3 lor 
a > 


61 
52 
53 
51 
55 


3.1104 0244 
3.1804 7862 
3.2520 3020 
3.3262 1017 
3.4000 2740 


3.5230 3044 
3.0111 1235 
3.7013 0010 
3.7030 2401 
3.8887 7303 


3.0800 8502 
4.0087 8547 
4.2115 0208 
4.3273 1838 
4.4463 1004 


4.5154 2320 
4.6508 8500 
4.700-1 1247 
4.0341 2485 
5.0821 4850 


5.7803 0030 
5.0827 1327 
0. 1921 0824 
0.4088 3202 
0.0331 4114 


5G 
57 

5S 
5ft 
60 


3.4766 2802 
3.6547 4000 
3.6347 3177 
3.7106 1324 
3.8001 3470 


3.0850 0230 
4.0850 4217 
4.1877 8322 
4.2024 7780 
4.3997 8975 


4.5685 0343 
4.0042 2075 
4.8233 2107 
4.0550 0230 
5.0922 5130 


5.2340 1305 
5.3010 5144 
5.5534 0008 
5.7200 0301 
5.8010 0310 


0.8053 0108 
7.1055 8002 
7.3542 8215 
7.61 1U 8203 
7.8780 0090 


61 
02 
63 
64 
65 


3.8860 3782 
3.0730 6407 
4.0624 6802 
41538 0304 
4.2473 2688 


4.5007 8440 
4.0225 2010 
4.7380 0233 
4.8565 4404 
4.9770 5826 


5.2322 8827 
5.3761 7020 
5.5240 2105 
5.0750 3102 
5.8320 1074 


0.0083 5120 
0.2504 0173 
6.4379 1370 
0.6310 5120 
6.8299 8273 


8.1538 2408 
8.4392 0703 
8.7345 8020 
9.0402 9051 
0.3507 0008 


66 
67 
68 
69 
70 


4.3428 9071 
4.4400 0676 
4.6406 1030 
4.0426 8107 
,4.7471 4140 


5.1024 0721 
5.2200 6739 
5.3007 1668 
5.4047 3440 
5.0321 0280 


5.0924 0020 
0.1571 0130 
0.3205 1400 
0.5004 0310 
6.0702 6076 


7.0343 8222 
7.2450 2808 
7.4033 0054 
7.0872 0674 
7.9178 2191 


0.6841 8520 
10.0231 3168 
10.3730 4129 
10.7370 2924 
11.1128 2520 


71 
72 
73 
74 
75 


4.8630 6208 
4,0631 C(iOO 
fi.0748 3723 
5.1890 2107 
C.3057 7405 


5.7729 0543 
5.0172 2806 
0.0651 6876 
0.2107 8773 
0.3722 0743 


0.8020 3032 
7.0516 6700 
7.2455 8701 
7.4448 4158 
7.0496 7472 


8.1553 5657 
8.4000 1727 
8.0520 1778 
8.0115 7832 
0,1780 2607 


11.5017 7414 
11.9043 3G24 
12.3200 8801 
12.7622 2250 
13.1985 5038 


76 

77 
78 
79 
80 


6.4251 6300 
5.5472 1093 
5.6720 3237 
5.7000 5310 
5.0301 4530 


0.6315 1261 
0.0948 0043 
0.8021 7044 
7.0337 2470 
7.2005 0782 


7.8500 3802 
8.0700 8032 
8.2981 7800 
8.5203 7801 
8.7008 5402 


9.4542 0344 
9.7370 2224 
10.0300 6001 
10.3300 0171 
10.0408 0050 


13.0604 0004 
14.1380 1713 
14.0334 0873 
15.1450 4013 
16.6767 3754 


81 

. 82 
83 
84 
85 


6.0035 7357 
6.2000 0307 
0.3395 0406 
6,4821 4290 
Q.6270 0112 


7.3898 0701 
7.5745 5210 
7.7039 1609 
7,0680 13S9 
8.1560 0424 


9.0017 7761 
0.2403 2030 
0.5030 8286 
0.7050 3414 
10.0335 7268 


10.0001 1727 
11.2880 2070 
11.6275 8842 
. 11.0704 1607 
12.3357 0855 


16.2243 8835 
16.7922 4195 
17.3790 7041 
17.9882 6038 
18.0178 5S81 


86 
87 
88 
80 
00 


0.7771 2002 
6.0200 0014 
7.0866 2228 
7.2440 4053 
7.4070 6782 


8.3008 8834 
8.5000 1055 
8.7841 5832 
0.0037 0228 
0.2288 5033. 


10.3004 0583 
10,5930 0000 
10,8843 1405 
11.1836 3331 
11.4911 8322 


12.7057 7081 
13.0800 6320 
13.4705 0180 
13.8830 4805 
14.3004 6711 


10.2604 8387 
10.0430 IfiSO 
20.0410 5285 
21.3044 2120 
22.1121 7595 


01 
02 
02 
04 
05 


7.6740 3088 
7.7460 OC21 
7.0103 3020 
8.0076 1B12 
8.2707 OW21 


0.4505 7774 
9.6060 0718 
0.0384 0886 
10.1800 3068 
10.4410 0385 


11.8071 0070 
12.1318 8861 
12.4055 1544 
12.8083 1711 
13.1006 4684 


14.7204 8112 
15.1713 6556 
15.6265 0652 
10.0053 0172 
10.6781 6077 


22.8801 0210 
23.6871 150H 
24.5101 6473 
25.3742 3040 
20.2023 2856 


OG 
07 
08 
00 
100 


8.4000 0267 
8.6664 8773 
8:8B12 f>871 
0.0504 1203 
0.2540 4030 


10.7020 4305 
10.0702 1004 
11.2444 0530 
11.5265 7003 
11,8137 1035 


13.5224 0085 
13.8043 2852 
14,2704 2255 
14.0000 2417 
10.0724 2234 


17.0765 0550 
17.5877 7070 
18.1154 0388 
18.0588 0600 
19.2180 3108 


27.1815 1000 
28.1328 0201 
20.1175 1311 
30.1360 2807 
31.1914 0798 



31 



TABLE V COMPOUND AMOUNT OF 1 

(l + 0" 



n 


4% 


4%' 


6% 


5|% 


6% 


1 

2 
3 
4 
5 


1.0400 0000 
1.0816 0000 
1.1248 6400 
1.1008 5850 
1.2166 6200 


1.0450 0000 
1.0920 2500 
1.1411 6613 
1.1025 1860 
1.2461 8104 


1.0500 0000 
1.1025 0000 
1.1576 2600 
1.2155 0625 
1.2762 8150 


1.0560 0000 
1.1130 2500 
1.1742 4138 
1.2388 2465 
1.3000 6001 


1.0600 0000 
1.1236 0000 
1.1910 1600 
1.2624 7606 
1.3382 2558 


6 

7 
8 
9 
10 


1.2663 1002 
1.3150 3178 
1.3685 6905 
1.4233 1181 
1.4802 4428 


1.3022 6012 
1,3608 6183 
1.4221 0061 
1.4860 0514 
1.6520 6042 


1.3400 0564 
1,4071 0042 
1.4774 6644 
1.5513 2822 
1.0288 9463 


1.3788 4281 
1.4546 7916 
1.5346 8651 
1.0100 0427 
1.7081 4446 


1.4186 1011 
1.6036 3026 
1.5038 4807 
1.6804 7806 
1.7008 4770 


11 
12 
13 
11 
15 


1.5304 6406 
1.0010 3222 
1.6650 7351 
1.7316 7645 
1.8000 4351 


1.6228 6305 
1.6058 8143 
1.7721 9610 
1.8510 4402 
1.9352 8244 


1.7103 3936 
1.7958 5633 
1.8856 4914 
1.9709 3160 
2.0780 2818 


1.8020 0240 
1.0012 0740 
2.0067 7390 
2.1160 9146 
2.2324 7040 


1.8082 0856 
2.0121 0647 
2.1320 2826 
2,2600 0306 
2.3065 5810 


16 
17 
18 
19 
20 


1.8720 8125 
1.0470 0050 
2.0258 1652 
2.1068 4018 
2.1011 2314 


2.0223 7015 
2.1133 7681 
2.2084 7877 
2.3078 6031 
2.4117 1402 


2,1823 7459 
2.2020 1832 
2.4066 1923 
2.5209 5020 
2.6532 0771 


2.3662 6270 
2.4848 0216 
2.0214 6627 
2.7650 4601 
2,9177 6749 


2.6403 5168 
2.6027 7270 
2.8643 3015 
3.0255 0050 
3.2071 3647 


21 
22 
23 
24 
25 


2.2787 6807 
2,3690 1879 
2.4647 1554 
2.5633 0416 
2.6658 3633 


2.5202 4116 
2.6336 5201 
2.7521 6635 
2.8760 1383 
3.0054 3446 


2.7850 6250 
2.0252 0072 
3.0716 2376 
3.2250 9994 
3.3863 6494 


3.0782 3415 
3.2475 3703 
3.4261 6157 
3.6145 8990 
3.8133 9236 


3.3096 6360 
3.6035 3742 
3.8107 4960 
4.0480 3464 
4.2918 7072 


26 
27 
28 
29 
30 


2.7724 6078 
2.8833 6858 
2.9087 0332 
3.1186 6145 
3.2433 9761 


3.1406 7001 
3.2820 0956 
3.4296 9000 
3.5840 3640 
8.7453 1813 , 


3.5556 7269 
3.7334 5632 
3.9201 2914 
4.1161 3660 
4.3219 4238 


4.0231 2803 
4.2444 0102 
4.4778 4307 
4.7241 2444 
4.0839 5120 


4.6493 8296 
4.8223 4594 
6.1110 8670 
5.4183 8790 
5.7434 0117 


31 
32 
33 
94 
35 


3.3731 3341 
3.5080 5876 
3.6483 8110 
3.7043 1634 
3.0460 8800 


3.9138 6745 
4.0800 8104 
4.2740 3018 
4.4603 6154 
4.6673 4781 


4.5380 3949 
4.7649 4147 
5,0031 8854 
5.2633 4797 
6.5160 1537 


5.2680 6861 
6.6472 6238 
5.8623 6181 
6.1742 4171 
6.5138 2501 


6.0881 0064 
6,4533 '8068 
6.8405 8088 
7.2610 2528 
7.6860 8679 


36 
37 

38 
39 
40 


4.1030 3255 
4.2680 8086 
4,4388 1345 
4.0163 6590 
4.8010 2063 


4.8773 7846 
5.0968 6040 
5.3262 1921 
6.5658 9908 
5.8163 6454 


5.7918 1614 
6.0814 0604 
6.3854 7729 
6,7047 6115 
7,0399 8871 


6.8720 8538 
7.2500 6008 
7.6488 0283 
8.0604 8600 
8.5133 0877 


8.1472 5200 
8.6360 8712 
0.1542 5235 
9.7035 0749 
10.2857 1794 


41 
42 
43 
44 
45 


4.9930 6145 
5.1927 8391 
5.4004 9527 
5.6165 1508 
5.8411 7568 


6.0781 0094 
6.3616 1648 
6.6374 3818 
6.9361 2290 
7.2482 4843 


7.3919 8815 
7.7615 8766 
8.1496 6693 
8.5571 5028 
8.9850 0770 


8.0815 4076 
0.4755 2550 
9.9066 7040 
10.6464 9677 
11.1265 5400 


10.9028 6101 
11.5570 3267 
12.2504 5463 
12.0854 8191 
13.764(3 1083 


46 
47 
48 
49 
50 


6.0748 2271 
6.3178 1562 
6.5705 2824 
6.8333 4937 
7.1066 8335 


7.5744 1961 
7.9152 6849 
8.2714 5567 . 
8.6436 7107 
9.0326 3627 


9.4342 5818 
9.0059 7109 
10.4012 6065 
10.0213 3313 
11.4673 9970 


11.7385 1466 
12.3841 3287 
13.0652 6017 
13.7838 4048 
14.6410 6120 


14.5004 8748 
16,4069 1673 
16.3938 7173 
17.3775 0403 
18.4201 5427 




32 



TABLE V COMPOUND AMOUNT OP 1 



n 


4% 


4j% 


6% 


65% 


6% 


51 
52 
53 
54 
55 


7.3009 5068 
7.6865 8871 
7.0040 5220 
8.3138 1435 
8.6403 6002 


0.4301 0490 
0.8638 0403 
10.3077 3863 
10.7715 8077 
11.2563 0817 


12.0407 0978 
12.6428 0826 
13.2740 4868 
13.0386 9011 
14.0350 3002 


16.3417 6007 
10.1865 0037 
17.0767 7252 
18.0140 4001 
10.0057 0171 


10.5253 0353 
20.0968 8634 
21.0386 9840 
23.2650 2037 
24 0503 2150 


56 
57 

58 
59 
60 


8.9022 21*30 
0.3510 1040 
0.7250 8088 
10.1150 2035 
10.5106 2741 


11.7028 4204 
12.2021 0093 
12.8453 1758 
13.4233 5687 
14.0274 0703 


15.3074 1240 
16.1357 8309 
16.0426 7224 
17.7897 0085 
18.6701 8580 


20.0510 7860 
21.1538 8703 
22.3173 6170 
23.5448 0611 
24.8307 7045 


26.1203 4080 
27.6B71 0134 
20.3680 2742 
31.1204 0307 
32.9876 0085 


61 
62 
63 
64 
65 


10.0404 1250 
11.3780 2000 
11.8331 5010 
12.3004 7017 
12.7087 3522 


14.0586 4129 
15.3182 8014 
16.0076 0275 
10.7270 4487 
17.4807 0230 


10.6131 4619 
20.5038 0245 
21.0234 0257 
22.7040 6720 
23.8300 0056 


26.2059 5782 
27.0472 8650 
20.1678 8620 
30.7721 1004 
32.4645 8654 


34.0609 5230 
37.0649 6944 
30.2888 0761 
41.0461 0067 
44.1440 7165 


60 
67 
68 
69 
70 


13.3106 8403 
13.8431 1201 
14.3068 3640 
14.0727 0905 
15.5716 1835 


18.2073 3400 
10.0893 0403 
10.9483 8541 
20.8460 6276 
21.7841 3558 


25.0318 9659 
26.2834 0037 
27.5076 6488 
28.0775 4813 
30.4264 2554 


34.2501 3880 
30.1338 0643 
38.1212 0074 
40.2170 3008 
42.4200 1023 


40.7036 0094 
40.0012 0014 
, 62.5773 6766 
65.7320 0060 
50.0750 3018 


71 
. 72 
73 
74 
75 


16.1044 8308 
16.8422 6241 
17.5150 5200 
18.2165 0102 
18.0452 5460 


22.7644 2168 
23.7888 2006 
24.8503 1769 
25.0770 8088 
27.1460 0620 


31.9477 4681 
33.5451 3415 
35.2223 0080 
36.0835 1040 
38.8320 8502 


44.7635 6163 
47.2255 5761 
40.8229 6318 
52.6632 2615 
55.4542 0350 


62.0204 8609 
00.3777 1516 
70.3003 7806 
74.6820 0074 
70.0669 2070 


76 
77 

78 
79 
80 


10.7030 6485 
20.4011 8744 
21.3108 3404 
22.1032 6834 
23.0407 0007 


28.3686 1112 
20.6451 0862 
30.0702 3250 
32.3732 0802 
33.8300 0643 


40.7743 2022 
42.8130 3023 
44.0630 8804 
47.2013 7244 
40.5014 4107 


68.6041 8470 
61.7219 1495 
65.1166 2027 
68.0980 3430 
72.4764 2628 


83.8003 3603 
88.8283 5620 
94.1680 6757 
00.8075 4102 
105.7950 0348 


81 

82 
83 
84 
85 


23.0717 0103 
24.0306 6207 
25.0278 8018 
26.0650 0475 
28,0436 0404 


35.3524 5077 
36.0433 1106 
38.6057 0000 
40.3430 1026 
42.1584 6513 


52.0305 1312 
64.6414 8878 
67.3735 0322 
00.2422 4138 
03,2543 6344 


76.4626 2973 
80.6680 7436 
85.1048 1845 
80.7855 8347 
04.7237 0066 


112.1437 5309 
118.8723 7828 
126.0047 2007 
133.5650 0423 
141,6780 0440 


86 
87 
88 
89 
99 


20.1053 4014 
30.3310 6310 
31.5452 4103 
32,8070 5120 
34.1103 3334 


44.0555 8601 
46.0380 8006 
48.1008 0087 
50.2747 410.1 
62.6371 0630 


60.4170 7112 
60.7370 2467 
73.2248 2001 
70.8860 0106 
80.7303 0605 


00.0336 0004 
105.4209 4098 
111.2285 0407 
117.3461 0674 
123.8002 0601 


160.0736 3875 
160.0780 6708 
168.6227 4060 
178.7401 0403 
180.4646 1123 


91 
93 
93 
94 
95 


35.4841 0068 
36.0034 7004 
38.3706 0078 
30.9147 0417 
41.5113 8504 


64.9012 7603 
67.3718 3241 
60.0536 0487 
62.0514 7520 
66.4707 0108 


84.7668 8330 
80.0052 2747 
93.4554 8884 
08.1282 0328 
103.0346 7645 


130.0002 1724 
137.7927 2419 
145,3713 2402 
163.3667 4684 
101.8019 1791 


200,8323 8100 
212.8823 2482 
225.6652 0431 
230.1046 8017 
263.6462 408 


96 
97 
98 
99 
100 


43.1718 4138 
44.8087 IfiOS 
46.6046 6363 
48.5624 6018 
50.5040 4818 


68.4169 7730 
71.4957 4128 
74.7130 4904 
78.0751 3087 
81.6885 1803 


108.1864 1027 
113.5957 3078 
119.2785 1732 
125.2302 0319 
131.5012 5786 


170.7010 2340 
180.0806 7909 
189.9046 0057 
200.4442 0443 
211.4686 3667 


268.7600 3028 
284.8846 7200 
301.9776 4042 
320.0963 0620 
330,3020 8351 



33 



TABLE V COMPOUND AMOUNT OF 1 

(1 + i) 



n 


6|% 


7% 


7|% 


8% 


8l% 


1 

2 
3 
4 
5 


1.0650 0000 
1.1342 2600 
1.2070 4963 
1.2864 0636 
1.3700 .8066 


1.0700 0000 
1.1449 0000 
1.2250 4300 
1.3107 9601 
' 1.4025 6173 


1.0750 0000 
1.1556 2500 
1.2422 0888 
1.3354 0914 
1.4350 2933 


1.0800 0000 
1.1604 0000 
1.2697 1200 
1.3004 8SUG 
1.4093 2808 


1.0850 0000 
1.1772 2500 
1.2772 8013 
1.3868 6870 
1.5036 0000 


6 

7 
8 
9 
10 


1.4501 4230 
1.5539 865C 
1.0549 9507 
1.7025 7039 
1.8771 3747 


1.5007 3035 
1.6057 8148 
1.7181 8018 
1.8384 5921 
1.9671 6136 


1.5433 0153 
1.6590 4914 
1.7834 7783 
1.9172 3860 
2.0010 3156 


1.5868 7432 
1.7138 2427 
1.8509 3021 
1.0990 0463 
2.1589 2500 


1.6314 6751 
1.7701 4225 
1.0206 0434 
2.0838 5571 
2.2609 8344 


11 
12 
13 - 
14 
15 


1.9901 5140 
2.1200 9024 
2.2074 8750 
2.4148 7418 
2.5718 4101 


2.1048 5195 
2.2521 9159 
2.4098 4500 
2.5785 3415 
2.7590 3154 


2.2156 0893 
2.3817 7900 
2.6004 1307 
2.7624 4406 
2.9588 7736 


2.3316 3900 
2.5181 7012 
2.7196 2373 
2.9371 9362 
3.1721 6011 


2.4531 6703 
2.6616 8623 
2.8879 2956 
3.1334 0367 
3.3997 4288 


16 
17 
18 
19 
20 


2.7390 1067 
2.9170 4637 
3.1066 5438 
3.3085 8691 
3.5236 4506 


2.9521 6375 
3.1588 1521 
3.3799 3228 
3.6165 2764 
3.8696 8446 


3.1807 9315 
3.4193 5204 
3.6758 0409 
3.9514 8940 
4.2478 5110 


3.4259 4264 
3.7000 1806 
3.9960 1050 
4.3157 0100 
4.6609 5714 


3.6887 2102 
4.0022 6231 
4.3424 5461 
4.7116 6325 
5.1120 4612 


21 
22 
23 
24 
25 


3.7526 8199 
3.9066 0632 
4.2563 8573 
4.5330 5081 
4.8276 9911 


4.1405 6237 
4.4304 0174 
4.7406 2986 
5.0723 6695 
5.4274 3264 


4.5664 3993 
4.9089 2293 
5.2770 9215 
5.6728 7400 
6.0983 3961 


5.0338 3372 
5.4305 4041 
5.8714 0365 
6.3411 8074 
6.8484 7520 


6.5465 7005 
6.0180 2850 
6.5205 6002 
7.0845 7360 
7.6867 6236 


28 
27 
28 
29 
30 


5.1414 9955 
5.4756 9702 
5.8316 1733 
8.2106 7245 
6.6143 6616 


5.8073 6292 
6.2138 6763 
6.6488 3836 
7.1142 6705 
7.6122 5504 


6.5657 1508 
7.0473 9371 
7,5759 4824 
8.1441 4436 
8.7549 5519 


7.3963 5321 
7.0880 6147 
8.0271 0039 
9.3172 7490 
10.0026 5089 


8.3401 3716 
9.0490 4881 
9.8182 1796 
10.6527 6649 
11.5582 5164 


31 
32 
38 
34 
35 


7.0442 9996 
7.5021 7946 
7.9808 2113 
8.5091 5950 
9.0622 5487 


8.1451 1290 
8.7162 7080 
9.3253 3975 
9.9781 1354 
10.6765 8148 


9.4116 7683 
10.1174 4509 
10.8762 5347 
11.6919 7248 
12.5688 7042 


10.8676 6944 
11.7370 8300 
12.6760 4964 
13.6901 3361 
14.7853 4429 


12.5407 0303 
13.6060 6279 
14.7632 2913 
10.0181 0300 
17.3796 4241 


36 
37 
38 
39 
40 


9.6513 0143 
10.2786 3603 
10.9467 4737 
11.6582 8596 
12.4160 7453 


11.4239 4210 
12.2236 1814 
13.0792 7141 
13.9948 2041 
14.9744 5784 


13.5115 3570 
14.5249 0088 
15.6142 6844 
16.7853 3858 
18.0442 3897 


15.9681 7184 
17.2456 2558 
18.6252 7563 
20.1152 9768 
21.7246 2150 


18.8560 1201 
20.4597 4053 
22.1088 2824 
24.0857 2800 
26.1330 1558 


41 

42 
43 
44 
45 


13.2231 1938 
14.0826 2214 
14.9979 9258 
15.9728 6209 
17.0110 9813 


16.0226 6989 
17.1442 6678 
18.3443 5475 
19.6284 6959 
21.0024 5176 


19.3975 5689 
20.8523 7306 
22.4163 0168 
24.0975 2431 
25.9048 3863 


23.4624 8322 
25.3394 8187 
27.3666 4042 
29.5659 7160 
31.9204 4939 


28,3543 2190 
30.7644 3927 
33.3794 1000 
36.2166 6702 
39.2950 8371 


46 
47 

48 
49 
N 


18.1168 1951 
19.2044 1278 
20.5485 4961 
21.8842 0533 
23.3066 7868 


22.4726 2338 
24.0457 0702 
25.7289 0651 
27.5299 2997 
29.4570 2506 


27.8477 0163 
20.9362 7915 
32.1815 0008 
34.5951 1259 
37.1897 4603 


34.4740 8534 
37.2320 1217 
40.2106 7314 
43.4274 1899 
46.9016 1261 


42.6351 6583 
46.2591 5402 
50.1911 8309 
54.4574 3365 
59.0863 1551 



34 



TABLE VI PRESENT VALUE OF 1 

" = (! + i)~ n 



n 


a% 


\% 


5% 


!% 


1% 


1 

2 
3 

4 
5 


0.0068 5002 
0.9917 1846 
0.9876 0345 
0.0835 0551 
0.9704 2457 


0.9050 2488 
0.9900 7450 
0.9851 4876 
0.9802 4752 
0.9763 7067 


0.9942 0060 
0.0844 3463 
0.0827 0220 
0.0770 0302 
0.0713 3G88 


0.0025 5583 
0.0861 0708 
0.0778 3333 
0.0706 5417 
0.0033 2020 


0.0900 9001 
0.0802 0005 
0.9705 9015 
0.9000 8034 
0.9514 6500 


6 
7 

8 
9 
10 


0.0753 6057 
0.9713 1343 
0.0672 8308 
0.9632 6946 
0.9692 7240 


0.0706 1808 
0.9656 8963 
0.0(108 8520 
0.0561 0468 
0.9513 4794 


0.0657 0301 
0.0601 0301 
0.0545 3480 
0.0480 0007 
0.0434 0534 


0.0501 5802 
0.0400 4022 
0.9419 7540 
0.0349 6318 
0.0280 0315 


0.0420 4524 
0.9327 1805 
0.0234 8322 
0.0143 3082 
0.9052 8605 


11' 
12 
13 
14 
15 


0.9562 9211 
0.9513 2824 
0.9473 8082 
0,9434 497S 
0.9396 3505 


0.0466 1489 
0.0419 0634 
0.0372 1024 
0.0325 5046 
0.0270 1688 


0.9380 2354 
0.0326 8347 
0.0271 7406 
0.9217 0780 
0.0164 5183 


0.9210 9404 
0.9142 3815 
0.9074 3241 
0.0008 7733 
0.8030 7254 


0.8063 2372 
0.8874 4023 
0.8786 6260 
0.8600 6297 
0.8613 4947 


IG 
17 

18 
19 
20 


0.9356 3656 
0.9317 6425 
0.9278 8805 
0.9240 3789 
0.9202 0371 


0.9233 0037 
0.9187 0084 
0.0141 3016 
0.0005 8822 
0.0050 0290 


0.0111 3686 
0.0058 5272 
0.9005 9023 
0.8953 7620 
0.8001 3346 


0.8873 1700 
0.8807 1231 
0.8741 6014 
0.8676 4878 
0.8611 8985 


0.8528 2120 
0.8443 7749 
0.8300 1731 
0.8277 3992 
0.8195 4447 


21 
22 
23 
24 
25 


0.9163 8544 
0.0125 8301 
0.9087 9036 
0.9060 2542 
0.0012 7012 


0.9005 6010 
0.8000 7071 
0.8016 2160 
0.8871 8567 
0.8827 7181 


0.8850 2084 
0.8708 8816 
0.8747 8525 
0.8607 1103 
0.8046 6803 


0.8547 7901 
0.8484 1680 
0.8421 0014 
0.8358 3140 
0.8206 0933 


0.8114 3017 
0.8033 0621 
0.7954 4170 
0.7875 6013 
0.7707 6844 


20 
27 
28 
29 
30 


. 0.8976 3041 
0.8938 0022 
0.8000 0748 
0.8864 0413 
0.8827 2610 


0.8783 7001 
0.8740 0086 
0.8096 6155 
0.8063 3488 
0.8010 2973 


0.8506 5330 
0.8546 6782 
0.8497 1118 
0.8447 8327 
0.8398 8305 


0.8234 3358 
0.8173 0330 
0.8112 1000 
0.8051 8080 
0.7901 8600 


0.7720 4706 
0.7644 0392 
0.7508 3657 
0.7493 4215 
0.7419 2202 


31 
82 
33 
34 
35 


0.8700 6334 
0.8754 1677 
0.8717 8334 
0.8681 0599 
0.8646 6364 


0.8507 4000 
0.8524 8358 
0.8482 4237 
0.8440 2226 
0.8308 2314 


0.8350 1304 
0.8301 7038 
0.8263 5681 
0.8205 6915 
0.8158 1020 


0.7032 3762 
0.7873 3262 
0.7814 7158 
0.7756 5418 
0.7608 8008 


0.7345 7715 
0.7273 0411 
0.7201 0307 
0.7129 7334 
0.70J30 1420 


30 
37 
38 
39 
40 


0.8600 7024 
0.8574 0372 
0.8538 4603 
0.8603 0310 
0.8407 7487 


0.8356 4492 
0.8314 8748 
0.8273 5073 
0.8232 3455 
0.8101 3880 


0.8110 7807 
0.8003 7511 
0.8010 9854 
0.7070 4008 
0.7924 2660 


0.7641 4800 
0.7684 6061 
0.7528 1440 
0.7472 1032 
0.7416 4700 


0.0089 2405 
0.6020 0490 
0.6861 3337 
0.6783 6067 
0.6710 6314 


41 
42 
43 
44 
45 


0.8432 0128 
0.8397 6227 
0.8362 7778 
0.8328 0776 
0.8203 6211 


0.8150 6354 
0.8110 0850 
0.8009 7363 
0.8020 5884 
0.7989 6402 


0.7878 3002 
0.7832 0189 
0.7787 1936 
0.7742 0317 
0.7607 1318 


0.7301 2701 
0.7306 4716 
0.7262 0809 
0.7108 0052 
0.7144 6114 


0.6660 0311 
0.6584 1802 
0.6518 0092 
0.0464 4546 
0.6390 6402 


46 
47 

48 
49 
50 


0.8259 1082 
0.8224 8380 
0.8190 7100 
0.816B 7237 
0.8122 8784 


0.7940 8907 
0.7910 3300 
0.7870 0841 
0.7831 82SO 
0.7702 8607 


0.7652 4023 
0,7608 1116 
Q.7G03 9884 
0.7C20 1210 
0.7476 6080 


0.7001 3264 
0.7038 5374 
O.H086 1414 
0.6034 1353 
0.6882 C165 


0.6327 2704 
0.6204 6301 
0.0202 0041 
0.6141 1021 
0.6080 3882 



TABLE VI PRESENT VALUE OF 1 



n 


&* 


1% 


** 
12/o 


!% 


1% 


51 
52 
53 
54 
55 


0.8089 1735 
0.8055 6084 
0.8022 1827 
0.7088 8066 
0.7955 7467 


0.7764 0002 
0.7716 5127 
0.7677 1270 
0.7638 9324 
0.7600 9277 


0.7433 1480 
0.7390 0394 
0.7347 1809 
0.7304 6700 
0.7262 2080 


0.6831 2819 
0.6780 4286 
0.6720 9540 
0.6079 8651 
0.6030 1291 


0.6020 1864 
0.5960 5800 
0.5001 5649 
0.5843 1330 
0.5785 2808 


56 
57 
58 
59 
60 


0.7922 7363 
0.7880 8608 
0.7857 1228 
0.7824 5207 
0.7792 0538 


0.7663 1122 
0.7525 4847 
0.7488 0445 
0.7450 7906 
0.7413 7220 


0.7220 0008 
0.7178 2179 
0.7136 6878 
0.7096 1991 
0.7054 0505 


0.6680 7733 
0.6531 7849 
0.6483 1612 
0.6434 8995 
0.6386 0970 


0.5728 0008 
0.6071 2879 
0.5015 1305 
0.5559 6411 
0.5604 4962 


61 
62 
63 
64 
65 


0.7759 7216 
0.7727 6236 
0.7695 4591 
0-7063 6278 
0.7631 7289 


0.7376 8378 
0.7340 1371 
0.7303 6190 
0.7267 2826 
0.7231 1209 


0.7013 1405 
0.6972 4678 
0.6932 0310 
0.6891 8286 
0.6861 8694 


0.6339 4511 
0.6292 2692 
0.6245 4185 
0.6198 9260 
0.6162 7807 


0.5449 9962 
0.5300 0358 
0.5342 0097 
0.6289 7120 
0.5237 3392 


66 
67 
68 
69 
70 


0.7600 0620 
0.7568 5265 
0.7537 1218 
0.7605 8474 
0.7474 7028 


0.7195 1512 
0.7159 3644 
0.7123 7357 
0.7088 2943 
0.7053 0291 


0.6812 1221 
0.6772 0151 
0.6733 3373 
0.6694 2873 
0.6655 4038 


0.6106 9784 
0.6061 6170 
0.6016 3940 
0.5971 6070 
0.5927 1533 


0.5185 4844 
0.5134 1429 
0.5083 3099 
0.5032 9801 
0.4983 1480 


71 
72 
73 
74 
75 


0.7443 6874 
0.7412 8008 
0.7382 0423 
0.7361 4114 
0.7320 9076 


0.7017 9394 
0.6983 0243 
0.6048 2829 
0.6913 7143 
0.6879 3177 


0.6616 8654 
0.6578 4909 
0.6540 3389 
0.6502 4082 
0.6464 6975 


0.6883 0306 
0.5839 2363 
0.5796 7681 
0.5752 6234 
0.6709 7999 


0.4933 8105 
0.4884 9009 
0.4830 5949 
0.4788 7078 
0.4741 2949 


76 
77' 

78 
79 
80 


0.7290 5304 
0.7260 2792 
0.7230 1536 
0.7200 1529 
0.7170 2768 


0.0845 0923 
0.6811 0371 
0.6777 1513 
0.6743 4342 
0.6709 8847 


0.6427 2064 
0.6389 9308 
0.6352 8724 
0.6316 0289 
0.6279 3991 


0.6667 2052 
0.5026 1009 
0.5583 2326 
0.5541 6701 
0.5500 4170 


0.4004 3514 
0.4047 8726 
0.4001 8541 
0,4566 2912 
0.4511 1794 


81 
82 
83 
84 
85 


0.7140 6246 
0.7110 8959 
0.7081 3901 
0,7062 0067 
0.7022 7453 


0.6676 5022 
0.6643 2868 
0.6610 2346 
0.6577 3479 
0,6544 6248 


0.6242 9817 
0.6206 7755 
0.6170 7793 
0.6134 9919 
0.6099 4120 


0.5459 4710 
0.5418 8297 
0.5378 4911 
0.6338 4527 
0.5298 7123 


0.4400 5142 
0.4422 2913 
0.4378 5003 
0.4335 1547 
0.4292 2324 


86 
87 
88 
89 
90 


0.6993 6052 
0.6964 5861 
0.6935 6874 
0.6906 9086 
. 0.6878 2493 


0.6512 0644 
0.6479 6661 
0.6447 4290 
0.6415 3622 
0.6383 4350 


0.6064 0384 
0.6028 8700 
0,5993 9066 
0.5969 1439 
0.5924 6838 


0.5259 2678 
0.5220 1169 
0.5181 2675 
0.6142 6873 
0.5104 4043 


0.4249 7350 
0,4207 0585 
0.4165 9985 
0.4124 7510 
0.4083 0119 


91 
92 
93 
94 
95 


0.6849 7088 
0.6821 2868 
0.6792 9827 
0.6764 7960 
0.6736 7263 


' 0,6351 6766 
Q.6320 0763 
0.6288 6331 
0.6257 3464 
0.6226 2163 


0.5890 2242 
0.5866 0638 
0.5822 1015 
0,5788 3303 
0.5764 7668 


0.5066 4063 
0,5028 6911 
0.4991 2667 
0.4964 1009 
0.4917 2217 


0.4043 4771 
0.4003 4427 
0.3903 8040 
0.3924 5590 
0.3885 7020 


96 
97 
98 
99 
100 


0.6708 7731 
0.6680 9359 
0.6663 2141 
0.6626 6074 
0.6598 1163 


0.6196 2391 
0.6164 4170 
0.6133 7483 
0.6103 2321 
0.6072 8678 


0.5721 3020 
0.5088 2108 
0,5656 2220 
0.5622 4245 
0.5589 8172 


0.4880 6171 
0.4844 2860 
0.4808 2233 
0.4772 4301 
0.4736 9033 


0.3847 2297 
0.3809 1383 
0.3771 4241 
0.3734 0832 
0.3697 1121 



TABLE VI PRESENT VALUE OP 1 



n 


H 


|% 


le, 
la % 


!% 


1% 


101 
102 
103 
104 
105 


0.6570 7372 
0.0643 4727 
0.0510 3214 
0.0489 2827 
0.6402 3502 


0.6042 0545 
0.6012 5015 
0.5082 6781 
0.6052 0130 
0.5023 2071 


0.5557 3901 
0.5525 1080 
0.5403 1257 
0.5461 2083 
0.5420 5957 


0.4701 0410 
0.4000 6412 
0.4031 0019 
0.4597 4213 
0.4503 1073 


0.3660 6071 
0.3624 2644 
0.3588 3806 
0.3552 8521 
0.3517 6753 


106 
107 
108 
109 
110 


0.0435 415 
0.6408 8380 
0.0382 2453 
0.6355 7030 
0.6320 3006 


0.5803 8279 
0.58G4 5054 
0.6836 3288 
0.5806 2973 
0.5777 4102 


0,5308 1067 
0.5300 8004 
0.5335 6756 
0.5304 7313 
0.5273 9065 


0.4520 2281 
0.4405 5117 
0.4402 0404 
0.4428 8302 
0.4305 8012 


0.3482 8400 
0.3448 3632 
0.3414 2210 
0.3380 4168 
0.3346 9474 


111 
112 
113 
114 
115 


0.0303 1276 
0.0270 9734 
0.0250 9270 
0.0224 9004 
0.6109 1006 


0.5748 6669 
0.5720 0600 
0.5601 0085 
0.5603 2021 
0.5635 1105 


0.5243 3801 
0.5212 9711 
0.5182 7385 
0.5162 0812 
0.5122 7982 


0.4303 1377 
0.4330 6677 
0.4298 4190 
0.4200 4124 
0.4234 6616 


0.3213 8003 
0.3280 0003 
0.3248 5141 
0.3210 3500 
0.3184 5056 


116 
117 
118 
119 
120 


0.0173 4379 
0.0147 8220 
0.0122 3123 
0.6090 9080 
0.6071 6102 


0.5G07 0811 
0.6579 1852 
0.5551 4280 
0.5523 8000 
0.6400 3273 


0.5003 0885 
0.5003 5512 
0.5034 1851 
0,6004 9893 
0.4975 9029 


0.4203 1379 
0.4171 8491 
0.4140 7031 
0.4100 0083 
0.4079 3730 


0.3152 0758 
0.3121 7582 
0.3000 8407 
0.3000 2473 
0.3029 9478 


121 
122 
123 
124 
125 


0.0040 4168 
0.0021 327G 
0.5990 3431 
0.5971 4620 
0.5946 0842 


0.5408 0824 
0.6441 7730 
0.5414 7001 
0.5387 7012 
0.6300 0565 


0.4947 1047 
0.4018 4140 
0,4889 8896 
0.4801 5307 
0.4833 3363 


0.4040 0055 
0.4018 8040 
0.3988 9409 
0.3050 2625 
0.3929 7702 


0.2009 9483 
0.2970 2459 
0.2940 8375 
0.2011 7203 
0,2882 8014 


126 
127 
128 
129 
130 


0.5922 0091 
0.5897 4366 
0.5872 9658 
0.5848 5060 
0.5824 3286 


0,5334 2850 
0.5307 7463 
0.5281 3300 
0.5266 0643 
0,5228 9107 


0.4805 3063 
0.4777 4369 
0.4740 7302 
0.4722 1841 
0.4604 7978 


0.3900 5252 
0.3871 4801 
0.3842 6601 
0.3814 0630 
0.3785 6711 


0.2864 3470 
0.2820 0870 
0.2798 1060 
0.2770 4019 
0.2742 0722 


131 
132 
133 
134 
135 


0.6800 1613 
0.5770 0942 
0.5762 1270 
0.5728 2503 
0.5704 4906 


0.5202 9052 
0.6177 0201 
0.6151 2637 
0.5125 0350 
0.5100 1349 


0.4667 5703 
0.4640 6007 
0.4613 6881 
0.4686 8310 
0.4560 2303 


0.3757 4800 
0.3729 6185 
0.3701 7553 
0.3674 1988 
0.3646 8475 


0.2716 8141 
0.2688 0248 
0.2662 3018 
0.2636 9424 
0.2609 8430 


136 
137 
138 
139 
140 


0,5630 8205 
0.5657 2486 
0.5633 7746 
0.5010 3979 
0.5587 1182 


0.5074 7611 
0.6040 5136 
0.5024 3016 
0.4009 3040 
0.4074 5220 


0.4633 7832 
0.4607 4805 
0.4481 3483 
0.4455 3587 
0.4420 5108 


0.3619 G997 
0.3502 7541 
0.3666 0000 
0.3539 4630 
0.3513 1147 


0.2584 0030 
0.2658 4197 
0.2533 0888 
0.2508 0087 
0.2483 1770 


141 
142 
143 
144 
145 


0.5503 9351 
0.5540 8483 
0.6617 8672 
0.6494 0015 
0.6472 1009 


0.4049 7731 
0.4025 1474 
0.4000 6442 
0.4870 2628 
0.4862 0028 


0.4403 8308 
0.4378 2908 
0.4352 8980 
0.4327 0642 
0.4302 5560 


0.3480 0625 
0.3401 .0040 
0.3435 2406 
0.3400 0681 
0.3384 2800 


0.2458 5011 
0.2434 2486 
0.2410 1471 
0.2380 2843 
0.2362 6577 


146 
147 
148 
149 
150 


0.5449 4548 
0,5420 8429 
0,5404 3249 
0.5381 0003 
0.6360 6688 


0.4827 8635 
0.4803 8443 
0.4779 9446 
0.4760 1637 
0,4732 5012 


0.4277 0033 
0,4252 7953 
0,4228 1312 
0.4203 0102 
0.4170 2313 


0,3350 0028 
0.3334 0871 
0.3300 2670 
0.3284 6320 
0.3200 1815 


0.2330 2050 
0.2310 1040 
0.2293 1723 
0.2270 4070 
0.2247 0877 



37 



TABLE VI PRESENT VALUE OP 1 



n 


1|% 


l\% 


1|% 


1|% 


2% 


i 

2 
3 
4 
5 


0.9888 7615 
0.9778 7407 
0.9669 9537 
0.9662 3770 
0.9455 9970 


0.9876 5432 
0.9754 6106 
0.9634 1833 
0.9515 2428 
0.9397 7706 


0.9852 2107 
0.9706 6176 
0.9563 1099 
0.9421 8423 
0.9282 6033 


0.9828 0098 
0.9058 9777 
0.9402 8528 
0.9329 5851 
0.9109 1254 


0.9803 0216 
0.0011 0878 
0.9423 2233 
0.9238 4543 
0.0057 3081 


6 
7 
8 
9 
10 


0.9350 8005 
0.9248 7743 
0.9143 9054 
0.9042 1808 
0.8941 5881 


0.9281 7488 
0.0107 1693 
0.9053 9845 
0.8942 2069 
0.8831 8093 


0.9145 4219 
0.9010 2070 
0.8877 1112 
0.8745 9224 
0.8610 6723 


0.0011 4254 
0.8856 4378 
0.8704 1167 
0.8554 4136 
0.8407 2800 


0.8870 7138 
0.8705 0018 
0.8534 9037 
0.8307 5527 
0.8203 4830 


11 
12 
13 
14 
15 


0.8842 1142 
0.8743 7470 
0.8646 4742 
0,8550 2835 
0.8455 1629 


0.8722 7740 
0.8015 0860 
0.8508 7269 
0.8403 6809 
0.8299 9318 


0.8489 3323 
0.8363 8742 
0.8240 2702 
0.8118 4928 
0.7998 5150 


0.8262 0889 
0.8120 5788 
0.7980 9128 
0.7843 6490 
0.7708 7469 


0.8042 0304 
0.7884 0318 
0.7730 3253 
0.7578 7502 
0.7430 1473 


10 
17 
18 
19 
20 


0.8361 1005 
0.8268 0846 
0.8176 1034 
0.8085 1455 
0.7995 1995 


0.8197 4635 
0.8096 2002 
0.7996 3064 
0.7897 6860 
0.7800 0856 


0.7880 3104 
0.7763 8620 
0.7649 1159 
0.7536 0747 
0.7424 7042 


0.7576 1631 
0.7445 8605 
0.7317 7090 
0.7191 9401 
0.7008 2468 


0.7284 4581 
0.7141 0250 
0.7001 5937 
0.0864 3070 
0.0720 7133 


21 
22 
23 
24 
25 


0.7906 2542 
0.7818 2983 
0.7731 3210 
0.7645 3112 
0.7560 2583 


0.7703 7881 
0.7608 0796 
0.7614 7453 
0.7421 9707 
0.7330 3414 


0.7314 9795 
0.7206 8703 
0,7100 3708 
0.0995 4392 
0.0892 0683 


0.0040 6789 
0.6827 2028 
0.6709 7817 
0.0594 3800 
0.6480 9032 


0.0507 7682 
0.0408 3904 
0.0341 5602 
0.0217 2149 
0.6095 3087 


26 
27 
28 
29 
30 


0.7476 1616 
0.7392 9806 
0.7310 7348 
0.7229 4040 
0.7148 9780 


0.7239 8434 
0.7150 4028 
0.7062 1853 
0.6974 9978 
0.6888 8867 


0.0790 2052 
0.6689 8674 
0.0590 0925 
0.6493 6887 
0.6397 6243 


0.0369 4070 
0.6259 9479 
0.6152 2829 
0.0040 4697 
0.5042 4764 


0.5976 7928 
0.6858 6204 
0.6743 7465 
0.6631 1231 
0,5620 7080 


31 
32 
33 
34 
35 


0.7069 4467 
0.6990 8002 
0.6913 0287 
0.6836 1223 
0.6760 0716 


0.6803 8387 
0.6719 8407 
0.6630 8797 
0.6564 9429 
0.6474 0177 


0.6303 0781 
0.6209 9292 
0.6118 1668 
0.6027 7407 
0.5938 6608 


0.5840 2710 
0.5739 8247 
0.5641 1063 
0.5544 0830 
0.6448 7311 


0.5412 4507 
0.5300 3330 
0.5202 2873 
0.6100 2817 
0.6000 2701 


30 
37 

40 


0.6684 8667 
0.6610 4986 
1 0.6536 9578 
0.6464 2352 
0.6392 3216 


0.6394 0910 
0.6316 1622 
0.6237 1873 
0.6160 1850 
0.6084 1334 


0.6850 8074 
0.6764 4309 
0.6679 2423 
0.6595 3126 
0.6612 6232 


0.5356 0183 
0.5202 9172 
0.5172 4002 
0.5083 4400 
0.4996 0098 


0.4902 2318 
0.4806 1003 
0.4711 8719 
0.4619 4822 
0.4528 9042 


41 
42 
43 
44 
45 


0.6321 2080 
0.6250 8855 
0.6181 3454 
0.6112 789 
0.6044 5774 


0.6009 0206 
0.5934 8352 
0.6881 5656 
0.5789 2006 
0.6717 7290 


0.6431 1560 
0.6350 8Q25 
0.6271 8163 
0.5193 9067 
0.5117 1494 


0.4910 0834 
0.4826 6348 
0.4742 6386 
0.4661 0690 
0.4580 0040 


0,4440 1021 
0.4353 0413 
0.4207 0875 
0.4184 0074 
0.4101 0680 


40 
47 

48 
49 
00 


0.6977 3324 
0.6910 8356 
0.5845 0784 
0.6780 0528 
0.6715 7506 


0.6647 1397 
0,5577 4219 
0.5608 5649 
0.5440 5579 
0.6373 3906 


0.5041 6266 
0.4967 0212 
0.4893 6170 
0.4821 2976 
0.4760 0468 


0.4502 1170 
0.4424 6850 
0.4348 5848 
0.4273 7934 
0.4200 E883 


0.4021 5373 
0.3942 0830 
0.3865 3701 
0.3780 6844 
0.3716 2788 



38 



TABLE VI PRESENT VALUE OF 1 

IX* = (1 + ~ n 



n 


1|% 


ll% 


1|% 


lf% 


2% 


51 
58 
53 
54 
55 


0.5662 1637 
0.5580 2843 
0.5527 1044 
0.5405 (1102 
0.5404 8120 


0.5307 0624 
0.5241 5332 
0.517G 8220 
0.5112 9115 
0.5040 7802 


0.4679 8401 
0.4010 0887 
0.4542 5605 
0.4475 4192 
0.4400 2800 


0.4128 0475 
0.4057 0492 
0.3087 2719 
0.3018 6047 
0.3861 2970 


0.3642 4302 
0.3571 0100 
0.3500 9902 
0.3432 3433 
0.3305 0425 


56 
57 
58 
50 
60 


0.5344 0843 
0.5285 2250 
0.5220 4282 
0.5168 2850 
0.5110 7887 


0.4087 4401 
0.4925 8727 
0.4S05 0594 
0.480-1 0970 
0.474C 0700 


0.4344 1182 
0.4270 0104 
0.4210 0094 
0.4154 3541 
0.4092 9607 


0.3785 0685 
0.3710 9592 
0.3655 0796 
0.3593 1003 
0.3531 3025 


0.3200 0613 
0.3234 3738 
0.3170 9547 
0.3108 7791 
0.3047 8227 


61 
03 
63 
64 
65 


0.5053 0319 
0.4997 7077 
0.4942 1000 
0.4887 1288 
0.4832 7002 


0.4087 0874 
0.4020 2222 
0.4572 0713 
0.4515 (1250 
0.4450 8775 


0.4032 4720 
0.3972 8704 
0.3014 1000 
0.3S5G 3221 
0.3799 3321 


0.3470 5676 
0.3410 8772 
0.3352 2135 
0.3204 5687 
0.3237 8056 


0.2988 0014 
0.2920 4720 
0.2872 0314 
0.2816 7170 
0.2700 5009 


66 
67 
68 
69 
70 


0.4778 9905 
0.4725 8300 
0.4673 2568 
0.4021 2076 
0.4500 8500 


0.4404 8173 
0.4350 4303 
0.4200 7277 
0.4243 0817 
0.4101 2005 


0.3743 1843 
0.3687 8003 
0.3033 3058 
0.3670 0708 
0.3520 7002 


0.3182 2009 
0.3127 4701 
0.3073 6866 
0.3020 8222 
0.2008 8070 


0.2706 3703 
0.2053 3130 
0.2001 2873 
0.2550 2817 
0.2500 2701 


71 
78 
73 
74 
75 


0.4610 0177 
0.4408 7443 
0.4410 0302 
0.4300 8G02 
0.4321 2551 


0.4130 5402 
0.4088 4407 
0.4037 9001 
0.3988 1147 
0.3038 8787 


0.3474 0495 
0.3423 3000 
0.3372 7093 
0.3322 8003 
0.3273 7600 


0.2917 8064 
0.2867 0221 
0.2818 3018 
0.2709 8208 
0.2722 1914 


0.2451 2611 
0.2403 1874 
0.2350 0061 
0.2300 8087 
0.2204 6771 


76 

77 
78 
70 
80 


0.4273 1818 
0.4225 0433 
0.4178 0337 
0.4132 1470 
0.4080 1775 


0.3800 2500 
0.3842 2228 
0.3704 7870 
0,3747 9387 
0.3701 0079 


0.3225 3793 
0.3177 7130 
0.3130 7C23 
0.3084 4860 
0.3038 9015 


0.2675 3724 
0.2620 3686 
0.2584 1362 
0.2539 0916 
0.2490 0114 


0.2220 1737 
0.2170 0408 
0,2133 0616 
0.2092 1192 
0.2051 0073 


81 
88 
83 
84 
85 


0.4040 7104 
0.3005 7070 
0.3061 3148 
0.3907 3570 
0,3803 8882 


0.3055 0083 
0.3010 8329 
0.3500 2547 
0.3522 2208 
0.3478 7420 


0.2993 9010 
0.2949 7454 
0.2900 1631 
0.2803 2Q50 
0.2820 8017 


0.2463 0825 
0.2410 8010 
0.2309 4269 
0.2328 6761 
0.2288 0242 


0.2010 8707 
0.1971 4607 
0.1032 7048 
0.1894 8068 
0.1867 7420 


86 
87 
88 
89 
00 


0.3820 0031 
0.3778 3001 
0.3730 3621 
0.3004 70(50 
0.3053 0010 


0.3436 7051 
0.3303 3770 
0.3351 4843 
0.3310 1080 
0.3200 2425 


0.2770 2036 
0.2738 1310 
0.2007 6000 
0.2057 7907 
0.2018 5218 


0.2240 2621 
0.2210 5770 
0.2172 6672 
0.2135 1014 
0.2008 4082 


0.1821 3167 
0.1786 6036 
0.1760 5018 
0.1716 2065 
0.1682 6142 


91 
08 
03 
94 
95 


0.3013 0448 
0.3572 8503 
0.3533 1020 
0.3403 7070 
0.3454 0207 


0.3228 8814 
0.3180 0187 
0.3140 0481 
0.3110 7030 
.0.3072 3601 


0.2579 8245 
0.2541 0000 
0.2504 1300 
0.2407 1300 
0.2430 0000 


0.2002 3766 
0.2026 0067 
0.1002 0450 
0.1957 7837 
0.1024 1118 


0.1040 0217 
0.1017 2762 
0.1585 5049 
0.1564 4754 
0,1623 0055 


96 
07 
08 
90 
100 


0.3410 4041 
0.3878 4801 
0.3340 9010 
0.3303 7340 
0.32GO 0805 


0.3034 4287 
0.2000 0006 
0.2059 0070 
0.2023 4242 
0.2887 8320 


0.2304 7487 
0.2350 3583 
0.2324 4000 
0.2200 1389 
0.2260 2044 


. 0.1801 OlflO 
0.1868 4053 
0.1820 6310 
0.1705 1105 
0.1764 2422 


0,1494 1132 
0,1404 8109 
0.1436 0060 
0.1407 0363 
0.1380 3297 




TABLE VI PRESENT VALUE OP 1 



n 


2j% 


* 


2 -Of 
t' 


3% 


s|% 


i 

2 
3 

4 
5 


0.9779 9511 
0.9564 7444 
0.9354 2732 
0.9148 4335 
0.8047 1232 


0.0750 0970 
0.0518 1440 
0.9285 9941 
0.0050 5004 
0.8838 5420 


0.0732 3001 
0.0471 8833 
0.9218 3770 
0.8971 0573 
0.8731 5400 


0.0708 7379 
0,0425 0601 
0.9151 4100 
0.8884 8705 
0.8026 0878 


0.0001 8357 
0.9335 1070 
0.0010 4271 
0.8714 4223 
0.8410 7317 




7 
8 
9 
10 


0.8750 2427 
0.8557 0040 
0.8300 3835 
0.8185 2101 
0.8005 1013 


0.8022 0687 
0.8412 0524 
0.8207 4657 
0.8007 2830 
0.7811 0840 


0.8407 8401 
0.8270 4128 
0.8040 0035 
0.7833 0385 
0.7023 0791 


0.8374 8426 
0.8130 0151 
0.7804 0923 
0.7004 1073 
0.7440 0801 


0.8135 0004 
0.7850 0000 
0.7894 1156 
0.7337 3007 
0.7080 1881 


11 
12 
13 
14 
15 


0.7828 0400 
0.7650 6748 
0.7488 1905 
0.7323 4137 
0.7102 2028 


0.7021 4478 
0.7435 5589 
0.7254 2038 
0.7077 2720 
0.0904 0550 


0.7410 9310 
0.7221 3440 
0.7028 0720 
0.0830 0728 
0.0060 9078 


0.7224 2128 
0.7013 7988 
0.0800 5134 
0.0011 1781 
0.0418 0105 


0.0840 4671 
0.0017 8330 
0.0394 0415 
0.0177 8170 
0.5008 9002 


10 
17 


0.7004 6580 
0.6850 5212 


0.0730 2493 
0.6571 0500 


0*0478 7424 
O.G305 3464 


0.0231 0094 
0.0050 1046 


0.5707 0501 
5572 0378 


18 
10 
20 


0.6699 7763 
0.6552 3484 
0.6408 1047 


0.6411 6591 
0.6255 2772 
0.0102 7004 


0.0130 6802 
0.6072 3400 
0.6812 6057 


0.5873 0401 
0.6702 8003 
0.6536 7675 


(X5383 0114 
0.6201 5609 
0.5026 0688 


21 


0.0207 1538 


0.5953 S020 


0.5656 0308 


0.5375 4928 


0,4855 7090 


22 
28 
24 


0.6129 2457 
0.6994 3724 
0.6802 4008 


0.6808 0407 
0.5000 0724 
0.5528 7535 


0.5505 5376 
0.5358 1874 
0.5214 7800 


0.5218 0260 
0.6066 9175 
0.4019 3374 


0*4691 5003 
0.4532 8503 
0,4379 5713 


25 


0.5733 4039 


0.5303 0050 


0.5076 2120 


0.4770 0667 


0*4231 4099 


20 
27 


0.5007 2997 
0.5483 9117 


0.5202 3472 
0.5133 0073 


0.4030 3700. 
0.4807 1821 


0.4630 0473 
0.4501 8006 


0.4088 3767 
0.3050 1224 


28 


0.5363 2388 


0.5008 7778 


0.4078 6227 


0.4370 7075 


0*3816 5434 


20 
80 


0,5245 2213 
0.5129 8008 


0,4880 0125 
0.4707 4209 


0.4563 3008 
0.4431 4421 


0.4243 4030 
0.4110 8676 


0*3087 4816 
0.3502 7841 


81 


0.5010 9201 


.0.4651 1481 


0,4312 8301 


0.3009 8715 


0.3442 3035 


32 
33 
34 
35 


0.4000 5233 
0.4708 5558 
0.4692 0641 
0.4580 0900 


0.4537 7055 
0,4427 0208 
0.4310 0534 
0.4213 7107 


0.4107 4103 
0,4086 0708 
0.3075 7380 
0.3800 3314 


0.3883 3703 
0.3770 2625 
0.3600 4490 
0.3553 8340 


0.3325 8071 
0.3213 4271 
0.3104 7005 
0.2009 7080 


30 


0.4488 7002 


0.4110 0372 


0.3705 7727 


0.3460 3243 


0.2898 3272 


37 


0.4389 9208 


0.4010 0705 


0.3064 0860 


0.3340 8294 


0.2800 3101 


88 
80 


0.4293 3270 
0.4108 8528 


0.3012 8402 
0,3817 4139 


0.3506 8059 
0.3471 4310 


0.3262 2016 
0.3167 5355 


0.2705 0194 
0.2014 1250 


40 


0,4100 4575 


0.3724 3002 


0.3378 5222 


0.3005 5084 


0.2525 7247 


41 


0.4016 0954 


0.3633 4005 


0.3288 0005 


0.2076 2800 


0,2440 3137 


42 


0.3927 7210 


0.3544 8483 


0.3200 0008 


0.2880 6022 


0,2357 7910 


43 


0,3841 2925 


0.3458 3880 


0.3114 4405 


0.2805 4294 


0.2278 0590 


44 


0.3750 7053 


0.3374 0370 


0.3031 0044 


0.2723 7178 


0.2201 0231 


45 


0.3074 0981 


0.3291 7440 


0.2040 0702 


0.2644 3862 


0.2126 6924 


40 


0.3593 2500 


0,3211 4670 


0.2871 0172 


0.2507 3663 


0.2054 6787 


. 47 


0.3514 1809 


0.3133 1204 


0.2704 1773 


0.2492 5870 


0.1085 1068 


48 


0.3430 8518 


0.3056 7116 


0.2710 3040 


0,2410 9880 


0.1018 0045 


40 


0.3301 2242 


0.2082 1579 


0.2040 0122 


0.2340 6020 


0.1863 2024 


50 


0.3287 2008 


0.2000 4221 


0.2576 7783 


0.2281 0708 


0.1790 6337 



TABLE VI PRESENT VALUE OF 1 



n 


2|% 


2 -Of 
a % 


2 -or 
i% 


3% 


s|% 


51 
53 
53 
54 
55 


0.3214 9250 
0.3144 1810 
0.3074 0930 
0.3007 3287 
0.2041 1528 


0.2838 4600 
0.2709 2208 
0.2701 0870 
0.2335 7028 
0.2671 6052 


0.2600 8402 
0.2439 7471 
0.2374 4497 
0.2310 0000 
0.2240 0(511 


0.2214 6318 
0.2150 1280 
0.2087 5029 
0.2020 7010 
0.1967 0717 


0.1720 9843 
0.1671 4824 
0.1014 9689 
0.1580 3407 
0.1507 6814 


50 
57 
58 
59 
00 


0.2870 4330 
0.2813 1374 
0.2731-2347 
0.2600 0040 
0.2031 4850 


0.2508 7855 
0.2447 5056 
0.2387 8082 
0.2329 0568 
0.2272 8350 


0.2188 8575 
0,2130 2740 
0.2073 2003 
0.2017 7710 
0.1003 7070 


0.1910 3600 
0.1854 7103 
0.1800 0084 
0.1748 2508 
0.1007 3309 


0.1456 0004 
0.1407 3433 
0.1369 7520 
0.1313 7701 
0.1260 3431 


61 
02 
03 
64 
05 


0.2573 C801 
0.2516 0487 
0.2401 6035 
0.2407 3971 
0.2354 4220 


0.2217 4009 
0.2103 3170 
0.2110 5541 
0.2059 0771 
0.2008 8557 


0.1911 2097 
0.1860 0581 
0.1810 2755 
0.1701 8263 
0.1714 0718 


0.1647 8941 
0.1509 8072 
0.1553 2082 
0.1508 0505 
0.1464 1325 


0.1220 4184 
0.1184 0453 
0.1144 8747 
0.1100 1501 
0.1008 7628 


00 
07 
08 
09 
70 


0.2302 0138 
0.2261 9450 
0.2202 3012 
0.2153 9278 
0.2106 5300 


0.1050 8593 
0.1012 0578 
0.1865 4223 
0.1810 0241 
0,1775 6368 


0.1068 7804 
0.1624 1172 
0.1680 6403 
0.1538 3448 
0.1407 1720 


0.1421 4879 
0.1380 0853 
0.1330 8887 
0.1300 8028 
0.1262 0730 


0.1032 0114 
0.0097 0022 
0.0963 0638 
0.0031 3503 
0.0890 8012 


71 

78 
78 
74 
75 


0.2060 1700 
0.2014 8420 
0.1070 5066 
0.1027 1458 
0.1884 7301 


0.1732 2300 
0.1089 0806 
0.1048 701(5 
0.1008 5478 
0.1500 3140 


0.1457 1023 
0.1418 1044 
0.1380 1603 
0.1343 2119 
0.1307 2622 


0.1220 1880 
0.1100 4737 
0.1155 7098 
0.1122 1357 
0.1080 4521 


0.0860 4311 
0.0840 0300 
0.0811 6232 
0.0784 1770 
0.0767 6590 


70 
77 

78 
79 
80 


0.1843 2657 
0.1802 7048 
0.1703 0365 
0.1724 2411 
0.1686 2093 


0.1531 0380 
0.1403 6065 
0.1457 2040 
0.1421 7218 
0.1387 0457 


0.1272 2747 
0.1238 2235 
0.1206 0837 
0.1172 8309 
0.1141 4412 


0,1057 7205 
0.1026 9131 
0.0907 0030 
0.0007 0641 
0.0939 7710 


0.0732 0376 
0.0707 2827 
0.0683 3650 
0.0600 2060 
0.0037 9285 


81 
82 
83 
84 
85 


0.1640 1025 
0.1012 0022 
0.1577 4105 
0.1542 6097 
0.1508 7528 


0.1353 2153 
0.1320 2101 
0.1288 0008 
0.1256 5040 
0.1225 9463 


0.1110 8017 
0.1081 1608 
0.1052 2237 
0.1024 0620 
0.0006 6540 


0.0012 3000 
0.0885 8243 
0.0800 0230 
0.0834 9743 
0.0810 6547 


0.0610 3561 
0.0596 5131 
0.0576 3750 
0.0555 0178 
0.0537 1187 


80 
87 
88 
89 
90 


0.1475 5528 
0.1443 0835 
0.1411 3286 
0.1380 2724 
0.1340 8007 


0.1106 0452 
0,1106 8733 
0.1138 4130 
0.1110 6468 
0.1083 5579 


0.0069 9706 
0.0044 0100 
0.0018 7533 
0.0804 1038 
0.0870 2324 


0.0787 0434 
0.0764 1108 
0.0741 8630 
0.0720 2502 
0.0000 2770 


0.0518 9553 
0.0501 4000 
0.0484 4J503 
0.0408 0079 
0.0452 2395 


91 
918 
93 
94 
95 


0.1320 1053 
0.1201 1445 
0.1262 7331 
0.1234 9468 
0.1207 7719 


0.1057 1200 
0.1031 3460 
0.1006 1012 
0.0981 0600 
0.0057 7073 


0.0846 9415 
0.0824 2740 
0.0802 2131 
0.0780 7427 
0,0750 8400 


0.0678 9105 
0.0059 1364 
0.0039 9383 
0.0621 2903 
0.0603 2032 


"0.0430 9464 
0.0422 1704 
0.0407 8941 
0.0304 1000 
0.0380 7735 


90 
97 
98 
99 
100 


0.1181 1050 
0.1155 2029 
0.1120 7828 
0.1104 9221 
0.1080 6084 


0.0034 3486 
0.0011 5506 
0.0880 3264 
0.0867 6356 
0.0846 4737 


0.0730 6104 
0.0710 7181 
0.0700 4550 
0.0081 7080 
0.0663 4634 


0.0685 6342 
0.0668 5769 
0.0552 0104 
0.0685 9383 
0.0620 3284 


0.0307 8071 
0,0356 4562 
0.0343 4350 
0.0331 8221 
0,0320 6011 



41 



TABLE VI PRESENT VALUE OP 1 



n 


4% 


4f% 


5% 


5|% 


6% 


1 


0.9615 3846 


0.0569 3780 


0.9523 8005 


0.0478 0730 


0.0433 002,'i 


% 


0.0245 5621 


0.91 57 2995 


0.0070 2048 


0.8984 5242 


0.8800 9044 


3 


0.8889 9636 


0.8762 0060 


0.8638 3760 


0.8510 1300 


O.&'iOO 1028 


4 


0.8548 0410 


0.8385 6134 


0.8227 0247 


0.8072 1074 


0.71)20 03UI1 


5 


0.8219 2711 


0.8024 5105 


0.7835 2017 


0.7051 3435 


0.7472 5817 


6 


0.7903 1463 


0.7678 9574 


0.7402 1640 


0.72C2 4683 


0.7040 (IOM 


7 


0.7699 1781 


0.7348 2840 


0.7106 8133 


0.0874 3081 


0.0(150 5711 


8 


0.7306 9021 


0.7031 8513 


0.6708 393(1 


0.0615 0887 


0.0274 12H7 


9 


0.7026 8674 


0.6729 0443 


0.6440 0892 


0.0170 2H20 


0.51)18 1)84(1 


10 


0.6765 6417 


0.6439 2768 


0.6139 1325 


0.5854 3058 


0.5683 0478 


11 


0.6495 8093 


0.6161 9874 


0.5846 7029 


0.5540 1050 


0.5207 876:1 


12 


0.6245 9705 


0.5896 6380 


0.6568 3742 


0.5250 81. r )2 . 


0,41)00 Ol);t(! 


13 


0.6005 7409 


0.5642 7164 


0.6303 2135 


0.4085 (1008 


0.4088 31)02 


14 


0.5774 7508 


0.5390 7286 


0.5060 0795 


0.4725 0037 


0.4423 001)1) 


15 


0.5552 6450 


0.5187 2044 


0.4810 1710 


0.4470 3305 


0.4172 OfiOO 


16 


0.5339 0818 


0.4944 6932 


0.4581 1152 


0.4245 8100 


0.3030 4028 


17 


0.5133 7325 


0.4731 7639 


0.4362 9669 


0.4024 453 


0,3713 0442 


18 


0.4936 2812 


0.4528 0037 


0.4155 2005 


0.3814 0500 


0.3608 4371) 


19 


0.4746 4242 


0.4333 0179 


0.3057 3300 


0.3016 7900 


0.3805 1111)1 


20 


0.4563 8695 


0.4146 4286 


0.3768 3948 


0.3427 280G 


0.3118 0473 


21 


0.4388 3360 


0.3967 8743 


0.3589 4230 


0.3248 6158 


0.2041 6540 


22 


0.4219 5539 


0.3797 0089 


0.3418 4087 


0.3070 2507 


0.2775 0610 


23 


0.4057 2633 


0.3633 5013 


0.3265 7131 


0.2918 7207 


0.2017 0720 


24 


0.3901 2147 


0.3477 0347 


0.3100 0701 


0.2700 5050 


0.2400 7865 


25 


0.3751 1680 


0.3327 3060 


0.2953 0277 


0.2022 3370 


0.2320 9803 


26 


0.3606 8923 


0.3184 0248 


0.2812 4073 


0.2485 6275 


0.2108 1003 


27 


0.3468 1657 


0.3046 9137 


0.2678 4832 


0.2350 0450 


0.2073 0705 


28 


0.3334 7747 


0.2915 7069 


0.2560 9364 


0.2233 2181 


0.1050 3014 


29 


0.3206 5141 


0.2790 1502 


0.2429 4032 


0.2116 7044 


0.1845 6074 


80 


0.3083 1867 


0.2670 0002 


0.2313 7746 


0.2000 4402 


0.1741 1013 


81 


0.2964 6026 


0.2555 0241 


0.2203 5047 


0.1001 8300 


0.1042 5484 


32 
33 


0.2850 5794 
0.2740 0417 


0.2444 9991 
0.2339 7121 


0.2008 6017 
0.1998 7264 


0.1802 0910 
0.1708 7119 


0.1640 5740 
0.1401 8022 


34 


0.2635 5209 


0.2238 9589 


0.1003 5480 


0.1010 0321 


0.1370 11(53 


35 


0.2534 1547 


0.2142 5444 


0.1812 9029 


0.1535 1903 


0.1801 0522 


86 
37 
38 
39 
40 


0.2436 6872 
0.2342 9685 
0.2252 8543 
0.2166 20G1 
0.2082 8904 


0.2050 2817 
0.1001 0921 
0.1877 5044 
0.1706 6549 
0.1719 2870 


0.1726 5741 
0.1644 3563 
0.1566- 0536 
0.1401 4707 
0.1420 4568 


0.1465 1024 
0.1370 3008 
0.1307 3041 
0.1230 2302 
0.1174 0314 


0,1227 4077 
0.1167 0318 
0.1002 3K8fi 
0.1030 6552 
0,0072 2210 


41 
42 
43 
44 
45 


0.2002 7703 
0.1925 7493 
0.1851 6820 
0.1780 4635 
0.1711 0841 


0.1645 2507 
0.1574 4026 
0.1506 6054 
0.1441 7276 
0.1379 6437 


0.1352 8100 
0.1288 3962 
0.1227 0440 
0.1168 0133 
0.1112 9061 


0.1113 3047 
0.1055 3504 
0.1000 3322 
0.0048 1822 
0.0898 7500 


0.0017 1005 
0.0805 2740 
0.0810 2002 
0.0770 0908 
0.0720 15007 


46 
47 

48 
49 
50 


0.1546 1386 
0.1582 8256 
0.1521 8476 
0.1463 4112 
0.1407 1262 


0.1320 2332 
0.1263 3810 
0.1208 S771 
0.1166 9158 
0.1107 0965 


0.1050 9668 
0.1009 4921 
0,0961 4211 
0.0915 6391 
0.0872 0373 


0.0861 8905 
0.0807 4840 
0.0765 3885 
0.0725 4867 
0.0687 0052 


0.0085 37H1 
0.0640 5831 
0.0600 0840 
0.0575 4500 
0.0542 8830 



TABLE VI PRESENT VALUE OF 1 
n = (1 + f)-n 



n 


4% 


4% 


6% 


6l% 


6% 


51 
52 
53 
54 
55 


0.1353 0059 
0,1300 0072 
0.1250 9300 
0.1202 8173 
0.1150 5551 


0.1050 4225 
0.1013 8014 
0.0070 1440 
0.0928 3(583 
0.0888 3007 


0.0830 5117 
0.0790 0036 
0.0753 2080 
0.0717 4272 
0.0683 2040 


O.OG51 8153 
0.0617 8344 
0.0585 6250 
0.0565 0948 
0.0526 1562 


0.0512 1544 
0.0483 1046 
0.0455 8150 
0.0430 0147 
0.0405 6742 


56 
57 
58 
59 
60 


0.1112 0722 
0.10GO 3002 
0.1028 1733 
0.0088 6282 
0.0050 0040 


0.0850 1347 
0.0813 5200 
0.0778 4938 
0.0744 9701 
0.0712 8001 


0.0060 7270 
0.0619 7400 
0.0500 2291 
0.0562 1230 
0.0535 3552 


0.0498 7263 
0.0472 7263 
0.0448 0818 
0.0424 7221 
0.0402 6802 


0.0382 7115 
0.0301 0480 
0.0340 6119 
0.0321 3320 
0.0303 1434 


61 
62 
03 
64 
65 


0.0014 0423 
0.0878 8808 
0.0845 0835 
0.0812 5803 
0.0781 3272 


0.0082 1915 
0.0652 8148 
0.0024 7032 
0.0507 8021 
0.0572 0594 


0.0600 8021 
0.0485 5830 
0.0402 4600 
0.0440 4381 
0.0419 4648 


0.0381 6026 
0.0301 6092 
0.0342 8428 
0.0324 9695 
0.0308 0270 


0.0285 9843 
0.0269 7UU5 
0.0254 5250 
0.0240 1179 
0.0226 5264 


66 
67 
68 
69 
70 


0.07C1 2702 
0.0722 3809 
0.0004 5070 
0.0067 8818 
0.0042 1040 


0.0547 4253 
0.0523 8510 
0.0501 2037 
0.0479 7069 
0.0459 0497 


0.0300 4003 
0.0380 4670 
0.0302 3495 
0.0345 0048 
0.0328 6017 


0.0201 0600 
0.0270 7485 
0.0262 3208 
0.0248 0453 
0.0235 6828 


0.0213 7041 
0.0201 0077 
0.0100 1050 
0.0170 4301 
0.0160 2737 


71 
72 
73 
74 
75 


0.0617 4042 
0.0603 7445 
0.0570 0081 
0.0548 0501 
0.0527 8367 


0.0430 2820 
0.0420 3055 
0.0402 2637 
0.0384 9413 
0.0368 3640 


0.0313 0111 
0.0208 1058 
0.0283 0103 
0.0270 3008 
0.0267 5150 


0.0223 3960 
0.0211 7498 
0.0200 7107 
0.0190 2471 
0.0180 3200 


0.0150 0021 
0.0150 6530 
0.0142 1254 
0.0134 0800 
0.0126 4011 


76 

77 
78 
79 
80 


0,0507 5353 
0.0488 0147 
0.0460 2440 
0.0451 1070 
0.0433 8433 


0.0352 5023 
0.0337 3228 
0.0322 7960 
0.0308 8005 
0.0205 5048 


0.0245 2524 
0.0233 5737 
0.0222 4512 
0.0211 8582 
0.0201 7098 


0.0170 9279 
0.0102 0170 
0.0163 6706 
0.0145 5646 
0.0137 9759 


0.0119 3313 
0.0112 6767 
0.0106 2044 
0.0100 1928 
0.0094 5216 


81 
82 
83 

84 
85 


0.0417 1570 
0.0401 1125 
0.0385 6861 
0.0370 8510 
0.0350 5875 


0.0282 8058 
0.0270 0850 
0.0250 0287 
0.0247 8744 
0.0237 2003 


0.01TB2 1017 
0.0183 0111 
0.0174 2003 
0.0165 0005 
0.0158 0019 


0.0130 7828 
0.0123 0648 
0.0117 5022 
0.0111 3706 
0.0106 5701 


0.0089 1713 
0.0084 1238 
0.0070 3021 
0.0074 8609 
0.0070 6320 


86 
87 
88 
89 
90 


0.0342 8720 
0.0320 6852 
0,0317 0050 
0.0304 8125 
0.0203 0800 


0.0220 9800 
0.0217 2115 
0.0207 8579 
0.0198 0070 
0.0100 3417 


0.0150 6037 
0.0143 3940 
0.0130 5657 
0,0130 0020 
0.0123 8(501 


0.0100 0004 
0.0004 8407 
0.0080 0040 
0.0085 2180 
0.0080 7763 


0.0000 6340 
0.0062 8022 
0.0050 3040 
0.0055 0472 
0.0062 7803 


91 
92 
93 
M 
95 


0.0281 8163 
0.0270 9772 
0.0200 5550 
0.0260 6337 
0.0240 8078 


0.0182 1451 
0.0174 3016 
0.0100 7058 
O.OlfiO-0132 
0.0162 7399 


0.0117 9706 
0.0112 3530 
0.0107 0028 
0.0101 9074 
0.0007 0547 


0.0076 5043 
0.0072 6728 
0.0008 7804 
0.0066 2032 
0.0001 8040 


0.0040 7028 
0.0040 9743 
0.0044 3154 
0.0041 8070 
0.0030 4405 


96 
97 
98 
99 
190 


0.0231 63215 
0.0222 7235 
0,0214 1572 
0.0205 0204 
0.0108 0004 


0.0146 1626 
0.0139 8085 
' 0.0183 8454 
0.0128 0817 
0.0122 5063 


0.0002 4331 
0.0088 0315 
0.0083 8306 
0.0079 8471 
0.0070 0440 


0.0058 5820 
0.0065 5279 
0.0052 6331 
0.0049 8802 
0.0047 2883 


0.0037 2081 
0.0035 1019 
0.0033 1160 
0.0031 2400 
0.0029 4723 



43 



TABLE VI PRESENT VALUE OP 1 



n 


6|% 


7% 


7j% 


8% 


8|% 


1 
9 
3 
4 
ff 


0.0389 0714 
. 0.8810 5928 
0.8278 400fl 
0.7773 2309 
0.7298 808-1 


0.9345 7944 
0.8734 3873 
0.8162 9788 
0.7028 9531 
0.7129 8018 


0.9302 3250 
0.8053 3201 
0.8040 0057 
0.7488 (H163 
0.0005 5803 


0.0251) 2503 
0.8573 3882 
0.7038 3221 
0.7350 20S5 
0.0805 Site!) 


0.1)2111 fiHOil 
0.8-Ul.l flflao 
0.7S211 (IS10 
0.7l!1.1 7-128 
(l.tlOfil) 4. r >11! 


C 
7 
8 
9 
10 


0.0853 3412 
0.0435 0021 
0.0042 3110 
0.5073 5323 
0.5327 2004 


0.0003 4222 
0.0227 497-1 
0.r>S20 0010 
0.5430 337-t 
0.5083 4020 


0.0-17!) (1162 
0.0027 5400 
0.5007 0223 
0.5215 8347 
0.4851 0303 


0. llUDt (111(13 

0.5S3-1 iioio 

0./540S! 0888 
0.5002 -IS'.)? 

o,-i<m '.wit) 


0.15] 21) .15011 

o./in-iii 2ii3. r > 
o.fwoii on ir. 

0.-179H 70(18 

0.-M22 85-ia 


11 

12 
13 
11 
15 


O.C002 1224 
0.4000 8286 
0.4410 1070 
-0.4141 0025 
0.3888 2052 


0.4750 9280 
0.4440 110(1 
0.4149 0445 
0.3878 1724 
0.3024 4002 


0.4513 431!) 
0.4108 5-113 
0.3006 ClOH 
0.3033 1347 
0.3370 0002 


0.428S 82SO 
0.31)71 1370 
0.307(1 0702 
0.3401 0101 
0.3152 4170 


o.d07 3033 

0.3757 0108 
0.3-102 0883 
0.31SI1 4178 
O.tllMl 3UKO 


16 
17 

18 
10 
20 


0.3050 9533 
0.3428 1251 
0.3218 8909 
0.3022 4384 
0.2837 9703 


0.3387 3-100 
0.3106 7-130 
0.2958 0302 
0.2705 0832 
0.2584 1000 


0.3143 8000 
0.2024 5302 
0.2720 4032 
0.2530 00i;t 
0.2354 1315 


0.2018 0017 
0.2703 (18SI5 

0.2502 40o;< 

0.2317 1200 
0.2145 -1N21 


0.2710 UU07 
0.2-11IM 58110 
0.2302 8-160 
0.2122 -1378 
0.105(1 1I33U 


21 
22 
23 
24 
25 


0.2004 7008 
0.2502 1228 
0.2349 4111 
0.2200 0198 
0.2071 3801 


0.2415 1300 
0.2257 1317 
0.2100 4088 
0.1971 4002 
0.1842 4018 


0.2180 8807 
0.2037 1007 
0.1804 0830 
0.17(52 7740 
0.1039 700H 


0.1080 6575 
0.1831) ll)fll 
0.1703 1C28 
0.1570 ODH'l- 
().14() 1700 


0.1802 OHIO 
0.10111 (t738 
0.1531 4005 
O.H11 5I7I 
0.1300 0378 


26 
27 

28 
20 
30 


0.1944 9570 
0.1826 2515 
0.1714 7902 
0.1610 1316 
0.1511 8607 


0.1721 9540 
0.1009 3037 
0.1504 0221 
0.1405 0282 
0.1313 0712 


0.1525 3800 
0.1418 0043 
0.1310 0008 
0.1227 87(11 
0.1142 2103 


0.1362 0170 
0.1251 8(182 
0.1151) 1372 
0.1073 2762 
0.0003 7733 


0.1101) 0210 
0.1105 0885 
0.1018 51-18 
0.0038 7233 
O.OKlin 182H 


31 
32 
33 
34 
85 


0.1419 5875 
0.1332 9400 
0.1251 5925 
0.1175 2042 
0.1103 4781 


0.1227 7301 
0.1147 4113 
0.1072 3470 
0.1002 1934 
0.0930 6204 


0.1002 5212 
0.0988 3018 
0.0010 4343 
0.0855 2H77 
0.0795 0104 


0.0920 1006 
0.0852 0005 
0.0788 8N03 
0.07,40 -IfiUl 
0.0070 34M 


0.0707 40fi 
0.0734 OiMl 
0.0077 :)58il 
0.0(12-1 2(13(1 
0.0575 ,'18.18 


86 
37 

38 
30 
40 


0,1030 1207 
0.0972 8917 
0.0913 5134 
0.0857 7590 
0.0805 4075 


0.0875 3540 
0.0818 0884 
0.0764 5080 
0.0714 5501 
0.0607 8038 


0.0740 1083 
0.0088 4720 
0.0040 4300 
0.0505 7580 
0.0554 1035 


0.0020 2458 
0.0670 H572 
0.0630 1)018 
0.041)7 1,'M1 

o.o4m> 3003 


0.0630 3005 
0.0-188 7Mfi 
0.0460 4742 
0.0-11 fi 1830 
0.0382 0677 


41 
42 
43 
44 
45 


0.0760 2612 
0.0710 0950 
0.0060 7550 
0.0026 0610 
0.0587 8515 


0.0624 1157 
0.0583 2867 
0.0545 1268 
0.0509 4043 
0.0476 1349 


0.0515 5288 
0.0479 5017 
0.0440 1030 
0.0414 9804 
0.0380 0283 


0.0420 2123 
0.0304 (till 
0.0305 4084 
0.0338 15411 

0.0313 27S8 


0.0362 07UO 
0.0326 0500 
0.0200 6868 
0.0271J 1100 
0,0264 4H4H 


40 
47 
48 
40 
50 


0.0551 9733 
0.0518 2848 
0.0480 0524 
0.0466 9506 
0.0429 0610 


0,0444 9869 
0.0415 8747 
0.0388 0079 
0.0303 2410 
0.0339 4770 


0.0359 0901 
0.0334 0438 
0.0310 7375 
0.0289 0682 
0.0268 8913 


0.0200 07.10 
0.0208 6801 
0.0248 0008 
0.0230 2(103 
0.0213 2123 


O.OarU 5-1K2 

o.oairt 17;M 
o.oioo a:ma 

0.018.1 02U7 
0.0100 24311 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 



n 


5% 


1% 


H% 


S* 


1% 


1 
2 
3 

4 
5 


1.0000 0000 
2.0041 6667 
3.0125 1738 
4.0260 0052 
5.0418 4064 


1.0000 0000 
2.0050 0000 
3.0150 2500 
4.0301 0013 
5.0502 5003 


1.0000 0000 
2.0058 3333 
3.0175 8403 
4.0351 3031 
5.0580 7460 


1.0000 0000 
2.0075 0000 
3.0225 5625 
4.0452 2542 
5.0755 6461 


1.0000 0000 
2.0100 0000 
3.0301 0000 
4.0604 0100 
5.1010 0601 


6 

7 
8 

10 


0.0628 4831 
7.0881 1018 
8.1176 43fl7 
9.1G14 6749 
10.1805 0860 


0.0755 0188 
7.1058 7939 
8.1414 0870 
9.1821 1583 
10.2280 2641 


6.0881 8364 
7.1230 9794 
8.1052 5284 
9.2128 8349 
10.2660 2531 


6.1136 -3136 
7.1594 8358 
8.2131 7971 
0.2747 7866 
10.3443 3940 


6.1620 1506 
7.2135 3621 
8.2856 7050 
9.3685 2727 
10.4622 1254 


11 
12 
13 
14 
15 


11.2320 5520 
12.2788 5549 
13.3300 1739 
14.3855 5013 
15.4454 0896 


11.2791 6654 
12.3355 6237 
13.3972 4018 
14.4642 2039 
15.5305 4752 


11.3265 1398 
12.31)25 8529 
13.4048 7537 
14.5434 2048 
15.6282 5710 


11.4219 2194 
12.6075 8036 
13.0013 9325 
14.7034- 0370 
15.8136 7923 


11.5668 3467 
12.6825 0301 
13.8093 2804 
14.0474 2132 
10.0968 9664 


16 
17 
18 
19 
20 


10.5098 5520 
17.5780 4027 
18.6518 9063 
19.7290 0684 
20.8118 1353 


16.0142 3020 
17.0973 0141 
18.7867 8791 
19.8797 1685 
20.9791 1544 


16.7194 2193 
17.8109 5189 
18.9208 8411 
20.0312 5593 
21.1481 0493 


16.0322 8183 
18.0692 7304 
19.1947 1849 
20.3386 7888 
21.4912 1897 


17.2678 0449 
18.4304 4314 
19.6147 4767 
20.8108 9504 
22.0190 0399 


21 
22 
23 
24 
25 


21.8985 2942 
22.9897 7330 
24.0855 6402 
25.1859 2054 
26.2908 6187 


22.0840 1101 
23.1944 3107 
24.3104 0322 
25.4319 5524 
26.5591 1502 


22.2714 6887 
23.4013 8577 
24.5378 9386 
26.6810 3157 
26.8308 3759 


22.6524 0312 
23.8222 9614 
25.0009 0336 
26.1884 7059 
27.3848 8412 


23.2391 9403 
24.4715 8598 
25.7163 0183 
26.0734 6485 
28.2431 9950 


26 
27 

28 
29 
30 


27.4004 0713 
28.5145 7549 
29.6333 8622 
30.7508 6867 
31.8850 1224 


27.6919 1059 
28.8303 7015 
29.9745 2200 
31.1243 9461 
32.2800 1658 


27.987a 5081 
29.1500 1036 
30.3206 6558 
31.4976 2607 
32.0812 6164 


28.5902 7075 
29.8046 9778 
31.0282 3301 
32.2009 4476 
33.6029 0184 


29.5256 3150 
30.8208 8781 
32.1200 '9609 
33.4503 8760 
34.7848 9153 


31 
32 
33 
34 
36 


33.0178 6646 
34.1554 4090 
35,2977 5524 
30.4448 2922 
37.5960 8268 


33.4414 1666 
34.6086 2375 
35.7816 6086 
36.9605 7520 
38.1463 7807 


33.8719 0233 
35.0094 8843 
36.2740 6045 
37.4860 5913 
38.7043 2648 


34.7541 7381 
36.0148 2991 
37.2849 4113 
38.5645 7819 
39.8538 1253 


36.1327 4045 
37.4940 6785 
38.8690 0853 
40.2576 9802 
41.6602 7560 


36 
87 
38 
30 
40 


38.7533 3552 
39.9148 0775 
41.0811 1945 
42.2522 9078 
43.4283 4199 


39.3361 0496 
40.5327 8549 
41.7354 4042 
42.9441 2066 
44.1588 4730 


39,9301 0071 
41.1630 2030 
42.4031 4395 
43.6504 9502 
44.0051 2352 


41.1527 1612 
42.4613 6149 
43.7798 2170 
45.1081 7037 
46.4404 8164 


43.0768 7830 
44.5076 4714 
45.9527 2361 
47.4122 6086 
48.8863 7336 


41 
42 
48 
44 
45 


44.6092 0342 
45.7961 6548 
46.9859 7866 
48.1817 5358 
49.3825 1088 


45.3706 4153 
46.6065 3974 
47.8395 7244 
40.0787 7030 
50.3241 6415 


46.1670 7007 
47.4363 7798 
48.7130 9018 
49.9972 4988 
61.2889 0050 


47.7948 3026 
49.1532 9148 
60.5210 4117 
61.9008 5573 
63.290X 1215 


50.3762 3709 
51.8789 8946 
63.3077 7936 
54.9317 5715 
56.4810 7472 


46 
47 
48 
49 
60 


50.5882 7134 
51.7990 5581 
53.0148 8521 
54.2357 8056 
55.4017 6298 


51.5767 8407 
52.8336 6390 
54.0978 3222 
55.3683 2138 
56.6451 6299 


52.5880 8575 
53.8048 495D 
55.2092 3621 
66.6312 9009 
57.8610 5596 


54.6897 8799 
56.0990 6140 
57.5207 1111 
58.9521 1644 
60.3942 5732 


58.0458 8547 
50.6263 4432 
61.2226 0777 
62.8348 3385 
64.4631 8218 



TABLE Vn AMOUNT OF ANNUITY OF 1 PER PERIOD 

, a + f) n - 1 

i 



n 


sk* 


1% 


5% 


! 


1% 


51 
52 
53 
54 
55 


56.8928 5366 
57,9290 7388 
69.1704 4603 
60.4189 8855 
61.6687 2600 


57.0283 8880 
59.2180 3075 
60.5141 2090 
61.8166 9150 
63.1257 7496 


59.1985 7877 
60.5430 0381 
61.8070 7059 
63.26S1 4287 
64.6271 4870 


01.8472 1424 
03.3110 (1835 
04.7850 0130 
8H.2717 052 
67.7088 3400 


60.1078 1401 
07.7BHS 1)215 
00.4405 8107 
71.1410 4088 
72.8524 5735 


56 
57 

58 
59 
60 


62.9256 7902 
64.1878 6935 
65.4563 1881 
66.7280 4930 
68.0060 8284 


64.4414 0384 
65.7636 1080 
67.0924 2891 
68.4278 9105 
69.7700 3051 


86.0041 4040 
67.3891 6465 
68.7822 6801 
70.1834 0701 
71.6929 0165 


00.2771 0035 
70.7900 7800 
72.3270 5301) 
73.8701 1100 
75.4241 3003 


74./380I) ll2 
70.3207 174 
78.0000 fiOOfl 
7U.870I) 0025 
81.0800 0080 


61 
62 
63 
64 
65 


69.2894 4162 
70.5781 4753 
71.8722 2314 
73.1716 9074 
74.4765 7278 


71.1188 8086 
72.4744 7607 
73.8368 4744 
76.2060 3168 
76.6820 6184 


73.0105 2001 
74.4364 2165 
75.8706 3411 
77.3132 1281 
78.7642 0666 


70.9898 1705 
78.5072 4159 
80.1564 9690 
81.7670 0062 
83.3708 5214 


83.480.3 flflfifl 
85.3212 3022 
87.1744 4852 
80.0401 809/5 
00.0300 4882 


66 
67 
68 
68 
70 


75.7868 9184 
77.1026 7055 
78.4239 3168 
79.7506 9806 
81.0829 9264 


77.0649 7215 
79.3547 9701 
80.7615 7099 
82.1553 2885 
83.5661 0549 


80.2236 6442 
81.8916 3579 
83.1681 7034 
84.6533 1800 
88.1471 2902 


84.0001 3353 
80.0336 0453 
88.2833 5657 
80.0454 8174 
91.0200 7286 


02.8400 1531 
04.7744 7540 
00.7222 2021 
08.0804 4242 
100.G73 3084 


71 
72 
73 
74 
75 


82.4208 3844 
83.7642 5860 
85.1132 7634 
86.4679 1500 
87.8281 9797 


84,9839 3602 
86.4088 5570 
87.8408 9908 
80.2801 0448 
Q0.7265 0500 


87.6406 5394 
89.1609 4359 
90.6810 4000 
92.2100 2188 
93.7479 1367 


03.3072 2340 
05.0070 2768 
96.7105 802S 
98.4440 7714 
100.1833 1440 


102.0831 0021 
104.7000 3121 
100.7570 3052 
108.8246 008 
110.9128 4084 


76 
77 

78 
79 
80 


89.1941 4880 
90.6657 9109 
91.9431 4855 
93.3262 4500 
94.7151 0436 


92.1801 3762 
93.0410 3821 
95.1092 4340 
96.5847 8962 
98.0677 1357 


95.2947 7650 
96.8506 6270 
98.4166 2490 
99.9897 1604 
101.5729 8038 


101.0346 8032 
103.0991 9940 
105.4760 4340 
107.2080 2066 
100.0725 3072 


113.0210 7530 
115.1521 9500 
117.3037 1701 
110.47(37 5418 
121.0715 2172 


81 
82 
83 
84 
85 


96.1097 5062 
97.6102 0792 
98.9165 0046 
100.3286 6254 
101.7466 8859 


90.5680 5214 
101.0568 4240 
102.5611 2161 
104.0739 2722 
105.5942 9685 


103.1654 9849 
104.7672 9723 
106.3784 3980 
107.9989 8070 
109.8286 7475 


110.8905 7470 
112.7222 5401 
114.5676 7091 
116.4209 2845 
118.3001 3041 


123.8882 3004 
120.1271 1031 
128.3883 0050 
130.0722 7440 
132.0780 0718 


86 
87 
88 
89 
90 


103.1706 3312 
104.6005 1076 
106.0363 4622 
107.4781 6433 
108.9259 9002 


107.1222 6834 
108.0578 7968 
110.2011 6908 
111.7621 7492 
113.3109 3580 


111.2884 7710 
112.9175 4322 
114.5762 2889 
110.2445 9022 
117.0226 8367 


120.1873 8130 
122.0887 8675 
124.0044 5206 
125.9344 8004 
127.8789 0400 


135.3087 8712 
137.0018 74flO 
140.0384 0874 
142.4388 7808 
144.8032 0740 


91 
93 
93 
94 
95 


110.3708 4831 
111.8397 6434 
113.3057 6336 
114.7778 7071 
116.2661 1184 


114.8774 9048 
116.4618 7793 
118.0341 3732 
119.6243 0800 
121.2224 2964 


119.6106 6699 
121.3082 9429 
123.0159 2001 
124.7335 1891 
126.4611 3110 


129.8380 8715 
131.8118 7280 
133.8004 6186 
136.8039 6631 
137.8224 9506 


147.3110 0014 
140.7850 1014 
152,2828 0033 
164,8056 080S 
157.3537 6501 


96 
97 
98 
99 
100 


117.7406 1230 
119.2310 9777 
120.7278 9401 
122.2309 2690 
123.7402 2243 


122.8285 4169 
124.4426 8440 
126.0648 9782 
127.6952 2231 
129,3336 9842 


128.1988 2103 
129.9466 4749 
131.7046 6960 
133.4729 4684 
135.2616 3903 


139.8561 6377 
141.0060 8499 
143.0693 7313 
146.0491 4343 
148.1446 1201 


160.0272 0200 
102.5285 0548 
165.1518 3114 
167.8033 4045 
170.4813 8204 



46 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 

( S at fl - <* + *)" " 1 
n] 



n 


3% 


i% 


5% 


> 


1% 


101 
102 
103 
104 
105 


125.2558 0000 
120.7777 0580 
128.3060 4033 
120.8405 5444 
131.3815 5075 


i:i0.0803 0602 
132.0352 0875 
134.20S4 4501) 
135.9H09 3732 
137.0407 8701 


137.0405 0034 
138.8300 0020 
140.0408 0870 
142.4702 0508 
144.3013 4253 


150.2555 0585 
162.3825 1281 
154.5253 8100 
150.6843 2202 
158.8504 5444 


173.1801, 0677 
175.9180 5874 
178.0772 3033 
181.4040 1172 
18-1.2780 6184 


100 
107 
108 
100 
110 


132.0280 7000 
13-1.4828 5005 
130.0431 0580 
137.0100 4251 
130.1834 1709 


130.3380 3594 
141.0347 2012 
142.7308 0076 
144.4536 0025 
140.1758 0725 


140.1431 0030 
147.0050 0178 
140.8580 0940 
151.7330 8043 
153.0181 0010 


101.0500 0035 
103.2587 8210 
105.4832 2200 
167.7243 4714 
160.0822 7974 


187.1214 3830 
189.0020 5274 
102.8025 7927 
105.8215 0500 
108.7707 2011 


111 
112 
113 
114 
115 


140.7033 4800 
142.3408 6255 
143.0420 8008 
145.5427 4942 
147.1401 7754 


147.0007 4058 
140.0-102 8032 
151.3SJ45 1172 
153.1514 8428 
154.0172 4170 


155.5143 0225 
157.4214 0001 
150.3307 0001 
101.2002 4285 
103.2000 8010 


172.2571 4084 
174.5490 7544 
176.8581 9351 
170.1846 2006 
181.5285 1408 


201.7075 1731 
204.7851 0248 
207.8330 4441 
210.9113 7485 
214.0204 8800 


110 
117 
118 
119 
120 


148.7022 9012 
150.3821 4203 
152.0087 3420 
153.0421 0401 
155.2822 7045 


15(1.0018 2701 
158.4752 8704 
100.2070 0348 
102.0000 0180 
103.8703 4081 


105.1020 3832 
107.1254 8354 
109.1003 8210 
171.0808 0109 
173.0848 0743 


183.8800 7854 
186.2001 6338 
188.0001 7203 
101.0811 0832 
193.5142 7708 


217.1006 0340 
220.3323 0042 
223.5350 2343 
226.7709 7000 
230.0386 8046 


121 
122 
123 
124 
125 


150.0202 8805 
158.5831 0098 
100.2430 2415 
101.0110 0717 
103.5802 3887 


105.0087 4354 
107.5272 3720 
109.3048 7344 
171.2110 9781 
173.0077 5030 


176.0944 0881 
177.1158 5321 
170.1490 2002 
181.1940 0502 
183.2510 3040 


105.9066 3410 
198.4363 7042 
200.9236 4174 
203.4305 0905 
206.9662 9832 


233.3300 7035 
236.6724 6712 
240.0301 0179 
243.4305 8370 
246.8739 7054 


120 
127 

128 
129 
130 


105.2078 4810 
100.0504 0423 
108.6521 1010 
170.3548 3331 
172.0040 4512 


174.9330 0508 
170.8077 0060 
178.0017 0030 
180.5S52 5830 
182.4881 8405 


185.3199 0474 
187.4010 2805 
180.4042 0071 
101.5005 8355 
103.7172 4778 


208.5009 7050 
211.0047 2784 
213.0477 1330 
210.2500 7115 
218.8710 4668 


250.3427 1034 
253.8401 4053 
257.3840 0800 
200.0584 5408 
204.5080 3862 


181 
132 
133 
134 
135 


173.7816 8114 
176,5050 7106 
177.2360 4400 
178.0764 3100 
180.7211 0203 


184.4000 2567 
180.3220 2870 
188.2542 4184 
100.1055 1305 
102.1404 0002 


105.8472 0500 
107.0807 0744 
200.1446 4740 
202.3121 5785 
204.4023 1210 


221.5134 8628 
224.1748 3743 
226.8561 4871 
220.5575 6082 
232.2792 5100 


208.2137 1000 
271.8958 6619 
275.6148 1476 
279.3700 0200 
283,1046 7263 


130 
137 
138 
139 
140 


182.4741 0777 
184.2344 7081 
180.0021 2040 
187.7771 2020 
180.5505 3400 


104.1072 2307 
100.0777 5010 
108.0681 4708 
200.0484 3872 
202.0480 8002 


200.0851 8302 
208.8008 4740 
211.1093 7744 
213.3408 4881 
215.5853 3709 


235.0213 4598 
237.7840 0608 
240.5073 8012 
243.3710 4152 
240.1960 2883 


286.0003 1020 
200.8062 8245 
204.7740 4527 
208.7226 9473 
302.7009 2107 


141 
142 
143 
144 
145 


191.3403 0530 
198.1400 5441 
104.0514 3214 
100.7037 2077 
108.5836 7805 


204.0580 2432 
200.0702 1804 
208.1000 1504 
210.1501 (1311 
212.2000 1303 


217.8429 1822 
220.1130 0858 
222.3070 0408 
224.0940 8400 
227.0057 0544 


249.0434 0580 
251.9112 3134 
254.8005 0558 
257.7115 6082 
200.04-44 0659 


300.7370 2089 
310.8043 0110 
314.0124 3501 
310.0615 5030 
323.2521 7405 


146 
147 

148 
149 
150 


200.4110 1023 
202.2400 5010 
20-1.0887 4800 
205.0301 1770 
207.7071 0744 


214.2010 1850 
2111,3332 2K09 
218.4148 0423 
220.5000 (1870 
222.0005 0354 


220.3290 0538 
231.0070 0317 
234.0100 5787 
230.3841 0004 
238.7030 7000 


203.5092 3064 
200.5702 3304 
209.5755 5569 
272.6073 7236 
275,6418 6205 


327.4846 0070 
331.7695 4367 
330.0771 3011 
340.4379 1050 
344,8422 8000 



47 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 

. (1 + 0" - 1 
i 



n 


ll% 


lj% 


l|% 


lj% 


2% 


1 

2 
3 
4 
5 


1.0000 0000 
2.0112 5000 
3.0338 7066 
4.0080 0767 
6.1137 7278 


1.0000 0000 
2.0125 0000 
3.0376 5625 
4.0756 2605 
5.1265 7229 


1.0000 0000 
2.0150 0000 
3.0462 2500 
4.0000 0338 
5.1522 6693 


1.0000 0000 
2.0175 0000 
3.0528 0625 
4.1002 3030 
5.1780 8038 


1.0000 0000 
2.0200 0000 
3.0004 0000 
4.1210 0800 
5.2040 4010 


6 

7 
8 
9 
10 


6.1713 0270 
7.2407 2980 
8.3221 8807 
0.4168 1260 
10.6217 4068 


6.1906 6444 
7.2680 3762 
8.3688 8809 
0.4633 7420 
10.6810 6637 


6.2295 6003 
7.3229 0410 
8.4328 3011 
9.5593 3160 
10.7027 2107 


6.2087 0590 
7.3784 0831 
8.5075 3045 
0.0504 1224 
10.8253 0945 


6.3081 2090 
7.4342 8338 
8.6829 0905 
0.7540 2843 
10.0407 2100 


11 
12 
13 
14 
15 


11.6401 1016 
12.7710 6140 
13.9147 3684 
16.0712 7062 
10.2408 2848 


11.7130 3720 
12.8603 6142 
14.0211 1604 
16.1003 7088 
10.3803 3463 


11.8632 0240 
13.0412 1143 
14.2368 2960 
15.4603 8205 
10.0821 3778 


12.0148 4304 
13.2251 0371 
14.4505 4303 
15.7095 3253 
16.9844 4935 


12,1687 1542 
13.4120 8973 
14.0803 3152 
15.0730 3815 
17.2934 1002 


16 
17 
18 
19 
20 


17.4236 3780 
18.0106 '5260 
19.8290 2267 
21.0620 9907 
22.2889 3619 


17.5911 0382 
18.8110 5336 
20.0461 0153 
21.2067 6893 
22.5629 7864 


17.0323 6984 
10.2013 5630 
20.4893 7672 
21.7967 1030 
23.1236 6710 


18.2816 7721 
19.0010 0056 
20.0446 3408 
22.3111 0578 
23.7016 1110 


18.6302 8525 
20.0120 7096 
21.4123 1238 
22.8405 5803 
24.2973 0080 


21 
22 
23 
24 
25 


23.6396 8671 
24.8046 0717 
26.0836 6788 
27.3769 0790 
28:6849 8913 


23.8460 1577 
26.1430 7847 
20.4573 6605 
27.7880 8403 
29.1364 3508 


24.4705 2211 
25.8376 7904 
27.2251 4364 
28.0335 2080 
30.0630 2361 


25.1103 8038 
20.5559 2620 
28.0206 5400 
29.5110 1037 
31.0274 6015 


26.7833 1719 
27.2089 8354 
28.8449 0321 
30.4218 6247 
32.0302 9972 


26 
27 
28 
29 
80 


30.0076 9626 
31.3462 8183 
32.6979 1625 
34.0667 6781 
36.4490 0769 


30.4996 2802 
31.8808 7337 
33.2793 8420 
34.6963 7069 
36.1290 6880 


31.5139 0896 
32.9866 7850 
34.4814 7867 
35.9987 0085 
37.6386 8137 


32.5704 3900 
34.1404 2238 
35.7378 7977 
37.3032 0267 
30.0171 5020 


33.6709 0572 
35.3443 2383 
37.0612 1031 
38.7022 3451 
40.6680 7021 


31 
32 
33 
34 
85 


30.8478 0903 
38.2623 4688 
39.6927 9829 
41.1393 4227 
42.0021 5987 


37.5806 8210 
39.0604 4069 
40.5385 7120 
42.0463 0334 
43.5708 0903 


39.1017 6159 
40.0882 8801 
42.2986 1233 
43.0330 9152 
45.5020 8780 


40,6099 5042 
42.4121 9955 
44.1544 1305 
45.0271 1527 
47.7308 3979 


42.3794 4070 
44.2270 2961 
46.1115 7020 
48.0338 0100 
40.0044 7703 


36 
87 
38 
39 
40 


44.0814 3417 
46.6773 6030 
47.0000 9649 
48.6198 5906 
60.1668 3248 


45.1155 0550 
46.6794 4932 
48.2926 4243 
49.8862 2921 
51.4895 6708 


47.2750 6921 
48.9861 0874 
50.7108 8638 
52.4806 8306 
54.2678 9301 


40.5661 2049 
51.4335 3675 
53.3336 2305 
55.2669 6200 
57.2341 3300 


51.9943 6710 
54.0342 5453 
56.1149 3002 
58.2372 3841 
60.4019 8318 


41 
42 
43 
44 
45 


61.7312 0934 
53.3131 8546 
64.9129 5879 
66.5307 2957 
58.1007 0028 


53.1331 7654 
54.7973 4125 
50,4823 0801 
.68.1883 3087 
50.9156 9108 


56.0819 1232 
57.9231 4100 
50.7919 8812 
61.6888 6794 
63.6142 0096 


59.2357 3124 
01.2723 5654 
63.3446 2278 
65.4531 5367 
67.5985 8386 


62.6100 2284 
64.8022 2330 
67.1594 0777 
69.5020 6712 
71.8927 1027 


46 
47 
48 
49 
50 


59.8210 7566 
61.4940 6276 
63.1858 7097 
64.8967 1201 
66.6268 0002 


61.6646 3721 
63.4354 4518 
65.2283 8824 
67.0437 4310 
08.8817 8980 


65.5084 1398 
67.5519 4018 
69,5662 1920 
71.6086 9758 
73.6828 2804 


69,7815 5908 
72.0027 3637 
74.2627 8426 
76.5623 8298 
78.9022 2468 


74.3305 0447 
76.8171 7676 
79.3535 1027 
81.9405 8906 
84,5704 0145 



48 



TABLE Vn AMOUNT OF ANNUITY OF 1 PER, PERIOD 

. a + o n - 1 
f 



n 


1|% 


l|% 


1|% 


1*0, 

J- 4 % 


2% 


si 

53 
53 
54 
55 


68.3703 5152 
70.1455 8548 
71.0347 2332 
73.7439 8895 
76.5736 0883 


70.7428 1226 
72.6270 9741 
74.5340 3613 
76.4000 2283 
78.4224 5502 


75.7880 7046 
77.0248 9152 
80.0937 6489 
82.2951 7136 
84.5295 0803 


81.2830 1301 
83.7054 6635 
86.1703 1201 
88.0782 9247 
91.2301 6259 


87.2700 8948 
90.0164 0927 
02.8107 3746 
95.6730 7221 
98.5865 3365 


56 
57 

58 
59 
60 


77.4238 1193 
70.2048 2081 
81.1868 0065 
83.1002 4023 
85.0361 2704 


80.4027 3631 
82.4077 7052 
84.4378 6765 
80.4933 4000 
88.5745 0776 


88.7075 4202 
80.0005 0600 
91.4359 0865 
03.8075 3863 
06.2140 6171 


03.8266 9043 
96.4680 5752 
00.1508 5002 
101.8921 0405 
104.0752 1588 


101.6582 6432 
104.6894 2961 
107.6812 1820 
110.8348 4257 
114.0515 3042 


61 
62 
63 
64 
65 


86.0917 7222 
88.0704 2066 
00.0713 4009 
91.0047 7464 
95.0409 6586 


90.0810 8010 
92.8152 1022 
94.0754 0034 
97.1625 0285 
99.3771 2520 


08.6578 7149 
101.1377 3956 
103.6548 0565 
100.2006 2774 
108.8027 7216 


107.5070 3216 
110.3884 0522 
113.3202 0231 
116.3033 0585 
119.3386 1370 


117.3326 7021 
120.6702 2161 
124.0928 0604 
127.5740 6216 
131.1201 5541 


66 
67 
68 
69 
70 


07.1101 7672 
90.2026 6021 
101.3186 0021 
103.4585 3154 
105.0224 4002 


101.6103 3933 
103.8896 8107 
100.1882 0083 
108.5155 6334 
110.8719 0776 


111.4348 1374 
114.1063 3504 
110.8179 3098 
119.5701 0005 
122.3037 5205 


122.4270 3944 
125.5605 1263 
128,7669 7010 
132.0204 0124 
135.3307 5826 


134.7480 7852 
138.4430 5209 
142.2125 2513 
140.0507 7663 
149.9779 1114 


71 
72 
73 
74 
75 


107.8106 9247 
110.0235 6276 
112.2013 2784 
114.6242 6778 
110.8126 6570 


113.2578 9773 
116.6730 2145 
118.1106 4172 
120.5050 3600 
123.1034 8644 


126.1092 0024 
128.0771 0738 
130.9083 6534 
133.9033 3007 
136.9727 8003 


138.6990 4653 
142.1262 7084 
145.6134 8074 
140.1817 2581 
152.7720 6601 


163.9774 6937 
158.0570 1875 
162.2181 5913 
166.4625 2231 
170.7917 7276 


76 

77 
78 
79 
80 


110.1268 0828 
121.4600 8487 
123.8334 8845 
120.2260 1520 
128.0466 6462 


125.6422 8002 
128.2128 0852 
130.8154 6863 
133.4506 6100 
136.1187 0526 


140.0273 7234 
143.1277 8202 
140.2740 0067 
140.4088 2010 
162.7108 5247 


156.4455 6609 
160.1833 6441 
103.0865 7329 
167.8563 3832 
171,7938 2424 


176.2070 0821 
179.7117 6038 
184.3059 9558 
188.0921 1649 
193,7719 6780 


81 

82 
83 
84 
85 


131.0039 3060 
133.5687 4642 
130.0713 0481 
138.0021 0801 
141.1614 7273 


138.8202 8020 
141.5555 3370 
144.3249 7787 
147.1200 4010 
149.0681 5310 


150.0015 1525 
150.3415 3708 
102.7310 0106 
160.1720 3507 
160.6652 2651 


176.8002 1017 
179.8707 1996 
184.0245 6255 
188.2440 9239 
192.5302 7976 


198.6473 9696 
203.6203 4400 
208,6927 6180 
213.8666 0683 
210.1439 3807 


86 
87 
88 
89 
90 


143.7495 3930 
140.3607 2102 
140.0133 4724 
151.0807 4739 
154.3962 5705 


152.8427 5501 
155.7632 8045 
168.7002 0557 
161.0830 5814 
164.7050 0702 


173.2102 0380 
170.8083 5605 
180.4004 8230 
184.1073 8054 
187.0200 0038 


196.9087 1716 
201.3646 1971 
205.8783 2556 
210.4811 0625 
215.1646 1718 


224.5268 1775 
230.0173 6411 
235.6177 0110 
241.3300 5521 
247.1506 5632 


91 
92 
93 
94 
95 


157.1332 1404 
150.0000 6301 
102.0098 4045 
165.6302 2276 
168.3024 3776 


167.7038 2021 
170.8608 6700 
173.0066 2881 
177.1715 8067 
180.3802 3151 


191.7488 4880 
195.6260 8102 
100.5694 5784 
203.5528 4071 
207.0061 4240 


210.9209 0708 
224.7787 7205 
229.7124 0148 
234.7323 6850 
239.8401 8496 


253.0097 8944 
259.1617 8523 
205.3450 2094 
271.0610 2136 
278.0840 5078 


96 
97 
98 
99 
100 


171.2808 5269 
174.2138 2078 
177.1737 3537 
180.1069 3089 
183.1038 1706 


183.6410 6040 
186.0305 7264 
100.2732 7080 
103.6510 0580 
107.0723 4200 


211.7202 3459 
215.8000 3811 
220.1344 7808 
224.4364 9680 
228.8030 4330 


245.0373 8819 
250.3255 4248 
255.7062 3947 
201.1810 0866 
266.7517 6780 


284.6400 6898 
201.3305 0210 
298.1603 8400 
305.1207 1108 
312.2323 0501 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 



n 


2|% 


2ior 
2% 


2f% 


3% 


3|% 


i 

2 
3 

4 
5 


1.0000 0000 
2.0225 0000 
3.0680 0025 
4.1370 3839 
6.2301 1971 


1.0000 0000 
2.0250 0000 
3.0756 2600 
4.1525 1603 
5.2503 2852 


1.0000 0000 
2.0276 0000 
3.0832 5625 
4.1080 4580 
5.2826 6706 


1.0000 0000 
2.0300 0000 
3.0000 0000 
4.1836 2700 
5.3091 3581 


1.0000 0000 
2.0350 0000 
3.1062 2500 
4.2140 4288 
5.3024 0588 


6 

7 
8 
9 
10 


6.3477 9740 
7.4900 2284 
8.6591 0180 
0.8539 9300 
11.0757 0784 


6.3877 3873 
7.5474 3016 
8.7301 1690 
9.9545 1880 
11.2033 8177 


0.4279 4040 
7.0047 0870 
8.8138 3825 
10.0602 1880 
11.3327 0482 


6.4684 0088 
7.6024 0218 
S.S923 3005 
10.1591 0613 
11.4038 7931 


0.5501 5218 
7.7704 0761 
0.0516 8077 
10.3084 0681 
11.7313 0316 


11 
13 
13 
14 
Iff 


12.3249 1127 
13.6022 2177 
14.9082 7176 
16.2437 0788 
17.6091 9130 


12.4834 6831 
13.7956 5297 
15.1404 4170 
16.5189 5284 
17.9310 2680 


12.0444 1585 
13.0921 3720 
15.3709 2107 
10.7007 8039 
18.2017 8062 


12.8077 0669 
14.1020 2950 
16.0177 0045 
17.0803 2410 
18.5089 1380 


13.1410 0102 
14.0010 0104 
10.1130 3030 
17.0700 8030 
19.2956 8088 


16 
17 

18 
19 
20 


19.0053 9811 
20.4330 1957 
21.8927 0251 
23.3853 4966 
24.9116 2003 


19.3802 2483 
20.8047 3045 
22.3863 4871 
23.0460 0743 
25.5446 5701 


10.7039 7948 
21.3074 8892 
22.8934 4487 
24.5230 1460 
26.1073 9750 


20.1568 8130 
21.7015 8774 
23.4144 3537 
25.1108 0844 
20.8703 7449 


20.9710 2971 
22.7060 1575 
24.4996 9130 
26.3671 8050 
28.2700 8181 


21 
22 
23 
24 
25 


26.4720 2923 
28.0676 4989 
29.0991 7201 
31.3074 0338 
33.0731 0996 


27.1832 7405 
28.8628 5500 
30.5844 2730 
32.3490 3798 
34.1577 0393 


27.9178 2593 
29.0856 0015 
31.5010 1921 
33.3082 2190 
35.2858 4810 


28.6704 8672 
30.5367 8030 
32.4528 8370 
34.4204 7022 
30.4592 0432 


30.2094 7068 
32.3280 0215 
34.4604 1373 
30.6605 2821 
38.9498 5660 


26 
27 
28 
29 
30 


34.8173 1628 
36.0007 0590 
38.4242 2178 
40.2887 0077 
42.1952 0402 


36.0117 0803 
37.9120 0073 
39.8598 0075 
41.8662 9577 
43.9027 0316 


37.2502 0802 
39.2807 5407 
41.3609 7542 
43.4984 0224 
45.0940 0830 


38.3530 4225 
40.7000 3352 
42.0300 2252 
45.2188 5020 
47.5754 1671 


41.3131 0168 
43.7500 6024 
46.2906 2734 
48.9107 0930 
51.8226 7728 


31 
32 
33 
34 
35 


44.1446 5746 
46.1379 1226 
48.1760 1528 
50.2599 7503 
52.3908 2508 


46.0002 7074 
48.1502 7761 
50.3540 3445 
52.6128 8531 
54.9282 0744 


47.9612 1003 
50.2608 6831 
52.0522 8900 
55.1002 2706 
57.0154 8391 


60.0020 7818 
52.5027 5852 
66.0778 4128 
57.7301 7062 
60.4020 8181 


54.4294 7008 
57.3345 0247 
60.3412 1005 
63.4631 6240 
00.6740 1274 


86 
37 
38 
39 
40 


54.5696 1804 
56.7974 3500 
59.0763 7735 
61.4045 7334 
63.7861 7024 


57.3014 1263 
59.7339 4794 
02.2272 9064 
04.7829 7900 
07.4025 5354 


00.1900 0972 
02,8554 0724 
65.5839 3094 
88.3874 8904 
71.2681 4490 


83.2760 4427 
60.1742 2259 
60.1594 4927 
72.2342 3275 
75.4012 5973 


70.0070 0318 
73,4678 6930 
77.0288 0472 
80.7240 0004 
84.5602 7775 


41 
42 
43 
44 
45 


60.2213 6521 
68.7113 4592 
71.2673 6121 
73.8606 4161 
76.5225 0605 


70.0876 1737 
72.8398 0781 
75.6008 0300 
78.5523 2308 
81.5101 3116 


74.2280 1898 
77.2092 8950 
80.3941 9496 
83.0060 3532 
86.9041 7379 


78.6632 9753 
82.0231 9645 
85.4838 9234 
89.0484 0911 
92.7198 0130 


88,5005 3747 
92.6073 7128 
90.8486 2028 
101.2383 3130 
105.7816 7290 


46 
i 47 
48 
49 
50 


79.2442 6243 
82.0272 5834 
84.8728 7165 
87.7826 1126 
90.7576 1776 


84.6640 3443 
87.6678 8530 
90.8695 8243 
94.1310 7199 
97.4843 4879 


90.2940 3867 
93.7771 2463 
07.3559 9566 
101,0332 8544 
104.8117 0079 


06.6014 5723 
100.3905 0095 
104.4083 0698 
108.5406 4785 
112.7968 6729 


110.4840 3145 
115.3500 7265 
120.3882 6650 
125.6018 4557 
130.9979 1016 



50 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 

fe fit iT = (1 H~ *')" - 1 



n 


2-w 
i/o 


2|% 


2|% 


3% 


3|% 


51 
58 
53 
54 
55 


03.7096 6416 
06.0101 5661 
100.0006 3513 
103.3426 7442 
106.6678 8460 


100.0214 57S1 
104.4444 9305 
108.0558 0629 
111.7560 0645 
115.5500 2136 


108.6940 2256 
112.6831 0818 
110.7818 0305 
120.0933 0573 
125.3207 1411 


117.1807 7331 
121.6001 0651 
126.3470 8240 
131.1374 0488 
130.0716 1072 


136.5828 3702 
142.3632 3831 
148.3459 4058 
154.6380 5782 
100.0468 8084 


50 
57 

58 
50 
60 


110.0670 1200 
113.5444 4002 
117.0001 8002 
120.7330 2160 
124.4504 3403 


110.4306 9440 
123.4256 8676 
127.6113 2893 
131.6901 1215 
135.0915 8005 


129.7070 3375 
134.3356 2718 
139.0208 5602 
143.8531 7700 
148.8091 4038 


141.1537 6831 
140.3883 8136 
151.7800 3280 
157.3334 3379 
163.0534 3680 


107.5800 3009 
174.4463 3207 
181.6500 1809 
188.9052 0085 
100.5108 S288 


61 

en 

03 
04 
65 


128.2505 6072 
132.1362 0764 
136.1002 7221 
140.1717 3083 
144.3255 9477 


140.3013 7070 
144.0011 6419 
140.5236 9330 
154.2617 8503 
150.1183 3027 


153.0013 0174 
150.1336 8002 
104.5098 5022 
170.0338 7726 
176.7098 0880 


168.9450 3001 
175.0133 9110 
181.2037 9284 
187.7017 0662 
104.3327 5782 


204.3949 7378 
212.5487 9780 
220.0880 0570 
229.7225 8609 
238.7028 7650 


66 
67 
68 
69 
70 


148.5720 2066 
] 52.0168 1137 
157.3664 1713 
161.8060 3651 
166.6396 1758 


164.0062 8853 
109.1986 0574 
174.4286 6314 
170.7893 7971 
185.2841 1421 


181.5418 2863 
187.5342 2892 
103.6014 2021 
200.0179 3427 
206.5184 2746 


201.1027 4055 
208.1970 2277 
215.4435 5145 
222.0068 5800 
230.5040 6374 


248.1105 7718 
257.8037 8238 
267.8268 9400 
273.2008 3535 
288.0378 6450 


71 

72 
73. 
74 
75 


171.2867 5898 
176.1407 1100 
181.1038 7705 
180.1787 1420 
191.3677 3536 


100.0162 1706 
106.6891 2240 
202.0063 6055 
208.6715 0031 
214.8882 9705 


213.1076 8422 
220.0006 2064 
227.1122 8700 
234.3578 7561 
241.8027 1709 


238.5118 8565 
246.0072 4222 
255.0672 5049 
263.7102 7727 
272.0308 6569 


300.0508 8085 
311.5524 6400 
323.4568 0024 
335.7777 8824 
348.5300 1083 


76 

77 
78 
79 
80 


100.6735 0041 
202.0986 6337 
207.6458 8320 
213.3179 1667 
210.1175 0877 


221.2605 0447 
227.7020 1709 
234.4868 1751 
241.3489 8706 
2-18.3827 1265 


249.4522 9181 
257.3122 2083 
265.3883 1615 
273.0804 0485 
282.2128 7345 


281.8007 8126 
291.2040 7460 
301.0019 9603 
311.0320 5084 
321.3630 1855 


361.7288 8121 
375.3890 6085 
389.5276 7708 
404.1011 4071 
410.3067 8086 


81 

82 
83 

84 
85 


226.0477 1407 
231.1112 8703 
237.3112 0160 
243.0607 9667 
250.1329 3867 


255,5022 8047 
262.0820 8748 
270.5560 3906 
278.3205 5566 
286.2785 6955 


200.9737 2747 
200,0755 0408 
300.2248 3137 
318.7285 1423 
328.4935 4837 


332.0030 0010 
342.9640 2038 
354.2529 4717 
365.8805 3558 
377.8560 5168 


434.0825 2430 
451.2009 1274 
407.0091 5409 
485.3791 2610 
603.3873 9448 


86 

87 
88 
89 
90 


256.7609 2060 
263.5380 5060 
270.4676 6674 
277.5531 7902 
284.7081 2555 


294.4355 3379 
302.7064 2213 
311.3663 3268 
320.1604 9100 
329.1542 5328 


338.5271 2005 
348.8366 1678 
350.4200 2374 
370.3139 3830 
381.4976 7170 


390.1026 6020 
402.8084 4001 
416.9853 9321 
420.4040 5500 
443.3489 0365 


521.9852 5320 
541.2547 3715 
501.1080 5295 
581.8400 0581 
603.2060 2701 


91 
92 
93 
94 
95 


202.2060 8337 
299.7807 2025 
307.5267 8045 
315,4451 1066 
323.6420 3177 


338.3831 0961 
347.8420 8735 
357.5387 6453 
387.4772 2339 
377.6041 5308 


302.0887 5492 
404.7059 4568 
416.0278 3418 
420.3933 4962 
442.2016 0674 


457.6493 7076 
472.3788 5189 
487.5602 1744 
503.1767 2397 
510.2720 2669 


625.3172 0206 
648.2033 0506 
071,8904 2073 
696.4065 8540 
721.7808 1506 


96 
97 
98 
99 
100 


331.8223 4090 
340.2883 4360 
348.9448 3130 
357.7060 9010 
366.8465 0213 


388.1057 5783 
398.8084 0177 
400.7786 1182 
421.0230 7711 
432.6486 5404 


455.3622 1257 
468.8846 7342 
482.7790 0104 
497.0554 2449 
511,7244 4867 


535.8501 8645 
552.9250 0205 
670.5134 3281 
588.0288 6000 
607.2877 3270 


748.0431 4451 
775.2246 5457 
803.3575 1748 
832.4750 3059 
862.6116 5666 



61 



TABLE YD AMOUNT OF ANNUITY OF 1 PER PJJRIOD 

' (r-fo-Ci + y-i 

"I i 



n 


4% 


4% 


5% 


6|% 


6% 


l 
9 

3 
4 

5 


I.OOOO 0000 
2.0400 0000 
3.1216 0000 
4.2464 61400 
.4163 2266 


1.0000 0000 
2.0450 0000 
3.1370 2500 
4.2781 9113 
5.4707 0973 


1.0000 0000 
2.0500 0000 
3.1525 0000 
4.3101 2500 
5.5256 3125 


1.0000 0000 
2.0550 0000 
3.1080 2500 
4.3422 6638 
5.5810 9103 


1.0000 0000 
2.0600 0000 
3.1830 0000 
4.3740 1600 
5.6370 9290. 


6 

7 
8 
9 
10 


0.6329 7540 
7.8982 0448 
9.2142 2020 
10.6827 0631 
12.0001 0712 


6.7108 9166 
8.0191 6170 
9.3800 1362 
10.8021 1423 
12.2882 0937 


6.8019 1281 
8.1420 0845 
9.5491 0888 
11.0205 6432 
12.5778 9254 


6.8880 5103 
8.2608 0384 
9.7215 7300 
11.2502 5051 
12.8753 5370 


G.0753 1864 
8.3038 3705 
9.8974 0791 
11.4013 1508 
13.1807 9494 


11 
It 
13 
14 
15 


13.4803 5141 
15.0268 0546 
16.0268 3768 
18.2919 1110 
20.0235 8764 


13.8411 7879 
15.4650 3184 
17.1599 1327 
18.9321 0937 
20.7840 5429 


14.2067 8716 
15.9171 2652 
17.7129 8285 
19.5986 3199 
21.5785 6359 


14.5834 0825 
10.3855 0065 
18.2807 9814 
20.2925 7203 
22.4080 0350 


14.9710 4264 
16.8600 4120 
18.8821 3767 
21.0150 0503 
23,2750 0988 


16 
17 

18 
19 
20 


21.8245 3114 
23.0975 1239 
25.6454 1288 
27.0712 2940 
29.7780 7858 


22.7103 3073 
24.7417 0089 
26.8550 8370 
29.0635 6246 
31.3714 2277 


23.0674 9177 
25.8403 6636 
28.1323 8467 
30.5390 0391 
33.0650 5410 


24.6411 3099 
26.9904 0269 
29.4812 0483 
32.1020 7110 
34.8683 1801 


25.6725 2808 
28.2128 7970 
30.9050 5255 
33.7590 9170 
36.7855 9120 


21 
23 
23 
24 
25 


31.0092 0172 
34.2479 6979 
36.6178 8858 
30.0820 0412 
41.6459 0820 


33.7831 3080 
36.3033 7796 
38.9370 2996 
41.0891 9631 
44.6652 1015 


35.7192 5181 
38.6052 1440 
41.4304 7512 
44.5019 9887 
47.7270 9882 


37.7860 7560 
40,8643 0965 
44.1118 4609 
47.5379 9825 
51.1525 8816 


39.0927 2068 
43.3022 9028 
46.9058 2700 
50.8155 7735 
64.8645 1200 


28 
27 
28 
20 
30 


44.3117 4462 
47.0842 1440 
40.0075 8298 
52.9602 8630 
66.0840 3775 


47.5706 4460 
50.7113 2301 
53.9933 3317 
57.4230 3316 
61.0070 6966 


61.1134 5376 
54.0691 2645 
68.4025 8277 
62.3227 1191 
66.4388 4750 


54.0659 8051 
68.0891 0943 
63.2335 1046 
67.7113 6353 
72.4364 7797 


69.1563 8272 
63.7057 0608 
68.5281 1102 
73.0397 9832 
79.0581 8022 


31 
32 
83 
34 
35 


59.3283 3526 
62.7014 6807 
66.2095 2742 
60.8579 0851 
73.6522 2486 


64.7623 8779 
68.6602 4524 
72,7562 2628 
77.0302 6640 
81.4066 1800 


70.7607 8988 
75.2988 2937 
80.0637 7084 
85,0009 5938 
90.3203 0735 


77.4104 2926 
82.6774 9787 
88.2247 6026 
04.0771 2207 
100.2513 6378 


84.8016 7739 
90.8897 7803 
97.3431 0471 
104,1837 5460 
111.4347 7987 


86 
37 
38 
39 
.40 


77.5083 1386 
81.7022 4640 
85.0703 3026 
90.4091 4971 
96.0255 1570 


86.1630 6581 
91.0413 4427 
96.1382 0476 
101.4644 2398 
107.0303 2306 


05.8363 2272 
101.6281 3886 
107.7095 4580 
114,0950 2309 
120.7907 7424 


106.7661 8879 
113.6372 7417 
120.8873 2425 
128.6301 2708 
136.6056 1407 


119.120S 0000 
127.2681 18(10 
135.0042 0678 
145,0584 5813 
154.7619 6502 


41 
42 
43 
44 
45 


90.8265 3033 
104.8195 9778 
110.0123 8169 
115.4128 7696 
121.0293 9204 


112.8466 8760 
118.9247 8854 
126.2764 0402 
131.9138 4220 
138.8490 6510 


127.8397 6295 
135.2317 6110 
142.9933 3866 
151.1430 0550 
159.7001 5587 


145,1189 2285 
154.1004 6360 
163.5759 8910 
173.5726 6850 
184.1191 6527 


165.0470 8356 
176.0605 4457 
187.5075 7724 
199,7680 3188 
212.7435 1370 


46 
47 
48 
49 
50 


126.8705 6772 
132.9453 9043 
139.2632 0604 
145.8337 3429 
152.8670 8366 


146.0982 1363 
153.6726 3314 
161.5879 0163 
109.8503 5720 
178.5030 2828 


168.6851 6366 
178.1194 2186 
188.0253 0204 
108.4266 6259 
209.3479 9572 


105.2457 1936 
206.9842 3302 
219.3683 6679 
232.4336 2696 
246.2174 7645 


226.5081 2462 
241.0086 1210 
256.5645 2882 
272.0584 0055 
290,3369 0458 



52 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 



n 


4% 


4% 


5% 


6|% 


6% 


51 
52 
53 
54 
55 


169.7737 6700 
107.1647 1708 
174.8513 0630 
182.8453 5805 
101.1501 7200 


187.6356 0455 
190.9747 0040 
200.8380 3408 
217.1403 7262 
227.0170 5038 


220.8163 0550 
232.8501 6528 
245.4980 7354 
258.7739 2222 
272.7120 1833 


200.7604 3706 
270.1012 0072 
292.2807 7309 
300.3025 4601 
327.3774 8562 


308.7500 5880 
328.2814 2230 
348.0783 0773 
370.9170 0020 
394.1720 2057 


50 
57 

58 
59 
60 


100.8055 3001 
208.7077 Olfil 
218.1406 7197 
227.8756 5885 
237.0006 8520 


230.1742 0750 
250.9371 0080 
203.2202 7053 
276.0745 9711 
280.4970 5398 


287.3482 4924 
302.7150 0171 
318.8514 4470 
335.7040 1703 
353.5837 1788 


340.3832 4733 
360.4343 2503 
387.5882 1386 
409.9056 0502 
433.4503 7173 


418.8223 4810 
444.0510 8905 
472.0487 0040 
602.0077 1782 
533.1281 8080 


01 
02 
63 
04 
65 


248.5103 1261 
250.4507 2511 
270.8287 6412 
282.6610 0428 
204.0683 8045 


303.5253 0190 
318.1840 0319 
333.5022 8333 
349.5098 8008 
366.2378 3096 


372.2029 0378 
391.8700 4807 
412.4008 5141 
434.0933 4308 
458.7080 1118 


458.2001 4217 
484,4900 9999 
512.1433 8540 
541.3112 7170 
572.0S33 0104 


500.1168 7174 
001.0828 2405 
038.1477 9349 
077.4360 0110 
719.0828 6070 


06 
07 
08 
09 
70 


307.7671 1507 
321.0778 0030 
334.0200 1231 
340.3177 4880 
364.2004 5876 


383.7185 3335 
401.9858 0735 
421.0752 3138 
441.0236 1670 
461.8696 7055 


480.6379 1174 
505.6698 0733 
531.9532 9770 
650.5500 0258 
588.5285 1071 


004.6470 7818 
638.7081 1608 
674.0320 1341 
713.0532 7415 
753.2712 0423 


703.2278 3241 
810.0215 0230 
850.6227 9250 
012.2001 0005 
007.0321 0065 


71 
72 
73 
74 
75 


370.8620 7711 
396.0505 6010 
412.8088 2200 
430.4147 7550 
448.0313 6052 


483.6538 1513 
606.4182 3081 
630.2070 6747 
555.0663 7505 
581.0443 0103 


018.0540 3025 
050.9020 8300 
084.4478 1721 
710.0702 0807 
750.6537 1848 


705.7011 2046 
840.4640 8209 
887.0002 3000 
937.6132 0278 
000.0764 2803 


1027.0030 0083 
1089.0285 8582 
1150.0063 0097 
1220.3060 7903 
1300.0480 7077 


76 
77 

78 
. 79 
80 


467.6760 2118 
487.2706 3003 
507.7708 7347 
520.0817 0841 
551.2440 7675 


008.1913 6822 
030.5599 0934 
000.2051 6790 
607.1844 0052 
720.6570 9864 


795.4864 0440 
830.2007 2402 
879.0737 0085 
024.0274 4889 
971.2288 2134 


1045.5300 3252 
1104.0348 1731 
1165.7567 3226 
1230.8733 5264 
1299.5713 8093 


1380.0060 0055 
1403.8050 3059 
1552.0342 9278 
1040.7023 5035 
1746.5998 9137 


81 
82 
83 
84 
85 


574.2947 7582 
508.2005 6085 
623.1972 2052 
640.1251 1870 
076.0001 2345 


703.3877 9497 
708.7402 4576 
835.0835 5680 
874.2893 1680 
914.0323 3012 


1020.7002 6240 
1072.8207 7552 
1127.4712 0430 
1184.8448 2752 
1245.0870 6880 


1372.0478 1321 
.1448.5104 4204 
1520.1786 1730 
1014.2833 3575 
1704.0639 1921 


1852.3058 8485 
1964.5306 3794 
2083.4120 1622 
2200.4167 3719 
2342.0817 4142 


80 
87 
88 
89 
00 


704.1337 2839 
733.2090 7753 
763.0310 4063 
796.1762 8226 
827.0833 3354 


960.7007 0125 
1000.8463 7086 
1046.8844 0381 
1004.0042 0468 
1145.2600 0050 


1308.3414 2234 
1374.7684 0345 
1444.4064 1812 
1617.7212 3903 
1504.6073 0008 


1708.7927 0977 
1808.7203 0881 
2004.1502 5570 
2115.3848 4980 
2232.7310 1000 


2484.5000 4591 
2634,6342 8400 
2793.7123 4174 
2002.3360 8225 
3141.0761 8718 


91 
92 
93 
94 
95 


862.1020 0688 
807.5807 7356 
034.4002 4450 
972.8608 5428 
1012.7840 4846 


1107.8061 1180 
1252.7073 8692 
1310.0792 1933 
1370.0327 8420 
1432.0842 5049 


1676.3376 0003 
1760.1045 4033 
1849.1007 7080 
1042.5052 6564 
2040.0036 2892 


2350.5312 2252 
2487.1404 3970 
2024.9331 0304 
2770.3044 8796 
2923.0712 3480 


3330.5390 9841 
3531.3720 8032 
3744,2644 0514 
3069.0000 0044 
4209.1042 4901 


90 
97 
98 
99 
100 


1054.2060 3430 
1097.4678 7677 
1142.3005 0080 
1180.0012 5443 
1237.6237 0461 


1498.1660 5117 
1600.6720 2847 
1038.0677 0976 
1712.7808 1930 
1700.8559 5027 


2143.7282 0537 
2251.9140 1504 
2305.6103 4642 
2484.7858 0374 
2010.0251 5003 


3085.4731 5271 
3250.1741 7011 
3430.2037 5580 
3020.2582 6237 
3820.7024 0080 


4402.6506 0469 
4731.4095 3486 
5010.2041 0090 
6318.2717 6337 
6638.3080 5857 



63 



TABLE VH AMOUNT OF ANNUITY OF 1 PER PERIOD 

: (1 '+ 0" ~ 1 

i 



n 


6|% 


7% 


7|% 


8% 


8l% 


1 

2 
3 
4 



1.0000 0000 
2.0060 0000 
3.1992 2500 
4.4071 7463 
5.0930 4098 


1.0000 0000 
2.0700 0000 
3.2149 0000 
4.4399 4300 
5.7507 3901 


1.0000 0000 
2.0750 0000 
3.2300 2500 
4.4729 2188 
6.8083 9102 


1.0000 0000 
2.0800 0000 
3.2404 0000 
4.5001 1200 
5.8800 0006 


1.0000 0000 
2.0850 0000 
3.2022 2500 
4.6305 1413 
6.0253 7283 



7 

8 
9 
10 


7.0637 2764 
8.6228 0094 
10.0768 5648 
11.7318 6215 
13.4944 2254 


7.1532 9074 
8.0540 2109 
10.2598 0267 
11.9779 8876 
13.8104 4706 


7.2440 2034 
8.7873 2187 
10.4403 7101 
12.2298 4883 
14.1470 8750 


7.3350 2004 
8.9228 0330 
10.0300 2703 
12.4875 5784 
14.4806 0247 


7.4290 2052 
0.0004 0702 
10.8300 3027 
12.7512 4301 
1-1.8360 0032 


11 
12 
13 
14 
Iff 


15.3715 8001 
17.3707 1141 
19.4998 0765 
21.7672 9515 
24.1821 6933 


15.7835 9932 
17.8884 5127 
20.1406 4286 
22.5504 8780 
25.1290 2201 


10.2081 1900 
18.4237 2799 
20.8055 0759 
23.3659 2000 
20.1183 0470 


10.6454 8740 
18.9771 2040 
21.4902 0058 
24.2149 2030 
27.1521 1303 


17.0000 8270 
10.6402 4079 
22.2100 3003 
25.0988 0559 
28.2322 0016 


16 
1? 
18 
10 
20 


26.7540 1034 
20.4930 2101 
32.4100 0738 
35.5167 2176 
38.8253 0807 


27.8880 5355 
30.8402 1730 
33.9990 3251 
37.3789 6479 
40.9954 0232 


29.0772 4200 
32.2580 3621 
35.0773 8786 
39.3531 0194 
43.3046 8134 


30.3242 8304 
33.7502 2509 
37.4502 4374 
41.4402 0324 
45.7019 6430 


31.0320 1204 
35.3207 3300 
39.3220 0638 
43.0054 4908 
48.3770 1323 


21 
22 
23 
24 
2S 


42.3489 5373 
46.1016 3573 
00.0982 4205 
54.3546 2778 
58.8876 7859 


44.8651 7878 
49.0057 3910 
53.4361 4090 
68.1706 7070 
63.2490 3772 


47.5525 3244 
52.1189 7237 
57.0278 0530 
62.3049 8744 
67.0778 0150 


50.4220 2144 
65.4507 6610 
00.8932 0557 
00.7047 5022 
73.1060 3996 


53.4800 5R30 
59.0360 2940 
05.0530 6700 
71.6832 1882 
78.0077 0242 


26 
27 
28 
29 
30 


63.7153 7769 
68.8588 7725 
74.3325 7427 
80.1041 9159 
86.3748 6405 


68.6764 7030 
74.4838 2328 
80.6976 9091 
87.3466 2927 
94.4007 8632 


74.0702 0112 
80.0319 1020 
87.0703 0901 
06.2552 5810 
103.3994 0252 


70.0544 1515 
87.3507 0836 
96.3388 2083 
103.0659 3622 
113.2832 1111, 


80.3545 6478 
04.0040 0103 
103.7437 4075 
113.5010 5871 
124.2147 2520 


31 
32 
33 
34 
35 


92.9892 3021 
100.0335 3017 
107.5357 0903 
115.5255 3076 
124.0340 9026 


102.0730 4137 
110.2181 5426 
118.9334 2506 
128.2587 0481 
138.2308 7835 


112.1543 5771 
121.6050 3454 
131.6833 7003 
142.5590 3310 
154.2510 0558 


123.3458 0800 
134.2136 3744 
145.0506 2044 
158.626(5 7007 
172.3168 0308 


136.7720 7084 
148.3130 7087 
161.0203 4261) 
170.08JJ6 7170 
103.7010 71530 


30 
37 

38 
39 
40 


133.0969 4513 
142.7482 4650 
153.0268 8259 
103.9736 2995 
175.8319 1590 


148.9134 5084 
100.3374 0202 
172.5010 2017 
185.6402 9158 
109.0351 1199 


100.8204 7000 
180,3320 1170 
194.8669 1258 
210.4711 8102 
227.2585 1060 


187.1021 4707 
203.0703. 1081 
220.3150 4540 
238.0412 2103 
259.0666 1871 


210.0813 1780 
228.0382 2081 
2411.31)79 7035 
271-.5008 0750 
205.0825 3024 


41 
42 
43 
44 
45 


188.0479 9044 
201.2711 0981 
215.3537 3195 
230,3517 2463 
240.3245 8602 


214.6006 6983 
230.6322 3972 
247.7764 9050 
268,1208 5125 
286,7493 1084 


245.3007 5857 
204.6983 1546 
285.5600 8012 
307.9009 0080 
332.0(345 1511 


280.7810 4081 
304.2435 2342 
320.5830 0530 
356-9406 4572 
386.5050 1788 


321.8155 5182 
350,1008 7372 
380.0343 1200 
414.3137 2959 
450,5303 0061 


46 
47 
48 
49 
50 


263,3356 8475 
281.4525 0426 
300.7489 1704 
321.2954 6606 
343.1796 7198 


300,7617 0260 
329,2243 8598 
353.2700 9300 
378.9989 9051 
406.6289 2047 


367.0603 5376 
385.8170 5528 
415.7533 3442 
447.9348 3451 
482.6209 4709 


418.4280 8677 
452.9001 0211 
490.1321 6428 
530,3427 3742 
573.7701 5642 


480.8254 8032 
632.4008 4015 
678.7108 0107 
028.9100 8418 
6B3.3884 1782 



TABLE Vm PRESENT VALUE OF ANNUITY OF 1 PER, PERIOD 



1 - (1 + f)- 
z 




n 


5% 


!% 


5* 


1% 


1% 


1 

2 
3 

4 
5 


0.0058 5062 
1.0875 0908 
2.9761 7253 
3.0580 7804 
4.9381 0201 


0.0060 2488 
1.0850 9038 
2.0702 4814 
3.0504 0560 
4.0258 6633 


0.9042 0050 
1.0820 3513 
2.0063 3733 
3.0423 4034 
4.0130 7723 


0.9925 5583 
1.0777 221)1 
2.0555 5024 
3.9261 1041 
4.8894 3901 


0.9900 0901 
1.9703 9500 
2.9409 8521 
3.9019 6555 
4.8634 3124 



7 

8 

10 


C.9134 6318 
0.8847 7061 
7.8520 5969 
8.8163 2915 
0.7740 0104 


6.8963 8441 
0.8620 7404 
7.8229 5924 
8.7790 6302 
9.7304 1180 


5.8703 8084 
0.8394 8385 
7.7040 1876 
8.7430 1781 
9.6805 1316 


5.8455 0703 
0.7946 3785 
7.7366 1325 
8.6715 7042 
9.5005 7058 


5.7954 7047 
6.7281 0453 
7.6610 7775 
8.5600 1768 
0.4713 0463 


11 
12 
13 
11 
15 


10.7298 9374 
11.0812 2198 
12.0280 0280 
13.6720 5267 
14.6115 8702 


10.0770 2073 
11.0189 3207 
12.5501 5131 
13.4887 0777 
14.4166 2466 


10.6245 3009 
11.5571 2010 
12.4842 0511 
13.4060 9201 
14.3226 4473 


10.5200 7452 
11.4340 1207 
12.3423 4608 
13.2430 2242 
14.1360 9405 


10.3078 2825 
11.2650 7747 
12.1337 4007 
13.0037 0304 
13.8050 5252 


16 
17 

18 
19 
20 


15.4472 2418 
16.3789 7843 
17.3068 G048 
18.2300 0438 
19.1611 0809 


15.3309 2602 
10.2586 3186 
17.1727 6802 
18.0823 5624 
18.9874 1915 


15.2336 8100 
10.1305 3432 
17.0401 3354 
17.0365 0974 
18.8266 0320 


15.0243 1201 
15.0050 2402 
10.7701 8107 
17.0468 2084 
18.5080 1000 


14.7178 7378 
15.5622 5127 
16.3082 6858 
17.2260 0850 
18.0456 5297 


21 
22 
23 
24 
25 


20.0674 0352 
20.9800 7053 
21.8888 7289 
22.7938 9831 
23.6051 6843 


19.8879 7025 
20.7840 5800 
21.0750 8056 
22.5628 0622 
23.4450 3803 


10.7107 1404 
20.5906 0220 
21.4053 8745 
22.3350 0038 
23.1997 0741 


10.3027 9870 
20.2112 1459 
21.0533 1473 
21.8891 4614 
22.7187 5547 


18.8569 8313 
10.0603 7934 
20.4558 2113 
21.2433 8720 
22.0231 5570 


26 
27 

28 
29 
30 


24.5020 0884 
25.4805 0500 
20.3766 0254 
27.2630 0008 
28.1457 3278 


24.3240 1704 
25.1980 2780 
26.0076 8033 
26.0330 2423 
27,7040 5307 


24.0594 2070 
24.0140 8802 
25.7037 9970 
26.6085 8307 
27.4484 0702 


23.6421 8005 
24.3504 0280 
25.1707 1251 
26.0758 0331 
26.7760 8021 


22.7952 0300 
23.5590 0759 
24.3104 4310 
25.0057 8530 
25.8077 0822 


31 
32 
33 
34 
35 


20.0247 9612 
20.9002 1189 
30.7710 0524 
31.6401 0122 
32.5047 2-180 


28.0/507 9907 
20.5032 8355 
30.3615 2502 
31.1955 4818 
32.0353 7132 


28.2a34 8000 
20.1130 5044 
20.0300 0025 
30.759C 7540 
31.5753 8506 


27.5083 1783 
28.3550 5045 
29.1371 2203 
20.0127 7021 
30.6826 5020 


26.5422 8537 
27.2605 8047 
27.9800 9256 
28.7020 6580 
29.4085 8000 


36 
37 

38 
39 
40 


33.3057 0109 
34.221)1 0181 
35.0760 5084 
36.9272 53U4 
30.7740 2881 


32.8710 1024 
33.7025 0372 
34.5298 5445 
35.3630 8000 
30.1722 2780 


32.3804 0463 
33.1028 3074 
33.00.15 3828 
34.7915 8736 
35.5840 1300 


31.4408 0525 
32.2062 6570 
32.0580 8010 
33.7052 9048 
34.4460 3344 


30.1075 0504 
30.7905 0094 
31.4840 6330 
32.1630 3208 
32.8340 8611 


41 
42 
43 
44 
45 


37.0172 0009 
38.4570 5236 
30.2033 3013 
40.1261 3788 
40.9554 8090 


30.0872 0141 
37.7082 0091 
38.0052 7364 
30.4082 3238 
40.2071 0640 


36.3718 4487 
37.1561 0670 
37.0338 2012 
38.7080 2020 
30.4777 4248 


35.1830 6646 
35.0137 1200 
30.6380 2070 
37.3587 3022 
38.0731 8130 


33.4090 8922 
34.1581 0814 
34.8100 0800 
35.4554 5352 
30.0045 0844 


40 

47 
48 
40 
50 


41.7814 0081 
42.0038 8461 
43.4220 5562 
44.2380 2700 
46.0500 1682 


41.0021 8547 
41.7032 1037 
42.6803 1778 
43.3035 0028 
44.1427 8635 


40.2429 0170 
41.0038 0287 
41.7602 0170 
42.5122 1380 
43.2508 0460 


38.7823 1401 
39.4801 0774 
40.18-17 8189 
40.8781 0642 
41.5804 4707 


36.7272 3008 
37.3530 0000 
37.0730 5040 
38.5880 7871 
30.1001 1753 



TABLE Vm PRESENT VALUE OF ANNUITY OF 1 PER PERIOD 



n 


55* 


! 
3% 


i* 


!% 


1% 


51 

S& 
64 
55 


46.8598 3317 
46.6363 0401 
47.4676 1228 
48.2665 0184 
49.0620 7661 


44.9181 9537 
46.6897 4664 
46.4674 5934 
47.2213 5258 
47.9814 4635 


44.0031 7940 
44.7421 8336 
45.4709 0144 
46.2073 6853 
46.0335 7933 


42.2403 7525 
42.0270 1S12 
43.6000 13C1 
44.2685 0002 
44.0310 1103 


30.7081 3617 
40.3041 0423 
40.1)843 5072 
41.5(580 (1408 
42.1471 0210 


50 
57 

58 
59 
60 


49.8643 6003 
60.6433 3612 
61.4290 4840 
62.2116 0046 
52.9907 0684 


48.7377 5057 
49.4903 0605 
50.2391 0950 
50.9841 8855 
61.7256 6075 


47.6555 8841 
48.3734 1020 
40.0870 6898 
40.7005 8880 
50.6010 9304 


45.5890 8020 
46.2428 0770 
46.8011 8388 
47.6340 7382 
48.1733 7352 


42.7100 9224 
43.2871 2102 
43.8480 3408 
44.40-15 8870 
44.0550 3841 


01 
69 
63 
M 
65 


53.7668 7800 
64.6394 3036 
55.3089 7627 
56.0753 2905 
56.8386 0194 


52.4632 4453 
53.1972 5824 
53.9276 2014 
54.6643 4839 
55.3774 6109 


51.2033 0800 
51.0005 5478 
62.5937 6787 
53.2829 4073 
63.0681 2668 


48.8073 1863 
49.4365 4455 
60.0610 8040 
50.6809 7006 
51.2962 5713 


46.6000 3803 
40.0306 4161 
46.5730 0258 
47.1028 7385 
47.0200 0777 


60 
67 
68 
69 
70 


67.6985 0814 
68.3653 6078 
59.1090 7296 
69.8696 5770 
60.6071 2798 


56.0969 7621 
56.8129 1166 
57.5252 8522 
58.2341 1405 
58.9394 1766 


54.3403 3888 
55.3266 0040 
55.0900 3413 
56.0693 6287 
57.3349 0925 


51.0060 5407 
62.5131 0007 
63.1147 4007 
63.7110 0077 
64.3040 2210 


48.1451 5021 
48.0585 7060 
40,1660 0140 
40.0701 0040 
60.1086 1435 


71 

n 

73 
71 
75 


61.3614 9672 
62.0027 7680 
62.8309 8103 
63.5661 2216 
64.2982 1292 


59.6412 1151 
60.3395 1394 
61.0343 4222 
61.7267 1366 
62.4136 4643 


67.9965 0579 
58.6644 4488 
59.3084 7877 
50.0587 1959 
60.6061 8934 


64.8920 2516 
55.4708 4880 
56.0564 2501 
66.6316 8706 
57.2020 6704 


60.6018 OG39 
61.1603 0148 
51.0340 6007 
62.1120 2175 
52.5870 5124 


70 
77 
78 
79 
80 


66.0272 6696 
65.7632 9388 
66.4763 0924 
67.1963 2453 
67.9133 6221 


63.0981 5460 
63.7792 5836 
64.4569 7350 
65.1313 1691 
65.8023 0638 


61.2479 0988 
61.8860 0207 
62.5221 9021 
63.1537 9310 
63.7817 3301 


57.7603 9740 
58.3310 0815 
68.8002 3141 
50.4443 0842 
50.0044 4012 


63,0504 8637 
53.6212 7304 
53.0814 6006 
64.4370 8817 
54.8882 0011 


81 

89 
83 
84 
85 


68.6274 0467 
69.3384 9426 
70.0466 3326 
70.7518 3393 
71.4541 0846 


66.4699 5561 
67.1342 8419 
67.7963 0705 
68.4630 4244 
69.1075 0491 


64.4060 3118 
65.0267 0874 
65.6437 8667 
66.2572 8585 
66.8672 2705 


60.5403 8722 
61.0822 7010 
61.6201 1030 
62.1530 6460 
62.0838 3570 


55.3348 6753 
56.7770 8008 
50.2140 3720 
60.0484 6270 
67,0776 7000 


80 

87 
88 
89 
90 


72.1634 6898 
72.8499 2759 
73.5434 9633 
74:2341 8720 
74.9220 1212 


69.7687 1136 
70.4066 7796 
71.0514 2086 
71.6929 5608 
72.3312 9058 


67.4736 3080 
68.0766 1789 
68.6760 0845 
69.2718 2283 
69.S642 8121 


63.2007 0257 
63.7317 7427 
64.2400 0002 
64.7041 6875 
65.2746 0018 


57.5020 4961 
57.9234 1636 
58.3400 1520 
58.7524 9030 
50,1008 8148 


91 
92 
98 
94 
95 


75.6069 8300 
76.2891 1168 
76.9684 0995 
77.6448 8955 
78.3185 6218 


72.9664 6725 
73.5984 7487 
74.2273 3818 
74.8530 7282 
75.4756 9434 


70.4533 0363 
71.0380 1001 
71.6211 2017 
72.1000 6370 
72.7754 3047 


65.7812 4081 
66.2841 1802 
66.7832 4458 
67.2780 5467 
67.7703 768B 


fiO,50C2 2010 
50.0055 7340 
60.3610 5302 
60.7644 0082 
01.1420 8002 


90 
97 
98 
99 
160 


78.9894 3960 
70.6575 3308 
80.3228 5450 
80.9854 1524 
81.6452 2677 


76.0952 1825 
76.7116 6995 
77.3250 3478 
77,9353 6799 
78.5426 4477 


73.3475 6967 
73.9163 9076 
74.4810 1204 
75.0441 5530 
75.6031 3712 


68.2584 3856 
68.7428 6705 
69,2236 8038 
60.7009 3230 
70.1746 2272 


61.5277 0200 
61.0086 1082 
62.2857 5023 
62.6591 6756 
63.0288 7877 



56 



TABLE Yin PRESENT VALUE OF ANNUITY OF 1 PER PERIOD 

1 - (1 + f)-" 
z 



(a^ati) 



n 


or 


.|% 


4% 


i% 


1% 


i 


82.3023 0049 
82.9600 4777 
83.6082 7991 
84.2572 0818 
84.0034 4381 


79.1409 1021 
79.7481 0937 
80.3404 3718 
80.9417 2864 
81.5340 5825 


70.1588 7702 
70.7113 9392 
77.2007 0048 
77.8008 3331 
78.3497 9288 


70.6447 8682 
71.1114 6094 
71.6746 4113 
72.0343 8325 
72.4907 0298 


63.3949 2947 
63.7673 6691 
64.1101 9397 
64.4714 7918 
04.8232 4071 


100 
107 
108 
100 
110 


85.5409 9796 
80.1878 8175 
80.8201 0028 
87.4610 8258 
88.0040 2163 


82.1234 4104 
82.7098 9158 
83.2934 2440 
83.8740 5410 
84.4517 9522 


78.8896 0355 
79.4202 8350 
79.9598 5115 
80.4903 2428 
81.0177 2093 


72.0436 2670 
73.3931 7696 
73.8393 8160 
74.2822 6461 
74.7218 5073 


66.1715 3140 
66.5163 6772 
66.8577 8983 
06.1968 3151 
60.5305 2025 


111 
112 
113 
114 
115 


88.7249 3437 
89.3520 3171 
89.9777 2460 
00.0002 2364 
91.2201 3059 


85.0206 6191 
85.5980 0850 
80.1078 2942 
80.7341 5802 
87.2976 7027 


81.5420 5895 
82.0033 5600 
82.5810 2991 
83.0908 0803 
83.0091 7785 


76.1581 6450 
76.6012 3027 
70.0210 7223 
70.4477 1437 
76.8711 8052 


00.8619 0718 
67.1900 0710 
07.5148 6852 
07.8364 9358 
68.1540 4414 


116 
11? 
118 
119 
120 


91.8374 8338 
92.4522 0658 
03.0644 9081 
03.0741 8767 
04.2813 4800 


87.8683 7838 
88.4162 9G90 
88.9714 3970 
89.5238 2059 
90.0734 5333 


84.1184 8071 
84.0248 4182 
85.1282 6033 
85.0287 6920 
86.1263 6554 


77.2914 9431 
77.7086 7922 
78.1227 5863 
78.5337 6536 
78.9410 9207 


68.4702 4172 
08.7824 1755 
00.0015 0252 
00.3075 2725 
09.7005 2203 


121 

123 
124 
125 


94.8859 9030 
95.4881 2315 
98.0877 5747 
96.6849 0367 
97.2795 7209 


00.6203 6167 
01.1045 2892 
91.7059 9893 
92.2447 7505 
02.7808 7070 


86.6210 6602 
87.1129 0742 
87.6018 0038 
88.0880 4946 
88.5713 8308 


79,3405 9322 
79,7484 7962 
80.1473 7432 
80.5432 9957 
80.9362 7749 


70.0005 1080 
70.2975 4145 
70.5010 2520 
70.8827 9722 
71.1710 8036 


120 
127 
128 
120 
130 


97.8717 7301 
98.4015 1666 
99.0488 1324 
99.0336 7290 
100.2161 0670 


03.3142 0920 
03.8450 7384 
04.3732 0780 
04.8087 1422 
05.4216 0619 


80.0510 1361 
89.5296 5731 
00.0040 3032 
00.4708 4873 
00.9463 2851 


81.3203 3001 
81.7134 7802 
82.0977 4583 
82.4701 6219 
82.8577 1929 


71.4506 2115 
71.7391 2985 
72.0189 4045 
72.2959 8064 
72.6702 7786 


131 
133 
133 
, 134 
135 


100.7901 2189 
101.3737 3131 
101.0480 4401 
102.5217 6994 
103.0922 1800 


95.0418 9071 
96.4595 9872 
90.9747 2509 
97.4872 8865 
97.0973 0214 


01.4130 8564 
01.8771 3661 
92,3384 9442 
02.7971 7758 
03.2532 0000 


83.2334 6828 
83.6064 2013 
83.9706 0506 
84.3440 1554 
84.7087 0029 


72.8418 5927 
73.1107 6175 
73.3709 8193 
73.0405 7017 
73.0015 6058 


130 
137 
138 
130 
140 


103,0003 0104 
104.2260 2590 
104.7894 0335 
105.3504 4314 
105.0091 6490 


08.5047 7825 
00.0097 2900 
99.5121 0875 
100.0121 0821 
100.5095 0041 


03.7005 7892 
04.1673 2787 
04.0054 0270 
05.0500 0857 
05,4030 5050 


86.0706 7020 
85.4299 4667 
85.7805 4657 
80.1404 0288 
80.4018 0434 


74.1599 6095 
74.4158 0293 
74.0601 1181 
74.0199 1268 
75.1682 3038 


141 
143 
143 
144, 
145 


100.4655 4847 
107.0190 3330 
107.5714 1902 
108,1209 1517 
108,0081 3120 


101,0045 3772 
101,4970 6246 
101.9871 1088 
102.4747 4310 
102.9500 4344 


05.9343 3304 
00.3721 0272 
00.8074 6201 
07.2402 1804 
07.0704 7304 


86.8405 0059 
87.1806 0108 
87.5301 2514 
87.8710 0195 
88.2005 2055 


75.4140 8948 
75.6676 1434 
75.8985 2905 
76.1371 5747 
70.3734 2324 


140 
147 
148 
140 
150 


109.2130 7074 
100.7657 0103 
110.2901 9363 
110.8343 8350 
111.3703 4044 


103.4427 2979 
103.0231 1422 
104.4011 0808 
104.8707 2505 
105.3499 7518 


08.0082 3307 
08.5235 1360 
08,0403 2003 
90.3006 8765 
00.7840 1078 


88.5464 2082 
88.8788 3864 
80.2007 0630 
80.6382 2858 
80.8642 4073 


70.6073 4974 
70.8389 0014 
77.0082 7737 
77.2053 2413 
77.6201 2290 



67 



TABLE VD3 PRESENT VALUE OF ANNUITY OP 1 PER PERIOD 

.!-(!+ Q-" 



n 


1|% 


l\% 


1|% 


' lf% 


2% 


l 
2 
3 
4 
5 


0.9888 7515 
1.9667 4923 
2.9337 4460 
3.8890 8230 
4.8355 8200 


0.0876 5432 
1.9631 1538 
2.0265 3371 
3.8780 5708 
4.8178 3504 


0.9852 2107 
1.0558 8342 
2.0122 0042 
3.8543 8465 
4.7826 4497 


0.0828 0008 
1.9486 0875 
2.8079 8403 
3.8300 4254 
4.7478 5508 


0.0803 0216 
1.9415 6004 
2.8838 8327 
3.S077 2870 
4.7134 5061 


G 

7 
8 
9 
10 


6.7706 6205 
6.6953 3048 
7.6097 3002 
8.5130 4810 
9.4081 0690 


5.7460 0902 
6.6627 2585 
7.5681 2420 
8.4623 4408 
0.3455 2501 


5.0071 8717 
0.5982 1306 
7.4859 2508 
8.3605 1732 
0.2221 8455 


5.0489 9702 
0.5346 4139 
7.4050 6297 
8.2604 0432 
0.1012 2291 


5.0014 3080 
0.471!) 0107 
7.3254 8144 
8.1622 3071 
8.9825 8501 


11 
13 
IB 
14 
15 


10,2923 1832 
11.1666 9302 
12.0313 4044 
12.8863 6880 
13.7318 8509 


10.2178 0337 
11.0793 1197 
11.0301 8466 
12.7706 6276 
13.6005 4502 


10.0711 1770 
10.0075 0521 
11.7315 3222 
12.5433 8150 
13.3432 3301 


0.9274 0181 
10.7396 4000 
11.5370 4097 
12.3220 0587 
13.0028 8046 


9.7808 4805 
10.6753 4122 
11.3483 7375 
12.1002 4877 
12.8402 0350 


10 
17 
18 
IB 
20 


14.C679 0514 
15.3948 0360 
16.2124 1395 
17.0209 2850 
17.8204 4845 


14.4202 9227 
15.2290 1829 
16.0295 4893 
16.8193 0769 
17.6093 1613 


14.1312 6405 
14.9070 4031 
15.6725 6080 
10.4261 6837 
17.1680 3879 


13.8504 0677 
14.5950 8282 
15.3268 6272 
10.0400 6073 
16.7528 8130 


13.5777 0031 
14.2918 7188 
14.0920 3125 
16.6784 6201 
16.3514 3334 


21 
22 
23 
24 
25 


18.0110 7387 
10.8920 0371 
20.1660 3580 
20.0305 6603 
21.6865 0276 


18.3696 0495 
19.1305 6291 
19.8820 3744 
20.6242 3451 
21.3572 6865 


17.9001 3073 
18.0208 2437 
19.3308 6145 
20.0304 0537 
20.7190 1120 


17.4475 4010 
18.1302 0948 
18.8012 4764 
10.4006 8505 
20.1087 8106 


17.0112 0010 
17.0580 4820 
18.2922 0412 
18.0130 2600 
19.5234 5047 


20 
27 

30 


22.4342 0792 
23.1735 0508 
23.9045 7940 
24.6275 1986 
25.3424 1766 


22.0812 5290 
22.7062 9025 
23.5025 1778 
24.2000 1756 
24.8889 0623 


21.3980 3172 
22.0670 1746 
22.7207 1071 
23.3700 7668 
24.0168 3801 


20.7457 3106 
21.3717 2644 
21.9860 6474 
22.5910 0171 
23.1858 4034 


20.1210 3570 
20.7068 0780 
21.2812 7230 
21.8443 8400 
22.3004 6555 


31 
32 
33 
34 
35 


26.0493 6233 
26.7484 4236 
27.4307 4522 
28.1233 5745 
28.7003 6460 


26.5692 9010 
26.2412 7418 
26.9049 6215 
27.5604 5644 
28.2078 5822 


24.6461 4582 
25.2671 3874 
25.8789 6442 
26.4817 2840 
27.0755 9468 


23.7698 7650 
24.3438 5807 
24.0079 0951 
25.4023 7780 
20.0072 6100 


22.0377 0162 
23.4083 3482 
23.0885 0355 
24.4086 0172 
24.0986 1933 


38 
37 
38 
39 
40 


20.4378 127 
30.1289 0114 
30.7825 0692 
31,4290 2044 
82.0682 5260 


28.8472 6737 
29.4787 8259 
30.1026 0133 
30.7185 1983 
31.3269 3316 


27.6606 8431 
28.2371 2740 
28.8050 5103 
29.3645 8288 
20.0158 4520 


26.6427 6283 
27.0690 4456 
27.5862 8457 
28.0046 2867 
28.5942 2055 


25.4888 4248 
25.0604 5341 
26.4406 4060 
26.0025 8883 
27.3554 7024 


41 
42 
43 
44 
45 


32.7903 7340 
33.3254 6195 
33.0435 9649 
34.5548 5438 
35.1593 1212 


31.9278 3622 
32.5213 1874 
33.1074 7530 
33.6863 9536 
34.2581 6825 


30.4580 0079 
30.9040 5004 
31.5212 3157 
32.0406 2223 
32.6623 3718 


20.0862 3780 
20.6078 0135 
30.0420 6522 
30.5081 7221 
30.0602 0261 


27.7994 8045 
28.2347 0358 
28.0615 0233 
20.0790 0307 
20.4901 6987 


46 
47 
48 
49 

M 


36.7570 4536 
36.3481 2891 
36.0326 3674 
37.5106 4202 
38.0822 1708 


34.8228 8222 
36.3806 2442 
35.9314 8091 
36.4755 3670 
37.0128 7574 


33.0564 8083 
33.5531 0195 
34.0425 6305 
34.5246 8339 
34.9006 8807 


31.4164 7431 
31.8589 4281 
32.2038 0120 
32.7211 8063 
33.1412 0046 


29.8023 1360 
30.2805 8100 
30.6731 1067 
31.0620 7801 
31.4236 0589 



68 



TABLE VHI PRESENT VALTTE OF ANNUITY OF 1 PER PERIOD 



n 


1|% 


1|% 


1|% 


l|% 


2% 


51 
52 
53 
51 
55 


38.0474 3345 
30.2003 01 88 
30.7CflO 7232 
40.3050 3304 
40.8401 1514 


37.5435 8000 
38.0077 3431 
38.5854 1000 
39.0007 0770 
30.0010 8007 


35.4076 7208 
35.0287 4185 
30.3820 0000 
30.8305 3882 
37.2714 0081 


33.5540 1421 
33.0597 1013 
34.3584 4033 
34.7503 1579 
35.1354 4550 


31.7878 4892 
32.1449 4992 
32.4950 4894 
32.8382 8327 
33.1747 8752 


5G 

57 
58 
59 
00 


41.3805 8358 
41.0001 0013 
42.4317 4806 
42.0485 7740 
43.4500 5033 


40.1004 3128 
40.6030 1855 
41.0705 2449 
41.5000 2410 
42.0345 9170 


37.7058 7803 
38.1338 7058 
38.6565 3761 
38.9700 7202 
30.3802 0880 


35.5130 5135 
35.8850 4727 
30.2515 4623 
30.6108 5520 
30.9030 8552 


33.5040 0305 
33.8281 3103 
34.1452 2050 
34.4501 0441 
34.7008 8008 


01 

ca 

03 
01 
05 


43.0050 4052 
44.4048 2020 
44.0600 3110 
43.4477 4407 
45.0310 2000 


42.5033 0054 
42.0002 2275 
43.4234 2088 
43.8749 0247 
44.3200 8022 


30.7835 1614 
40.1808 0408 
40.5722 2077 
40.0678 5298 
41.3377 8018 


37.3110 4228 
37.0521 3000 
37.9873 5135 
38.3108 0723 
38.0405 9678 


35.0590 0282 
35,3520 4002 
35.0398 4310 
35.9214 1480 
30.1974 0555 


00 
07 
08 
00 
70 


40.4080 1075 
40.8815 0284 
47.3488 2852 
47.8100 5627 
48.2070 4004 


44.7014 0105 
45.1005 0503 
45.0201 7840 
40.0505 4050 
40.4006 7602 


41.7121 0401 
42.0808 0125 
42.4442 2783^' 
42.8021 0400 
43,1548 7183 


38.0588 1748 
39.2715 0509 
30.5780 3375 
30.8810 1597 
40,1779 0207 


30.4081 0348 
30.7334 3478 
30.9035 0351 
37.2485 9108 
37.4986 1020 


71 
78 
73 
74 
75 


48.7108 4270 
40,1007 1714 
40.0086 2010 
50.0450 0708 
50.4777 32BO 


40.8830 3024 
-47,2024 7431 
47.0002 7003 
48.0050 8240 
48.4880 7027 


43.5023 3078 
43.84-10 0077 
44.1810 3771 
44.5142 2434 
44,8410 0034 


40.4000 8321 
40.7504 4542 
41.0382 7500 
41.3162 5867 
41.5874 7771 


37.7437 4441 
37.9840 0314 
38.2100 0975 
38.4506 5002 
38.0771 1433 


7G 
77 

78 
79 
80 


50.0050 5077 
51.3270 1510 
51.7464 78-17 
52.1580 0317 
52.5073 1002 


48.8770 0533 
40.2022 1701 
40.0410 0040 
50.0104 0027 
50.3800 5700 


45.1041 3820 
45.4810 0002 
45.7040 8485 
40.1034 3335 
40.4073 2340 


41.8550 1495 
42.1179 5081 
42.3703 0443 
42.0303 3359 
42.8709 3474 


38.8091 3170 
39.1107 9578 
39.3301 0194 
30.5304 0380 
39.7445 1359 


81 

88 
83 
84 
85 


52.0713 8280 
63.3709 5057 
53.7000 0104 
54.1508 2074 
54.5432 1557 


50,7522 5380 
51.1133 3717 
51,4099 0204 
51.8221 8532 
62.1700 5058 


40,7007 2205 
47.0010 0720 
47.2023 1251 
47.5780 3301 
47.8007 2218 


43,1252 4298 
43.3003 3217 
43,0032 7480 
43.8301 4237 
44.0060 0470 


30.0450 0160 
40.1427 40B3 
40.3300 2011 
40.5255 1570 
40.7112 8090 


80 
87 
88 
89 
90 


54.0253 0588 
55.3031 4540 
65.0707 8100 
50.0402 0120 
50.4110 30-11 


62.5130 3000 
52.8520 7088 
53.1881 2631 
53.5101 3011 
53.8400 0035 


48.1380 4254 
48.4124 5571 
48.0822 2237 
48.0480 0234 
40.2008 5452 


44.2809 3090 
44.5109 8869 
44.7282 4441 
44.9417 0355 
45.1510 1037 


40.8034 2156 
41.0710 8192 
41.2470 4110 
41.4180 0774 
41.5860 2010 


91 

99 
93 
94 
95 


50.7720 3400 
57.1302 1002 
57.4835 3021 
57.8320 0007 
58.1784 0204 


54.1080 4850 
54.4878 5037 
54.8028 151S 
55.1138 0154 
55.4211 2744 


40.4078 3000 
40.7220 0080 
40,0724 2065 
50.2101 3365 
50.4022 0054 


45.3578 4803 
45.6005 3800 
45,7507 4310 
45.0655 2147 
40.1479 3205 


41.7618 9133 
41.9130 1895 
42.0721 7645 
42.3270 2200 
42. .'1800 2264 


90 
97 
98 
90 
190 


58.5200 5235 
58.8579 0000 
50.1010 0100 
50.5223 0440 
50.8400 0251 


55.7245 7031 
60.0242 0008 
50.3202 0308 
60.0126 0010 
60.0013 3030 


50.7010 7541 
50.0370 1124 
61.1700 0034 
61.3000 7422 
61.0247 03(17 


40.3370 3455 
40.5228 8408 
40.7065 3718 
40.8850 4882 
47.0014 7304 


42.5204 3380 
42.0760 1555 
42.8105 2506 
42.9003 1807 
43.0983 5104 



59 



TABLE vm PRESENT VALUE OF ANNUITY OF i PER PERIOD 

.!-(!+ i)-* 



n 


2 -or 
4% 


2|% 


Q?0/ 

"l'0 


3% 


8l% 


l 
2 
3 
4 
5 


0.9779 9511 
1.9344 6955 
2.8698 9687 
3.7847 4021 
4.6794 5253 


0.9766 0976 
1.9274 2415 
2.8660 2356 
3.7019 7421 
4.6458 2850 


0.9732 3001 
1.9204 2434 
2.8422 6213 
3.7394 2787 
4,6125 8136 


0.9708 7379 
1.9134 6970 
2.8280 1135 
3.7170 9840 
4.6797 0719 


0.9661 8357 
1.8996 0428 
2.8010 3608 
3.6730 7921 
4.6150 5238 


6 
7 
8 
9 
10 


6.5544 7680 
6.4102 4626 
7.2471 8461 
8.0657 0622 
8.8602 1635 


6.5081 2536 
6.3493 9060 
7.1701 3717 
7.9708 6653 
8.7520 6393 


5.4623 6078 
6.2894 0800 
7.0943 1441 
7.8770 7820 
8.6400 7616 


5.4171 9144 
6.2302 8200 
7.0196 9219 
7.7861 0892 
8.5302 0284 


C.3285 5302 
0.1146 4398 
0.8730 G6G4 
7.0070 8051 
8.3106 0532 


ii 

12 
13 
14 
15 


9.6491 1134 
10.4147 7882 
11.1635 9787 
11.8959 3924 
12.6121 6551 


9.5142 0871 
10.2577 6460 
10.9831 8497 
11.6909 1217 
12.3813 7773 


0.3820 6920 
10.1042 0360 
10.8070 1080 
11.4910 0814 
12.1500 9892 


9.2520 2411 
9.9540 0399 
10.0349 6633 
11.2900 7314 
11.9379 3509 


9.0015 5104 
9.0033 3433 
10,3027 3849 
10.9205 2028 
11.6174 1000 


16 
17 
18 
19 
20 


13.3126 3131 
13.9976 8343 
14.6676 6106 
15.3228 9590 
15.9637 1237 


13.0550 0206 
13.7121 0772 
14.3633 6363 
14.9788 9134 
15.5891 6229 


12.8046 7316 
13.4361 0769 
14.0487 0061 
14.0400 0157 
15.2272 5213 


12.6011 0203 
13.1061 1847 
13.7535 1308 
14.3237 9911 
14.8774 7486 


12.0941 1681 
12.6513 2059 
13.1896 8173 
13.7098 3742 
14.2124 0330 


21 
22 
23 
24 
25 


16.6904 2776 
17.2033 6232 
17.8027 8955 
18.3890 3624 
18.9623 8263 


16.1845 4857 
16.7654 1324 
17.3321 1048 
17.8849 8583 
18.4243 7642 


16.7929 4612 
16.3434 9987 
10.8793 1861 
17.4007 9670 
17.9083 1795 


15.4150 2414 
15.9369 1664 
16.4430 0839 
10.9355 4212 
17.4131 4709 


14.0979 7420 
15.1071 2484 
16.0204 1047 
10,0683 0760 
10.4815 1450 


26 
27 
28 
29 
30 


19.5231 1260 
20.0716 0376 
20.0078 2764 
21.1323 4977 
21.6453 2985 


18.9506 1114 
19.4640 1087 
19.9648 8866 
20.4536 4991 
20.9302 9269 


18.4022 5592 
18.8829 7413 
19.3508 2040 
19.8061 5708 
20.2493 0130 


17.8768 4242 
18.3270 3147 
18.7641 0823 
19.1884 5459 
19.6004 4135 


10.8903 5220 
17.2853 6451 
17.6070 1886 
18.0367 6700 
18.3620 4541 


31 
82 
33 
34 
35 


22.1470 2186 
22.6376 7419 
23.1175 2977 
23.5868 2618 
24.0467 9577 


21.3954 0741 
21.8491 7796 
22.2918 8094 
22.7237 8628 
23.1451 5734 


20.6805 8520 
21.1003 2023 
21.5088 3332 
21.9064 0712 
22,2933 4026 


20.0004 2849 
20.3887 0553 
20.7067 9178 
21.1318 3068 
21.4872 2007 


18.7302 7670 
19.0088 6547 
19.3902 0818 
19.7006 8423 
20.0000 6110 


36 
37 
38 
89 
40 


24.4946 6579 
24.9336 5848 
25.3629 9118 
25.7828 7646 
26.1936 2221 


23.5662 5107 
23.9573 1812 
24.3486 0304 
24.7303 4443 
26.1027 7505 


22.6099 1753 
23.0364 1009 
23.3931 0568 
23.7402 4884 
24.0781 0106 


21.8322 5250 
22.1672 3544 
22.4924 6159 
22.8082 1513 
23,1147 7197 


20.2004 0381 
20.5705 2542 
20.8410 8736 
21.1024 0087 
21.3550 7234 


41 
42 
43 
44 
45 


26.5951 3174 
26.9879 0390 
27.3720 3316 
27.7477 0969 
28.1161 1950 


25.4661 2200 
25.8206 0683 
26.1604 4569 
26.5038 4945 
26.8330 2386 


24,4009 1101 
24.7269 2069 
25.0383 0563 
25.3414 7607 
25.0304 7209 


23.4123 9997 
23.7013 5920 
23.9819 0213 
24.2542 7392 
24.5187 1254 


21.6901 0371 
21.8348 8281 
22.0626 8870 
22.2827 0102 
22.4964 5026 


46 
47 
48 
49 
50 


28.4744 4450 
28.8258 6259 
29.1696 4777 
29.5066 7019 
29.8343 9627. 


27.1541 6962 
27.4674 8256 
27.7731 6371 
28.0713 6947 
28.3623 1168 


26.9236 7381 
26.2029 9154 
26.4749 3094 
26.7396 9215 
26.9971 6998 


24.7764 4907 
25.0247 0783 
25,2667 0664 
26,5016 5693 
25.7297 6401 


22.7000 1813 
22.8994 3780 
23.0912 4425 
23.2766 0450 
23.4550 171J7 



60 



TABLE Vm PRESENT VALUE OF ANNUITY OF 1 PER PERIOD 

. 1 - (I + z)- n 



n 


2j% 


2|% 


2-07 
4/0 


3% 


3|% 


51 
52 
53 
54 
55 


30.16fi8 8877 
30.4703 0687 
30.7778 0023 
31.0786 3910 
31.3726 6438 


28.6461 5774 
28.0230 8072 
29.1032 4048 
20.4508 2876 
20.7130 7928 


27.2478 5400 
27.4018 2871 
27.7202 7368 
27.9603 6368 
28.1852 0870 


25.9512 2710 
26.1602 3990 
26.3740 0028 
20.5776 6047 
26.7744 2764 


23.6280 1630 
23.7957 6454 
23.9572 6043 
24.1132 9510 
24.2040 5323 


50 
57 
58 
59 
60 


31.6602 0708 
31.0416 1142 
32.2167 3489 
32.4868 0420 
32.7480 5285 


20.9048 5784 
30.2090 1740 
30.4484 0722 
30.6813 7200 
30.0086 5649 


28.4041 5454 
28.0171 8203 
28.8245 0800 
20.0262 8522 
20.2226 6201 


26.0654 6373 
27.1500 3506 
27.3310 0540 
27.5058 3058 
27.6765 0307 


24.4097 1327 
24.5504 4700 
24.6864 2281 
24.8177 9981 
24.9447 3412 


61 
02 
03 
04 
65 


33.0063 1086 
33.2680 0673 
33.5041 6208 
33.7440 0170 
33.0803 4406 


31.1303 0667 
31.3467 2836 
31.5577 8377 
31.7636 0148 
31.9045 7705 


20.4137 8298 
20.5007 8870 
20.7808 1634 
20.0560 9887 
30.1284 6605 


27.8403 5307 
28.0003 4270 
28.1656 7201 
28.3064 7826 
28.4528 0152 


25.0073 7596 
25.1858 7049 
25.3003 5700 
25.4109 7388 
25.5178 4010 


00 
07 
08 
09 
70 


34.2106 0643 
34.4367 0003 
34.6560 3005 
34.8714 3183 
35.0820 8402 


32.1605 0208 
32.3517 6876 
32.5383 1009 
32.7203 0340 
32.8078 6608 


30.2053 4400 
30.4577 5581 
30.0168 2074 
30.7000 5522 
30.0193 7247 


28.5050 4031 
28.7330 4884 
28.8670 3771 
28.9071 2390 
29.1234 2135 


25.6211 1030 
25.7208 7051 
25.8172 7480 
25.9104 1052 
26.0003 9604 


71 

72 
73 
74 
75 


35.2881 0261 
35.4805 8001 
35.6866 3750 
35.8703 5214 
36.0678 2605 


33.0710 7098 
33.2400 7803 
33.4049 5417 
33.5658 0805 
33.7227 4044 


31.0060 8270 
31.2068 9314 
31.3440 0816 
31.4792 2036 
31.0099 5568 


20.2400 4015 
20.3650 8752 
29.4806 6750 
20.5028 8106 
29.7018 2628 


28.0873 3975 
20.1713 4275 
26.2525 0508 
26.3309 2278 
26.4066 8868 


70 

77 
78 
79 
80 


36.2521 5262 
36.4324 2310 
36.6087 2675 
36.7811 5085 
30.0407 8070 


33.8758 4433 
34.0252 1398 
34.1700 4047 
34.3131 1205 
34.4518 1722 


31.7371 8304 
31.8010 0540 
31.0815 1377 
32.0087 0085 
32.2120 4008 


29.8076 9833 
29.9102 8004 
30.0099 8004 
30.1067 8035 
30.2007 6346 


26.4708 9244 
26.5506 2072 
26.6189 5721 
26.6840 8281 
20.7487 7667 


81 
82 
83 
84 
85 


37.1147 0004 
37.2750 0020 
37.4337 3130 
37.6880 0127 
37.7388 7655 


34.6871 3876 
34.7191 5070 
34.8479 6074 
34.0730 2023 
35.0062 1486 


32.3240 3015 
32.4321 4613 
32.6373 6850 
32.6307 7469 
32.7304 4000 


30.2020 0335 
30.3805 8577 
30.4666 8813 
30.5600 8556 
30.6311 5103 


26.8104 1127 
26.8600 6258 
20.9275 0008 
26.9830 9186 
27.0368 0373 


80 
87 
88 
89 
90 


37.8864 3183 
38.0307 4018 
38.1718 7304 
38.3000 0028 
38.4448 0025 


35.2158 1938 
35.3325 0671 
35.4463 4801 
35.5674 1269 
35.6657 6848 


32.8364 3804 
32.0308 3094 
33.0227 1627 
33.1121 3165 
33.1901 5480 


30.7008 5637 
30.7862 6735 
30.8604 5374 
30.9324 7036 
31.0024 0714 


27.0886 0020 
27.1388 3986 
27.1872 8480 
27.2340 9108 
27.2703 1664 


91 
92 
93 
94 
95 


38.5760 0078 
38.7060 2423 
38.8322 0754 
38.0557 0221 
30,0765 6040 


35.7714 8144 
35.8746 1604 
35.0752 3510 
36.0734 0010 
36.1601 7080 


33.2838 4005 
33.3602 7644 
33,4404 9776 
33.6245 7202 
33.6005 5671 


31.0702 9820 
31.1362 1184 
31.2002 0567 
31.2623 3560 
31.3220 5502 


27.3230 1028 
1 27.3652 2732 
27.4000 1673 
27.4454 2680 
27.4835 0415 


90 
97 
98 
99 

too 


30.1046 8800 
30.3102 0020 
30.4231 8748 
30.6336 7068 
30.6417 4052 


30.2626 0574 
38.3637 6170 
36.4420 0434 
36.6294 5790 
36.6141 0520 


33.674B 0775 
33.7404 7056 
33.8105 2612 
33.8846 0508 
33.0510 4232 


31.3812 1034 
31.4380 7703 
31.4032 7807 
31.6408 7250 
31.5089 0534 


27.6202 9387 
27.5558 3048 
27.5001 8308 
27.0233 6529- 
27.6554 2540 



TABLE Vm PRESENT VALUE OP ANNUITY OP 1 PER PERIOD 



n 


4% 


4|% 


5% 


6 1% 


6% 


1 
a 

3 

4 
5 


0.0615 3846 
1.8860 9467 
2.7760 0103 
3.6298 9622 
4.4518 2233 


0.0560 3780 
1.8726 6775 
2.7480 6435 
3.5876 2570 
4.3800 7674 


0.0523 8005 
1.8594 1043 
2.7232 4803 
3.5450 5050 
4.3204 7607 


0.9478 0730 
1.8403 1071 
2.0970 3338 
3.5051 6012 
4.2702 8448 


0.0433 0623 
1.8333 9207 
2.0730 1105 
3.4051 0501 
4.2123 0370 


6 
7 

8 
9 
10 


5.2421 3686 
6.0020 5407 
6.7327 4487 
7.4353 3161 
8.1108 9578 


5.1678 7248 
5.8927 0094 
6.5958 8007 
7.2687 9050 
7.0127 1818 


5.0750 0206 
5.7863 7340 
6.4632 1276 
7.1078 2108 
7.7217 3493 


4.0955 3031 
6,6820 0712 
6.3345 6500 
6.9521 0525 
7.5376 2583 


4.0173 2433 
5.5823 8144 
0.2097 0381 
0.8010 0227 
7.3000 8705 


11 

in 

IS 
11 
16 


8.7604 7671 
9.3850 7376 
9.0856 4785 
10.5631 2293 
11.1183 8743 


8.5280 1692 
9.1185 8078 
9.6828 5242 
10.2228 2528 
10.7305 4573 


8.3064 1422 
8.8632 5164 
0.3935 7299 
9.8986 4094 
10.3706 5804 


8.0925 3033 
8.0185 1786 
9.1170 7853 
9.5806 4790 
10.0375 8004 


7.8808 7458 
8.3838 4304 
8.8520 8290 
9.2040 8303 
0.7122 4800 


16 
17 
18 
19 
99 


11.6522 9561 
12.1656 6885 
12.6592 9607 
13.1339 3940 
13.5903 2634 


11.2340 1505 
11.7071 9143 
12.1599 9180 
12.5032 9359 
13.0070 3645 


10.8377 6050 
11.2740 6626 
11.6805 8690 
12.0853 2086 
12.4622 1034 


10.4621 6203 
10:8046 0856 
11.2400 7447 
11.6070 5352 
11.0503 8240 


10.1058 0527 
10.4772 6000 
10.8270 0348 
11.1581 1040 
11,4609 2122 


21 
22 
23 
24 
26 


14.0201 5095 
14.4611 1533 
14.8568 4167 
15.2460 6314 
15.6220 7904 


13.4047 2388 
13.7844 2476 
14.1477 7489 
14.4954 7837 
14.8282 0896 


12.8211 5271 
13.1630 0268 
13.4885 7388 
13.7080 4170 
14.0030 4457 


12.2752 4400 
12.5831 0073 
12.8760 4240 
13.1516 9806 
13.4139 3206 


11.7640 7002 
12.0415 8172 
12.3033 7808 
12.6503 5763 
12.7833 5010' 


28 
27 
28 
29 
30 


15.9827 6918 
16.3295 8575 
16.6630 6322 
16.0837 1463 
17.2020 3330 


15.1466 1145 
15.4513 0282 
15.7428 7361 
16.0218 8853 
16.2888 8854 


14.3751 8530 
14.6430 3362 
14.8981 2726 
15.1410 7358 
15.3724 5103 


13.6624 9541 
13.8080 9091 
14.1214 2172 
14.3331 0116 
14.5337 4517 


13.0031 6019 
13.2105 3414 
13.4061 0428 
13.5007 2102 
13.7048 3115 


31 
32 
38 
34 
36 


17.5884 9356 
17.8735 5150 
18.1476 4687 
18.4111 9776 
18.6646 1323 


16.5443 9095 
16.7888 9086 
17.0228 6207 
17.2467 5796 
17.4610 1240 


15.5028 1050 
15.8026 7667 
16.0025 4921 
10.1020 0401 
16.3741 0420 


14.7230 2907 
14.0041 0817 
15.0750 6936 
15.2370 3257 
15.3005 5220 


13.0200 8509 
14.0840 4330 
14.2302 2001 
14,3081 4114 
14.4082 4(530 


39 
37 
38 
39 
10 


18.9082 8195 
19.1426 7880 
19.3678 6423 
19.5844 8484 
19.7927 7388 


17.6660 4058 
17.8622 3970 
18.0400 0023 
18.2296 5572 
18.4015 8442 


16.5468 5171 
16.7112 8734 
10.8678 0271 
17.0170 4067 
17.1590 8635 


15.5360 6843 
15.6739 0851 
15.8047 3793 
15.0286 6154 
16.0461 2469 


14.0200 8713 
14.7307 8031 
14.8460 1010 
14.9400 7408 
15.0402 0087 


41 
42 
48 
44 
46 


19.9930 5181 
20.1856 2674 
20.3707 9494 
20.5488 4120 
20.7200 3970 


18.5661 0040 
18.7235 4075 
18.8742 1029 
19.0183 8305 
19.1563 4742 


17.2943 6706 
17.4232 0758 
17.5450 1108 
17.6627 7331 
17.7740 6082 


16.1574 6416 
16.2620 0920 
16.3630 3242 
16.4578 6003 
16.5477 2572 


15.1380 1592 
15.2245 4332 
16.3001 7294 
15.3831 8202 
15.4558 3200 


4ft 
47 
48 
49 
50 


20.8846 5356 
21.0429 3612 
21.1951 3088 
21.3414 7200 
21.4821 8462 


19.2883 7074 
10.4147 0884 
10.5356 0654 
10.6512 0813 
19.7620 0778 


17.8800 6650 
17.9810 1571 
18.0771 6782 
18,1687 2173 
18.2550 2546 


16.6329 1537 
16.7136 6386 
16.7902 0271 
16.8027 5139 
16.9315 1700 


15.5243 0000 
15.5800 2821 
15.6500 2061 
15.7075 7227 
15.7618 6064 



62 



TABLE Vm PRESENT VALUE OF ANNUITY OF 1 PER PERIOD 



n 


4% 


4|% 


6% 


6-<y 

"3 JO 


6% 


51 
52 
63 
54 
55 


21.6174 8521 
21.7476 8103 
21.8720 7403 
21.0029 6007 
22.1080 1218 


10.8070 5003 
10.0003 3017 
20.0003 4400 
20.1501 8140 
20.2480 2057 


18.3380 7663 
18.4180 7298 
18.4034 0284 
18.5051 4566 
18.6334 7100 


10.0066 0943 
17.0584 8287 
17.1170 4538 
17.1725 5486 
17.2251 7048 


16.8130 7007 
15.8013 0252 
15.9009 7408 
15.9409 7554 
15.9006 4207 


50 
57 

58 
59 
60 


22.2180 1040 
22.3267 4043 
22,4205 0076 
22.5284 2057 
22.0234 8007 


20.3330 3404 
20,4143 8004 
20.4922 3002 
20.5007 3303 
20.6380 2204 


18.6085 4473 
18.7006 1870 
18.8195 4170 
18.8767 5400 
18.0292 8952 


17.2760 4311 
17.3223 1575 
17.3071 2393 
17.4006 9614 
17.4498 5416 


10.0288 1412 
16.0649 1808 
16.0080 8017 
10.1311 1337 
10.1614 2771 


61 
62 
63 
64 
65 


22.7148 0421 
22.8027 8280 
22.8872 0124 
22.9086 4027 
23.0400 8109 


20.7062 4118 
20.7715 2266 
20.8330 0298 
20.8037 7310 
20.0500 7913 


18.9802 7674 
19.0288 3404 
19.0750 8003 
10.1101 2384 
10.1010 7033 


17.4880 1343 
17.5241 8334 
17.6684 0762 
17.5900 6457 
17.0217 0737 


16.1000 2614 
10.2170 0579 
10.2424 5829 
16.2604 7009 
10.2801 2272 


66 
67 
68 
69 
70 


23.1218 0001 
23.1040 4770 
23.2035 0740 
23.3302 0558 
23.3045 1408 


21.0057 2106 
21.0581 0084 
21.1082 3021 
21.1502 0000 
21.2021 1187 


19.2010 1930 
19.2300 0000 
10.2753 0101 
10.3008 1048 
10.3420 7005 


17.6600 6433 
17.6780 3017 
17.7048 7125 
17.7297 3570 
17.7533 0400 


10.3104 0314 
16.3306 5300 
10.3406 7349 
16.3076 1650 
16.3845 4387 


71 
72 
73 
74 
75 


23.4502 6440 
23.5150 3885 
23.6727 2060 
23.0270 2408 
23.0804 0834 


21.2460 4007 
21.2880 7662 
21.3283 0298 
21.3067 0711 
21.4030 3360 


10.3730 7776 
19.4037 8834 
10.4321 7037 
10.4502 1845 
19.4840 0005 


17.7750 4300 
17.7008 1804 
17.8168 8970 
17.8350 1441 
17.8539 4731 


16.4005 1308 
10.4155 7838 
10.4207 0093 
16.4431 0899 
10.4558 4810 


76 
77 

78 
79 
80 


23.7311 0187 
23.7700 0333 
23.8268 8782 
23.8720 0752 
23.0163 0185 


21.4388 8383 
21.4720 1011 
21.5048 0579 
21.5357 8545 
21.5003 4403 


19.5094 0510 
19.5328 5257 
19.5560 0708 
10.5762 8361 
10.5004 0048 


17.8710 4010 
17.8872 4180 
17.9026 9887 
17.9171 6532 
17.0300 6291 


16.4077 8123 
16.4790 3889 
16.4800 6033 
16.4096 7802 
16.5091 3077 


81 

82 
83 

84 
85 


23.0571 0754 
23.0072 1870 
24.0357 8730 
24.0728 7240 
24.1085 3110 


21.6936 3151 
21.0207 0001 
21.6466 0288 
21.6713 0032 
21.0951 1035 


10.6156 7606 
10.0339 7776 
10.0614 0730 
19.0080 0704 
10.0838 1623 


17.9440 3120 
17.9564 2708 
17.0081 7789 
17.9703 1554 
17.0808 7256 


16.5180 4700 
16.5264 6028 
16.5343 0640 
16.6418 8348 
16.5480 4608 


86 
87 
88 
80 
90 


24.1428 1842 
24.1767 8094 
24,2074 8746 
24,2370 6870 
24.2872 7750 


21.7178 0806 
21.7306 3000 
21.7003 1588 
21.7802 0658 
21.7002 4075 


19.0088 7200 
10.7132 1200 
19.7208 6857 
19.7308 7483 
10.7522 6174 


17.0008 7910 
18.0003 6410 
18.0183 5400 
18.0208 7645 
18.0340 5308 


10.6550 1008 
16.5018 9030 
10.5678 2670 
16.5734 2141 
16.5780 0944 


91 
92 
93 
94 
95 


24.2954 5023 
24,3226 6005 
' 24.3480 1245 
24.3730 0682 
24.3077 5650 


21.8174 5626 
21.8348 8542 
21.8515 0400 
21.8075 2631 
21.8828 0030 


10.7040 5880 
10.7752 0410 
10.7850 0438 
10.7001 8612 
10.8058 0059 


18.0426 1041 
18.0498 0700 
18.0507 4062 
18.0632 0094 
18.0604 4734 


10.5830 7872 
16,5883 7615 
16.5028 0700 
16.6060 8830 
16.0000 3244 


96 
97 
98 
99 
100. 


24.4200 1884 
24.4431 0110 
24.4640 0002 
24.4851 08tt 
24,5049 0000 


21.8074 1065 
21.0114 0340 
21.0247 8704 
21,0375 0012 
21.0408 5274 


10.8151 3300 
10.8230 3705 
10.8323 2100 
10.8403 0571 
10,8470 1020 


18.0753 0553 
18.0808 5833 
18.0801 2104 
18.0911 1055 
18.0058 3039 


16.0040 5325 
16.6081 0344 
10,6114 7494, 
16.0145 9000 
10.6175 4623 



TABLE VET PRESENT VALUE OP ANNUITY OF 1 PER PERIOD 



z) 



1 - (1 + 



n 


B|% 


7% 


7|% 


8% 


8|% 


1 

3 
3 
4 
5 


0.9389 6714 
1.8200 2642 
2.0484 7551 
3.4257 0800 
4,1556 7044 


0.9345 7944 
1.8080 1817 
2.6243 1604 
3.3872 1120 
4.1001 9744 


0.0302 325(5 
1.7056 0517 
2.6005 2574 
3.3403 2027 
4.0458 8490 


0.0250 2503 
1.7832 0475 
2.5770 9090 
3.3121 2084 
3.0027 1004 


0.9210 5890 
1.7711 1427 
2.6640 2237 
3.2756 0606 
3.0406 4208 


6 
7 
8 

10 


4.8410 1356 
5.4845 1977 
0.0887 5000 
0.6501 0419 
7.188S 3022 


4.7065 3900 
5.3802 8040 
6.9712 9851 
6.5152 3225 
7.0235 8154 


4.6038 4042 
5.2066 0132 
5.8673 0355 
6.3788 8703 
6.8640 8000 


4.0228 7006 
5.20U3 7006 
5.7400 3804 
0.2468 8701 
0.7100 8140 


4.6535 8717 
5.1185 1352 
5.6301 8207 
6.1100 6264 
6.5613 4800 


11 
13 
13 
14 
15 


7.6800 4240 
8.1587 2532 
8.5907 4208 
9.0138 4233 
9.4026 6S85 


7.4086 7434 
7.9426 8630 
8.3676 5074 
8.7464 0799 
9.1079 1401 


7.3164 2415 
7.7352 7827 
8.1258 4020 
8.4801 5373 
8.8271 1074 


7.1389 6426 
7,5300 7802 
7.0037 7694 
8.2442 3098 
8.5604 7800 


6.0680 8430 
7.3440 8607 
7.6000 6400 
8.0100 0668 
8.3042 3658 


16 
17 
18 
19 
20 


9.7677 6418 
10.1105 7670 
10.4324 6638 
10.7347 1022 
11.0185 0725 


9.4466 4860 
9.7032 2299 
10.0590 8691 
10.3355 0524 
10.5940 1425 


0.1415 0674 
9.4330 5976 
9.7000 0008 
0.0590 7821 
10.1044 0130 


8.8513 6016 
0.1216 3811 
9.3718 8714 
0.6035 0020 
0.8181 4741 


8.5753 3326 
8.8251 0104 
0,0554 7044 
0.2077 2022 
0.4633 3001 


21 
22 
23 
24 
25 


11.2840 8333 
11.5351 9562 
11.7701 3673 
11.9007 3871 
12,1078 7672 


10.8355 2733 
11.0612 4050 
11.2721 8738 
11.4693 3400 
11.6535 8318 


10.4134 8033 
10.0171 0101 
10.8060 8931 
10.0829 6680 
11.1460 4586 


10.0168 0310 
10.2007 4300 
10.3710 5805 
10.6287 5828 
10.6747 7019 


0,6436 2821 
0.8097 0550 
0.0629 4524 
10.10-10 0700 
10.2341 0078 


26 
Vt 
28 
29 
30 


12.3023 7251 
12./5749 9766 
12.7464 7668 
12.9074 8084 
13.0580 7591 


11.8257 7867 
11.9867 0904 
12.1371 1125 
12.2776 7407 
12.4000 4118 


11.2094 8452 
11.4413 8005 
11.6733 7703 
11.6961 6524 
11.8103 8627 


10.8090 7706 
10.9361 0477 
11.0510 7840 
11.1684 0001 
11.2577 8334 


10.3540 0288 
10.4646 0174 
10.5004 5321 
10.0003 2564 
10.7408 4382 


81 
82 
33 
84 
35 


13.2006 3465 
13.3339 2025 
13.4590 8850 
13.6766 0892 
13.0869 5673 


12.5318 1410 
12.6465 5532 
12.7537 0002 
12.8540 0936 
12.9470 7230 


11.9166 3839 
12.0154 7767 
12.1074 2009 
12.1929 4076 
12.2725 1141 


11.3407 0030 
11.4340 0944 
11.5138 8837 
11.5800 3307 
11.6545 0822 


10,8205 8410 
10.0000 7767 
10.0078 1343 
11.0302 4270 
11,0877 8137 


36 
37 
38 
39 
40 


13.7905 6970 
13.8878 5887 
13.9792 1021 
14.0049 8611 
14.1455 2687 


13.0352 0776 
13.1170 1660 
13.1934 7345 
13.2649 2846 
13.3317 0884 


12.3465 2224 
12.4153 6063 
12.4704 1361 
12.5380 8031 
12.6044 0860 


11.7171 0270 
11.7751 7851 
11.8288 6809 
11.8785 8240 
11.0246 1333 


11.1408 1233 
11.1806 8878 
11.2347 3020 
11.2702 6467 
11.3145 2034 


41 
42 
43 
44 
45 


14.2211 5190 
14.2921 6149 
14.3588 3708 
14.4214 4327 
14.4802 2842 


13.3941 2041 
13.4524 4898 
13.5069 6167 
13.5579 0810 
13.6055 2159 


12.6459 6155 
12.6939 1772 
12.7885 2811 
12.7800 2015 
12.8186 2898 


11.0672 3467 
12.0000 0867 
12.0432 3951 
12.0770 7302 
12.1084 0160 


11.3407 8833 
11.3822 0339 
11.4122 5107 
11.4308 0367 
11.4653 1206 


46 
47 
48 

50 


14.5354 2575 
14.5872 5422 
14.6350 1046 
14.0816 1451 
14.7246 2067 


13.6500 2018 
13.6916 0764 
13.7304 7443 
13.7667 9853 
13.8007 4629 


12.8645 3858 
12,8879 4287 
12.0190 1662 
12.9479 2244 
12.9748 1167 


12.1374 0880 
12.1642 0741 
12.1801 3649 
12.2121 6341 
12.2334 8464 


11.4887 6086 
11,5103 8420 
11.6303 0802 
11.6486 7099 
11.6656 0538 



64 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 = 1 = ; , 1 

(a-, ati) 1 - (1 + i)- (s-, at 2) 

n| * n/ 



n 


fi% 


|% 


% 


1% 


1% 


1 
9 

3 

4 
5 


1.0041 6687 
0.6031 2717 
0.3361 1406 
0.2626 0068 
0.2026 0693 


1.0050 0000 
O.C037 6312 
0.3360 7221 
0.2531 3279 
0.2030 0097 


1.0058 3333 
0.5043 7024 
0.3372 2976 
0.2536 5044 
0.2035 1367 


1.0075 0000 
0.5050 3200 
0.3383 4579 
0.2547 0501 
0.2045 2242 


1.0100 0000 
0.5075 1244 
0.3400 2211 
0.2502 8109 
0.2060 3980 


G 

7 
8 

10 


0.1601 0564, 
0.1452 4800 
0.1273 5512 
0.1134 3876 
0.1023 0506 


0.1605 0646 
0.1457 2864 
0.1278 2886 
0.1139 0738 
0.1027 7057 


0.1700 8504 
0.1462 0086 
0.1283 0351 
0.1143 7608 
0.1032 3632 


0.1710 6891 
0.1471 7488 
0.1202 5552 
0.1153 1020 
0.1041 7123 


0.1725 4837 
0.1486 2828 
0.1306 0020 
0.1167 4037 
0.1056 8208 


11 
12 
13 
14 
15 


0.0031 9767 
0.0866 0748 
0.0791 8532 
0.0736 8082 
0.0680 1045 


0.0038 5903 
O.OSOO 6643 
0.0700 4224 
0.0741 3BOO 
0.0003 6436 


0.0941 2176 
0.0865 2675 
0.0801 0004 
0.0746 0205 
0.0608 1990 


0.0950 5094 
0.0874 5148 
0.0810 2188 
0.0755 1146 
0.0707 3639 


0.0964 6408 
0.0888 4870 
0.0824 1482 
0.0769 0117 
0.0721 2378 


16 
17 

18 
19 
20 


0.0647 3055 
0.0010 5387 
0,0677 8063 
0.0548 5191 
0.0522 1630 


0.0651 8037 
0.0015 0579 
0.0582 3173 
0.0553 0253 
0.0520 6645 


0.0656 4401 
0.0610 5960 
0.0586 8409 
0.0657 5532 
0.0531 1889 


0.0665 6879 
0.0628 7321 
0.0595 0768 
0.0608 6740 
0.0540 3063 


0.0679 4460 
0.0642 5808 
0.0600 8205 
0.0580 5175 
0.0564 1632 


21 
22 
23 
24 
25 


0.0408 3183 
0.0476 6427 
0.0456 8531 
0.0438 7130 
0.0422 0270 


0.0502 8163 
0.0481 1380 
0.0461 3465 
0.0443 2061 
0.0420 5180 


0.0507 3383 
0.0485 6585 
0.0405 8603 
0.0447 7258 
0.0431 0388 


0.0616 4543 
0.0494 7748 
0.0474 9846 
0.0466 8474 
0.0440 1650 


0.0630 3076 
0.0608 6371 
0.0488 8684 
0.0470 7347 
0.0464 0676 


26 
27 
28 
29 
30 


0.0406 6247 
0.0302 3646 
0,0379 1230 
0.0366 7074 
0.0355 2036 


0.0411 1163 
0.0308 8565 
0.0383 6107 
0.0371 2014 
0.0350 7802 


0.0415 6376 
0.0401 3793 
0.0388 1415 
0.0375 8186 
0.0304 3191 


0.0424 7093 
0.0410 6176 
0.0307 2871 
0.0384 0723 
0.0373 4816 


0.0438 6888 
0.0424 4553 
0.0411 2444" 
0.0308 9502 
0.0387 4811 


31 
32 
33 
34 
35 


0.0344 5330 
0.0334 4458 
0.0324 9708 
0.0316 0540 
0.0307 6476 


0.0349 0304 
0.0338 0453 
0.0320 4727 
0.0320 5580 
0.0312 1550 


0.0353 5633 
0.0343 4816 
0.0334 0124 
0.0326 1020 
0.0316 7024 


0.0362 7352 
0.0352 6834 
0.0343 2048 
0.0334 3053 
0.0326 0170 


0.0376 7673 
0.0306 7080 
0.0367 2744 
0.0348 3907 
0.0340 0308 


36 
37 
38 
39 
40 


0.0299 7000 
0.0202 2003 
0.0286 0875 
0.0278 3402 
0.0271 9310 


0.0304 2104 
0.0206 7130 
0.0280 6045 
0.0282 8607 
0.0276 4552 


0.0308 7710 
0.0301 2698 
0.0294 1649 
0.0287 4258 
0.0281 0251 


0.0317 0973 
0.0310 5082 
0.0303 4157 
0.0298 0893 
0.0200 3016 


0.0332 1431 
0.0324 6805 
0.0317 6150 
0.0310 0160 
0.0304 6560 


41 
42 
43 
44 
45 


0.0263 8352 
0.0260 0303 
0.0254 4061 
0.0249 2141 
0.0244 1675 


0.0270 3631 
0,0264 5622 
0.0250 0320 
0.0253 7541 
0.0248 7117 


0.0274 0379 
0.0260 1420 
0.0203 6170 
0.0258 3443 
0.0253 3073 


0.0284 2276 
0.0278 4452 
0.0272 0338 
0.0207 0761 
0.0262 6521 


0.0298 5102 
0,0202 7563 
0,0287 2737 
0.0282 0441 
0.0277 0505 


46 
47 

48 
49 
50 


0.0239 3409 
0.0234 7204 
0.0230 2020 
0.0226 0468 
0.0221 9711 


0.0243 8804 
0.0239 2733 
0.0234 8503 
0.0230 0087 
0.0220 5376 


0.0248 4005 
0.0243 8708 
0.0230 4024 
0.0235 2205 
0.0231 Iflll 


0.0257 8405 
0.0253 2532 
0.0248 8504 
0.0244 0202 
0,0240 5787 


0.0272 2775 
0.0207 7111 
0.0203 3384 
0,0260 1474 
0.0255 1273 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 * _.- , 1 



n 


n% 


|% 


h% 


!% 


1% 


51 


0.0218 0657 


0.0222 6260 


0.0227 2663 


0.0230 0888 


0.0251 2680 


Kt 


0.0214 2010 


0.0218 8075 


0.0223 5027 


0.0232 0503 


0.0247 5603 


53 


0.0210 6700 


0.0215 2507 


0.0210 8910 


0.0220 3546 


0.0243 0050 


54 


0.0207 1830 


0.0211 7086 


0.0210 4167 


0.0225 8038 


0.0240 5058 


56 


0.0203 8234 


0.0208 4130 


0.0213 0071 


0.0222 6005 


0.0237 2637 


56 


0.0200 5843 


0.0205 1707 


0.0200 8300 


0.0210 3478 


0.0234 0823 


57 


0.0107 4503 


0.0202 0508 


0.0206 7261 


0.0216 2400 


0.0231 0156 


58 


0.0104 4426 


0.0100 0481 


0.0203 7100 


0.0213 2507 


0.0228 0573 


50 


0.0101 5287 


0.0100 1302 


0.0200 8170 


0.0210 3727 


0.0225 2020 


00 


0.0188 7123 


0.0103 3280 


0.0108 0120 


0.0207 6836 


0.0222 4445 


61 


0.0185 0888 


0.0100 6006 


0.0105 2000 


0.0204 8873 


0.0219 7800 


62 


0.0183 8636 


0.0187 0706 


0.0192 0762 


0.0202 2706 


0.0217 2041 


63 


0.0180 8026 


0.0185 4337 


0.0100 1368 


0.0100 7500 


0.0214 7125 


64 


0.0178 3316 


0.0182 0681 


0.0187 0773 


0.0107 3127 


0.0212 3013 


65 


0.0175 0371 


0.0180 5780 


0.0185 2046 


0.0194 0460 


0.0200 0667 


66 


0.0173 6166 


0.0178 2027 


0.0182 0848 


0.0102 0524 


0.0207 7062 


67 


0.0171 3639 


0.0176 0163 


0.0180 7440 


0.0100 4280 


0.0205 6130 


68 


0.0169 1788 


0.0173 8366 


0.0178 6710 


0.0188 2716 


0.0203 3888 


69 


0.0167 0674 


0.0171 7206 


0.0176 4622 


0.0186 1786 


0.0201 3280 


70 


0.0164 0971 


0.0160 6657 


0.0174 4138 


0.0184 1404 


0.0100 3282 


71 


0.0162 0052 


0.0167 6603 


0.0172 4230 


0.0182 1728 


0.0107 3870 


72 


0.0161 0403 


0.0165 7280 


0.0170 4901 


0.0180 2554 


0.0106 5010 


73 


0.0150 1572 


0.0163 8422 


0.0168 6100 


0.0178 3017 


0.0103 6700 


74 


0.0157 3166 


0.0162 0070 


0.0166 7814 


0.0176 6706 


0.0191 8010 


75 


0.0155 5253 


0.0160 2214 


0.0165 0024. 


0.0174 8170 


0.0100 1600 


76 


0.0163 7816 


0.0158 4832 


0.0163 2709 


0.0173 1020 


0.0188 4784 


77 


0.0152 0836 


0.0156 7008 


0.0161 5851 


0.0171 4328 


0.0186 8416 


78 


0-0150 4206 


0.0156 1423 


0.0150 0432 


0.0160 8074 


0.0185 2488 


79 


0.0148 8177 


0.0153 5360 


0.0158 3436 


0.0108 2244 


0.0183 6084 


80 


0.0147 2464 


0.0151 0704 


0.0156 7847 


0.0106 6821 


0.0182 1886 


81 


0.0145 7144 


0.0150 4430 


0.0165 2650 


0.0105 1700 


0.0180 7180 


82 


0.0144 2200 


0.0148 0562 


0,0153 7830 


0.0163 7136 


0.0170 2851 


88 


0.0142 7620 


0.0147 5028 


0.0152 3373 


0.0102 2847 


0.0177 8880 


84 


0.0141 3301 


0.0140 0855 


0.0160 0268 


0.0100 8008 


0.0176 6273 


85 


0.0130 0600 


0.0144 7021 


0.0149 5501 


0.0150 5303 


0.0175 1008 


86 


0.0138 6035 


0.0143 3513 


0.0148 2000 


0.0158 2034 


0.0173 0060 


87 


0.0137 2685 


0.0142 0320 


0.0146 8036 


0.0156 0076 


0.0172 6417 


89 


0.0135 0740 


0.0140 7431 


0.0145 6115 


0.0155 0423 


0.0171 4080 


89 


0.0134 7088 


0.0130 4837 


0.0144 3688 


0.0154 4064 


0.0170 2050 


90 


0.0133 4721 


0.0138 2527 


0.0143 1347 


0.0153 1080 


0.0100 0300 


91 


0.0132 2620 


0.0137 0403 


0.0141 0380 


0.0152 0100 


0.0167 8832 


92 


0.0131 0803 


0.0136 8724 


0.0140 7679 


0.0150 8657 


0.0166 7624 


98 


0.0120 0234 


0.0134 7213 


0.0130 6236 


0.0140 7382 


0.0165 0673 


94 


0.0128 7016 


0.0133 5060 


0.0138 5042 


0.0148 0356 


0.0164 6071 


95 


0.0127 6837 


0.0132 4030 


0.0137 4000 


0.0147 6571 


0.0163 5511 


96 


0.0126 5002 


0.0131 4143 


0.0136 3372 


0.0146 5020 


0.0102 5284 


97 


0.0125 5374 


0.0130 3583 


0.0135 2880 


0.0145 4606 


0.0161 5284 


98 


0.0124 4076 


0.0120 3242 


0.0134 2608 


0.0144 4602 


0.0160 5603 


99 


0.0123 4700 


0.0128 3115 


0.0133 2540 


0.0143 4701 


0.0150 5030 


100 


0.0122 4811 


0.0127 3104 


0.0132 2606 


0.0142 6017 


0.0158 6574 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

, f . 1 

C^tfQ 



n 


% 


|% 


-% 
12 10 


!% 


1% 


101 
102 
103 
104 
105 


0.0121 5033 
0.0120 6440 
0.0119 6054 
0.01 IS 0842 
0.0117 7800 


0.0128 3473 
0.0125 3947 
0.0124 4611 
0.0123 6457 
0.0122 6481 


0.0131 3045 
0.0130 3587 
0.0120 4310 
0.0128 5234 
0.0127 6238 


0.0141 5633 
0.0140 0243 
0.0130 7143 
0.0138 8226 
0.0137 0487 


0.0157 7413 
0.0156 8440 
0.0155 0608 
0.0155 1073 
0.0154 2066 


10G 
107 
108 
100 
110 


0.0110 8948 
0.0110 0250 
0,0115 1727 
0.0114 3358 
0,0113 5143 


0.0121 7679 
0.0120 9045 
0.0120 0575 
0.0119 2264 
0.0118 4107 


0.0120 7594 
0.0125 0020 
0.0125 0028 
0.0124 2385 
0.0123 4208 


0.0137 0022 
0.0130 2524 
0.0136 4201 
0.0134 6217 
0.0133 8296 


0.0153 4412 
0.0152 6336 
0.0151 8423 
0.0151 0609 
0.0150 3009 


111 
112 
113 
114 
115 


0.0112 7079 
0,0111 9161 
0.0111 1386 
0.0110 3750 
0.0100 6240 


0.0117 6102 
0.0116 8242 
0.0116 0626 
0.0115 2048 
0.0114 6500 


0.0122 6301 
0.0121 8671 
0.0121 0023 
0.0120 3414 
0.0110 6041 


0.0133 0527 
0.0132 2905 
0.0131 5425 
0.0130 8084 
0.0130 0878 


0.0149 5320 
0.0148 8317 
0-0148 1166 
0.0147 4133 
0.0146 7245 


116 
117 
118 
110 
120 


0.0108 8880 
0,0108 1030 
0.0107 4524 
0.0106 7630 
0.0106 0655 


0.0113 8105 
0.0113 1013 
0.0112 3060 
0.0111 7021 
0.0111 0205 


0.0118 8709 
0.0118 1680 
0.0117 4098 
0.0110 7832 
0.0116 1085 


0.0120 3803 
0.0128 6857 
0.0128 0037 
0.0127 3338 
0.0120 0758 


0.0146 0488 
0.0145 3860 
0.0144 7356 
0.0144 0973 
0.0143 4709 


131 
122 
123 
124 
125 


0.0105 3800 
0.0104 7261 
0.0104 0715 
0,0103 4288 
0.0102 7965 


0.0110 3505 
0,0109 0918 
0.0100 0441 
0.0108 4072 
0.0107 7808 


0.0115 4464 
0.0114 7030 
0.0114 1528 
0.0113 5228 
0.0112 0033 


0.0120 0294 
0.0125 3042 
0.0124 7702 
0.0124 1568 
0.0123 6540 


0.0142 8501 
0.0142 2525 
0.0141 0590 
0.0141 0780 
0.0140 5065 


126 
127 
128 
120 
130 


0.0102 1746 
0.0101 5625 
0.0100 9603 
0.0100 3077 
0.0090 7844 


0.0107 1047 
0.0106 5586 
0.0105 0023 
0.0105 3755 
0.0104 7081 


0.0112 2040 
0.0111 6948 
0.0111 1054 
0.0110 5265 
0.0100 9560 


0.0122 9614 
0.0122 3788 
0.0121 8000 
0.0121 2428 
0.0120 0888 


0.0130 9462 
0.0130 3030 
0.0138 8624 
0.0138 3203 
0.0137 7975 


131 
132 
133 
134 
135 


0.0090 2102 
0.0008 0440 
0.0008 0883 
0.0007 5403 
0.0007 0005 


0.0104 2208 
0,0103 6704 
0.0103 1107 
0.0102 677B 
0.0102 0430 


0,0100 3035 
0.0108 8410 
0.0108 2072 
0.0107 7619 
0,0107 2340 


0.0120 1440 
0.0110 0080 
0.0110 0808 
0.0118 6621 
0.0118 0510 


0.0137 2837 
0.0130 7788 
0.0130 2825 
0.0135 7947 
0.0135 3151 


136 
137 
138 
130 
140 


0.0006 4080 
0.0005 9463 
0,0005 4205 
0.0004 0213 
0.0094 4205 


0.0101 5179 
0.0101 0002 
0.0100 4002 
0.0009 0870 
0.0099 4030 


0,0100 7101 
0.0106 2052 
0.0105 7021 
0.0105 2007 
0.0104 7187 


0.0117 5493 
0,0117 0550 
0.0110 5684 
0-0116 0804 
0.0116 0170 


0.0134 8437 
0.0134 3801 
0.0133 9242 
0.0133 4759 
0.0133 0340 


141 
142 
143 
144 
145 


0,0003 9271 
0,0003 4408 
0,0002 0015 
0.0002 4800 
0.0002 0233 


0-0099 0056 
0.0008 5250 
0.0008 0610 
0.0007 6860 
0.0007 1252 


0.0104 2380 
0.0103 7044 
0.0103 2078 
0.0102 8381 
O.OL02 3851 


0.0115 1536 
0,0114 6966 
0.0114 2464 
0.0113 8031 
0.0113 3064 


0.0132 6012 
0.0132 1746 
0,0131 7640 
0.0131 3419 
0.0130 0366 


146 
147 
148 
140 
ISO 


0.0001 5Q41 
0.0001 1114 
0.0000 0050 
0.0000 2247 
0.0089 7005 


0.0000 0710 
0.0000 2250 
0.0005 7844 
0.0005 3500 
0.0004 0217 


0.0101 0380 
0,0101 4086 
0.0101 0040 
0.0100 6373 
0.0100 2150 


0.0112 9364 
0.0112 5127 
0.0112 0063 
0.0111 0841 
0.0111 2700 


0.0130 5358 
0.0130 1423 
0.0120 7651 
0,0120 3739 
0.0128 0088 



67 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 



(a^ati) l-(l+i)- n 



. , 



n 


1|% 


1|% 


ll% 


lf% 


2% 


i 

3 
3 

4 
5 


1.0112 5000 
0.6084 5323 
0.3408 0130 
0.2670 7058 
0.2068 0034 


1.0126 OOOO 
0.5003 0441 
0.3417 0117 
0.2678 8102 
0.2076 0211 


1.0150 0000 
0.5112 7792 
0.3433 8206 
0.2504 4478 
0.2000 8032 


1.0175 0000 
0.6131-0295 
0.3460 0746 
0.2010 3237 
0.2100 2142 


1.0200 0000 
0.6150 4950 
0.3407 5407 
0.2620 2375 
0.2121 5830 


6 

7 
8 
9 
10 


0.1732 9034 
0.1403 6762 
0.1314 1071 
0,1174 6432 
0.1002 9131 


0.1740 3381 
0.1500 8872 
0.1321 3314 
0.1181 7055 
0.1070 0307 


0.1765 2521 
0.1516 5010 
0.1336 8402 
0.1106 0082 
0.1084 3418 


0.1770 2250 
0.1530 3059 
0.1350 4202 
0.1210 5813 
0.1098 7634 


0.1785 2681 
0.1545 1106 
0.1365 0080 
0.1225 1644 
0.1113 2063 


11 
12 
13 
14 
15 


0.0071 5084 
0.0805 5203 
0.0831 1020 
0.0770 0138 
0.0728 2321 


0.0978 6839 
0.0002 5831 
0.0838 2100 
0-0783 0616 
0.0735 2646 


0.0002 9384 
0.0018 7090 
0.0852 4030 
0.0797 2332 
0.0749 4430 


0.1007 3038 
0.0931 1377 
0.0800 7283 
0.0811 5602 
0.0703 7730 


0.1021 7704 
0.0045 6060 
0.0881 1835 
0.0820 0107 
0.0778 2547 


16 
17 
13 
10 
20 


0.0086 4363 
0.0040 5008 
0.0610 8113 
0.0587 5120 
0.0561 1531 


0.0603 4072 
0.0650 6023 
O.OG23 8479 
0.0504 5648 
0.0508 2039 


0.0707 6508 
0.0670 7006 
0.0638 0578 
0.0008 7847 
0.0682 4574 


0.0721 9058 
0.0685 1023 
0.0652 4402 
0.0623 2061 
0.0590 0122 


0.0730 5013 
0.0000 0984 
0.0067 0210 
0.0637 8177 
0.0011 5072 


21 
22 
23 
24 
25 


0.0537 3145 
0.0515 6626 
0.04Q5 8833 
0.0477 7701 
0.0461 1144 


0.0544 3748 
0.0522 7238 
0.0502 0666 
0.0484 8005 
0.0468 2247 


0.0558 0550 
0.0537 0331 
0.0517 3075 
0.0400 2410 
0.0482 0346 


0.0573 1404 
0.0551 5038 
0.0581 8706 
0.0513 8665 
0.0497 2052 


0.0587 8477 
0.0566 3140 
0.0540 0810 
0.0528 7110 
0,0512 2044 


26 
27 

28 
20 
30 


0.0445 7479 
0.0431 5273 
0.0418 3209 
0.0406 0408 
0.0304 5063 


0.0452 8729 
0.0438 6677 
0.0425 4863 
0.0413 2228 
0.0401 7854 


0.0407 3196 
0.0463 1627 
0.0440 0108 
0.0427 7878 
0.0416 3010 


0.0482 0200 
0.0407 0070 
0.0454 8151 
0.0442 0424 
0.0431 2076 


0.0406 9923 
0.0482 9309 
0.0460 8907 
0.0457 7830 
0.0440 4902 


31 
32 
33 
34 
35 


0.0383 8866 
0.0373 8535 
0.0364 4349 
0.0355 5763 
0.0347 2200 


0.0391 0942 
0.0381 0791 
0.0371 6786 
0.0362 8337 
0.0354 5111 


0.0405 7430 
0.0305 7710 
0.0380 4144 
0.0377 0180 
0.0309 3303 


0.0420 7005 
0.0410 7812 
0.0401 4770 
0.0302 7303 
0.0384 5082 


0.0435 0635 
0.0426 1001 
0.0416 8053 
0,0408 1807 
0.0400 0221 


36 
37 
38 
3ft 
40 


0.0330 3529 
0.0331 0072 
0.0324 8589 
0.0318 1773 
0.0311 8349 


0.0340 0533 
0.0330 2270 
0.0332 1083 
0.0326 6365 
0.0310 2141 


0.0301 5240 
0.0364 1437 
0.0347 1613 
0.0340 5463 
0.0334 2710 


0.0370 7507 
0.0360 4257 
0.0362 4090 
0.0356 0399 
0.0349 7200 


0.0302 3286 
0.0386 0078 
0.0378 2067 
0.0371 7114 
0.0305 6575 


41 
42 
43 
44 
45 


0.0305 8009 
0.0300 0709 
0.0294 0064 
0.0289 3949 
0.0284 4197 


0.0313 2068 
0.0307 4906 
0.0302 0466 
0.0290 8557 
0.0291 0012 


0.0328 3106 
0.0322 0420 
0.0317 2405 
0.0312 1038 
0.0307 1970 


0.0343 8170 
0.0338 2057 
0.0332 8606 
0.0327 7810 
0.0322 9321 


0.0359 7188 
0.0354 1720 
0.0348 8003 
0.0343 8704 
0.0339 0062 


46 
47 
48 
49 
50 


0.0279 6652 
0.0275 1173 
0.0270 7032 
0.0266 5010 
0.0262 5898 


0.0287 1676 
0.0282 6406 
0.0278 3075 
0.0274 1563 
0.0270 1763 


0.0302 5125 
0.0208 0342 
0.0203 7500 
0.0280 6478 
0.0286 7108 


0.0318 3043 
0.0313 8830 
0.0300 0569 
0.0305 6124 
0.0301 7391 


0.0334 5342 
0.0330 1702 
0.0320 0184 
0.0322 0306 
0.0318 2321 



TABLE 



PERIODICAL PAYMENT OF ANtttJITY "WHOSE 
PRESENT VALUE IS 1 

1 i 1 



n 


ll% 


ll% 


l|% 


lf% 


2% 


61 
6% 
63' 
64 
66 


0.0258 7404 
0.026C 0006 
0.0251 6149 
0.0248 1043 
0.0244 8213 


0.0266 3571 
0.0202 6807 
0.0259 1653 
0.0266 7760 
0.0252 5146 


0.0281 0460 
0.0278 3287 
0.0274 8537 
0.0271 6138 
0.0268 3018 


0.0208 0209 
0.0204 4665 
0.0201 0492 
0.0287 7672 
0.0284 6120 


0.0314 6856 
0.0311 0909 
0.0307 7392 
0.0304 5226 
0.0301 4337 


66 
67 
68 
69 
60 


0.0241 6502 
0.0238 6116 
0.0236 6726 
0.0232 8366 
0.0230 0085 


0.0240 3730 
0.0246 3478 
0.0243 4303 
0.0240 6158 
0.0237 8003 


0.0265 2106 
0.0262 2341 
0.0250 3661 
0.0250 6012 
0.0253 9343 


0.0281 5705 
0.0278 0606 
0.0275 8503 
0.0273 1430 
0.0270 5336 


0.0298 4656 
0.0295 6120 
0.0202 8667 
0.0200 2243 
0.0287 6707 


61 
63 
63 
64 
66 


0.0227 4534 
0.0224 8060 
0.0222 4247 
0.0220 0320 
0.0217 7178 


0.0235 2758 
0.0232 7410 
0.0230 2004 
0.0227 0203 
0.0225 6268 


0.0251 3604 
0.0248 8751 
0.0246 4741 
0.0244 1534 
0.0241 0004 


0.0268 0172 
0.0265 5892 
0.0263 2455 
0.0260 9821 
0.0258 7952 


0.0285 2278 
0.0282 8643 
0.0280 5848 
0.0278 3855 
0.0276 2624 


66 
67 
68 
60 
70 


0.0215 4758 
0.0213 3037 
0.0211 1085 
0.0200 1671 
0.0207 1760 


0.0223 4065 
0.0221 2560 
0.0210 1724 
0.0217 1627 
0.0215 1041 


0.0230 7386 
0.0237 6370 
0.0235 6033 
0.0233 6320 
0.0231 7235 


0.0266 6813 
0.0254 8372 
0.0252 6506 
0.0260 7459 
0.0248 8030 


0.0274 2122 
0.0272 2316 
0.0270 3173 
0.0268 4665 
0.0260 6766 


71 

73 
78 
74 

75 


0.0205 2552 
0.0203 3806 
0.0201 6770 
0.0100 8177 
0.0198 1072 


0,0213. 2041 
0.0211 4501 
0.0200 6600 
0.0207 9215 
0.0206 2325 


0.0229 8727 
0.0228 0779 
0.0226 3368 
0.0224 6473 
0.0223 0072 


0.0247 0985 
0.0245 3600 
0.0243 6750 
0.0242 0413 
0.0240 4670 


0.0264 0440 
0.0263 2683 
0.0261 6454 
0.0260 0736 
0.0268 5608 


76 
77 

78 
79 
80 


0.0100 4442 
0.0104 8260 
0.0103 2536 
0.0101 7226 
0.0100 2323 


0.0204 5010 
0.0202 0053 
0.0201 4435 
0.0100 0341 
0.0108 4652 


0.0221 4146 
0.0210 8676 
0.0218 3045 
0.0216 0036 
0,0215 4832 


0.0238 0200 
0.0237 4284 
0.0236 0806 
0.0234 5748 
0.0233 2003 


0.0257 0751 
0.0255 6447 
0.0254 2576 
0.0252 0123 
0.0251 6071 


81 
82 
83 
84 
85 


0.0188 7812 
0.0187 3678 
0.0185 0008 
0.0184 6480 
0.0183 3400 


0.0107 0356 
0.0106 6437 
0.0104 2881 
0.0102 0675 
0.0101 6808 


0.0214 1019 
0.0212 7583 
0.0211 4500 
0.0210 1784 
0.0208 0306 


0.0231 8828 
0.0230 5036 
0.0220 3403 
0.0228 1223 
0.0226 0375 


0.0250 3405 
0.0240 1110 
0.0247 9173 
0.0246 7681 
0.0245 6321 


86 
87 
88 
80 
00 


0.0182 0654 
0,0180 8215 
0.0179 6081 
0.0178 4240 
0.0177 2684 


0.0100 4207 
0.0180 2041 
0.0188 0110 
0.0180 8400 
0.0185 7140 


0.0207 7333 
0.0206 6584 
0.0205 4138 
0,0204 2084 
0.0203 2113 


0.0225 7850 
0.0224 6636 
0.0223 5724 
0.0222 5102 
0.0221 4760 


0.0244 5381 
0.0243 4760 
0.0242 4416 
0.0241 4370 
0.0240 4602 


01 
99 
93 
04 
06 


0.0176 1403 
0.0176 0387 
0.0173 0020 
0.0172 0110 
0.0171 8851 


0.0184 6076 
0.0183 5271 
0.0182 4724 
0.0181 4426 
0.0180 4366 


0.0202 1516 
0.0201 1182 
0,0200 1104 
0.0190 1273 
0.0198 1681 


O.d220 4600 
0.0219 4882 
0.0218 6327 
0.0217 0017 
0.0216 6944 


0.0230 6101 
0.0238 5850 
0.0237 6868 
0.0236 8118 
0.0235 0602 


06 
07 
08 
00 
100 


0.0170 8810 
0.0160 0007 
0.0168 0418 
0.0168 0041 
0.0167 0870 


0.0170 4540 
0.0178 4041 
0.0177 5660 
0.0170 6391 
0.0176 7428 


0.0107 2321 
0.0196. 3186 
0.0105 4268 
0.0104 5560 
0.0103 7057 


0.0216 8101 
0.0214 9480 
0.0214 1074 
0.0213 2876 
0.0212 4880 


0.0235 1313 
0.0234 3242 
0,0233 6383 
0.0232 7720 
0.0232 0274 



69 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 _ i _.-, 1 



n 


2-07 
4% 


2l% 


2|% 


3% 


3l% 


1 

2 
3 

4 
5 


1.0225 0000 
0.5100 3768 
0.3484 4458 
0.2042 1803 
0.2137 0021 


1.0260 0000 
0.5188 2716 
0.3501 3717 
0.2668 1788 
0.2152 4086 


1.0275 0000 
0.6207 1825 
0.3518 3243 
0,2074 2059 
0.2107 0832 


1.0300 0000 
0.5220 1084 
0,3535 3030 
0.2000 2706 
0.2183 5457 


1.0350 0000 
0.5264 0049 
0.3500 3418 
0.2722 5114 
0.2214 8137 


6 
7 

8 

10 


0.1800 3400 
0.15GO 0026 
0.1370 8402 
0.1230 8170 
0.1127 8768 


0.1815 4007 
0.1574 9543 
0.1394 0735 
0.1254 5089 
0.1142 6876 


0.1830 7083 
0.1589 9747 
0.1409 6706 
0.1260 4005 
0.1157 3972 


0.1845 0750 
0.1005 0035 
0.1424 5030 
0.1284 3380 
0.1172 3051 


0,1870 0821 
0.1035 4440 
0.1464 7005 
0.1314 4001 
0.1202 4137 


11 
12 
18 
14 
15 


0.1030 3049 
0.0900 1740 
0.0805 7080 
0.0840 6230 
0.0702 8852 


0.1051 0596 
0,0974 8713 
0.0910 4827 
0.0855 3053 
0.0807 6640 


0.1005 8029 
0,0089 6871 
0.0025 3252 
0.0870 2457 
0.0822 5017 


0.1080 7745 
0.1004 6200 
0.0040 2054 
0.0885 2034 
0.0837 0068 


0.1110 0107 
0.1034 8305 
0.0070 0167 
0.0015 707H 
0.0808 2507 


1ft 
17 
18 
1 
20 


0.0751 1063 
0.0714 4039 
0.0681 7720 
0.0052 0182 
0.0026 4207 


0.0765 9899 
0.0720 2777 
0.0000 7008 
0.0607 0002 
0.0041 4713 


0.0780 9710 
0.0744 3180 
0.0711 8003 
0.0082 7802 
0,0066 7173 


0.0700 1086 
0.0750 6253 
0.0727 0870 
0.0098 1388 
0.0072 1571 


0.0820 8483 
0.0700 4313 
0.0758 1084 
0.0720 4033 
0.0703 0108 


21 
22 
28 
24 
25 


0.0002 7672 
0.0681 2821 
0.0661 7097 
0.0543 8023 
0.0527 3590 


0.0617 8733 
0.0506 4061 
0.0570 9638 
0.0550 1282 
0.0542 7592 


0.0033 1041 
0.0011 8040 
0,0592 4410 
0.0574 0803 
0.0568 3907 


0.0048 7178 
0.0027 4730 
0.0008 1300 
0.0500 4742 
0.0574 2787 


0.0080 3060 
0.0060 3207 
0.0640 1880 
0.0022 7283 
0.0600 7404 


20 
27 
28 
30 
80 


0.0512 2134 
0.0408 2188 
0.0485 2625 
0.0473 2081 
0.0401 9934 


0.0527 6875 
0.0513 7087 
0.0500 8793 
0.0488 9127 
0.0477 7764 


0.0543 4110 
0.0529 5770 
0.0616 7738 
0.0604 8936 
0.0493 8442 


0.0559 3820 
0.0545 0421 
0,0532 9323 
0.0521 1407 
0,0510 1020 


0.0592 0540 
0.0578 5241 
0.0500 0205 
0.0564 4538 
0.0543 7133 


31 
32 
83 
34 
35 


0.0451 5280 
0.0441 7415 
0.0432 5722 
0.0423 9665 
0.0416 8731 


0.0467 3000 
0.0457 6831 
0.0448 5938 
0,0440 0075 
0.0432 0558 


0.0483 5463 
0,0473 9203 
0.0464 0253 
0.0450 4875 
0.0448 6046 


0.0400 9803 
0.0400 4002 
0.0481 6012 
0.0473 2106 
0.0405 3920 


0.0533 7240 
0.0624 4160 
0.0516 7242 
0.0607 50(50 
0.0400 0835 


36 
37 
88 
80 
40 


0.0408 2622 
0.0401 0043 
0.0394 2753 
0.0387 8543 
0.0381 7738 


0.0424 5158 
0.0417 4090 
0.0410 7012 
0.0404 3015 
0.0398 3623 


0,0441 1132 
0.0434 0063 
0.0427 4704 
0.0421 2260 
0.0416 3161 


0.0468 0379 
0,0451 1102 
0.0444 5934 
0.0438 4385 
0.0432 0238 


0.0403 8416 
0.0480 1325 
0.0479 8214 
0.0473 8775 
0.0408 2728 


41 
42 
43 
44 
45 


0.0376 0087 
0.0370 5364 
0.0305 3364 
0.0360 3901 
0.0355 0805 


0.0392 6786 
0.0387 2870 
0.0382 1088 
0,0377 3037 
0.0372 0752 


0.0409 7200 
0.0404 4175 
0.0300 3871 
0,0394 6100 
0.0300 0093 


0,0427 1241 
0.0421 0107 
0.0410 9811 
0.0412 2085 
0.0407 8518 


0.0462 0822 
0.0467 9828 
0.0453 2630 
0.0148 7708 
0.0444 5343 


46 
47 
48 
40 
50 


0.0351 1921 
0.0340 9107 
0.0342 8233 
0.0338 9179 
0,0335 1836 


0.0368 2676 
0.0364 0609 
0.0300 0590 
0.0356 2348 
0.0352 6806 


0,0385 7403 
0,0381 6358 
0.0377 7158 
0.0373 9773 
0.0370 4092 


0.0403 0264 
0.0390 0051 
0.0395 7777 
0.0302 1314 
0.0388 6550 


0.0440 5108 
0,0430 0010 
0.0433 0040 
0,0420 0107 
0.0426 3371 



70 



TABLE IX PERIODICAL PAYMENT OP ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 i 



(a^ati) l-(l+f)- 



f 



n 


2|% 


Q-Q7 
-4 a % 


2|% 


3% 


3i% 


51 
58 
53 
51 
55 


0.0331 0102 
0.0328 1384 
0.0324 9004 
0.0321 7064 
0.0318 7480 


0.0340 0870 
0.0345 7440 
0.0342 5440 
0.0330 4790 
0.0336 5410 


0.0307 0014 
0.0363 7444 
0.0360 0207 
0.0367 0491 
0.0354 7063 


0.0386 3382 
0.0382 1718 
0.0379 1471 
0.0370 2568 
0.0373 4907 


0.0423 2166 
0.0420 2429 
0.0417 4100 
0.0414 7000 
0.0412 1323 


50 
57 

58 
50 
60 


0.0315 8530 
0.0313 0712 
0.0310 3977 
0.0307 8268 
0.0305 3533 


0.0333 7243 
0.0331 0204 
0.0328 4244 
0.0325 0307 
0.0323 5340 


0.0352 OG12 
0.0340 4404 
0.0346 0270 
0.0344 5153 
0.0342 2002 


0.0370 8447 
0.0368 3114 
0.0365 8848 
0.0363 5503 
0.0361 3200 


0.0409 6730 
0.0407 3245 
0.0405 0810 
0.0402 9366 
0.0400 8862 


61 
6% 
63 
64 
65 


0.0302 0724 
0.0300 0705 
0.0208 4704 
0.0206 3411 
0.0204 2878 


0.0321 2204 
0.0310 0126 
0.0316 8700 
0.0314 8240 
0.0312 8403 


0.0330 0707 
0.0337 8402 
0.0335 7860 
0.0333 8118 
0.0331 0120 


0.0350 1008 
0.0357 1386 
0.0356 1682 
0.0363 2760 
0.0351 4581 


0.0308 0249 
0.0397 0480 
0.0395 2513 
0.0303 6308 
0.0391 8826 


60 
67 
68 
00 
70 


0.0202 3070 
0.0200 3055 
0.0288 5500 
0.0286 7077 
0.0285 0458 


0.0310 0308 
0.0300 1021 
0.0307 3300 
0.0305 6206 
0.0303 0712 


0.0330 0837 
0.0328 3236 
0.0326 6285 
0.0324 0055 
0.0323 4218 


0.0340 7110 
0.0348 0313 
0.0346 4150 
0.0344 8618 
0.0343 3663 


0.0300 3031 
0.0388 7892 
0.0387 3376 
0.0385 0463 
0.0384 6005 


71 
73 
78 
74 
75 


0.0283 3810 
0.0281 7728 
0.0280 2160 
0.0278 7118 
0.0277 2654 


0.0302 3790 
0.0300 8417 
0.0200 3508 
0.0207 0222 
0.0206 5358 


0.0321 0048 
0.0320 4420 
0.0310 0311 
0.0317 6008 
0.0316 3560 


0.0341 9266 
0.0340 6404 
0.0330 2053 
0.0337 9191 
0.0336 0706 


0.0383 3277 
0.0382 0973 
0.0380 9160 
0.0379 7810 
0.0378 6010 


76 
77 

78 
79 
80 


0.0275 8467 
0.0274 4808 
0.0273 1680 
0.0271 8784 
0.0270 0370 


0.0205 1960 
0.0203 8007 
0.0202 0403 . 
0.0201 4338 
0.0200 2605 


0.0316 0878 
0.0313 8633 
0.0312 6800 
0.0311 5382 
0.0310 4342 


0.0336 4849 
0.0334 3331 
0.0333 2224 
0.0332 1610 
0.0331 1176 


0.0377 6460 
0.0376 6300 
0.0375 0721 
0.0374 7420 
0.0373 8480 


81 

83 
83 
84 
85 


0.0200 4350 
0.0268 2062 
0.0207 1387 
0.0200 0423 
0.0204 0787 


0.0280 1248 
0.0288 0254 
0.0286 0608 
0.0285 0208 
0.0284 0310 


0.0309 3074 
0.0308 3361 
0.0307 3380 
0.0306 3747 
0.0305 4420 


0.0330 1201 
0.0329 1570 
0.0328 2284 
0.0327 3313 
0.0320 4660 


0.0372 0804 
0.0372 1628 
0.0371 3070 
0.0370 6025 
0.0360 8062 


86 
87 
88 
80 
00 


0.0203 0467 
0.0202 0452 
0,0261 0730 
0.0201 0201 
0.0200 1120 


0.0283 0033 
0.0283 0255 

0.0282 linn 

0.0281 2363 
0.0280 3809 


0.0304 5307 
0.0303 6007 
0.0302 8210 
0-0302 0041 
0.0301 2126 


0.0326 6284 
0.0324 8202 
0.0324 0303 
0,0323 2848 
0.0322 6560 


0.0360 1576 
0.0308 47C6 
0.0367 8190 
0.0367 1868 
0.0366 6781 


91 
99 
92 
94 
05 


0.0250 2224 
0.0258 3577 
0,0267 5176 
0,0256 7012 
0.0255 0078 


0,0270 5523 
0.0278 7486 
0.0277 9000 
0.0277 2120 
0.0276 4786 


0.0300 4400 
0.0299 7038 
0.021)8 0860 
0.0208 2887 
0.0207 6141 


0.0321 81)08 
0.0321 1604 
0.0320 6107 
0.0319 8737 
0.0310 2677 


0.0365 0019 
0.0365 4273 
0.0364 8834 
0.0364 3594 
0.0363 8546 


06 
97 
98 
99 
100 


0,0265 1306 
0.0254 3868 
0.0253 6678 
0.0262 9480 
0,0252 2594 


0.0275 7602 
0,0276 0747 
0.0274 4034 
0.0273 7617 
0.0273 1188 


0.0200 0605 
0.0206 3272 
0.0206 7134 
0.0206 1185 
0.0294 5418 


0.0318 6610 
0.0318 0856 
0.0317 6281 
0.0316 9886 
0.0310 4667 


0.0363 3682 
0.0362 8905 
0.0362 4478 
0.0362 0124 
0.0361 6927 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 _ i 1 



(a^ati) l- 



n 


4% 


*!% 


6% 


5|% 


6% 


1 
2 
8 
4 
5 


1.0400 0000 
0.5301 9608 
0.3603 4864 
0.2754 9005 
0.2246 2711 


1.0450 0000 
0.5339 9756 
0.3637 7336 
0.2787 4365 
0.2277 9164 


1.0600 0000 
0.5378 0488 
0.3672 0856 
0.2820 1183 
0.2309 7480 


1.0650 0000 
0.5416 1800 
0.3706 5407 
0.2852 9449 
0.2341 7644 


1.0600 0000 
0.5454 3689 
0.3741 0981 
0.2885 9149 
0.2373 0640 


6 
7 
8 
9 
10 


0.1907 6190 
0.1666 0961 
0.1485 2783 
0.1344 9299 
0.1232 9094 


0.1938 7839 
0.1697 0147 
0.1516 0965 
0.1375 7447 
0.1263 7882 


0.1970 1747 
0.1728 1982 
0.1647 2181 
0.1406 9008 
0.1295 0458 


0.2001 7895 
0.1759 6442 
0.1678 6401 
0.1438 3946 
0.1326 6777 


0.2033 0263 
0.1701 3502 
0.1010 3604 
0.1470 2224 
0.1368 6706 


11 
12 
13 
U 
15 


0.1141 4904 
0.1065 5217 
0.1001 4373 
0.0946 6807 
0.0899 4110 


0.1172 4818 
0.1096 6619 
0.1032 7635 
0.0978 2032 
0.0931 1381 


0.1203 8889 
0:1128 2541 
0.1064 5577 
0.1010 2397 
0.0963 4229 


0.1235 7006 
0.1160 2923 
0.1096 8426 
0.1042 7012 
0.0996 2500 


0.1207 0294 
0.1192 7703 
0.1120 0011 
0.1075 8401 
0.1020 0276 


18 
17 
18 
19 
20 


0.0858 2000 
0.0821 9852 
0.0789 9333 
0.0761 3862 
0.0736 8176 


0.0890 1537 
0.0854 1758 
0.0822 3690 
0.0794 0734 
0.0768 7614 


0.0022 6991 
0.0886 9914 
0.0855 4622 
0.0827 4501 
0.0802 4259 


0.0955 8254 
0.0020 4197 
0.0889 1092 
0.0861 5006 
0.0830 7033 


0.0080 5214 
0.0054 4480 
0.0023 5654 
0.0800 2080 
0.0871 8456 


21 
22 
23 
24 
25 


0.0712 8011 
0.0691 9881 
0.0673 0906 
0.0655 8683 
0.0640 1190 


0.0740 0057 
0.0725 4565 
0.0706 8249 
0.0689 8703 
0.0674 3903 


0.0779 9611 
0.0759 7061 
0.0741 3082 
0.0724 7090 
0.0709 5246 


0.0814 6478 
0.0794 7123 
0.0776 6066 
0.0760 3580 
0.0745 4035 


0.0850 0455 
0.0830 4557 
0.0812 7848 
0.0790 7000 
0.0782 2072 


26 
27 
28 
29 
80 


0.0625 6738 
0.0612 3854 
0.0600 1298 
0.0588 7993 
0.0678 3010 


0.0660 2137 
0.0647 1946 
0.0635 2081 
0.0624 1461 
0.0613 9154 


0.0095 6432 
0.0082 0186 
0.0671 2253 
0.0600 4551 
0.0660 5144 


0.0731 9307 
0.0710 5228 
0.0708 1440 
0.0607 6867 
0.0088 0539 


0.0700 0436 
0.0750 0717 
0.0745 0255 
0.0736 7901 
0.0726 4801 


31 
82 
83 
34 
35 


0.0568 5535 
0.0559 4859 
0.0551 0357 
0.0543 1477 
0.0535 7732 


0.0604 4345 
0.0595 6320 
0.0587 4453 
0.0570 8191 
0.0572 7046 


0.0641 3212 
0.0632 8042 
0.0624 9004 
0.0617 5645 
0.0610 7171 


0.0079 1065 
0.0670 0519 
0.0663 3469 
0.0056 2958 
0.0649 7493 


0.0717 9222 
0.0710 0234 
0.0702 7203 
0.0605 0843 
0.0680 7380 


36 
37 
38 
39 
40 


0.0528 8688 
0.0522 3957 
0.0516 3192 
0.0510 6083 
0.0505 2349 


0.0566 0578 
0.0550 8402 
0.0554 0169 
0.0548 5567 
0.0643 4315 


0.0004 3446 
0.0598 3070 
0.0592 8423 
0.0587 6462 
0.0582 7816 


0.0643 603G 
0.0637 0903 
0.0032 7217 
0.0027 7901 
0.0623 2034 


0.0683 0483 
0.0078 5743 
0.0673 5812 
0.0068 9377 
0.0604 0154 


41 
42 
43 
44 
45 


0.0500 1738 
0.0495 4020 
0.0490 8989 
0.0486 0454 
0.0482 6246 


0.0538 0158 
0.0534 0868 
0.0520 8235 
0.0625 8071 
0.0522 0202 


0.0578 2220 
0.0573 0471 
0.0569 0333 
0.0566 1025 
0.0562 6173 


0.0618 0000 
0.0614 8027 
0.0611 1337 
0.0607 0128 
0.0604 3127 


0.0000 5880 
0.0066 8342 
0.0653 3312 
0.0650 0000 
0.0047 0050 


46 
47 
48 
49 
50 


0.0478 8205 
0.0475 2189 
0,0471 8065 
0.0468 5712 
0.0466 6020 


0.0518 4471 
0.0615 0734 
0.0611 8858 
0.0508 8722 
0.0606 0215 


0.0559 2820 
0.0656 1421 
0.0653 1843 
0.0550 3065 
0.0547 7074 


0.0601 2175 
0.0598 3120 
0.0595 5854 
0.0503 0230 
0.0590 6145 


0.0044 1485 
0.0041 4708 
0.0038 9706 
0.0630 0356 
0,0634 4429 



72 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 _ i . = i -I ^ 



n 


4% 


4% 


5% 


ftt 


6% 


51 
52 
53 
54 
55 


0.0402 5885 
0.045D 8212 
0.0457 1015 
0.0454 0010 
0.0452 3124 


0.0503 3232 
0.0500 7070 
0.0498 3400 
0.0490 0510 
0.0493 8754 


0.0545 2807 
0.0542 0450 
0.0540 7334 
0.0538 (M38 
0.0530 0080 


0.0588 3405 
0.058(1 218(1 
0.05H4 21HO 
0.0582 3245 
0.0580 5458 


0.0032 3880 
O.OIJ30 4(117 
0.0028 (1551 
0.0020 IXJ02 
0.0(125 3(100 


56 
57 
58 
59 
00 


0.0450 0487 
0.0447 8032 
0.0445 8401 
0.0443 8830 
0.0442 0185 


0.0401 8105 
0.0489 8500 
0.0487 0807 
0.0480 2221 
0.0484 5420 


0.0534 8010 
0.0533 03-13 
0.0531 3020 
0.0520 7802 
0.0528 2818 


0.0678 8008 
0.0677 21100 
0.0675 8000 
0.0674 3051) 
0.0673 0707 


0.0023 8705 
0.0(122 4744 
0.0021 1574 
0.0010 0200 
0.0018 7572 


61 
62 
03 


0.0440 2308 
0.0438 5430 
0.0436 0237 


0.0482 0402 
0.0481 4284 
0.0479 0848 


0.0620 8027 
0.0625 5183 
0.0524 2442 


0.0671 8202 
0.0670 (1400 
0.05110 5258 


0.0(117 (1042 

o.ooio 0:100 

0.0015 (1704 


04 
05 


0.0435 3780 
0.0433 0019 


0.0478 0115 
0.0477 3047 


0.0523 0806 
0.0521 8015 


0.0608 4737 
0.0607 4800 


0.0014 7016 
0.0013 9000 


66 
67 
08 


0.0432 4921 
0.0431 1451 
0.0420 8578 


0.0470 0008 
0.0474 8705 
0,0473 7487 


0.0520 8057 
0.0519 7757 
0.0518 7080 


0.0600 6-113 
0,0605 (1544 
0.05(1.1 8103 


0.0013 1022 
0.0012 3454 
0.0(111 (11130 


09 
79 


0.0428 6272 
0.0427 4500 


0.0472 0745 
0.0471 0611 


0.0617 8716 
0.0510 91G 


0.06(14 0242 
0.0603 2754 


0.0010 0025 
0.0010 3313 


71 


0.0420 3253 


0.0470 0750 


0.0510 1503 


0.0602 5076 


0.0000 7370 


72 


0.0425 2480 


0.0400 7405 


0,0515 8033 


O.Omil 8082 


0.0000 1774 


73 


0.0424 2100 


0.0408 8600 


0.0514 0103 


0.05(11 2(152 


0.0008 0505 


74 


0.0423 2334 


0.0408 0159 


0.0513 8053 


0.0500 0(1(15 


0.0008 1642 


75 


0.0422 2000 


0.0407 2104 


0.0613 2101 


0.06(10 1002 


0.0007 0807 


76 


0.0421 3809 


0.0400 4422 


0.0512 5700 


0.0650 5(145 


0.0007 2403 


77 


0.0420 5221 


0.0405 7004 


0.0511 0580 


0.0550 0677 


0.000(1 an 5 


78 


0.0419 0930 


0.0405 0104 


0.0611 3750 


0.0668 6781 


0.0000 4407 


79 


0.0418 0007 


0.0404 3434 


0.051O 8222 


0.0568 1243 


0.0000 0724 


89 


0.0418 1408 


0.0403 7069 


0.0510 2002 


0.0557 0048 


0.0005 7254 


81 


0.0417 4127 


0.0403 0995 


0.0509 7003 


0,0557 2884 


0.0005 3084 


82 


0.0410 7160 


0.0402 5107 


0.0500 3211 


0.0650 008(1 


0.0005 0003 


83 


0.0410 0403 


0.0401 0063 


0.0508 801)4 


0,0550 5:il)5 


0.0(104 70JI8 


84 


0.0415 4054 


0.0401 4379 


0.0608 4800 


0.0650 1047 


0.0(104 52(11 


85 


0.0414 7909 


0.0400 0334 


0.0508 0810 


0.0655 8(183 


0.0004 20H1 


86 


0.0414 2018 


0.0400 4510 


0.0507 0433 


0.0666 6503 


0.0004 0240 


87 


0.0413 0370 


0.0450 0915 


0,0507 2740 


0.0556 20(17 


0,(H!()3 706(1 


88 


0.0413 0953 


0.0459 6622 


0.0500 Oa28 


0.0664 0800 


0.0003 6706 


89 


0.0412 5V58 


0,0450 1325 


0.0500 5888 


0.0664 727.1 


0.0003 8767 


90 


0.0412 0776 


0.0468 7310 


0.0500 2711 


0,0554 4788 


0.0003 183(1 


91 


0.0411 5005 


0.0458 3480 


0.0505 0081) 


0.0554 2485 


0.0(103 0025 


92 


0.0411 1410 


0.0457 9827 


0.0505 OH15 


0,0554 0207 


0.0002 a*m 


93 


0.0410 7010 


0.0457 0331 


0.0505 4080 


0.056S 800(1 


0.0(102 0708 


94 


0,0410 2789 


0.0457 2001 


0.0505 1478 


0,0653 0007 


0.0002 5100 


95 


0.0409 8738 


0.0450 0709 


0.0504 0003 


0.05511 4204 


0.0002 3768 


96 


0.0409 4800 


0.0450 0740 


0.05O4 0048 


0.0653 2410 


0.0(102 2408 


97 


0.0409 1119 


0.0450 3834 


0.0504 4407 


0.0561) 0711 


0.01102 1135 


98 


0.0408 7538 


0.0450 1048 


0.0504 2274 


0.0652 0101 


0,0001 0035 


99 


0.0408 4100 


0.0465 8385 


0.0504 0245 


0.0552 7fl77 


0.0001 8803 


100 


0.0408 0800 


0.0455 6839 


0,0503 8314 


0.0662 0132 


0.0001 7730 



73 



TABLE IX PERIODICAL PAYMENT OF ANNUITY WHOSE 
PRESENT VALUE IS 1 

1 _ i 1 



(a^ati) !-(!+*)- 



(s^ati) 



n 


6|% 


7% 


7f% 


8% 


8|% 


i 

2 
3 

4 
5 


1.0050 0000 
0.5402 6150 
0.3775 7G70 
0.2010 0274 
0.2400 3454 


1.0700 0000 
0.5530 9170 
0.3810 5100 
0.2952 2812 
0.2438 0000 


1.0760 0000 
0.5500 2771 
0.3845 3703 
0.2085 0751 
0.2471 6472 


1.0800 0000 
0.5007 0023 
0.3880 3351 
0.3010 2080 
0.2504 5045 


1.0850 0000 
0.6040 1031 
0.3915 3925 
0.3052 8781) 
0.2537 0575 


6 

7 
8 
9 
10 


0.2005 0831 
0.1823 3137 
0.1042 3730 
0.1502 3803 
0.1301 0409 


0.2097 0580 
0.1855 5322 
0.1074 0770 
0.1534 8647 
0.1423 7750 


0.2130 4489 
0.1888 0032 
0.1707 2702 
0.1507 0710 
0.1450 8593 


0.2103 1539 
0.1920 7240 
0.1740 1470 
0.1000 7071 
0.1490 2040 


0.2100 0708 
0.1053 0022 
0.1773 3005 
0.1034 2372 
0.1524 0771 


11 
12 
13 
14 
15 


0.1300 5521 
0.1226 6817 
0.1162 82CO 
0.1109 4048 
0.1063 6278 


0.1333 5090 
0.1250 0199 
0.1100 5085 
0.1143 4494 
0.1007 9462 


0.1300 9747 
0.1292 7783 
0.1230 0420 
0.1177 0737 
0.1132 8724 


0.1400 7034 
0.1320 9502 
0.1206 2181 
0.1212 0085 
0.1108 2954 


0.1434 0293 
0.1301 5280 
0.1300 2287 
0.1248 4244 
0.1204 2040 


16 
17 

18 
19 
20 


0.1023 7767 
0.0089 0033 
0.0058 5401 
0.0931 5676 
0.0907 5040 


0.1058 5706 
0.1024 2610 
0.0094 1200 
0.0007 5301 
0.0943 0293 


0.1003 9110 
0.1000 0003 
0.1030 2890 
0.1004 1090 
0.0080 0219 


0.1120 7087 
0.1090 2943 
0.1007 0210 
0.1041 2703 
0.1018 5221 


0.1100 1354 
0.1133 1108 
0.1104 3041 
0.1079 0140 
0.1050 7007 


21 
22 
23 
24 
25 


0.0880 1333 
0.0800 0120 
0.0840 6078 
0.0833 9770 
0.0819 8148 


0.0022 8900 
0.0004 0677 
0.0887 1393 
0.0871 8002 
0.0858 1052 


0.0060 2937 
0.0941 8087 
0.0025 3528 
0.0010 5008 
0.0807 1067 


0.0908 3225 
0.0980 3207 
0.0004 2217 
0.0049 7700 
0.0030 7878 


0.1030 0541 
0.1019 3802 
0.1003 7193 
0.0989 0975 
0.0077 1108 


26 
27 
28 
29 
30 


0.0806 9480 
0.0705 2288 
0.0784 5305 
0.0774 7440 
0.0765 7744 


0.0845 6103 
0.0834 2573 
0.0823 9193 
0.0814 '4866 
0.0805 8040 


0.0884 0001 
0.0874 0204 
0.0804 0520 
0.0854 9811 
0.0840 7124 


0.0025 0713 
0.0914 4809 
0.0004 8801 
0.0896 1854 
0.0888 2743 


0.0005 8010 
0.0055 0025 
0.0046 3014 
0.0038 0577 
0.0030 5058 


31 
32 
33 
34 
35 


0.0757 5393 
0.0740 9005 
0,0742 0924 
0.0736 6010 
0.0730 6220 


0.0797 9001 
0.0700 7292 
0.0784 0807 
0.0777 0074 
0.0772 3300 


0.0839 1028 
0.0832 2699 
0.0825 9307 
0.0820 1401 
0.0814 8291 


0.0881 0728 
0.0874 5081 
0.0808 BIOS 
0.0803 0411 
0.0858 0320 


0.0023 0524 
0.0017 4247 
0.0011 7588 
0.0000 60K4 
0.0001 8037 


36 
37 
38 
39 
40 


0.0725 1332 
0.0720 0634 
0.0715 3480 
0,0710 9854 
0.0706 9373 


0.0707 1631 
0.0762 3086 
0.0757 0606 
0.0753 8070 
0.0760 0914 


0.0809 9447 
0.0805 4533 
0.0801 3197 
0.0797 6124 
0.0794 0031 


0.0853 4467 
O.OU40 2440 
0.0846 8HH4 
0.0841 8513 
0.0838 0010 


0.0807 0000 
0.0803 (1790 
0.0800 0066 
0.0886 8103 
0,0883 8201 


41 
42 
48 
44 
AS 


0.0703 1779 
0.0699 6842 
0.0696 4352 
0.0693 4119 
0.0690 5068 


0.0746 5962 
0.0743 3691 
0.0740 3600 
0.0737 6760 
0.0734 9957 


0.0790 7663 
0.0787 7789 
0.0785 0201 
0.0782 4710 
0.0780 1146 


0.0835 6149 
0.0832 8084 
0.0830 3414 
0.0828 0152 
0.0825 8728 


0.0881 0737 
0.0878 6570 
0.0876 2512 
0.0874 1363 
0.0872 1961 


46 
47 
48 
49 
50 


0.0687 9743 
0.0685 5300 
0.0683 2606 
0.0081 1240 
0.0679 1393 


0.0732 5996 
0.0730 3744 
0.0728 3070 
0.0726 3863 
0.0724 5985 


0.0777 9353- 
0.0775 9190 
0.0774 0827 
0.0772 3247 
0.0770 7241 


0.0823 8991 
0.0822 0799 
0.0820 4027 
0.0818 8567 
0.0817 4886 


0.0870 4154 
0.0808 7807 
0.0867 2796 
0.0866 9005 
0.0864 6334 



74 



TABLE X COMPOUND AMOUNT OP 1 FOR FRACTIONAL 

PERIODS 



p 


' n% 


\% 


S* 


! 


1% 


2 
3 

4 

e 

12 
18 
20 


1.0020 8117 
1.0013 8(100 
1.0010 4004 
1.0000 0324 
1.0003 4050 
1.0003 1000 
1.0001 6004 


1.0024 OOS8 
1.001(1 0300 
1.0012 4700 
1.0008 3100 
1.0004 1571 
1.0003 8373 
1.0001 0185 


1.0020 1243 
1.0010 4068 
1.0014 5515 
1.0000 01)87 
1.0004 8482 
1.0004 4751 
1.0002 2373 


1.0037 4200 
1.0024 0378 
1.0018 0075 
1.0012 4011 
1.0000 22SO 
1.0005 7404 
1.0002 8743 


1.0040 8750 
1.0033 2228 
1.0024 0008 
1.0010 5077 
1.0008 2054 
1.0007 0570 
1.0003 8270 


P 


lj% 


lj% 


1|% 


ll% 


2% 


2 
3 
4 

6 
12 
13 

26 


1.0050 0027 
1.0037 3002 
1.0028 0081 
1.0018 0027 
1.0001) 3270 
1.0008 0092 
1.0004 3037 


1,0002 3050 
1.0041 4043 
1.0031 1040 
1.0020 7257 
1.0010 3575 
1.0000 5004 
1.0004 7700 


1.0074 7208 
1.0040 7521 
1,0037 2900 
1.0024 8452 
1.0012 4149 
1.0011 4504 
1.0005 7280 


1.0087 1205 
1.0057 0003 
1.0043 4058 
1.0028 0502 
1.0014 4077 
1.0013 3540 
1.0000 0748 


1.0009 5050 - : 
1.0000 2271 
1.0040 0203 
1.0033 0580 
1.0010 5168 
1.0015 2444 
1.0007 0103 


P 


2|% 


2|% 


2-V 
4/0 


3% 


3|% 


2 
3 
4 
G 
12 
26 
52 


1.0111 8742 
1.0074 1444 
1.00H5 7816 
1.0037 1532 
1.001S 5S04 
1.0008 5016 
1.0004 2709 


L0124 2284 
1.0082 0484 
1.0001 0225 
1,0041 231)2 
1.0020 5084 
1.0000 5017 
1.0004 741)7 


1.0130 5076 
1.0(100 8300 
1.0008 0522 
1.00.16 3108 
1.0022 0328 
1.0010 4300 
1.0005 2184 


1,0148 8010 
1.0090 0103 
1.0074 1707 
1.00-10 3862 
1.0024 0027 
1.0011 3752 
1.0005 6800 


1.0173 4060 
1.0115 3314 
1.0080 8745 
1.0067 5004 
1.0028 7000 
1.0013 2401 
1.0000 0170 


P 


4% 


4-<5f, 

*JJ fO 


5% 


5\% 


6% 


2 
3 
4 

12 
20 
52 . 


1.01518 0300 
1,0131 5H41 
1.0008 5341 
1.0005 5K20 
1.0032 7374 
1.0015 0003 
1.0007 5453 


1.0222 5242 
1,0147 8040 
1.0110 0400 
1.0073 0312 
1.0030 7481 
1.0010 1)430 
1.0008 4084 


1.0240 0508 
1,0103 0030 
1.0122 7224 
1.0081 04H5 
1. 00-10 7412 
1.0018 7831 
1.0000 3871 


1.0271 3103 
1.0180 0713 
1,0134 7518 
1.0080 0340 
1.0044 7170 
1.0020 0138 
1.0010 3010 


1.0206 Q302 
1.011)0 1282 
1.014(5 7386 
1.0007 6880 
1.004S 0755 
1.0022 43(13 
1.0011 2118 


P 


6% 


7% 


7-% 

1 2 


8% 


8|% 


a 

4 


12 
2 
52 


1.0310 8837 
1.0212 1347 
1.0108 0828 
1.0105 5107 
1.0052 6109 
1.0024 2504 
1.0012 1179 


1.0344 0804 
1.0228 0012 
1.0170 5R53 
1.0113 4020 
1,0050 5415 
l.OOSfl 0564 
1.0013 0107 


1.0308 2207 
1.0243 0081 
1.0182 4400 
1.0121 2038 
l.OflflO 4402 
1.0027 8544 
1.0013 0176 


1.0302 3048 
1.0250 8557 
1.0104 2056 
1.0120 0040 
1.0004 3403 
1.0020 0443 
1.0014 8112 


1,0410 3333 
1.0275 6044 
1,0200 0440 
1,0130 8052 
1.0008 2140 
1.0031 4262 
1.0016 7008 



75 



TABLE XI NOMINAL RATE j WHICH IF CONVERTED 
TIMES PER YEAR GIVES EFFECTIVE RATE i 



P 


a% 


1% 


5% 


!% 


1% 


2 
3 
4 
6 
12 
13 
2ffi 


.0041 6234 
.0041 6089 
.0041 6017 
.0041 5945 
.0041 6873 
.0041 6808 
.0041 834 


.0049 9377 
.0049 9169 
.0049 9005 
.0049 8962 
.0049 8858 
.0049 8850 
.0049 8802 


.0058 2485 
.0058 2203 
.0068 2062 
.0068 1921 
.0058 1780 
.0068 1709 
.0058 1704 


.0074 8599 
.0074 8133 
.0074 7900 
.0074 7007 
.0074 7434 
.0074 7416 
.0074 7309 


.0099 7612 
.0099 0085 
.0090 0272 
.0099 5850 
.0090 5440 
.0099 5414 
.0009 5224 


P 


1|% 


1|% 


1|% 


l|% 


2% 


2 
3 
4 
6 
12 
13 
26 


.0112 1854 
.0112 0807 
.0112 0285 
.0111 9763 
.0111 9241 
.0111 9200 
.0111 8960 


.0124 0118 
.0124 4828 
.0124 4183 
.0124 3530 
.0124 2895 
.0124 2846 
.0124 2540 


.0149 4417 
.0149 2562 
.0149 1636 
.0149 0710 
.0148 9785 
.0148 9714 
.0148 9288 


,0174 2410 
.0173 9890 
.0173 8631 
.0173 7374 
.0173 0119 
.0173 0022 
.0173 5443 


.0109 0099 
.0198 6813 
.0198 5173 
.0198 3634 
.0108 1898 
.0198 1772 
.0198 1017 


^ 


2;% 


2|% 


2 -or 
4 % 


3% 


3*% 


2 
3 
4 
6 
12 
26 
52 


.0223 7484 
.0223 3333 
.0223 1261 
.0222 9192 
.0222 7125 
.0222 6013 
.0222 5537 


.0248 4667 
.0247 9451 
.0247 6809 
.0247 4349 
.0247 1804 
.0247 0434 
.0240 9848 


.0273 1349 
.0272 5170 
.0272 2087 
.0271 9009 
.0271 5936 
.0271 4283 
.0271 3575 


.0297 7831 
.0297 0490 
.0296 0829 
.0290 3173 
.0205 9524 
.0295 7561 
.0295 6721 


.0346 0809 
.0345 0043 
.0345 4078 
,0345 0024 
.0344 5078 
.0344 2420 
.0344 1281 


P 


4% 


4|% 


6% 


6|% 


6% 


2 
3 

4 
6 
12 
26 
52 


.0396 0781 
.0394 7821 
.0394 1363 
.0393 4918 
.0392 8488 
.0392 5031 
.0392 3561 


.0445 04S3 
.0443 4138 
.0442 5996 
.0441 7874 
.0440 9771 
.0440 5417 
.0440 3552 


.0493 9015 
.0491 8907 
.0490 8894 
.0489 8908 
.0488 8949 
.0488 3597 . 
.0488 1300 


.0542 3380 
.0540 2139 
. .0539 0070 
.0537 8036 
.0536 6039 
.0535 9593 
.0535 0834 


.0601 2003 
.0588 3847 
.0580 0538 
.0585 5277 
.0584 1001 
.0583 3426 
.0583 0157 


P 


6|% 


7% 


7|% 


8% 


8|% 


2 
3 
4 
6 
12 
28 
2 


.0639 7674 
.0636 4042 
.0634 7314 
.0633 0644 
.0631 4033 
.0630 5113 
.0630 1295 


.0688 1009 
.0684 2737 
.0682 3410 
.0680 4166 
.0678 4974 
.0677 4676 
.0677 0268 


.0736 4414 
.0731 9942 
.0729 7840 
.0727 5827 
.0725 3903 
.0724 2134 
.0723 7098 


.0784 0097 
.0779 6070 
.0777 0619 
.0774 5074 
,0772 0830 
.0770 7606 
.0770 1802 


.0832 0007 
, .0826 9033 
.0824 1768 
.0821 3712 
.0818 5702 
.0817 0811 
.0816 4401 



76 



TABLE XH THE VALUE OP THE CONVERSION FACTOR 

- f '-, tf-o 



p 


35% 


8% 


n% 


!% 


1% 


% 

3 

4 

12 
13 
26 


1.0010 4058 
1.0013 8701 
1.0015 811G 
1.0017 3471 
1.0010 0821) 
1.0010 2164 
1.0020 0170 


1.0012 4844 
1.0010 0482 
1.0018 7305 
1,0020 8131 
1.0022 8000 
1.0023 050 
1.0024 2182 


1.0014 5B21 
1.0010 4103 
1.0021 8485 
1.002-1 2781 
1.0020 7080 
1.0028 8050 
1.0028 0100 


1.0018 7150 
1.0024 0585 
1,0028 0812 
1.0031 2040 
1.0034 3280 
1.0034 50UO 
1.0036 0111 


1.0024 0378 
1.0033 2500 
1.0037 4223 
1.0041 f>8fll 
1.0045 7510 
1.004W 0714 
1.0047 1)041 


/ 


1|% 


lj% 


i-;-% 


l|% 


2% 


2 
3 
4 
6 
12 
13 
88 


1.0028 0403 
1.0037 4008 
1.0042 0802 
1.0040 7730 
1.0051 4583 
1.0051 8188 
1.0053 0818 


1.0031 1520 
1.0041 5510 
1.0046 7537 
1.00C1 0575 
1.0057 1032 
1.0057 5037 
1.0050 0000 


1.0037 3004 
1.0040 8340 
1.0050 0755 
1.0002 3101 
1,0008 5052 
1.0000 0458 
1.0071 0290 


1.0043 8176 
1.0058 1084 
1.0005 3878 
1.0072 0707 
1.0070 0571 
1.0080 5177 
1.0083 8820 


1,0040 7525 
l.OOHO 3733 
1.0074 H850 
1.0083 0125 
1.0001 3380 
1.0001 H700 
1.0005 8243 


/> 


2|% 


2 1% 


2|% 


3% 


3|% 


2 
3 
4 
6 
12 
588 
52 


LOOSE 0371 
1,0074 0202 
1.0083 0830 
1,0003 3444 
1.0102 7107 
1.0107 7505 
1.0109 0195 


1.0002 1142 
1.0082 8701 
1.0003 2677' 
1.0103 6006 
1.0114 0726 
1.0110 0781] 
1.0122 0810 


1.0008 2837 
1.001)1 1141 
1.0102 5422 
1.0113 0780 
1.0125 4243 
1.0131 5008 
1.0134 2343 


1.0074 4458 
1.0000 3431 
1.0111 8072 
1.0124 2S10 
1.0136 7002 
1.0143 4020 
1.0140 3757 


1.0080 7475 
1.0115 7748 
1.0130 3004 
1.0144 8578 
1.0151) 4203 
1.0107 2074 
1.0170 0310 


P 


4% 


4j% 


5% 


6f% 


6% 


2 
3 
4 
G 
12 
20 
52 


1.0000 0105 
1.0132 1713 
1.0148 7744- 
1,0165 3057 
1.0182 0351 
1.0101 0023 
1.0194 8470 


1.0111 2021 
1.0148 5328 
1.0107 2020 
1.0185 8053 
1.0204 0100 
1.0214 0080 
1.0210 0231 


1.0123 4754 
1.0184 8507 
1.0185 5042 
1.0200 3570 
1.0227 1470 
1.0238 3548 
1.0243 1002 


1.0135 R/506 
1.0181 1522 
1.0203 405 
1.0226 7810 
1.0240 0400 
1.0201 0720 
1.0207 2580 


1.0147 8151 
1.0107 4104 
1,0222 2GH8 
1.0247 1070 
1.0272 1070 
1.0385 5520 
1.0201 318(i 


J 


6|% 


7% 


1-% 

I j 70 


8% 


8*% 


2 
3 
4 
6 
12 
20 
S2 


1.0159 0419 
1.0213 0348 
1,0240 6523 
1.0267 5172 
1,0204 5204 
1,0300 0941 
1.0315 3404 


1.0172 0402 
1.0220 8254 
1.0258 8002 
1.0287 8208 
1.0316 9143 
1.0332 5078 
1.0330 3242 


1.0184 1103 
1,0245 0820 
1,0277 0120 
1.0308 1059 
1.0330 2617 
1.0356 0640 
1.0363 2706 


1.0100 1524 
1.0262 1065 
1.0)205 1004 
1.0328 3450 
1,0301 5721 
1.0379 4927 
1.0387 1704 


1.0208 1067 
1.0278 1074 
1.0313 3332 
1.0348 5402 
1.0383 8456 
1.0402 8846 
1.0411 Ofill 



77 



TABLE Xm AMERICAN EXPERIENCE TABLE OF MORTALITY 



Ag 
X 


Num- 
ber 
living 

, 


Num 
her 
of 
death 

* 


Yearly 
proba- 
bility of 
dying 

?* 


Yearly 
proba- 
bility of 
living 

P, 


Ag 
X 


Num- 
ber 
living 

' 


Num 
bor 
of 
deatbfl 

** 


Yearly 
proba- 
bility of 
dying 

Vx 


Yearly 
proba- 
bility of 
living 

Px 


10 


100,000 


749 


0.007 490 


0.002 510 


53 


66,797 


1,091 


0.010 333 


0.083 007 


11 


99,251 


746 


0-007 516 


0.002 484 


54 


05,706 


1,143 


0.017 300 


0.082 604 


12 


98,605 


743 


0.007 543 


0.992 457 


55 


64,563 


1,100 


0.018 571 


0.081 420 


13 


97,762 


740 


0.007 669 


0.992 431 


50 


63,364 


1,200 


0.019 885 


0.080 115 


14 


07,022 


737 


0.007 596 


0.002 404 


57 


62,104 


1,325 


0.021 335 


0.078 005 


15 


96,286 


735 


0.007 634 


0.002 366 


58 


60,779 


1,394 


0.022 936 


0.077 064 


16 


95,550 


732 


0.007 661 


0.992 330 


59 


50,385 


1,468 


0.024 720 


0.075 280 


17 


94318 


729 


0.007 688 


0.992 312 


60 


57,917 


1,640 


0.020 603 


0.973 307 


18 


94,080 


727 


0.007 727 


0.902 273 


61 


68,371 


1,628 


0.028 880 


0.071 120 


19 


93,362 


725 


0.007 766 


0.902 235 


62 


54,743 


1,713 


0.031 202 


0.068 708 


20 


92,637 


723 


0.007 805 


0.002 105 


63 


53,030 


1,800 


0.033 043 


0.006 067 


21 


91,914 


722 


0.007 855 


0.992 145 


64 


51,230 


1,880 


0.030 873 


0.003 127 


22 


01,102 


721 


0.007 906 


0.992 094 


65 


49,341 


1,980 


0,040 120 


0.050 871 


23 


90,471 


720 


0.007 968 


0.002 042 


66 


47,361 


2,070 


0.043 707 


0.056 203 


21 


89,751 


719 


0.008 Oil 


0.991 989 


67 


45,291 


2,168 


0.047 047 


0.052 353 


25 


80,032 


718 


0.008 065 


0.991 935 


68 


43,133 


2,243 


0.052 002 


0.047 008 


26 


88,314 


718 


0.008 130 


0.991 870 


69 


40,800 


2,321 


0.056 762 


0.043 238 


27 


87,596 


718 


0.008 197 


0.991 803 


70 


38,560 


2,391 


0.061 993 


0.038 007 


28 


86,878 


718 


0.008 264 


0.001 736 


71 


36,178 


2,448 


0.007 665 


0.032 335 


29 


86,160 


719 


0.008 345 


0.901 655 


72 


33,730 


2,487 


0.073 733 


0,020 207 


30 


85,441 


720 


0.008 427 


0.991 573 


73 


31.243 


2,505 


0.080 178 


0.010 822 


31 


84,721 


721 


0.008 610 


0.991 490 


74 


28,738 


2,501 


0.087 028 


0.012 972 


32 


84,000 


723 


0.008 607 


0.901 303 


75 


26,237 


2,470 


0.094 871 


0.005 620 


33 


83,277 


726 


0.008 718 


0.001 282 


76 


23,761 


2,431 


0.102311 


0,897 080 


34 


82,651 


729 


0.008 831 


0.991 169 


77 


21,330 


2,360 


0.111 004 


0,888 030 


36 


81,822 


732 


0.008 046 


0.001 054 


78 


18,061 


2,201 


0.120 827 


0.870 173 


30 


81,090 


737 


0.000 089 


0.900 911 


79 


10,670 


2,190 


0.131 734 


0.868 20(1 


37 


80,353 


742 


0.009 234 


0.090 776 


80 


14,474 


2,091 


0.144 400 


0.855 534 


38 


79,611 


749 


0.000 408 


0.900 592 


81 


12,383 


1,964 


0.158 005 


0.841 306 


39 


78,862 


756 


0.009 686 


0.900 414 


82 


10,410 


1,816 


0.174 297 


0,825 703 


40 


78,106 


765 


0.000 704 


0.090 206 


83 


8,603 


1,048 


0.191 501 


0.808 430 


41 


77,341 


774 


0.010 008 


0.980 092 


84 


0,055 


1,470 


0.211 359 


0.788 641 


42 


76,667 


785 


0.010 252 


0.989 748 


85 


5,485 


1,292 


0.236 652 


0.704 448 


43 


75,782 


707 


0.010 617 


0.980 483 


86 


4,193 


1,114 


0.205 081 


0,734 310 


44 


74,985 


812 


0.010 820 


0.989 171 


87 


3,070 


033 


0.303 020 


0.000 080 


45 


74,173 


828 


0.011 163 


0.988 837 


88 


2,140 


744 


0.340 602 


0.053 308 


46 


73,345 


848 


0.011 562 


0.988 438 


89 


1,402 


555 


0.305 863 


).0()4 137 


47 


72,407 


870 


0.012 000 


0.988 000 


90 


847 


385 


0.454 645 


0.54.1 4515 


48 


71,627 


896 


0.012 600 


0.987 491 


91 


462 


240 


0.532 468 


0.467 534 


49 


70,731 


927 


0.013 106 


0.980 804 


92 


210 


137 


0.634 250 


0.305 741 


50 
51 
52 


69,804 
68,842 
67,841 


962 
1,001 
1,044 


0.013 781 
0.014 641 
0.015 380 


0.086 210 
0.085 450 
0.084 611 


93 
94 
05 


70 
21 
3 


58 
18 
3 


0.734 177 
0.857 143 
1.000000 


0.265 823 
0,142857 
0.000 000 



78 



TABLE XIV COMMUTATION COLUMNS, AMERICAN 
EXPERIENCE TABLE, 8& 



Age 
X 


*>* 


NX 


M x 


Age 
X 


*, 


N x 


M x 


10 


70 801.0 


1 575 535.3 


17 012.01 


53 


10 787.4 


146 915.7 


5 853.005 


11 


67 081.6 


1 504 043.4 


17 000.89 


54 


10 252.4 


135 128.2 


5 682.801 


12 


05 180.0 


1 430 601.0 


16 606.20 


55 


9 733.40 


124 875.8 


5 510.644 


13 


02 00.4 


1 371 472.0 


10 131.12 


50 


229.00 


116 142.4 


5 335.808 


11 


50 038.4 


1 308 003,5 


15 073.00 


57 


8 740.17 


105 012.8 


6 168.C73 


15 


54 471.0 


1 240 025,0 


16 234.05 


58 


8 204.44 


97 172.64 


4 078.406 


10 


55 104.2 


1 101 653.4 


14 810.17 


50 


7 801.83 


88 008.20 


4 705.206 


17 


52 S32.0 


1 136 440.2 


14 402.30 


00 


7 351.65 


81 100.38 


4 008.020 


18 


GO 063.0 


1 033 G10.2 


14 000.83 


01 


013.44 


73 764.73 


4 410.322 


10 


48 562.8 


1 032 902.4 


13 031.08 


62 


480.75 


66841.28 


4 226.413 


20 


46 550,2 


084 300.6 


18 207.32 


63 


071.27 


60 364.64 


4 030.200 


21 


44 630,8 


037 843.4 


12 016.25 


64 


5 600.85 


64 283.27 


3 331.187 


22 


42 782.8 


803 212.6 


12 577.63 


65 


6 273.33 


48 010.41 


3 020.300 


23 


41 000.2 


850 420.0 


12 260.71 


GO 


4 800.66 


43 343.08 


3 424.843 


24 


39 307.1 


809 420.0 


11 035,38 


67 


4 518,05 


38 462.53 


3 218.321 


25 


37 073.6 


770 113.0 


11 031.14 


08 


4 167.82 


33 033.88 


3 010.290 


20 


30 106.1 


732 430.0 


11 337.50 


60 


3 808.32 


29 776.0(5 


2 SOl.SOfl 


27 


34 001.5 


600 333.8 


11 063.07 


70 


3 470.67 


25 067.74 


2 502.538 


28 


33 167.4 


001 732.4 


10 770.04 


71 


3 145.43 


22 497.07 


2384.657 


28 


31 771.3 


628 575.0 


10 515.18 


72 


2 833,42 


19 351.64 


2 170.018 


30 


30 440.8 


500 803.6 


10 250.02 


73 


2 535.75 


10 618.22 


1 077.107 


31 


20 103.5 


506 302.0 


10 011.17 


74 


2 253.57 


13 082.47 


1 780.731 


32 


27 937.5 


637 199.3 


9 771.375 


75 


1 087.87 


11 728.00 


1 501.240 


33 


26700.5 


609 261.8 


539.044 


76 


1 730.39 


741.028 


1 400.088 


31 


25 630.1 


482 501.3 


313.638 


77 


1 508.03 


8 001.633 


1 238.047 


35 


24544.7 


460 871.2 


004.065 


78 


1 205.73 


6 492.000 


1 070.158 


36 


23 502.6 


432 320.5 


8 882,708 


79 


1 100.65 


5 197.271 


024.803 7 


37 


22 501.4 


408 824.0 


8 070,415 


80 


023.338 


4 000.024 


784,804 


38 


21 630.7 


380 322.6 


8 475.068 


81 


703.234 


3 173,280 


655.024 5 


39 


20 015.5 


364 782.0 


8 270.860 


82 


620.405 


2 410.052 


538.005 7 


40 


10 727.4 


344 107.4 


8 088.915 


83 


404.995 


1 780.587 


434.477 


41 


18 873.0 


324 440.0 


7 902.231 


84 


386.641 


1 204.592 


342.862 4 


42 


18 052.0 


305 506.3 


7 710.738 


85 


204.610 


007.051 3 


203.005 


43 


17 203.0 


287 513,4 


7 540.910 


86 


217.598 


613.341 7 


100.850 


44 


10 504.4 


270 240.8 


7 365.480 


87 


154.383 


305.743 8 


141.000 3 


45 


15 773.0 


253 745,5 


7 102.800 


88 


103,903 


241.360 


05.801 07 


40 


15 070.0 


237 071.0 


7 022.083 


80 


65.023 1 


137.307 8 


00.070 82 


47 


14 302.1 


222 001.0 


fl 854.337 


90 


38.304 7 


71.774 70 


35.S77 52 


48 


13 738.5 


208 500,8 


087.406 


91 


20.180 


33.470 01 


10.056 00 


40 


13 107.0 


194 771.3 


6 621.410 


92 


0,11880 


13,283 00 


8.000 605 


50 


12408.0 


181 063.4 


355.436 


98 


3.222 30 


4.164 21 


3.081 545 


51 


11 000.6 


100 104.7 


6 180,012 


94 


0.827 Oil 


0.041 84 


.705 7G2 


&9 


11 330.5 


167 256.2 


021.006 


95 


0.114 232 


0.114 23 


.110 300 



79 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


if 


Vi 


VlOn 


1/n 


n 


R> 


V^ 


VlOn 


l/n 


1.00 


1.0000 


1.00000 


3.16228 


1.000000 


1.60 


2.2500 


1.22474 


3.87298 


.666667 


1.01 


1.0201 


1.00499 


3.17805 


.990099 


1.51 


2.2801 


1.22882 


3.S8587 


.662262 


1.02 


1.0404 


1.00095 


3.19374 


.080392 


1.52 


2.3104 


1.23288 


3.89872 


.657896 


1.03 


1.0009 


1.01489 


3.20936 


.970874 


1.53 


2.3409 


1.23003 


3.91152 


.653595 


1.04 


1.0S16 


1.01980 


3.22490 


.961538 


1.54 


2.3716 


1.24097 


3.02428 


.649351 


1.06 


1.1025 


1.02470 


3.24037 


.052381 


1.65 


2.4025 


1.24400 


3.03700 


.645161 


1.06 


1.1236 


1.02966 


3.25576 


.043306 


1.56 


2.4336 


1.24900 


3.04068 


.641020 


1.07 


1.1440 


1.03441 


3.27100 


.034579 


1.57 


2.4649 


1.26300 


3.00232 


.036943 


1.08 


1.1604 


1.03923 


3.28634 


.025920 


1.58 


2.4964 


1.25008 


3.07402 


.632011 


1.09 


1.1881 


1.04403 


3.30151 


.017431 


1.59 


2.5281 


1.26005 


3.08748 


.628031 


1.10 


1.2100 


1.04881 


3.31662 


.900001 


1.60 


2.6600 


1.20401 


4.00000 


.625000 


1.11 


1.2321 


1.05357 


3.33167 


.000901- 


1.61 


2.5921 


1.26886 


4.01248 


.621118 


1.12 


1.2544 


1.05830 


3.34664 


.892857 


1.62 


2.6244 


1.27279 


4.02402 


.617284 


1.13 


1.2769 


1.06301 


3.36155 


.884956 


1.63 


2.6569 


1.27071 


4.03733 


.613407 


1.14 


1.2906 


1.06771 


3.37639 


.877193 


1.64 


2.6806 


1.28062 


4.04060 


.600756 


1.15 


1.3225 


1.07238 


3.39116 


.860665 


1.65 


2,7225 


1.28452 


4.06202 


.606061 


1.16 


1.3456 


1.07703 


3.40588 


.862069 


1.66 


2.7550 


1.28841 


4.07431 


.602410 


1.17 


1.3689 


1.08167 


3.42053 


.854701 


1.67 


2.7889 


1.29228 


4.08650 


.508802 


1.18 


1.3924 


1.08628 


3.43511 


.847458 


1.68 


2,8224 


1.29616 


4.09878 


.505238 


1.10 


1.4161 


1.09087 


3.44964 


.840336 


1.69 


2.8561 


1.30000 


4.11006 


.501716 


1.20 


1.4400 


1.09545 


3.46410 


.833333 


1.70 


2.8900 


1.30384 


4.12311 


.588235 


1.21 


1.4041 


1.10000 


3.47851 


.826446 


1.71 


2.9241 


1.30707 


4.13621 


.584706 


1.22 


1.4884 


1.10454 


3.49285 


.819672 


1.72 


2.9584 


1.31149 


4.14729 


.581305 


1.23 


1.5129 


1.10905 


3.50714 


.813008 


1.73 


2.0029 


1.31529 


4.16033 


.578035 


1.24 


1.5376 


1.11355 


3.62136 


.806452 


1.74 


3.0276 


1.31909 


4.17133 


.574713 


1.25 


1.5625 


1.11803 


3.53653 


.800000 


1.76 


3.0625 


1.32288 


4.18330 


.671429 


1.26 


1.5876 


1.12260 


3.64966 


.793651 


1.76 


3.0976 


1.32666 


4.19624 


.568182 


1.27 


1.6129 


1.12694 


3.56371 


.787402 


1.77 


3.1320 


1.33041 


4.20714 


.564072 


1.28 


1.6384 


1.13137 


3.67771 


.781250 


1.78 


3.1684 


1.33417 


4.21900 


.561708 


1.20 


1.6641 


1.13578 


3.59166 


.776194 


1.79 


3.2041 


1.33791 


4.23084 


.568669 


1.30 


1.6900 


1.14018 


3.60555 


.769231 


1.80 


3.2400 


1.34164 


4.24264 


.555556 


131 


1.7161 


1.14455 


3.61939 


.763350 


1.81 


3.2761 


1.34536 


4.26441 


.552486 


1.32 


1.7424 


1.14891 


3.63318 


.757676 


1.82 


3.3124 


1.34907 


4.26615 


.540451 


1.33 


1.7689 


1.15326 


3.64692 


.751880 


1.83 


3.3489 


1.35277 


4.27786 


.546448 


1.34 


1.7956 


1.15768 


3.66060 


.746269 


1.84 


3.3856 


1.35647 


4,28052 


.643478. 


1.35 


1.8225 


1.16190 


3.67423 


.740741 


1.85 


3.4225 


1.36015 


4.30116 


..640541 


1.36 


1.8496 


1.16619 


3.68782 


.735294 


1.86 


3.4596 


1.36382 


4.31277 


.537634 


1.37 


1.8769 


1.17047 


3.70135 


.729927 


1.87 


3.4969 


1.36748 


4.82435 


.534750 


1.38 


1.0044 


1.17473 


3.71484 


.724638 


1.S8 


3.5344 


1.37113 


4.33590 


.531015 


1.30 


1.9321 


1.17898 


3.72827 


.719424 


1.89 


3.5721 


1.37477 


4.34741 


.529101 


1.40 


1.9600 


1.18322 


3.74166 


.714288 


1.90 


3.6100 


1.37840 


4.36890 


.526316 


1.41 


1.9881 


1.18743 


3.76600 


.700220 


1.91 


3.0481 


1.38203 


4.37036 


.523560 


1.42 


2.0164 


1.10164 


3.76829 


.704225 


1.Q2 


3.6864 


1.38564 


4.38178 


.620833 


1.43 


2.0449 


1.19583 


3.78153 


.699301 


1.03 


3.7249 


1.38024 


4.30318 


.518136 


1.44 


2.0736 


1.20000 


3.79473 


.694444 


1.04 


3.7636 


1.39284 


4.40464 


.515464 


1.45 


2:1025 


1.20416 


3.80780 


.689656 


1.95 


3.8025 


1.39642 


4.41588 


.612821 


1.46 


2.1316 


1.20830 


3.82099 


.684932 


1.96 


3.8416 


1.40000 


4.42719 


,510204 


1.47 


2.1609 


1.21244 


3.83406 


.680272 


1.07 


3.8809 


1.40357 


4,43847 


.507614 


1.48 


2.1004 


1.21655 


3.84708 


.675676 


1,08 


3.0204 


1.40712 


4.44972 


, .505051 


1.49 


2.2201 


1.22066 


3.86005 


.671141 


1.09 


3.9601 


1.41067 


4.46094 


.502513 


1.SO 


2J2500 


1.22474 


3.87298 


.666667 


9,00 


4,0000 


1.41421 


4.47214 


.600000 


n 


rt> 


v 


Vlon 


Vn 


n 


n 


V 


VfiF/i 


l/n 



80 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


n 


V 


vlon 


l/ 


n 


n? 


VS 


VlOn 


1/n 


2.00 


4.0000 


1.41421 


4.47214 


.600000 


2.50 


6.2500 


1.68114 


6.00000 


.400000 


2.01 


4.0401 


1.41774 


4.48330 


.407512 


2.51 


6.3001 


1.68430 


6.00090 


.398400 


2.02 


4.0804 


1.42127 


4.40444 


.406060 


2.52 


6.3604 


1.68746 


5.01090 


.396825 


2.03 


4.1200 


1.42478 


4.50555 


.402611 


2.53 


6.4000 


1.50060 


5.02091 


.396257 


2.04 


4.1616 


1.42820 


4.51664- 


.400106 


2.64 


6.4516 


1.50374 


5.03084 


.303701 


2.05 


4.2025 


1.43178 


4.52760 


.487805 


2.55 


6.5025 


1.50087 


5.04075 


.392157 


2.06 


4.2436 


1.43527 


4.53872 


.485437 


2.50 


0.6536 


1.00000 


5.05064 


.390025 


2.07 


4.2840 


1.43875 


4.54073 


.483002 


2.57 


6.0049 


1.60312 


5.00052 


.380105 


2.08 


4.3264 


1.44222 


4.50070 


.480760 


2.68 


6.0504 


1,60024 


5.07937 


.387597 


2.00 


4.3681 


1.44568 


4.67165 


.478400 


2.50 


6.7081 


1.60935 


6.08020 


.386100 


2.10 


4.4100 


1.44914 


4.68258 


.470190 


2.00 


0.7000 


1,61245 


6.00002 


.384015 


2.11 


4.4521 


1.45258 


4.50347 


.473034 


2,61 


0.8121 


1.61555 


6.10882 


.383142 


2.12 


4.4044 


1.45602 


4.60435 


.471698 


2.62' 


6.8644 


1.61804 


5.11850 


.381679 


2.13 


4.5369 


1.46045 


4.61610 


.400434 


2.63 


6.0160 


1.62173 


5.12835 


.380228 


2.14 


4.5706 


1.40287 


4.62601 


.407290 


2.04 


6.0606 


1.02481 


6.13800 


.378788 


2.15 


4.6225 


1.46620 


4.63081 


.486118 


2.65 


7.0225 


1.62788 


5.14782 


.377368 


2.10 


4.6656 


1.46060 


4.04758 


.432963 


2.66 


7.0750 


1.03095 


5.15752 


.375040 


2.17 


4.7080 


1.47300 


4.66833 


.400820 


2.07 


7.1289 


1.03401 


6.10720 


.374532 


2.18 


4.7524 


1.47048 


4.00905 


.458710 


2.G8 


7.1824 


1.63707 


6.17687 


.373134 


2.10 


4.7061 


1.47086 


4.67974 


.456621 


2.60 


7.2301 


1.64012 


5.18652 


.371747 


2.20 


4.8400 


1.48324 


4.60042 


.454545 


2.70 


7.2900 


1.64317 


6.10015 


.370370 


2.21 


4.8841 


1.48661 


4.70108 


.452489 


2.71 


7.3441 


1.04021 


5.20577 


.360004 


2.22 


4.0284 


1.48097 


4.71169 


.450450 


2.72 


7.3984 


1.64024 


5.21530 


.367047 


2.23 


4.0720 


1.40332 


4.72220 


.448430 


7.73 


7,4529 


1.05227 


5.22404 


.300300 


2.24 


5.0176 


1.40660 


4.73286 


.446429 


2.74 


7.5070 


1.05520 


5.23450 


.364064 


2.25 


5.0625 


1.50000 


4.74342 


.444444 


2.75 


7.6625 


1.66831 


5.24404 


.303036 


2.26 


5.1070 


1.50333 


4.75305 


.442478 


2.76 


7.0170 


1.66132 


5.26357 


.302310 


2.27 


5.1520 


1.50605 


4.70445 


.440529 


2.77 


7.0729 


1.60433 


5.20308 


.301011 


2.28 


5.1084 


1.60007 


4.77403 


.438500 


2.78 


7.7284 


1.66733 


5.27257 


.350712 


2.20 


5.2441 


1.51327 


4.78530 


.430681 


2.70 


7.7841 


1.67033 


5.28205 


.358423 


2.80 


5.2000 


1.51068 


4.70583 


.434783 


3.80 


7.8400 


1.67332 


5.20150 


.357143 


2.31 


5.3301 


1.51087 


4.80625 


.432000 


2.81 


7.8901 


1.07031 


5.30004 


.365872 


2.32 


5.3824 


1.62315 


4.81664 


.431034 


2.82 


7.9524 


1.67920 


6.31037 


,354010 


2.33 


5.4280 


1.52043 


4.82701 


.429185 


2.83 


8.0089 


1.68226 


5.31077 


.353367 


2.34 


5.4760 


1.62071 


4.83735 


.427350 


2.84 


8.0060 


1.08523 


6.32017 


.352113 


2.35 


5.5225 


1.53207 


4.84708 


.426632 


2.85 


8.1225 


1.08810 


5.33854 


.350877 


2.30 


6.5606 


1.63G23 


4.85708 


.423729 


2.86 


8,1796 


1.09116 


6.34700 


.349050 


2.37 


5.6100 


1.53048 


4.86826 


.421941 


2.87 


8.2369 


1.60411 


5.35724 


.348432 


2.38 


5.6644 


1.54272 


4.87852 


.420168 


2,88 


8.2044 


1.69700 


5.30056 


.347222 


2.30 


5.7121 


1.64500 


4.88876 


.418410 


2.80 


8.3621 


1.70000 


6.37687 


.346021 


2.10 


5.7000 


1.64019 


4.80808 


.410067 


2.00 


8.4100 


1.70204 


6.38516 


.344828 


2.41 


5.8081 


1.55242 


4.00918 


.414038 


2.01 


8,4681 


1.70587 


5.30444 


.343043 


2.42 


5.8564 


1.65563 


4.01935 


.413223 


2.02 


8.5204 


1.70880 


6.40370 


.342400 


2.43 


5,9049 


1.56885 


4.02050 


.411523 


2.93 


8.6849 


1.71172 


5.41295 


.341207 


2.44 


5.0536 


1.56205 


4.93004 


.400836 


2.94 


8.6430 


1.71404 


5.42218 


.340130 


2.45 


6.0026 


1.66525 


4,04075- 


.408163 


2.05 


8.7026 


1.71756 


5.43130 


.338983 


2.46 


6.0610 


1.50844 


4,05984 


.406504 


2.06 


8,7616 


1.72047 


6.44050 


.337838 


2.47 


6,1009 


1.67162 


4.06901 


.404858 


2.07 


8.8209 


1.72337 


5.44077 


.336700 


2.48 


6.1504 


1.67480 


4.97906 


.403220 


2.08 


8.8804 


1.72627 


5.45804 


.335570 


2.4Q 


6.2001 


1.67797 


4.08009 


.401600 


2.99 


8,0401 


1,72916 


6.40809 


.334448 


2.50 


6.2500 


1.68114 


6.00000 


.400000 


3.00 


0.0000 


1.73205 


6.47723 


.333333 


n 


If 


V^ 


ViOn 


1/n 


n 


rt> 


V^ 


Vlbln 


Vn 



81 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


ri> 


Vn" 


V5fn 


Vn 


n 


n 


Vn 


VlOn 


Vn 


3.00 


0.0000 


1.73205 


5.47723 


.333333 


3.50 


12.2500 


1.87083 


6.91008 


.286714 


3.01 


0.0601 


1.73404 


5.48635 


.332226 


3.51 


12.3201 


1.87360 


6.92453 


.284000 


3.02 


9.1204 


1.73781 


5,49545 


.331126 


3.52 


12.3904 


1.87017 


5.93206 


.284001 


3.03 


9.1809 


1.74060 


5.50454 


.330033 


3.63 


12.4609 


1.87883 


6.04138 


.283280 


3.04 


9.2416 


1.74356 


5.51362 


.328047 


3.54 


12.5316 


1.88140 


6.94070 


.282486 


3.05 


9.3025 


1.74642 


5.52268 


.327869 


3.55 


12.6026 


1.88414 


5.05810 


.281600 


3.06 


9.3636 


1.74029 


6.53173 


.326707 


3.56 


12.6730 


1.88680 


5.00657 


.280800 


3.07 


0.4240 


1.75214 


5.64076 


.325733 


3.57 


12.7440 


1.88044 


6.07406 


.280112 


3.08 


9.4864 


1.75400 


6.64977 


.324875 


3.58 


12.8164 


1.80200 


5.08331 


.270330 


3.00 


9.5481 


1.75784 


5.55878 


.323625 


3.50 


12.8881 


1.80473 


5.00100 


.278552 


3.10 


9.6100 


1.76068 


5.56776 


.322581 


3.00 


12.9000 


1.80737 


0.00000 


.277778 


3.11 


9.6721 


1.76362 


5.57674 


.321543 


3.01 


13.0321 


1.00000 


0.00833 


.277008 


3.12 


0.7344 


1.76636 


6.58570 


.320513 


3.62 


13.1044 


1.90203 


0.01064 


.276243 


3.13 


0.7969 


1.76918 


5.50464 


.310480 


3.03 


13.1700 


1.00526 


6.02405 


.276482 


3.14 


9.8506 


1.77200 


5,60357 


.318471 


3.64 


13.2400 


1.00788 


6.03324 


.274725 


3.15 


9.0225 


1.77482 


5.61249 


.317460 


3.65 


13.3226 


1.91050 


0.04152 


.273973 


3.10 


9.0856 


1.77764 


5.62130 


,316456 


3.66 


13.3950 


1.01311 


6.04070 


.273224 


3.17 


10.0489 


1.78045 


5.63028 


.315467 


3.67 


13.4689 


1.91672 


6.05805 


.272480 


3.18 


10.1124 


1.78326 


5.63015 


.314465 


3.68 


13.5424 


1.91833 


0.00030 


,271731) 


3.19 


10.1701 


1.78606 


5.64801 


.313480 


3.60 


13.6161 


1.02004 


6.07454 


.271003 


3.20 


10.2400 


1.78885 


6.65685 


.312500 


3.70 


13.6000 


1.02354 


6.08276 


.270270 


3.21 


10.3041 


1.70165 


5.66560 


.311526 


3.71 


13.7641 


1.02014 


6.00008 


,260542 


3.22 


10.3684 


1.79444 


5.67450 


.310659 


3.72 


13.8384 


1.92873 


6.00018 


.268817 


3.23 


10.4329 


1.7Q722 


6.68331 


.309598 


3.73 


13,9120 


1.03132 


6.10737 


.208007 


3.24 


10.4976 


1.80000 


6.69210 


.308042 


3.74 


13.9876 


1.03301 


6.11555 


.287380 






















3.25 


10.5626 


1.80278 


5.70088 


.307092 


3.75 


14.0625 


1.93640 


6.12372 


.200067 


3.26 


10.6276 


1.80555 


5.70964 


.308748 


3.76 


14.1376 


1.03007 


6.13188 


.266057 


3.27 


10.6929 


1.80831 


6.71839 


.305810 


3.77 


14.2129 


1.94105 


6.14003 


.265252 


3.28 


10.7584 


1.81108 


5.72713 


.304878 


3.78 


14.2884 


1.94422 


6.14817 


.264650 


3.20 


10.8241 


1.81384 


6.73685 


,303951 


3.79 


14.3641 


1.94079 


0.15630 


.263852 


3.30 


10.8900 


1.81650 


5,74456 


.303030 


3.80 


14.4400 


1.04936 


6.16441 


.263158 


3.31 


10,0561 


1,81034 


6.75326 


.302115 


3.81 


14.6161 


1.96102 


0.17252 


.262467 


3.32 


11.0224 


1.82209 


5.76104 


.301205 


3.82 


14.5024 


1.95448 


0.18081 


.261780 


3.33 


11.0880 


1.82483 


6.77002 


.300300 


3.83 


14.6680 


1.05704 


6.18870 


.261007 


3.34 


11,1556 


1.82757 


6.77027 


.209401 


3.84 


14.7450 


1.05060 


6.19677 


.260417 


3.35 


11.2225 


1.83030 


5.78702 


.298507 


3.85 


14.8225 


1.06214 


6.20484 


.250740 


3.36 


11.2896 


1.83303 


5.79655 


.207619 


3.86 


14.8006 


1.06460 


6.21280 


.250067 


3.37 


11.3560 


1,83576 


6.80517 


.206736 


3.87 


14.9769 


1.06723 


6.22003 


.258308 


3.38 


11.4244 


1.83848 


5.81378 


.295858 


3.88 


15.0544 


1.06077 


6.22896 


.257732 


3.39 


11.4921 


1.84120 


5.82237 


.294985 


3.89 


15.1321 


1.07231 


0.23000 


.257000 


3.40 


11.5600 


1.84391 


5.83005 


.294118 


3.00 


16.2100 


1.07484 


0.24500 


.266410 


3.41 


11.6281 


1.84662 


5.83052 


.293255 


3.01 


15,2881 


1.97737 


6.25300 


.255754 


3.42 


11.6064 


1.84932 


5.84808 


.292308 


3.02 


15.3064 


1.07900 


6.20000 


.255102 


4.43 


11.7640 


1.85203 


5.85662 


.291545 


3.93 


16.4449 


1.98242 


6.28897 


.264453 


3.44 


11.8336 


1.85472 


5.86615 


.200608 


3.94 


16.6236 


1.98494 


6.27694 


.253807 


3,45 


11.9025 


1.85742 


5.87367 


.289855 


3.95 


15.6026 


1.08746 


6.28400 


.263165 


3.46 


11.9718 


1.86011 


5,88218 


.280017 


3.90 


16.6816 


1.98097 


6.20285 


.252525 


3.47 


12.0409 


1.86270 


6.89067 


.288184 


3.97 


15.7609 


1.99249 


0.30070 


.261880 


3.48 


12.1104 


1.86548 


6.89015 


.287356 


3.98 


15.8408 


1.99400 


6.30872 


.261256 


; 3,49 


12,1801 


1.86815 


6.00762 


,286533 


3.99 


15,9201 


1.99750 


6.31064 


.260027 


3.50 


12.2500 


1.87083 


6.01608 


.285714 


4.00 


16.0000 


2.00000 


6.32456 


.250000 


'. n 


n 


Vn" 


Vl0n 


Vn 


n 


rt> 


Vn 


Vion 


Vn 



82 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



It 


if 


V^ 


VSTn 


Vn 


n 


a 


V^ 


VlOn 


Vn 


4.00 


lfi.0000 


2.00000 


6.32450 


.250000 


4.50 


20.2500 


2.12132 


0.70820 


.222222 


4.01 


10.0801 


2.00250 


0.33240 


.249377 


4.61 


20.3401 


2.12368 


6.71665 


.221729 


4.02 


10.1004 


2.00400 


6.34035 


.248750 


4.52 


20.4304 


2.12603 


6.72309 


.221230 


4.03 


10.2400 


2.00749 


6.34823 


.248130 


4.53 


20.5200 


2.12838 


0.73063 


.220751 


4.04 


10.3210 


2.00008 


0.35610 


.247525 


4.54 


20.0116 


2.13073 


6.73796 


.220264 


4.05 


10.4025 


2.01240 


0.36300 


.240014 


4.56 


20.7025 


2.13307 


6.74537 


.21078O 


4.00 


10.4830 


2.01404 


0.37181 


.240305 


4.50 


20.7030 


2.13542 


6.76278 


.210298 


4.07 


10.6040 


2.01742 


0.37000 


.245700 


4.57 


20.8840 


2:13776 


6.70018 


.218818 


4,08 


10.0404 


2.01000 


0,38740 


.245008 


4.58 


20.9764 


2.14000 


6.70767 


218341 


4.00 


10.7281 


2.02237 


0,30531 


.244400 


4.50 


21.0681 


2.14243 


6.77496 


.217866 


4.10 


10.8100 


2.02485 


0.40312 


.243002 


4.GO 


21.1000 


2.14476 


6.78233 


.217391 


4.11 


10.8021 


2.02731 


0.41003 


.243300 


4.01 


21.2521 


2.14709 


6.78970 


.21692O 


4.12 


10.0744 


2.02978 


0.41872 


.242718 


4.02 


21.3444 


2.14042 


6.79706 


.21645O 


4.13 


17.0500 


2.03224 


6.42051 


.242131 


4.03 


2i:4300 


2.16174 


6.80441 


.215983 


4.14 


17.1390 


2,03470 


6.43428 


.241540 


4.04 


21.6206 


2.15407 


6.81176 


.215617 


4.15 


17.2225 


2.03715 


6.44205 


.240004 


4.05 


21.6225 


2.15630 


0.81009 


.215054 


4,10 


17.30flO 


2.03001 


0.44081 


.240385 


4.06 


21.7166 


2.15870 


6.82642 


.214592 


4.17 


17.3880 


2.04200 


0.45755 


.230808 


4.07 


21.8080 


2.16102 


6.83374 


.214133 


4.18 


17.4724 


2.04450 


G.40520 


.230234 


4.08 


21.9024 


2.16333 


6.84105 


.213675 


4,10 


17.5501 


2.04605 


6.47302 


.238063 


4.00 


21.9961 


2.16604 


6.84836 


.213220 


4.20 


17.0400 


2.04030 


0.48074 


.238005 


4.70 


22.0000 


2.16795 


6.85665 


.212766 


4.21 


17,7241 


2.05183 


6.48845 


.237630 


4.71 


22.1841 


2.17025 


6.86294 


.212314 


4.22 


17.8084 


2.05426 


6.40615 


.230067 


4.72 


22.2784 


2.17256 


6.87023 


.211864 


4.23 


17.8920 


2.05070 


6.50384 


.230407 


4.73 


22.3720 


2.17486 


6.87750 


.211416 


4.24 


17.0770 


2.05013 


6.51153 


.235840 


4.74 


22.4076 


2.17715 


6.88477 


.21097O 


4.25 


18.0025 


2.00155 


6.51020 


.235204 


4.75 


22.5025 


2.17945 


6.89202 


.210526 


4.20 


18.1470 


2.00308 


6.62687 


.234742 


4.70 


22.0676 


2.18174 


6.89928 


.210084 


4.27 


18.2320 


2.00040 


6.5U52 


,234102 


4.77 


22.7629 


2.18403 


6.90652 


.209644 


4.28 


18.3184 


2.0G882 


6.S4217 


.233045 


4.78 


22.8484 


2.18632 


6,01376 


.2092O5 


4.20 


18.4041 


2.07123 


6.54081 


.233100 


4.79 


22.9441 


2.18801 


6.92008 


.208768 








V 














4.30 


18.4000 


2.07304 


6.55744 


.232558 


4.80 


23.0400 


2.10080 


6.92820 


.208333 


4.31 


18.5701 


2.07005 


0.50506 


.232010 


4.81 


23.1301 


2.19317 


6.93642 


.2070OO 


4.32 


18.0024 


2.07840 


8.67207 


.231481 


4.82 


23.2324 


2.19545 


0.94262 


.207469 


4.33 


18.7480 


2.08087 


6.C8027 


.230047 


4.83 


23.3280 


2.10773 


6.04982 


.207039 


4.34 


18.8356 


2.08327 


6.58787 


.230415 


4.84 


23.4250 


2.20000 


6.95701 


.206612 


4.35 


18.0225 


2.08507 


0.60545 


.220885 


4.85 


23.5226 


2.20227 


0.96410 


.206186 


4.30 


10.0006 


2.08806 


6.60303 


.220358 


4.80 


23,0100 


2,20454 


6.97137 


.205761 


4.37 


10.0060 


2.00045 


0,01000 


.228833 


4.87 


23.7109 


2.20081 


6.97864 


.206339 


4,38 


10.1844 


2.00284 


6.01810 


.228311 


4.88 


23.8144 


2.20907 


6.08670 


,204918 


4.30 


10.2721 


2.00523 


0.82571 


.227700 


4.80 


23.9121 


2,21133 


6.90285 


.204490 


4.40 


10.3600 


2,00762 


6.63325 


.227273 


4.90 


24,0100 


2.21360 


7.00000 


.204082 


4.41 


10.4481 


2.10000 


6.04078 


.226757 


4.01 


24.1081 


2,21585 


7.00714 


.203666 


4.42 


10.5304 


2.10238 


fl.64831 


.220244 


4.02 


24.2064 


2.21811 


7.01427 


.203252 


4.43 


10.0240 


2.10470 


0.65582 


.226734 


4.03 


24.3049 


2.22036 


7,02140 


.202840 


4.44 


10.7136 


2.10713 


0.00333 


.225225 


4.04 


24.4086 


2.22261 


7.02851 


.202429 


4.45 


10.8025 


2.10050 


0.07083 


.224710 


4.05 


24.6026 


2.22486 


7.03662 


.202020 


4.4U 


10.8010 


2.11187 


0.67832 


.224215 


4.00 


24.0016 


2.22711 


7.04273 


.201613 


4.47 


10.0800 


2.11424 


0.68B81 


.223714 


4.07 


24.7000 


2.22935 


7,04982 


.201207 


4.48 


20.0704 


2.11000 


0.00328 


.223214 


4.08 


24,8004 


2,23159 


7-06691 


.200803 


4.40 


20.1601 


2.11896 


0.70070 


.222717 


4.00 


24.0001 


2.23383 


7.06399 


,200401 


4.50 


20.2500 


2.12132 


0.70820 


.222222 


5.00 


26.0000 


2,23607 


7,07107 


.200000 


n 


n 


vS 


VlOn 


1/n 


n 


7 


V 


VlSn 


Vn 



83 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


n 


^ 


Vlfln 


Vn 


n 


n> 


^ 


ViOn 


Vn 


5.00 


25.0000 


2.23607 


7.07107 


.200000 


5.50 


30.2500 


2.34521 


7.41620 


181818 


6.01 


25.1001 


2.23830 


7.07814 


.190601 


5.61 


30.3801 


2.34734 


7.42294 


181488 


5.02 


25.2004 


2.24054 


7.08520 


.100203 


5.52 


30.4704 


2.34047 


7.42067 


181150 


5.03 


25.3000 


2.24277 


7.09225 


.198807 


5.53 


30.5809 


2.35100 


7.43640 


180832 


5.04 


25.4016 


2.24490 


7.00930 


.108413 


6.54 


30.6916 


2.35372 


7.44312 


.180505 


6.05 


25.5026 


2.24722 


7.10034 


.108020 


5.55 


30.8025 


2.35584 


7.44983 


.180180 


5.06 


25.6036 


2.24944 


7.11337 


.107628 


5.56 


30.9136 


2.36707 


7.45054 


.179856 


5.07 


25.7040 


2.26167 


7.12030 


.107239 


5.57 


31.0249 


2.30008 


7.40324 


.179533 


6.08 


25.8064 


2.25389 


7.12741 


.196850 


5.58 


31.1364 


2.36220 


7.40994 


.170211 


5.09 


25.0081 


2.25610 


7.13442 


.196464 


5.50 


31.2481 


2.36432 


7.47003 


.178801 


5.10 


20.0100 


2.25832 


7.14143 


.106078 


5.00 


31.3600 


2.30643 


7.48331 


.178571 


5.11 


26.1121 


2.26053 


7.14843 


.195095 


5.61 


31.4721 


2.36854 


7.48999 


.178253 


5.12 


20.2144 


2.26274 


7.15542 


.195312 


5.02 


31.5844 


2.37065 


7.49667 


.177936 


6 13 


28.3160 


2 26405 


7.16240 


.104032 


5.63 


31.0000 


2.37276 


7.50333 


.177020 


6.14 


26.4106 


2.26716 


7.16938 


.104553 


5.64 


31.8096 


2.37487 


7.60999 


.177306 


5.15 
6.10 
5.17 
6.18 
5.19 


26.5225 
26.6266 
26.7280 
26.8324 
26.0361 


2.26936 
2.27166 
2.27376 
2.27696 
2.27816 


7.17636 
7.18331 
7.10027 
7.19722 
7.20417 


.194175 
.193798 
.193424 
.193050 
.192078 


5.06 
5.00 
5.67 
5.08 
5.69 


31.0225 
32.0350 
32.1489 
32.2024 
32.3761 


2.37607 
2.37908 
2.38118 
2.38328 
2.38537 


7.51Q65 
7.52330 
7.52994 
7.53858 
7.64321 


.178991 
.178678 
.176387 
.176056 
.175747 


5.20 

5.21 
5.22 
5.23 
5.24 


27.0400 
27.1441 
27.2484 
17.3620 
27.4576 


2.28035 
2.28254 
2.28473 
2.28692 
2.28910 


7.21110 
7.21803 
7.22400 
7.23187 
7.23878 


.192308 
.191939 
.101571 
.191205 
.190840 


5.70 

5.71 
6.72 
5.73 
5.74 


32.4900 
32.6041 
32.7184 
32.8329 
32.0470 


2.38747 
2.38056 
2.30105 
2.39374 
2.39583 


7.64983 
7.55645 
7.6Q307 
7.50968 
7.57628 


.175439 
.17C131 
,174825 
.174620 
.174216 


5.26 
5.26 
6.27 
5.28 
5.29 


27.6025 
27.6676 
27.7729 
27.8784 
27.0841 


2.20120 
2.29347 
2.29565 
2.29783 
2.30000 


7.24600 
7.26259 
7.25048 
7.26636 
7.27324 


.100476 
.190114 
.180753 
.180304 
.180036 


5.75 
5.76 
5.77 
5.78 
6.79 


33.0625 
33.1770 
33.2020 
33.4084 
33.5241 


2.39702 
2.40000 
2.40208 
2.40410 
2.40624 


7.58288 
7.68947 
7.59005 
7.60263 
7.60920 


.173013 
.173611 
.173310 
.173010 
.172712 


5.30 

5.31 
5.32 
5.33 
5.34 


28.0000 
28.1001 
28.3024 
28.4080 
28.5150 


2.30217 
2.30434 
2.30651 
2.30808 
2.31084 


7.28011 
7.28697 
7.20383 
7.30068 
7.30753 


.188079 
.188324 
.187070 
.187017 
.187266 


5.80 

5.81 
5.82 
5.83 
5.84 


33.6400 
33.7561 
33.8724 
33.0880 
34.1050 


2.40832 
2.41039 
2.41247 
2.41454 
2.41661 


7.01577 
7.62234 
7.02889 
7.63644 
7.04199 


.172414 
.172117 
.171821 
.171527 
.171233 


5.35 
5.30 
6,37 
5.38 
5.30 


28.6225 
28.7206 
28.8360 
28.0444 
20.0521 


2.31301 
2.31517 
2.31733 
2.31048 
2.32164 


7.31437 
7.32120 
7.32803 
7.33486 
7.34100 


.186016 
.180567 
.136220 
.185874 
.185529 


5.85 
5.86 
5.87 
5.88 
5.89 


34.2225 
34.3306 
34.4560 
34.5744 
34.6021 


2.41868 
2.42074 
2.42281 
2.42487 
2.42003 


7.04863 
7.0S500 
7.00169 
7.00812 
7.67403 


.170940 
.170040 
.170358 
.170008 
.160770 


5.40 

5.41 
6.42 
5.43 
5.44 


20.1600 
20.2081 
29.3764 
20.4840 
20.5036 


2.32379 
2.32594 
2.32800 
2.33024 
2.33238 


7.34847 
7.35527 
7.36200 
7.30886 
7.37604 


.185185 
.184843 
.184502 
,184102 
.183824 


5.90 

6.01 
5.02 
5.03 
5.04 


34.8100 
34.9281 
35.0464 
35.1040 
35.2836 


2.42899 
2.43105 
2.43311 
2.43510 
2.43721 


7.08115 
7.08765 
7.00416 
7.70006 
7.70714 


.160402 
.100205 
.108019 
.168034 
.168350 


5.45 
5.40 
5.47 
5.48 
5.40 


20.7026 
20.8116 
20.0200 
30.0304 
30.1401 


2.33452 
2.33060 
2.33880 
2.34004 
2.34307 


7.38241 
7.38918 
7.39594 
7.40270 
7.40045 


.183480 
.183150 
.182815 
.182482 
.182149 


5.06 
5.06 
5.07 
5.08 
5.00 


35.4025 
35.5216 
35.6400 
35.7604 
35.8801 


2.43020 
2.44131 
2.44336 
2.44540 
2.44745 


7.71362 
7.72010 
7.72858 
7,73305 
7.73051 


.168067 
.107785 
.107504 
.107224 
.160045 


5.50 


30.2500 


2.34521 


7.41620 


.181818 


0.00 


30.0000 


2.44040 


7.74697 


.160607 


n 


n* 


V^ 


VlOn 


1/n 


n 


n 


V^ 


VlOn 


1/n 



84 



ABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



* 


v; 


VBT 


1/n 


n 


71" 


A 


VlOn 


1/n 


uo.oooo 


2.44040 


7.74B07 


.100007 


G.50 


42.2600 


2.54051 


8.00226 


.163846 


M0.1201 


2.451 fi3 


7.75242 


.160380 


0.51 


42.3801 


2.55147 


8.00846 


.153610 


.'HI.2404 


2.45H. r .7 


7.75887 


.106113 


6.52 


42.5104 


2.55343 


8.07465 


.163374 


30.3000 


2.455(11 


7.70531 


.106837 


6.53 


42.0400 


2.55539 


8.08084 


.163139 


30.4810 


2.45704 


7.77174 


.100501) 


6.64 


42.7710 


2.55734 


8.08703 


.152005 


30.0025 


2.45007 


7.77817 


.105280 


0.55 


42.0025 


2.55030 


8.09321 


.152072 


30.723(1 


2.40171 


7.78460 


.106017 


0.50 


43.0330 


2.50125 


8.00938 


'. 152439 


:ill.H14l) 


2.40374 


7.70102 


.104745 


0.57 


43.1040 


2.50320 


8.10555 


.152207 


30.0004 


2.40577 


7.70744 


.104474 


0.58 


43.2004 


2.50515 


8.11172 


151076 


37.0881 


2.40770 


7,80385 


.104204 


0.50 


43.4281 


2.50710 


8.11788 


.161745 


37.2100 


2.40082 


7.81026 


.103034 


0.60 


43.5000 


2.60005 


8.12404 


.151515 


37.3321 


2.471 H4 


7.81006 


.103060 


0.01 


43.0021 


2.67000 


8.13010 


.151286 


37.4544 


2.47:18(1 


7.82304 


.103300 


0.02 


43.8244 


2.57294 


8.13634 


.151067 


37.6700 


2.475H8 




.103132 


0.03 


43.0560 


2.57488 


8.14248 


.160830 


37.0000 


2.47700 


7'.83682 


.102800 


0.04 


44,0800 


2.57082 


8.14802 


.160602 


H7.8225 


2.47002 


7,84210 


.102(502 


0.05 


44,2225 


2.57870 


8.15475 


.160370 


37,045(1 


2.4X103 


7.84857 


.102338 


0.00 


44.355U 


2.58070 


8.10088 


.150160 


38.00KD 


2.4K305 


7.85403 


.102075 


0.07 


44.4880 


2.58263 


8.10701 


.149925 


38.1024 


2.48.100 


7.80130 


.101812 


O.OS 


44.0224 


2.58-157 


8.17313 


.140701 


38.3101 


2.48707 


7.80700 


.101551 


0.00 


4-1.7601 


2.58060 


8.17824 


.149477 


38.4400 


2.48008 


7.87401 


.101200 


6.70 


44.8000 


2.58844 


8.18635 


.149264 


3K.5041 


2.41)100 


7.88030 


.101031 


0.71 


45.0241 


2.50037 


8.10146 


.149031 


38.0884 


2.40300 


7.8S070 


. .100772 


0,72 


45.1684 


2.50230 


8.19758 


.148810 


38.8120 


2.40000 


7.80303 


.100514 


0.73 


45.2020 


2,60422 


8.20306 


.148588 


3S.0370 


2.40800 


7.80037 


,100250 


0.74 


45.4270 


2.50015 


8.20075 


.148368 


30.0025 


2.50000 


7.00fiOO 


.100000 


0.76 


45.6025 


2.50808 


8.21684 


.148148 


30.1870 


2.60200 


7.01202 


.160744 


0.70 


45.6070 


2.00000 


8.22192 


.147929 


iiO.3120 


2.5O400 


7.91&13 


.150400 


0.77 


45.8320 


2.00192 


8.22800 


.147710 


30 4384 


2.60.WO 


7.02406 


.159230 


0.78 


45.0084 


2.00384 


8.23408 


.147493 


8!SQ41 


2.50700 


7.03005 


.158983 


0.70 


40.1041 


2.00070 


8.24016 


.147276, 


30.0000 


2.50008 


7.03726 


.168730 


6.80 


46.2400 


2.00708 


8.24021 


.147050 


30.8101 


2.51107 


7.94.'155 


.158470 


0.81 


40.3761 


2.00000 


8.25227 


.140843 


30.0424 


2.51.-IOO 


7.0408-1 


,158228 


6.82 


40.6124 


2.01151 


8.25833 


.146028 


40.0080 


2.61fiOfi 


7.05013 


.157078 


0.83 


46,0480 


2.01348 


8.20438 


.146413 


40.105G 


2.51704 


7.0(1241 


,157720 


6.84 


40.7866 


2.01634 


8.27043 


.146109 


40.3225 


2.61002 


7.00809 


.157480 


0.85 


40.0225 


2.01725 


8.27047 


.145085 


40.4400 




7.07400 


.157233 


0.80 


47.0500 


2.01010 


8.28251 


.146773 


40.5700 


rt j|O'JU() 


7.08123 


.150080 


0,87 


47.1000 


2.02107 


8.28856 


.145560 


40.7044 


o '*" I |U7 


7.0H740 


.150740 


0.88 


47.3,'J44 


2.02208 


8.20458 


.146349 


40.8321 


2.527K4 


7.00375 


.150405 


0.80 


47.4721 


2.02488 


8.30060 


.145138 


40.0000 


2.52082 


8.00000 


.150250 


6.00 


47.0100 


2.02079 


8.30662 


.144928 


41.08K1 


2.531 HO 


8.00025 


.150000 


0.01 


47.7481 


2.02800 


8.31264 


.144718 


41.2154 


2.6UH77 


8.01240 


.155703 


0.02 


47,8804 


2.03050 


8.31806 


.144600 


41.3440 


2 63fi74 


8.01 873 


.155521 


0.03 


48.0240 


2,63240 


8.32406 


.144300 


41.4730 


2i6772 


8.02400 


.165280 


0.04 


48.1030 


2.03439 


8.33067 


.144092 


41.0025 


2 53001) 


8,03110 


.155030 


0,06 


48.3025 


2.03020 


8.33067 


.143885 


41.7310 


2>)41n 


8.03741 


.154700 


0.00 


48.4410 


2.03818 


8.34200 


.143678 


41.8000 




8.04303 


.154600 


0.07 


48.5800 


2.04008 


8.34805 


.143472 


41.0004 


B^45.TH 


8.04084 


.154321 


O.OS 


48.7204 


2,04107 


8.36404 


.143266 


42.1201 


2.54708 


8.05(106 


.154083 


6.00 


48.8001 


2,04880 


8.36002 


.143062 


42.2500 


2.54001 


8,00220 


.153846 


7.00 


40.0000 


2.04675 


8.30660 


.142857 


n' 


Vn 


vB5 


1/n 


n 


tfi 


vS 


vlon 


Vn 



85 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


**\ 


_ 

V " 


N/lOn 


1/n 


n 


na 


v 


VlOn 


1/n 


7.00 




*&*&_' 


8.36060 


142857 


7.50 


6.2500 


2.73801 


8.66025 


133333 


7.01 




IIWKD4 


8.37257 


142653 


7.51 


6.4001 


2.74044 


8.60603 


133166 


7.02 




''.QffX 8.37854 


142460 


7.52 


6.5504 


2.74226 


8.67179 


132979 


7.03 


9.4209 


irifcyBL 


B.38451 


142248 


7.53 


56.7009 


2.74408 


8.07756 


132802 


7.04 


9.5G1G 


2)Q5^m 


JL39047 


142046 


7.54 


6.8510 


2.74591 


8.68332 


132026 


7.05 


49.7025 


2.GrilC s 


^.30843 


141844 


7.55 


67.0026 


2.74773 


8.68907 


132450 


7.00 


49.8430 


2.65707 


8.41238 


141643 


7.56 


57.1530 


2.74955 


8.69483 


132275 


7.07 


49.0849 




8.41*33 


141443 


7.57 


57.3049 


2.75130 


8.70057 


.132100 


7.08 


60.1204 


2.60093 


SA&7 


.141243 


7.58 


57.4504 


2.75318 


8.70032 


.131920 


7.09 


60.2081 


2.60271 A 


,832021 


.141044 


7.59 


67.0081 


2.75500 


8.71206 


.131752 


7.10 


60.4100 


2.60458 


8.42615 


.140845 


7.60 


67.7000 


2.75081 


8.71780 


.131579 


7.11 


60.5521 


2.60646 


8.43208 


.140047 


7.61 


67.9121 


2.75802 


8.72363 


.131400 


7.12 


60.6944 


2.60833 


8.43801 


.140449 


7.62 


58.0644 


2.70043 


8.72920 


.131234 


7.13 


60.8369 


2.67021 


8.44393 


.140252 


7.63 


58.2109 


2.76226 


8.73499 


.131002 


7.14 


60.9796 


2.67208 


8.44985 


.140050 


7.64 


68.3090 


2.70405 


8.74071 


.130890 


7.16 


61.1226 


2.67306 


8.46577 


.139860 


7.65 


68.6225 


2.70580 


8.74043 


.130719 


7.10 


51.2056 


2.67682 


8.40108 


.139065 


7.06 


68.0750 


2.70707 


8.75214 


.130548 


7.17 


51.40S9 


2.07769 


8.40759 


.139470 


7.67 


68.8289 


2.76948 


8.75786 


.130378 


7.18 


51.5524 


2.67955 


8.47349 


.139276 


7.68 


58.9824 


2.77128 


8.70356 


.130208 


7.19 


51.6961 


2.68142 


8.47939 


.139082 


7.69 


69.1361 


2.77308 


8.70920 , 


.130039 


7.20 


51.8400 


2.68328 


8.48528 


.138889 


7.70 


59.2900 


2.77489 


8.77490 


.129870 


7.21 


51.9841 


2.68514 


8.49117 


.138690 


7.71 


59.4441 


2.77669 


8.78000 


.129702 


7.22 


52.1284 


2.68701 


8.49700 


.138604 


7.72 


59.5984 


2.77849 


8.78036 


.129534 


7.23 


52.2729 


2.68887 


8.50294 


.138313 


7.73 


69.7529 


2.78029 


8.79204 


.129366 


7.24 


52.4170 


2.69072 


8.50882 


.138122 


7.74 


59/9076 


2.78209 


8.79773 


.129199 


7.26 


52.5026 


2.69258 


8.51469 


.137931 


7.75 


60.0625 


2.78388 


8.80341 


.129032 


7.26 


52.7076 


2.69444 


8.52050 


.137741 


7.76 


60.2170 


2.78568 


8.80909 


.128800 


7.27 


52.8529 


2.69629 


8.52643 


.137562 


7.77 


60.3729 


2.78747 


8.81470 


.128700 


7.28 


52,9984 


2.69815 


8.53229 


.137363 


7.78 


00.6284 


2.78927 


8.82043 


.128635 


7.29 


63.1441 


2.70000 


8.53815 


.137174 


7.79 


00.6841 


2.79100 


8.82010 


.128370 


7.30 


63.2900 


2.70185 


8.54400 


.136986 


7.80 


00.8400 


2.79285 


8.83170 


.128205 


7.31 


53.4361 


2.70370 


8.54985 


.136799 


7.81 


00.9901 


2.79464 


8.83742 


.128041 


7.32 


53.5824 


2.70655 


8.56570 


.136612 


7.82 


61.1624 


2.79043 


8.84308 


.127877 


7.33 


53.7289 


2.70740 


8.56154 


.136426 


7.83 


61.3089 


2.79821 


8.84873 


.127714 


7.34 


53.8750 


2.70924 


8.50738 


.136240 


7.84 


61.4666 


2.80000 


8.85438 


.127651 


7.36 


54.0225 


2.71109 


8.57321 


.136054 


7.86 


61.6226 


2.80179 


8.80002 


.127389 


7.30 


54.1600 


2.71293 


8.57904 


.135870 


7.86 


61.7796 


2,80357 


8.80500 


.127220 


7.37 


54.3169 


2.71477 


8.58487 


.135685 


7.87 


01.9309 


2.80535 


8.87130 


.127005 


7.38 


54.4644 


2.71602 


8.59009 


.135601 


7.88 


62.0944 


2.80713 


8.87694 


.126904 


7.39 


54.612 


2.71846 


8.59651 


.135318 


7.89 


62.252 


2.80891 


8.88257 


.120743 


7.40 


54.760 


2.72029 


8.60233 


.135135 


7.0 


62.4100 


2.81069 


8.S8819 


.120582 


7.41 


54.00S 


2.72213 


8.60814 


.134953 


7.91 


02.568 


2.81247 


8.89382 


.126422 


7.42 


55.0564 


2.72307 


8.61394 


.13477 


7.92 


62.7204 


2.81425 


8.89944 


.126263 


7.43 


55.204 


2.72580 


8.61974 


.134590 


7.93 


02.884 


2.81603 


8.90506 


-.126103 


7.44 


55.353 


2.72764 


8.02554 


.13440 


7.94 


63.043 


2.81780 


8.91067 


.125945 


7.46 


55.502 


2.72947 


8.63134 


.13422 


7.96 


63.202 


2.81967 


8.91628 


.125780 


7.40 


55.651 


2.73130 


8.63713 


.13404 


7.96 


93.361 


2.82135 


8.92188 


.125628 


7.47 


66.800 


2.73313 


8.64292 


.13386 


7.97 


63.520 


2.82312 


8.92749 


.125471 


7.48 


55.0504 


2.73496 


8.64870 


.13369 


7.98 


63.6804 


2.82489 


8.93308 


.125313 


7.40 


56.100 


2.73679 


8.65448 


.13351 


7.99 


03.840 


2.82066 


8.93868 


.126150 


7.50 


56.250 


2.73861 


8.60025 


.13333 


8.00 


64.000 


2.82843 


8.04427 


.125000 


n 


n 


V^ 


v^On 


.1/n 


n 


n 


V^ 


VlOn 


1/n 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


n a 


v 


VlOn 


1/n 


n 


n* 


v^ 


VlOn 


1/n 


8.00 

8.01 
8.02 
8.03 
8.04 


04.0000 
04.1001 
04.3204 
04.4800 
04.0410 


2.82843 
2.83019 
2.83100 
2.83373 
2.83549 


8.04427 
8.04080 
8.05545 
8.00103 
8.06000 


.125000 
.124844 
.124088 
.124533 
.124378 


8.50 

8.51 
8.52 
S.53 
8.54 


72.2500" 
72.4201 
72.6004 
72.7009 
72.9310 


2.01548 
2.01719'. 
2.01890- 
2.02002 
2.02233 


9.21954 
0.22/07 
9.23038 
0.23580 
9.24121 


.117647 
.1J7609 
.117371 
.117233 
.117000 


8.05 
8.00 
8.07 
8.08 
8.09 


04.8025 
04.0030 
05.1249 
05.2804 
,05.4481 


2.83725 
2.83901 
2.84077 
2.84253 
2.84429 


8.07218 
8.97775 
8.08332 
8.08SSS 
8.00444 


.124224 
.124000 
.123910 
.123702 
.123009 


8.55 
8.56 
8.57 
8.58 
8.50 


73.1025 
73.2730 
73.4440 
73.0104 
73.7881 


2.92404 
2.92C76 
2.92746 
2.02910 
2.03087 


0.24602 
9.25203 
0.25743 
0.20283 
0.20823 


.110050 
.110822 
.110080 
.110550 
.116414 


8.10 

8.11 
8.12 
8.13 
8.14 


05.0100 
05.7721 
05.9344 
00.0900 
06.2590 


2.84005 
2.84781 
2.84050 
2.85132 
2.85307 


0.00000 
0.00555 
0.01110 
0.01006 
0.02210 


.123457 
.123305 
.123153 
.123001 
.122850 


8.60 

8.01 
8.02 
8.03 
8.04 


73.0000 
74.1321 
74.3044 
74.4709 
74.0490 


2.03258 
2.03428 
2.93508 
2.03700 
2.93039 


0.27362 
0.27901 
0.28440 
0.2897S 
9.20510 


.116270 
.110144 
.110000 
.115875 
.115741 


8.1S 
8.10 
8.17 
8.18 
8.10 


00.4225 
00.5850 
00.7480 
00.0124 
07.0701 


2.85482 
2.85657 
2.85832 
2.80007 
2.80182 


9.02774 
9.03327 
0.03881 
0.04434 
0.04080 


.122009 
.122540 
.122300 
.122240 
.122100 


S.05 
8.00 
8.07 
8.08 
8.00 


74.8225 
74.0050 
75.1080 
75.3424 
75.6101 


2.94109 
2.94279 
2.04449 
2.94018 
2.94788 


0.30054 
0.30591 
0.31128 
9.31605 
0.32202 


.115607 
.115473 
.115340 
.115207 
.116076 


8.90 

8.21 
8.22 
8.23 
8.24 


07.2400 
07.4041 
Q7.S084 
07.7329 
07.8070 


2.80350 
2.80531 
2.80706 
2.80880 
2.87054 


0.0/3530 
0.00001 
0.00042 
9.07103 
0.07744 


.121951 
.121803 
.121055 
.121507 
.121359 


8.70 

8.71 
8.72 
8.73 
8,74 


75.0000 
75.8041 
70.0384 
70.2129 
76.3870 


2.94058 
2.95127 
2.05200 
2.05400 
2.05035 


0.32738 
9.33274 
0.33800 
0.34345 
9.34880 


.114043 
.114811 
.114679 
.114548 
.114410 


8.25 
8.20 
8.27 
8.28 
8.29 


08.0625 
08.2270 
08.3920 
68.5584 
68.7241 


2.87228 
2.87402 
2.87570 
2.87750 
.2.87024 


0.08205 
0.08845 
9.00305 
9.00945 
0.10404 


.121212 
.121005 
.120010 
.120773 
.120027 


8.75 
8.70 
8.77 
8.78 
8.70 


70.5025 
70.7370 
76.0129 
77.0884 
77.2041 


2.95804 
2.05973 
2.00142 
2.06311 
2.00479 


9.35414 
9.35940 
9.30483 
0.37017 
9.37560 


.114280 
.114165 
.114025 
.113806 
.113700 


8.30 

8.31 
8.32 
8.33 
8.34 


68.8000 
00.0501 
00.2224 
09.3880 
00.5556 


2.88007 
2.88271 
2.88444 
2.88017 
2.88701 


0.11043 
fl.11502. 
9,12140 
0.12088 
0.13230 


.120482 
.120337 
.120102 
.120048 
.110004 


8.80 

8.81 
8.82 
8.83 
8.84 


77.4400 
77.8101 
77,7024 
77.0089 
78.1456 


2.90048 
2.00810 
2.00086 
2.07163 
2.07321 


0.38083 
0.38010 
0.30140 
0.30081 
0.40213 


.113030 
.113507 
.113370 
.113250 
.113122 


8.35 
8.30 
8.37 
8.38 
8.30 


09.7225 
09,8800 
70.0500 
70.2244 
70.3921 


2.88904 
2.80137 
2.80310 
2.80482 
2.80055 


0.13783 
0.14330 
0.14877 
0.15423 
9.15000 


.110700 
.110017 
.110474 
.110332 
.110100 


8.85 
8.80 
8.87 
8.88 
8.80 


78.3226 
78.4000 
78.0700 
78.8544 
70,0321 


2.07480 
2.07658 
2.07825 
2.07903 
2,08101 


0.40744 
0.41270 
0.41807 
0.42338 
0.42808 


.112094 
.112807 
.112740 
.112613 
.112480 


8.40 
8.41 
8,42 
8.43 
8.44 


70.5000 
70.7281 
70.8064 
71.0040 
71.2330 


2.80828 
2.90000 
2.00172 
2.00345 
2.00517 


0.10,11/5 
0.17001 
0.17000 
0.18150 
0.18006 


.110048 
.118000 
.118705 
.118024 
.118483 


8.00 

8.01 
8.02 
8.03 
8.04 


70.2100 
70.3881 
70.5004 
70.74-10 
70.0230 


2.08320 
2.08-100 
2.08004 
2.08831 
2.08098 


0.43398 
9.43028 
0.44468 
0.44087 
0.45510 


.112300 
.112233 
.112108 
.111082 
.111857 


8.45 
8.40 
8.47 
8.48 
8.40 


71.4025 
71.5710 
71.7409 
71.9104 
72.0801 


2,00080 
2.00801 
2.01033 
2.01204 
2.91370 


0.10230 
0.10783 
0,20320 
0.20800 
0.21412 


.118343 
.118203 
.118004 
.117926 
.117786 


8.0i> 
8.00 
8.07 
8.08 
8.00 


80.1025 
80.2810 
80,4000 
80.0404 
80.8201 


2,00100 
2.90333 
2.99500 
2,00006 
2,00833 


9.40044 
0.40673 
9.47101 
0.47620 
0,48160 


.111732 
.111007 
.111483 
.111369 
.111236 


8.50 


72.2500 


2.01548 


0.21054 


.117047 


0.00 


81.0000 


3.00000 


0.48083 


.111111 


n 


n 


V 


VlOn 


t/n 


n 


n 


Vn- 


VlOn 


Vn 



87 



TABLE XV SQUARES SQUARE ROOTS RECIPROCALS 



n 


n> 


v 


vTOn 


1/n 


n 


n 


V/z 


N/lOn 


1/n 


9.00 

9.01 
0.02 
9.03 
0.04 


81.0000 
81.1801 
81.3604 
81.5409 
81.7216 


3.00000 
3.00167 
3.00333 
3.00500 
3.00666 


9.48083 
9.49210 
9.49737 
9.50263 
9.50789 


.111111 
.110088 
.110865 
.110742 
.110010 


9.50 

0.51 
0.52 
0.53 
9.54 


90.2500 
90.4401 
90.6304 
00.8209 
91.0116 


3.08221 
3.08383 
3.08545 
3.08707 
3.08860 


0.74079 
0.76192 
9.75705 
0.70217 
0.76720 


.106263 
.105162 
.106042 
.104932 
.104822 


0.05 
9.06 
9.07 
0.08 
0.00 


81.9025 
82.0836 
82.2640 
82.44G4 
82.6281 


3.00832 
3.00998 
3.01164 
3.01330 
3.01406 


9.51315 
9.51840 
0.62365 
9.52890 
0.53415 


.110497 
.110376 
.110254 
.110132 
.110011 


9.55 
9.56 
9.57 
9.58 
9.50 


91.2025 
91.3930 
01.6840 
91.7704 
91.0681 


3.09031 
3.00102 
3.09354 
3.00616 
3.09677 


0.77241 
0.77753 
0.78264 
0.78776 
0.79285 


.104712 
.104003 
.104493 
.104384 
.104275 


9.10 

0.11 
0.12 
0.13 
9.14 


82.8100 
82.9021 
83.1744 
83.3560 
83.5396 


3.01662 
3.01828 
3.01993 
3.02150 
3.02324 


9.53939 
9.54463 
9.54087 
9.56510 
9,56033 


.109890 
,109769 
.109649 
.109520 
.109400 


9.60 

9.61 
9.62 
9.63 
9.64 


02.1600 
02.3521 
92.5441 
02.7369 
02.0206 


3.00830 
3.10000 
3.10161 
3.10322 
3.10483 


0.70790 
9.80306 
0.80816 
0.81326 
0.81835 . 


.104167 
.104058 
.103050 
.103842 
.103734 


0.15 
0.16 
0.17 
0.18 
0.10 


83.7225 
83.9056 
84.0889 
84.2724 
84.4561 


3.02490 
3.02655 
3.02820 
3.029&5 
3.03150 


9.56556 
9.67079 
9.57001 
9.58123 
0.68645 


.100200 
.100170 
.109061 
.108032 
.108814 


9.65 
9.66 
9.07 
9.68 
9.69 


03.1225 
03.3150 
03.5080 
03.7024 
93.8961 


3.10644 
3.10805 
3.10966 
3.11127 
3.11288 


0.82344 
0.82853 
0.83362 
0.83870 
0.84378 


.103627 
.103520 
.103413 
.103306 
.103100 


9.20 

0.21 
0.22 
0.23 
0.24 


84.6400 
84.8241 
85.0084 
85.1929 
85.3776 


3.03315 
3.03480 
3.03645 
3.03809 
3.03974 


9.50166 
9.50687 
9.60208 
0.60729 
9.61249 


.108606 
.108578 
.108400 
.108342 
.108225 


9.70 

9.71 
9.72 
9.73 
9.74 


94.0900 
04.2841 
04.4784 
94.0729 
94.8676 


3.11448 
3.11609 
3.11700 
3.11020 
3.12000 


0.84886 
9.85393 
0.85001 
0.80408 
0.86014 


.103003 
.102087 
102881 
.102775 
.102669 


0.25 
9.26 
9.27 
9.28 
9.29 


85.5625 
85.7476 
85.9320 
86.1184 
86.3041 


3.04138 
3.04302 
3.04467 
3.04631 
3,04795 


9.61769 
9.62289 
9.62808 
9.63328 
9.63846 


.108108 
.107001 
.107876 
.107750 
.107643 


9.75 
9.78 
9.77 
9.78 
9.79 


95.0025 
95.2576 
95.4620 
95.6484 
95.8441 


3.12250 
3.12410 
3.12570 
3.12730 
3.12800 


0.87421 
9.87027 
0.88433 
0.88039 
9.89444 


.102564 
.102450 
.102364 
.102240 
.102145 


9.30 

9.31 
0.32 
0.33 
0.34 


86.4900 
86,6761 
86.8624 
87.0489 
87.2356 


3.04060 
3.05123 
3.05287 
3.05450 
3.05614 


9.64365 
9.64883 
9.65401 
9.65919 
9.66437 


.107527 
.107411 
.107206 
.107181 
.107060 


9.80 

9.81 
9.82 
0.83 
0.84 


90.0400 
96.2361 
06.4324 
06.0289 
06.8256 


3.13050 
3.13209 
3.13369 
3.13528 
3.13088 


9.80049. 
9.00454 
0.00050 
0.01464 
0.01008 


.102041 
.101037 
.101833 
,101720 
.101020 


9.35 
9.36 
9.37 
9.38 
9.39 


87.4225 
87.6090 
87.7909 
87.9844 
88.1721 


3.05778 
3.05041 
3.06105 
3.00268 
3.06431 


0.66954 
9.67471 
0,67988 
9.68504 
0.60020 


.106052 
.100838 
.100724 
,100010 
.100406 


0.85 
0.86 
9.87 
0.83 
9.89 


07.0226 
97.2106 
97.4109 
97.6144 
97,8121 


3.13847 
3.14006 
3.14166 
3.14326 
3.14484 


0.92472 
0.02075 
0.03470 
0.93082 
0.04485 


.101523 
.101420 
.101317 
.101216 
.101112 


9.40 

9.41 
9.42 
0.43 
0.44 


88.3600 
88.5481 
88.7364 
88.9243 
89.1136 


3.06504 
3.00757 
3.00920 
3.07083 
3.07246 


0.09536 
0.70052 
0.70567 
0.71082 
0.71597 


.100383 
.100270 
.100167 
.100045 
.105032 


9.90 

9.91 
0.92 
0.03 
9.94 


98.0100 
08.2081 
08.4064 
98.0040 
98.8036 


3.14643 
3.14802 
3.14000 
3,16110 
3.16278 


0.94087 
9.95490 
9.95992 
0.00404 
9.96096 


.101010 
.100008 
.100800 
.100705 
.100604 


0.45 
9.46 
9.47 
9.48 
.9.49 


89.3026 
89.4916 
89.6809 
89.8704 
90.0601 


3.07409 
3.07571 
3.07734 
3.07896 
3.08058 


9,72111 
9.72625 
9.73139 
9.73653 
0.74166 


,105820 
.106708 
.105607 
.405486 
.105374 


9.95 
9.06 
9,97 
9.98 
9.00 


99.002C 
00.2016 
09.4000 
90,0004 
00.8001 


3.15430 
3,15595 
3.15763 
3.15011 
3.16070 


0.97497 
0.07008 
9.98499 
0.08000 
9.99500 


.100503 
.100402 
400301 
.100200 
.100100 


9.50 


90.2500 


3.08221 


9.74670 


.105263 


10.00 


100.000 


3.16228 


10.0000 


.100000 


n 


n* 


V 


v'lOn 


1/n 


n 


n> 


V^ 


VWn 


1/n 



88 



13841 

156