<f)> 00 ^
OU 164763
FINANCIAL
MATHEMATICS
BY
CLARENCE H. RICHARDSON, PH.D.
Professor of Mathematics, Bucknell University
AND
ISAIAH LESLIE MILLER
Late Professor of Mathematics, South Dakota State College
of Agriculture and Mechanic Arts
NEW YORK
D. VAN NOSTRAND COMPANY, INC.
250 FOURTH AVENUE
1946
COPY RIGHT, 1946
BY
D. VAN NOSTHAND COMPANY, INC.
All Rights Reserved
Thin book, or any parts thereof, may not be
reproduced in any form without written per-
mission from the authors and the publishers.
Based on Business \fathematics, I. L. Miller, copyright 1935; second edition copyright 1939;
and Commercial Algebra and Mathematics of Finance, I. L. Miller and C. H. Richardson,
copyright 1939 by D. Van Nostrand Company, Inc.
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
This text is designed for a three-hour, one-year course for students
who desire a knowledge of the mathematics of modern business and
finance. While the vocational aspects of the subject should be especially
attractive to students of commerce and business administration, yet an
understanding of the topics that are considered interest, discount, an-
nuities, bond valuation, depreciation, insurance may well be desirable
information for the educated layman.
To live intelligently in this complex age requires more than a super-
ficial knowledge of the topics to which we have just alluded, and it is pal-
pably absurd to contend that the knowledge of interest, discount, bonds,
and insurance that one acquires in school arithmetic is sufficient to under-
stand modern finance. Try as one may, one cannot escape questions of
finance. The real issue is: shall we deal with them with understanding and
effectiveness or with superficiality and ineffectiveness?
While this text presupposes a knowledge of elementary algebra, we
have listed for the student's convenience, page x, a page of important
formulas from Miller and Richardson, Algebra: Commercial Statistical
that should be adequate for the well-prepared student. Although we make
frequent reference to this Algebra in this text on Financial Mathematics,
the necessary formulas are found in this reference list.
In the writing of this text the general student and not the pure mathe-
matician has been kept constantly in mind. The text includes those tech-
niques and artifices that many years of experience in teaching the subject
have proved to be pedagogically fruitful. Some general features may be
enumerated here: (1) The illustrative examples are numerous and are
worked out in detail, many of them having been solved by more than one
method in order that the student may compare the respective methods of
attack. (2) Line diagrams, valuable in the analysis and presentation of
problem material, have been given emphasis. (3) Summaries of important
formulas occur at strategic points. (4) The exercises and problems are nu-
frierous, and they are purposely selected to show the applications of the
theory to the many fields of activity. These exercises and problems are
abundant, and no class will hope to do more than half of them. (5) Sets
iv Preface
of review problems are found at the ends of the chapters and the end of the
book.
A few special features have also been included: (1) Interest and dis-
count have been treated with unusual care, the similarities and differences
having been pointed out with detail. (2) The treatment of annuities is
pedagogical and logical. This treatment has been made purposely flexible
so that, if it is desired, the applications may be made to depend upon two
general formulas. No new formulas are developed for the solution of
problems involving annuities due and deferred annuities, and these special
annuities are analyzed in terms of ordinary annuities. (3) The discussion
of probability and its application to insurance is more extended than that
found in many texts.
In this edition we are including Answers to the exercises and problems.
While we have exercised great care in the preparation of this book, it
is too much to expect that it is entirely free from errors. For the notifica-
tion of such errors, we shall be truly grateful.
C. H. RICHARDSON.
Buckneli University,
Lewisburg, Pennsylvania,
1946.
CONTENTS
CHAPTER I, SIMPLE INTEREST AND DISCOUNT .
ART. PAGE
1. Interest 1
2. Simple Interest Relations 1
3. Ordinary and Exact Interest 3
4. Methods of Counting Time 4
5. The Six Per Cent Method of Computing Ordinary Interest 7
6. Present Value and True Discount 9
7. Bank Discount 12
8. Summary and Extension 15
9. Comparison of Simple Interest and Simple Discount Hates 17
10. Rates of Interest Corresponding to Certain Discount Rates in the Terms of
Settlement 20
11. Exchanging Debts 22
12. To Find the Date When the Various Sums (Debts) Due at Different Times
May Be Paid in One Sum 25
13. To Find the Equated Date of an Account 28
CHAPTER II, COMPOUND INTEREST AND COMPOUND
DISCOUNT
14. Compound Interest 35
15. Compound Interest Formula 36
16. Nominal and Effective Rates of Interest 38
17. Present Value at Compound Interest 42
18. Other Problems Solved by the Compound Interest Formulas 45
19. Equation of Value 48
20. Equated Time 50
21. Compound Discount at a Discount Rate 53
22. Summary of Interest and Discount 54
CHAPTER III, ANNUITIES CERTAIN
23. Definitions 57
24. Amount of an Annuity 58
25. Present Value of an Annuity 63
26. Relation between and 66
0rfl 8 n\
27. Summary. Formulas of an Ordinary Annuity of Annual Rent R Payable
Annually for n Years 67
28. Other Derivations of a^i and 8*\ 67
v
vi Contents
AKT. PAGE
29. Amount of an Annuity, Where the Annual Rent, R, is Payable in p Equal
Installments 69
30. Present Value of an Annuity of Annual Rent, R, Payable in p Equal Install-
ments 78
31. Summary of Ordinary Annuity Formulas 79
32. Annuities Due 83
33. Deferred Annuities 89
34. Finding the Interest Rate of an Annuity 92
35. The Term of an Annuity 95
36. Finding the Periodic Payment 97
37. Perpetuities and Capitalized Cost 100
38. Increasing and Decreasing Annuities 105
CHAPTER IV, SINKING FUNDS AND AMORTIZATION
39. Sinking Funds Ill
40. Amortization Ill
41. Book Value 113
42. Amount in the Sinking Fund at Any Time 113
43. Amount Remaining Due After the kth Payment lias Been Made 114
44. The Amortization and Sinking Fund Methods Compared 116
45. Retirement of a Bonded Debt 118
CHAPTER V, DEPRECIATION
46. Definitions 122
47. Methods of Treating Depreciation 123
48. The Straight Line Method 123
49. Fixed-Percentage-on-Decreasing- Value Method 125
50. The Sinking Fund Method 128
51. The Unit Cost Method 130
52. Depreciation of Mining Property 134
53. Composite Life of a Plant 136
CHAPTER VI, VALUATION OF BONDS
54. Definitions 141
55. Purchase Price 141
56. Premium and Discount 144
57. Amortization of Premium and Accumulation of Discount 147
58. Bonds Purchased Between Dividend Dates 150
59. Annuity Bonds 152
60. Serial Bonds 153
61. Use of Bond Tables 154
62. Determining the Investment Rate When the Purchase Price of a Bond is
Given 155
Contents vii
CHAPTER VII, PROBABILITY AND ITS APPLICATION IN
LIFE INSURANCE BA/SW
ART. PAGE
63. The History of Probabilities 161
64. Meaning of a priori Probability 162
65. Relative Frequency. Empirical Probability 164
66. Permutations. Number of Permutations of Things All Different 165
67. Combinations. Number of Combinations of Things All Different 167
68. Some Elementary Theorems in Probability 169
69. Mathematical Expectation 172
70. Repeated Trials 173
71. Meaning of Mortality Table 176
72. Probabilities of Life 178
CHAPTER VIII, LIFE ANNUITIES
73. Pure Endowments 182
74. Whole Life Annuity 185
75. Present Value (Cost) of a Life Annuity 185
76. Life Annuity Due 186
77. Deferred Life Annuity 186
78. Temporary Life Annuity 187
79. Forborne Temporary Life Annuity Due 189
80. Summary of Formulas of Life Annuities. Examples 1 90
81. Annuities Payable m Times a Year 193
CHAPTER IX, LIFE INSURANCE, NET PREMIUMS
(SINGLE AND ANNUAL)
82. Definitions 198
83. Whole Life Policy 199
84. Term Insurance 202
85. Endowment Insurance 204
86. Annual Premium Payable by m Equal Installment/a 205
87. Summary of Formulas of Life Insurance Premiums 207
88. Combined Insurance and Annuity Policies 208
CHAPTER X, VALUATION OF POLICIES. RESERVES
89. Meaning of Reserves 212
90. Computing Reserves, Numerical Illustration 213
91. Fackler's Accumulation Formula 214
92. Prospective Method of Valuation 216
93. Retrospective Method of Valuation 218
CHAPTER XI, GROSS PREMIUMS, OTHER METHODS OF
VALUATION, POLICY OPTIONS AND PROVISIONS,
SURPLUS AND DIVIDENDS
94. Gross Premiums 221
95. Surplus and Dividends 222
viii Contents
ART. PAGE
96. Policy Options 223
97. Surrender or Loan Value 223
98. Extended Insurance 224
99. Paid-up Insurance 225
100. Preliminary Term Valuation 227
101. Modified Preliminary Term Valuation 231
102. Concluding Remarks 235
Review Problems 237
Tables T-I-1 T-XIII-77
Answers 245
Index 261
USEFUL FORMULAS
From Miller and Richardson, Algebra: Commercial Statistical *
I. Logarithms
1. If a* = N, logaN = x
2. Iog MN = log a M + log a N
M
3. log a = loga M - log a N
N
4. loga M* = N loga M
II. Arithmetical Progression
1. I = a + (n - l)d
2. S n = - (a +
3. S n = - [2a + (n -
2
III. Geometrical Progression
1. i = ar n ~ L
2.
3.
4.
a ar n
1 -r
a rl rl a
r - 1
when r < \ 1
1 r
IV. Binomial Theorem
(a + b) n = a n + n(
n(n - 1)
a n ~ 2 b 2
PAGE
46
47
47
47
84
84
84
87
Article 60 (8) 87
Article 60 (9) 87
89
n(n - \)(n - 2)- (n - r + 1)
-: il- : : a n ~ r b r
rl
b n 42
V. Summation
H
90
* Miller and Richardson, Algebra: Commercial Statistical, D. Van Nostrand Co.,
Inc., New York, N. Y.
CHAPTER I
SIMPLE INTEREST AND DISCOUNT
1. Interest. Interest is the sum received for the use of capital. Ordina-
rily, the interest and capital are expressed in terms of money. The capital
is referred to as the principal. To determine the proper amount of interest
to be received for the use of a certain principal, we must know the time
that the principal has been in use and the rate of interest that is being
charged. The rate of interest is the rate per unit of time that the lender
receives from the borrower for the use of the money. The rate of interest
may also be defined as the interest earned by one unit of principal in one
unit of time. The unit of time is almost invariably one year, and the unit
of principal one dollar. The sum of the principal and interest is defined
as the amount.
When interest is paid only on the principal lent, it is called simple
interest. In case the interest is periodically added to the principal, and
the interest in the following period is each time computed on this principal
thus formed by adding the interest of the previous period, then we speak
of the interest as being compounded, and the sum by which the original
principal is increased at the end of the time is called the compound interest.
In this chapter only simple interest calculations will be considered.
2. Simple interest relations. Simple interest on any principal is
obtained by multiplying together the numbers which stand for the prin-
cipal, the rate, and the time in years.
If we let P = the principal,
i = the rate of interest (in decimal form),
n = the time (in years),
/ = the interest,
and S = the amount,
it follows from the definitions of interest and amount that:
/ = Pm, (1)
and S = P + I. (2)
2 Financial Mathematics
From relations (1) and (2), we get
S = P + Pni = P(l + ni),
p -
Relations (1) and (2) involve five letters (values). If we know any
three of the values, the other two may be found by making use of these
relations. Let us illustrate by examples.
Example 1. Find the interest on $700 for 4 years at 5%. Find the
amount.
Solution. Substituting in (1) the values, P = 700, n = 4, t = 0.05,
we obtain
/ = 700-4-0.05 = $140.00, interest.
And S = 700 + 140 = $840.00, amount.
Example 2. A certain principal in 5 years, at 5%, amounts to $625. Find
the principal.
Solution. S = 625, n = 5, i = 0.05.
Substituting in (3), we have
- $5 '
Example 3. Find the rate if $500 earns $45 interest in 18 months.
Solution. Here, P = 500, / = 45, n = 1J.
From relation (1) we have,
7 45
Example 4. In what time will $300 earn $81 interest at 6%?
Solution. Here, P = 300, i = 0.06, / = 81.
From relation (1) we have,
81
= 4J^ years.
Pi 300(0.06)
Simple Interest and Discount 3
Exercises
1. Making use of relations (1) and (2), express S in terms of 7, n, and i.
2. Find the interest on $5,000 for 2J^ years at 5%. Find the amount.
3. Find the simple interest on $350 for 7 months at 6J/%*
4. In what time will $750 earn $56.25 interest, if the rate is 5%?
6. At 4J^%, what principal will amount to $925 in 3J^ years?
6. In what time will $2,500 amount to $2,981.25 at 3>%?
7. $2,400 amounts to $2,526 in 9 months. Find the rate.
8. What is the rate of interest when $2,500 earns $87.50 interest in 6 months?
9. What principal will earn $300 interest in 16 months, at 5%?
10. In what time will $305 amount to $344.65 at 4% interest?
11. What is the rate when $355 amounts to $396.42 in 2 years and 4 months?
12. What sum must be placed at interest at 4% to amount to $299.52 in 4 years
and 3 months?
13. A building that cost $7,500, rents for $62.50 a month. If insurance and repairs
amount to 1% each year, what is the net rate of interest earned on the investment?
14. If the interest on a certain sum for 4 months at 5% is $7.54, what is the sum?
15. What principal in 2 years and 5 months, will amount to $283.84, at 4J/%?
16. At age 60 a person wishes to retire and invests his entire estate in bonds that
pay 4% interest. This gives him a monthly income of $87.50. What is the size of his
estate?
3. Ordinary and exact interest. Most of the problems considered in
simple interest involve intervals of time measured in days or parts of a
year. The general practice is to calculate the interest for a fractional
part of a year on the basis of 360 days in a year (12 months of 30 days
each). When 360 days is used as the basis for our calculations, we have
what is called ordinary simple interest. When the exact number of days
between two dates is counted and 365 days to a year is used as the basis of
our calculations, we have what is known as exact simple interest.
If we let d = the time in days,
P = the principal,
i = the rate,
I = ordinary interest,
and I e = exact interest, it follows that:
~ 360'
and /. = (5)
4 Financial Mathematics
If we divide the members of (5) by the corresponding members of (4),
we have
Ie ^360^ 72
I ~~ 365 ~ 73'
72 1
73 ' ^ * ~~ 73 ^ *
We notice from (6) that the exact interest for any number of days is
7^3 times the ordinary interest, or, in other words, exact interest is 7 /73
less than ordinary interest. Hence, we may find the exact interest by first
computing the ordinary interest and then diminishing it by K 3 of itself.
Example. What is the ordinary interest on $500 at 5% for 90 days?
What is the exact interest?
Solution. Substituting in (4), we get
500- 90- 0.05
'< - 360
6.25 ^ 73 = 0.085+ .
Hence, I, = 6.25 - 0.09 = $6.16.
Thus the ordinary interest is $6.25 and the exact interest is $6.16.
The exact interest could have been computed by applying (5), but the
method used above is usually shorter, as will be seen after the reading of
Art. 5.
4. Methods of counting time. In finding the time between two dates
the exact number of days may be counted in each month, or the time may
be first found in months and days and then reduced to days, using 30 days
to a month.
Example 1. Find the time from March 5 to July 8.
Solution. By the first method the time is 125 days. By the second
method we get 4 months and 3 days or 123 days.
Either of these methods of computing time may be used where ordinary
Merest is desired, but when exact interest is required the exact time must be
employed. Use of the following table will greatly facilitate finding the
exact number of days between two dates.
Simple Interest and Discount
TABLE SHOWING THE NUMBER OF EACH DAY OF THE YEAR COUNTING FROM
JANUARY 1
1
>>
Q S
d
S
1
M
c3
s
i
8?
s
I
I-D
k>
-3
bb
-
-4-3
a
j
>
o
fc
i
1%
>>
3 S
1
1
32
60
91
121
152
182
213
244
274
305
335
1
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
5
5
36
64
95
125
156
186
217
248
278
309
339
5
6
6
37
65
96
126
157
187
218
249
279
310
340
6
7
7
38
66
97
127
158
188
219
250
280
311
341
7
8
8
39
67
98
128
159
189
220
251
281
312
342
8
9
9
40
68
99
129
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
346
12
13
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
319
349
15
16
16
47
75
106
136
167
197
228
259
289
320
350
16
17
17
48
76
107
137
168
198
229
260
290
321
351
17
18
18
49
77
108
138
169
199
230
261
291
322
352
18
19
19
50
78
109
139
170
200
231
262
292
323
353
19
20
20
51
79
110
140
171
201
232
263
293
324
354
20
21
21
52
80
111
141
172
202
233
264
294
325
355
21
22
22
53
81
112
142
173
203
234
265
295
326
356
22
23
23
54
82
113
143
174
204
235
266
296
327
357
23
24
24
55
83
114
144
175
205
236
267
297
328
358
24
25
25
56
84
115
145
176
206
237
268
298
329
359
25
26
26
57
85
116
146
177
207
238
269
299
330
360
26
27
27
58
86
117
147
178
208
239
270
300
331
361
27
28
28
59
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
303
334
364
30
31
31
90
151
212
243
304
365
31
NOTE. For leap years the number of the day is one greater than the tabular] number after
February 28.
6 Financial Mathematics
Example 2. Find the exact interest on $450 from March 20 to August
10 at 7%.
Solution. The exact time is 143 days.
Substituting in (5), we have
7 450-143-0.07
7 < = 365
Example 3. Find the ordinary interest in the above exercise.
Solution. Either 143 days (exact time) or 4 months and 20 days (140
days) may be used for the time when computing the ordinary interest.
Using 143 days and substituting in (4), we have
450-143-0.07
'' = 360
Using 140 days and substituting in (4), we have
450-140-0.07
Io ""
360
Either $12.51 or $12.25 is considered the correct ordinary simple
interest on the above amount from March 20 to August 10. The com-
putation of ordinary interest for the exact time is said to be done by the
Bankers 1 Rule.
Exercises
1, Find the ordinary and exact interest on the following:
a. $300 for 65 days at 6%.
b. $475.50 for 49 days at 5%.
c. $58.40 for 115 days at 7%.
d. $952.20 for 38 days at
2. Find the ordinary and exact interest on $2,400 at 8% from January 12 to April 6.
Find the ordinary interest first and then use (6) to determine the exact interest.
3. Find the exact interest on $350 from April 10 to September 5 at 7%.
4. Find the ordinary interest on $850 from March 8 to October 5 at 6%.
6. How long will it take $750 to yield $6.78 exact interest at 6%?
6. How long will it take $350 to yield $3.65 ordinary interest at 5%?
7. The exact interest on $450 for 70 days is $7.77. What is the rate?
Simple Interest and Discount 7
8. If the exact interest on a given principal is $14.40, find the ordinary interest for
the same period of time by making use of (6).
9. The ordinary interest on a certain sum is $21.90. Find the exact interest for the
same period of time.
10. What is the difference between the ordinary and exact interest on $2,560 at 6%
from May 5 to November 3?
11. The difference between the ordinary and exact interest on a certain sum is $0.40.
Find the exact interest on this sum.
5. The six per cent method of computing ordinary interest.
Ordinary simple interest may be easily computed by applying the methods
of multiples and aliquot parts.
If we consider a year as composed of 12 months of 30 days each (360
days),
at 6%, the interest on $1 for 1 year is $0.06,
at 6%, the interest on $1 for 2 mo. (60 days) is $0.01,
at 6%, the interest on $1 for 6 days is $0.001.
That is, to find the interest on any sum of money at 6% far 6 days, point
off three places in the principal sum; and for 60 days, point off two places
in the principal sum.
By applying the above rule we may find the ordinary interest on any
principal for any length of time at 6%. After the ordinary interest at 6%
is found, it is easy to find it for any other rate. Also, by applying (6),
Art. 3, the exact interest may be readily computed.
Example 1. What is the ordinary interest on $3,754 for 80 days at 6%?
Solution. $37. 54 = interest for 60 days
12.51 " " 20 " (H-60days)
$50.05 " " 80 days
Example 2. What is the ordinary simple interest on $475.25 for 115
days at 6%?
Solution. $4.753 = interest for 60 days
2.376= " " 30 " (H-60days)
1.584= " " 20 " (K-60days)
0.396= " " 5 " (K-20days)
$9.11 = interest for 115 days.
8 Financial Mathematics
Example 3. Compute the ordinary interest on $865 for 98 days at 8%.
Solution. $8. 65 = interest for 60 days at 6%
4.325 = " " 30 " " " (Why?)
0.865= " " 6 " " "
0.288 = " " 2 " " " (Why?)
$14. 128 = interest for 98 days at 6%
4.709 = " " " " " 2% (H-6%)
$18 . 84 = interest for 98 days at 8%. (Why?)
Example 4. Find the simple interest on $580 for 78 days at
Solution. $5.80 = interest for 60 days at 6%
1.45 = " " 15 " " "
0.29 " " 3 " " "
$7.54 = interest for 78 days at 6%
1.885= " " " "
$5. 66 = interest for 78 days at 4^%. (Why?)
Example 5. Find the exact simple interest on $2,500 for 95 days at 7%.
Solution. $25. 00 = interest for 60 days at 6%
12.50 = " " 30 " " "
2.08- " " 5 " " "
$39. 58 = interest for 95 days at 6%
6.60 = " " 95 " " 1%
$46. 18 = ordinary interest for 95 days at 7% (Why?)
0.63 = 46. 18* 73
$45.55 = exact interest for 95 days at 7%. (Why?)
Exercises
1. Find the interest at 6% on:
$900 for 50 days, $365.50 for 99 days, $750 for 70 days, $870.20 for 126 days.
2. Solve 1, if the rate is 7H%-
3. Find the exact interest at 6% on:
$650 from March 3 to July 17
$800 from February 10, 1944, to May 5, 1944
$2,000 from August 10 to December 5.
4. Solve 3, if the rate is 8%.
Simple Interest and Discount 9
6. A person borrowed $250 from a bank on July 5 and signed a 7% note due
November 20. On September 10 he paid the bank $100. What was the balance (includ-
ing interest) due on the note November 20? (Use exact time.)
6. Solve 5, if 30 days is counted to each month.
7. Solve 5, if exact interest is used,
8. What is the difference between the exact and ordinary interest on $1,250 from
March 10 to October 3 at 7%?
6. Present value and true discount. In Art. 2 we found the relation
between the principal, P, and the amount, S, to be expressed by the equa-
tions:
and P =
m
1 + HI
We may look upon P and S as equivalent values. That is, P, the value at
the beginning of the period, is equivalent to S at the end of the period, and
vice versa. The following line diagram emphasizes these ideas.
P n years at i% S
I - , - 1
interest rate
P = T ^. S = P(l + m)
l + ni
The quantity S is frequently called the accumulated value of P, and
P is called the present value of S. Thus, the present value of a sum S due
in n years is the principal P that will amount to Sinn years. The quantity
P is also called the discounted value of S due in n years. The difference
between S and P, S P, is called the discount on S as well as the interest
on P. To distinguish it from Bank Discount (Art. 7) this discount on S
at an interest rate i% is called the true discount on S. We thus have the
several terms for P and S:
P S
Principal Amount
Present value of S Accumulated value of P
Discounted value of S Maturity value of P
S P = Interest on P at interest rate i
Discount on S at interest rate i
10 Financial Mathematics
Example 1. Find the present value of a debt of $250 due in 6 months
if the interest rate is 6%. Find the true discount.
Solution. Here, S = 250, n = l / 2 , and i = 0.06.
Substituting these values in formula (3), we get
250 250
6) 08
S - P = 250 - 242.72 = $7.28, true discount.
Example 2. A non-interest bearing note for $3,500, dated May 2 was
due in 6 months. Assuming an interest rate of 7^% find the value of the
note as of July 5.
Solution.
May 2 + 6 months = November 2, due date.
From July 5 to November 2 = 120 days.
The present value of the maturity value as of July 5 (or for 120 days)
is required and S = 3,500, n = J, and i = 0.075.
The following line diagram exhibits graphically the important rela*-
tionships of the example.
_ P _ $3500
May2 JijlyS Nov. 2
I [- - 120 days - >|
I I
(< - 6 months - >|
Example 3. On May 2, A loaned B $3,500 for 6 months with interest at
6% and received from B a negotiable note. On July 5, A sold the note to C
to whom money was worth 7)4%. What did C pay A for the note?
Solution.
Interest on $3,500 for 6 months at 6% = $105.00.
$3,500 + $105.00 = $3,605, maturity value.
May 2 + 6 months = November 2, maturity date.
From July 5 to November 2 = 120 days.
The present value of the maturity value as of July 5 (or for 120 days)
is required and S = 3,605, n = y 3 and i = 0.075.
Simple Interest and Discount 11
the value of the note as of July 5.
$3500 P $3605
|y 2 July 5 NoV. 2
1:
-120
6 months -
The student will notice that in the solution of a problem of the above
type we first find the maturity value of the note or debt and then find the
present value of this maturity value as of the specified date.
Exercises
1. Accumulate (that is, find the accumulated value of) $2,000 for 2 years at 5%
simple interest.
2. Accumulate $300 for 8 months at 6% simple interest.
3. At 6% simple interest find the present value of $6,000 due at the end of 8 months.
What is the discount?
4. Discount (that is, find the discounted value of) $2,000 for 2 years at 5% simple
interest.
6. Discount $300 for 8 months at 6% simple interest.
6. Draw graphs of the following functions using n as the horizontal axis and 8 as
the vertical axis:
(a) S = 100(1 + O.OGn) = 100 + 6n.
(b) S - 100(1 + 0.04n) == 100 + 4n.
7. Mr. Smith buys a bill of goods from a manufacturer who asks him to pay $1,000
at the end of 60 days. If Mr. Smith wishes to pay immediately, what should the manu-
facturer be willing to accept if he is able to realize 6% on his investments?
8. Solve Exercise 7 under the assumption that the manufacturer can invest his money
at 8%. Compare the results of Exercises 7 and 8 and note how the present value is
affected by varying the interest rate.
9. I owe $1,500 due at the end of two years and am offered the privilege of paying a
smaller sum immediately. At which simple interest rate, 5% or 6%, would my creditor
prefer to compute the present value of my obligation?
10.
$1,000.00 Lewisburg, Penna.
June 1, 1944.
Six months after date I promise to pay X, or order, one thousand
dollars together with interest from date at 7%.
Signed, Y.
12 Financial Mathematics
(a) What is the maturity value of the note?
(b) If X sold the note to W, to whom money was worth 6%, four months after
date, what did W pay X for the note?
(c) What rate of interest did X earn on the loan?
11. Solve Exercise 10 under the assumption that money was worth 8% to W.
7. Bank discount. Bank discount is simple interest, calculated on the
maturity value of a note from the date of discount to the maturity .date, and is
paid in advance. If a bank lends an individual $100 on a six months'
note, and the rate of discount is 8%, the banker gives the individual $96
now and collects $100 when the note becomes due. If one wishes to dis-
count a note at a bank, the bank deducts from the maturity value of the
note the interest (bank discount) on the maturity value from the date of
discount to the date of maturity. The amount that is left after deducting
the bank discount is known as the proceeds. The time from the date of
discount to the maturity date is commonly known as the term of discount.
An additional charge is usually made by the bank when discounting paper
drawn on some out-of-town bank. This charge is known as exchange.
The bank discount plus the exchange charge gives the bank's total charge.
The maturity value minus the total charge gives the proceeds (when an
exchange charge is made).
The terms face of a note and maturity value of a note need to be
explained. The maturity value may or may not be the same as the face
value. If the note bears no interest they are the same, but if the note
bears interest the maturity value equals the face value increased by^the
interest on the note for the term of the note.
The discount, maturity value, rate of discount, proceeds (when no
exchange charge is made), and the term of discount are commonly repre-
sented by the letters D, S, d, P, and n, respectively. From the definitions
of bank discount and proceeds we may write
D Snd, (7)
and P - S - D = S - Snd = S(l - nd). (8)
When applying formulas (7) and (8) we must express n in years and
d in the decimal form.
The quantity P is frequently called the discounted value of S at the
given rate of discount, and P is called the present value of S. S is also
called the accumulated value of P. The difference between S and P, S P,
is called both the discount on S and the interest on P. In each instance
Simple Interest and Discount 13
the calculation is at the discount rate d. The relations are pictured by the
line diagram.
P n years at d% S
I 1
discount rate
P = s(l - nd) S =
1 -nd
S P = Interest on P at discount rate d
= Discount on S at discount rate d
Example 1. A six months' note, without interest, for $375, dated
May 6, was discounted August 1, at 6%. Find the proceeds.
Solution.
May 6 + 6 mo. = Nov. 6, due date.
From August 1 to Novt 6 = 97 days, term of discount.
Discount on $375 for 97 days == $6.07, bank discount.
$375 $6.07 = $368.93, proceeds.
Example 2. If the above note were a 5% interest-bearing note, what
would be the proceeds?
Solution.
May 6 + 6 mo. = Nov. 6, due date.
From August 1 to Nov. 6 = 97 days, term of discount.
Interest on $375 for 6 mo. at 5% = $9.38.
$375.00 + $9.38 = $384.38, maturity value.
Discount on $384.38 for 97 days at 6% = $6.21, bank discount.
$384.38 - $6.21 = $378.17, proceeds.
$376
$378.17
$384.38
May 6
1
Avjg. 1
(^
<
Nok 6
97 days ' >|
i
!
Example 3. Solve Example 2, if K% of the maturity value were
charged for exchange.
Solution.
May 6 + 6 mo. = Nov. 6, due date.
From August 1 to Nov. 6 = 97 days, term of discount.
Interest on $375 for 6 mo. at 5% = $9.38.
14 Financial Mathematics
$375.00 + $9.38 = $384.38, maturity value.
Discount on $384.38 for 97 days at 6% = $6.21, bank discount.
34% of $384.38 = $0.96, exchange charge.
$6.21 + $0.96 = $7.17, total charge made by the banker.
$384.38 - $7.17 = $377.21, proceeds.
Example 4.
$500.00 Lewisburg, Penna.
February 1, 1944.
Ninety days after date I promise to pay X, or order,
five hundred dollars together with interest from date at 6%.
Signed, Y.
On March 10, X sold the note to banker B who discounted the note at
8%. What proceeds did X receive for the note?
Solution.
90 days after Feb. 1, 1944 is May 1, 1944, the due date.
From March 10 to May 1 is 52 days, the term of discount.
The interest on $500 for 90 days at 6% = $7.50.
$500.00 + $7.50 = $507.50, the maturity value.
The discount on $507.50 for 52 days at 8% = $5.86, the bank
discount.
$507.50 - $5.86 = $501.64, the proceeds.
$500 $501.64 $507.50
FeTbTl March 10 Mdyl
days -
-90 days-
In the solution of the above examples, certain fundamental facts have
been used, which we now point out.
If the note is given for a certain number of months, the maturity (due)
date is found by adding the number of months to the date of the note.
This is illustrated in Example 1. Thus, if a note for six months, is dated
May 6, it will be due on the corresponding (the 6th) day of the sixth month,
or November 6. November 30, would have been the due date of this note,
if it had been dated May 31. The correct date for three months after
November 30, 1930 is Feb. 28, 1931 and the correct date for three months
after November 30, 1931 is Feb. 29, 1932. What makes this difference?
Simple Interest and Discount 15
If the term of the note is a fixed number of days, the due date is found
by adding the number of days to the date of the note, using the exact
number of days of the intervening months. Thus, 90 days after Feb. 1,
1932 is May 1, for the 28 days remaining in February -f 31 days in March
+30 days in April + 1 day in May = May 1. What is the correct date
for 90 days after Feb. 1, 1931?
The term of discount is commonly found by counting the exact number
of days between the date of discount and the due date. Thus, the term of
discount in Example 1, is 97 days, being obtained as follows: 30 days
remaining in August + 30 days in September + 31 days in October +"6
days in November = 97 days. The date of discount is excluded but the
due date is included.
When February is an intervening month, use 28 days if no year date
is given, but if it occurs in a leap year use 29 days.
These four examples illustrate all the fundamental facts that are used
in" discounting a note. They merit a careful study by the student.
Simple discount,* like simple interest, is seldom used in computations
extending over a long period of time. In fact, the use of simple discount
leads to absurd results in long-term transactions.
Illustration. At 6% discount, the present value of $1,000 due at the
end of 20 years is, using P = 8(1 nd),
P = $1,000[1 - 20(0.06)] = - $200.
8. Summary and extension. We have used two methods to accumu-
late P and to discount S. The first method was based upon the simple
interest rate i and the second was based upon the simple discount rate d.
The relationships that we have developed are the following:
At simple interest. At simple discount.
/ = Pm D = Snd
S = P(l + ni) S
1 nd
Banks and individuals frequently lend money at a discount rate
instead of an interest rate. There are two reasons why the creditor may
* Bank discount is frequently referred to as simple discount.
16
Financial Mathematics
prefer to lend at a discount rate. First, the arithmetic is simplified when
the maturity value is known, and second, a larger rate of return is obtained.
Thus, if I request a loan of $100 from a bank for six months at 6%
discount, the banker actually gives me $97, collecting the discount of $3
in advance, and takes my non-interest-bearing note for $100. Note the
simplicity of the arithmetic: P = 100(1 - 0.06/2) = $97. Note also
that the rate of return (the interest rate) is larger than 6%. For we have
P = $97, n = y 2 , S = $100, i = ( ). Using S = P(l + raj, we obtain
from which
100 = 97 (l + ,
i = 0.0619 *= 6.19%.
However, the banker should not be accused of unfair dealing if he
quotes me the 6% discount rate or if he states that he charges 6% in advance.
He should be criticised if he quotes an interest rate and then charges a
discount rate. We shall return to the comparison of interest and discount
rates in Art. 9.
Example 1. I desire $900 as the proceeds of a 90 day loan from my
banker B who charges 5% discount. What sum will I pay at the end of
90 days?
Solution. We have P = $900, n = % d = 0.05. From P = S(l - nd)
we obtain
900 = 8(1 - 0.05/4).
Solving, we find S = $911.392.
Exercises
Find the proceeds of the following notes and drafts:
Face
Time
Date of
Paper
Hate of
Interest
Date of
Discount
Rate of
Discount
Rate of
Collection
1.
$1,500
3 mo.
January 1
Jan. 25
c%
x%
2.
380
90 days
March 10
5%
Apr. 20
6%
3.
2,000
6 mo.
August 1
6%
Nov. 10
7%
x%
4.
575
4 mo.
May 10
Aug. 1
7%
K%
5.
1,350
90 days
Feb. 1, 1928
6%
Mar. 7
8%
Mo%
6.
1,260
60 days
March 5
7%
April 1
6%
7.
2,500
2 mo.
April 10
May 1
6%
Ho%
Simple Interest and Discount 17
8. A $2,500 6% interest-bearing note dated February 10, 1944 was due Sept. 1,
1944. It was discounted July 10 at 7^%- What were the proceeds?
9. A person wishes to receive $250 cash from a bank whose discount rate is 6%.
lie gives the bank a note due in 4 months. What should be the face value of the note?
10. Solve formula (8) for n and d.
11. The proceeds on a $400 non-interest-bearing note discounted 78 days before
maturity were $394.80. What was the rate of discount?
12. A bank will loan a customer $1,000 for 90 days, discounting the note at 6%.
For what amount should the note be drawn?
13. How long before maturity was a $450 note discounted, if the proceeds were
$444.14, the discount rate being 7%?
14. A 90-day 6% note of $5,000, dated June 15, payable at a Louisville bank, was
discounted at a Chicago bank July 20, at 7%. If the exchange charge was $1.00, find
the proceeds.
15. A six months' note bearing 5% interest was dated March 7, 1935. It was dis-
counted at 6% on July 15, the bank charging $18.45 discount. Find the face of the note.
16. A man received $882 as the proceeds of a 90-day non-interest-bearing note.
The face of the note was $900. What was the rate of discount.
17. A bank's discount rate is 7%. What should be the face of the note if the pro-
ceeds of a 6 months' loan are to be $2,000?
18. A 4 months' note bearing 4J^% interest, dated August 15, was discounted Octo-
ber 11, at 6%. The proceeds were $791.33. Find the maturity value of the note.
Find its face value.
19. A 90-day 7% note for $1,200, dated April 1, was discounted June 10 at 6%.
Find the proceeds.
20. How long before maturity was a $500 6 months' 6% note discounted, if the
proceeds were $504.70, the discount rate being 8%?
21. The proceeds on a six months' 5% note, when discounted 87 days before maturity
at 6% were $1010.14. Find the face of the note.
22. Find the present value of $1,000 due at the end of 20 years if 5% discount rate
is used.
9. Comparison of simple interest and simple discount rates. In
Art. 8 we gave brief mention to the relation of interest rate to discount
rate. This relation is so important that we will consider the problem
more thoroughly at this point. We shall approach the question through
a series of examples.
Example 1. If $100, due at the end of one year, is discounted at 6%,
what is the corresponding rate of interest?
18 Financial Mathematics
Solution. We have S = $100, n = 1, d = 0.06. In order to find i t
we will first find P. Using P = 8(1 nd), we have
P = 100(1 - 0.06) = $94.
$94 1 year at i% $100
1 1
interest rate
Since S P is the interest on P, we may find i by using 7 = Pni.
We have / = $6, n = 1, P = $94. Hence,
i = fo = 0.06383 = 6.383%
We might have employed the relation S = P(l + ra) to obtain the
same result.
Example 2. If $100, due at the end of 6 months, is discounted at 6%,
what is the corresponding interest rate?
Solution. We have S = $100, n = y 2 , d = 0.06. From P = S (1 - nd),
we have
P = 100(1 - 0.06/2) = $97.
$97 6 months at i% $100
l__ 1
interest rate
Since S P is the interest on P, we may find i by using / = Pni.
We have / = $3, n = M, P = $97. Hence
97(t/2) = 3,
and i = 0.0619 = 6.19%.
Thus we notice that the interest rates corresponding to a given dis-
count rate vary with the term; the longer the term, the larger the interest
rate.
In general, we say that, for a given term, an interest rate i and a cor-
responding discount rate d are equivalent if the present values of S at i
and d are equal. Thus, if P is the present value of S due in n years,
P n years S
\ 1
we have
P - A- from (3),
and
Hence,
Solving we obtain
and
Simple Interest and Discount
P = Sd - nd)
nd).
19
from (8).
S
ra
i
d
l-nd
(9)
(10)
From (9) we observe that for a given d the values of i increase as n
increases. From (10) we observe that for a given i the values of d decrease
as n increases.
The student will also observe from (10) that i/(l + ni) is the present
value of i due in n years. That is, i/(l + ni) in advance is equivalent to i
at the end of the term. But i/(l + ni) equals d. Hence d is equal to i
paid in advance. Thus, we say discount is interest paid in advance.
Exercises
1. Solve Example 1 by using formula (3).
2. Solve Example 2 by using formula (3).
3. Employing equation (9) complete the table:
d
.08
.08
.08
.08
n
1
l /2
1 A
y*
i
4. Employing equation (10) complete the table:
i
.08
.08
.08
.08
n
1
Y*
X
1 A
d
5. A obtains $780 from Bank B. For this loan he gives his note for $800 due in 60
days. At what rate does Bank B discount the note? What rate of interest does A pay?
6. A note for $800, dated June 15, due in 90 days and bearing interest at 6%, was
sold on July 1 to a friend to whom money was worth 5%. What did the friend pay for
the note?
7. If the note described in Exercise 6 were sold to Bank B on July 5 at a discount
rate of 7%, what would Bank B pay for the note?
20 Financial Mathematics
8. $500.00 Pittsburgh, Penna.
May 15, 1945.
Ninety days after date I promise to pay John Jones, or order, five
hundred dollars together with interest at 6% from date.
Signed, Wm. Smith.
(a) Thirty days after date Jones sold the note to Bank B who discounted it at
7%. What did Jones receive for the note?
(b) Would it have been to Jones' advantage to have sold the note to friend C, to
whom money was worth 7%, rather than to Bank B?
9.
$1,000.00 Chicago, 111.
May 15, 1945.
Six months after date I promise to pay Joe Brown, or order, one thou-
sand dollars with interest from date at 5%.
Signed, Charles Paul.
(a) Two months after date Brown sold the note to Bank B who discounted it at
6%. What did Bank B pay for the note?
(b) Immediately after purchasing the note, Bank B sold the note to a Federal
Reserve Bank at a re-discount rate of 4%. How much did Bank B gain on
the transaction? [On transaction (b) use a 365-day year.]
10. Rates of interest corresponding to certain discount rates in the
terms of settlement. The subject of terms was discussed in Alg.: Com.
Stat.y p. 99.* An example will illustrate what is meant by the rates of inter-
est corresponding to the rates of discounts of the terms of settlement.
Example 1. On an invoice of $1,000, a merchant is offered the following
terms: 5, 3/30, n/90. What is the interest rate corresponding to each of
the rates of discount?
Solution.
I. If the buyer pays the account immediately, he receives a discount
of $50. That is, he settles the account for $950 which means
that he receives $50 interest on $950 for 90 days. We may
determine the interest rate by substituting in / = Pnij thus
obtaining:
/ 50 50
Pn 950(M) 237.50
= 0.2105 = 21.05%.
II. If the buyer settles the account at the end of 30 days, he receives
a discount of $30. That is, the account is settled for $970 which
* Miller and Richardson, Algebra: Commercial Statistical, D. Van Nostrand Co.
Simple Interest and Discount 21
means $30 interest on $970 for 60 days. We determine the
interest rate as in I and find,
30 ^
~ 970
= 0.1855 = 18.6%
The buyer may have his business so well organized that he knows about
what his money is worth to him in the running of the business. He can then
determine the best offer, in the terms of sale, to accept. An example will
illustrate.
Example 2. Assuming that money is worth 20% to the merchant in
his business, which is the best offer in Example 1?
Solution. To answer this question we must compare the present
values of the separate offers. That is, which offer has the least present
value assuming money worth 20%?
I. 5% discount on $1,000 means a discount of $50. Hence the
present value of this offer is $1,000 - $50 = $950.
II. 3% discount on $1,000 means a discount of $30 at the end of
30 days. Hence, $970 is required to settle the account at
the end of 30 days. Now, the present value of $970 is
_ 970 _ 970
P ** 1 + K 2 (0.20) = 1.0167 = $954 - 06 '
III. Here the present value of $1,000 for 90 days at 20% is required*
1,000 1,000
P = 1 + M(0.20) = L06
= $952.38.
We notice that the 5% cash discount is the best offer (assuming money
worth 20%) since it gives the least present value for the invoice.
Exercises
1. Determine the interest rates corresponding to bank discount rates of (a) 7% 90
days before maturity; (b) 7J^% 60 days before maturity; (c) 6% 6 months before
maturity; (d) 8% 4 months before maturity.
2. In discounting a 4 months' note a bank earns 9% interest. What rate of discount
does it use?
22 Financial Mathematics
3. What are the rates of discount corresponding to (a) 7% interest earned on a note
discounted 90 days before maturity; (b) 8% interest earned on a note discounted 4
months before maturity; (c) 6% interest earned on a note discounted 6 months before
maturity?
4. What rate of interest is earned on money used in discounting bills at a discount
rate of 9% per annum?
6. What is the rate of discount at which a bank may as well employ its funds as to
lend money at an interest rate of 8%?
6. A merchant has the privilege of 90 days credit or 3% off for cash: What rate of
interest does he earn on his money if he pays cash?
7. A merchant bought a bill of goods amounting to $2,500 and received the following
terms: 4, 3/10, n/90. What is the interest rate corresponding to each of the rates of
discount?
8. Assuming that money is worth 15% to the merchant in the conducting of his
business, which is the best offer in Exercise 7? (See illustrative Example 2, Art. 10.)
9. On an invoice of $4,200, a merchant is offered 60 days credit or a discount of 3%
for cash. Not having the money to pay cash, he accepts the credit terms. What rate
of interest does he pay on the net amount of the bill? How much would he have saved
if he had borrowed the money at 7% and paid cash?
10. 7% interest was earned in discounting a note 90 days before maturity; 6% was
earned in discounting a 4 months' note; and 5% was earned in discounting a 9 months'
note. What were the corresponding discount rates?
11. Assuming money worth 20% in one's business, which one of following offers is
the most advantageous to the buyer: 6, 5/30, n/4 mos.? (Assume an invoice of $100.)
12. Solve Exercise 11, assuming money worth 18%.
13. A bank used a discount rate of 6% in discounting a 4 months' note. What rate
of interest was earned on the transaction?
14. Assuming money worth 12%, which one of the following offers is the most advan-
tageous to the buyer: 6, 4/30, n/4 mos.?
11. Exchanging debts. When two or more debts (obligations) are to
be compared we must know when each debt is due and then compare their
values at some specified time. The value of a debt at a specified time
depends upon the rate of interest that is used. Let us suppose that a
debt of $200 is due in 2 months and one of $205 is due in 8 months. Assum-
ing money worth 6%, compare their values now. The value of the first
debt at this time is
200 200
= = $198.02 [(3), Art. 2]
1 + ^(0.06) 1.01
and the value of the second debt at this time is
205 205
1 + %(0.06) 1.04
= $197.12.
Simple Interest and Discount 23
Six months from now the first debt would be 4 months past due and
should draw interest for that time. The second debt would not be due
for 2 months and should be discounted for that time. Then their values
6 months from now would be
200[1 + H(0.06)] = 200(1.02) = $204.00
We notice that the first debt has a greater value on both dates of com-
parison. If 6% is used the value of the first debt will always be greater
than that of the second.
If 4% were used their values on the above dates would be $198.67,
$199.67 and $202.67, $203.64; respectively. That is, if 4% interest is
assumed the second debt has a greater value at all times.
If 6% interest is assumed, the sum of the values of the above debts at
the present is $395.14. This is shown by the equation
2 months >|
H - 8 months
We say that the sum of the values of $200 due in 2 months and $205
due in 8 months is equal to $395.14 due now, if money is assumed to be
worth 6%. Also, the sum of the values of $200 due in 2 months and $205
due in 8 months is equal to the sum of the values of $201.97 due in 3 months
and $201.97 due in 6 months, if 6% interest is assumed. This may be
shown by comparing the two sets of debts on some common date. Sup-
pose we take 8 months from now as a common date. Then
200(1.03) + 205 = 411.00
and 201.97(1.025) + 201.97(1.01) = 411.01.
Whenever the value of one set of obligations is equal to the value of
another set of obligations on a common date, the one set may be exchanged
for the other set, and the values of the two sets are said to be equivalent.
The common date used for the date of comparison is usually known as the
focal date, and the equality which exists, on the focal date, between the values
24 Financial Mathematics
of the two sets of obligations is called an equation of value. An example will
illustrate the meaning of focal date and equation of value.
Example 1. A person owes $600 due in 4 months and $700 due in 9
months. Find the equal payments necessary to equitably discharge the
two debts, if made at the ends of 3 months and 6 months, respectively,
assuming 6% simple interest.
Solution. We choose the end of 9 months for our focal date and set up
the equation of value.*
i :
! $600 i $700
j 1 j 1 1 1 1 1 1 1
012345678 9 months
Let x = the number of dollars in each of the equal payments.
The time from the date of making the first payment x until the focal
cfttte is 6 months and the payment will accumulate to
[1 + 34(0.06)]x = (1.03)z on the focal date.
The second payment is made 3 months before the focal date and it will
accumulate to
[1 + H(0.06)]s = (1.015)z on the focal date.
The $600 debt is due in 4 months, just 5 months before the focal date,
and will accumulate to
600[1 + % 2 (0.06)] = 615.00 on the focal date.
The $700 debt is due on the focal date and will be worth $700 on that
date.
The equation of value becomes
(1.03)z + (L015)z = 615 + 700,
(2.045)z = 1,315,
x = $643.03, the amount of each of the equal payments.
In setting up an equation of value, we assume that the equation is true
for any focal date. That is, we assume that if the value of one set of
debts is equal to the value of another set of debts on a given focal date,
then the values are equal on any other focal date. If in the above prob-
* In the construction of the line diagram, (a) place at the respective points the
maturity values, and (b) place the payments and the debts at different levels.
Simple Interest and Discount 25
lem we had taken 3 months from now for the focal date, we would have
obtained $643.07 for the amount of one of the equal payments. Using
5 months from now as focal date we obtain $643.02 as one of the equal
payments. We notice that a change in the focal date changes the values
of the payments, but this change is very slight and for short periods of
time we may neglect the small differences caused by different choices of
focal dates and choose the one that is most convenient. (In Art. 19 it
will be shown that the amount x is independent of the focal date when the
computations are based upon compound interest.) The last date occurring
seems to be the most convenient, for then no discount is involved.
Example 2. Solve Example 1, assuming that the original debts bear
7% interest to maturity. Choose 9 months from now as the focal date.
Solution. $600 at 7% amounts to $614 in 4 months and on the focal
date its amount is
614[1 + 5*2(0.06)] = 614(1.025) = $629.35.
$700 at 7% amounts to $736.75 in 9 months and on the focal date its
amount is this maturity value ($736.75).
I $614 ! $736.75
! 1 1 1 1 1 1 1 1 1
0123466789 montha
The equation of value becomes
(1.03)* + (1.015)x - 629.35 + 736.75,
(2.045)x = 1,366.10,
x = $668.02, the amount of one of the equal payments.
12. To find the date when the various sums (debts) due at different
times may be paid in one sum. A may owe B several sums (debts) due
at different times and may desire to cancel all of them at one time by paying
a single amount equal to the sum of the maturity values of the several
debts. The problem, then, is to find a date when the single amount may
be paid without loss to either A (debtor) or B (creditor). Evidently, this
should be at a time when the total interest gained by the debtor on the
sums past due would balance the total interest lost on the sums paid before
they are due. The date to be found is known as the equated date.
The solutions of problems of this character may be effected by either
26 Financial Mathematics
of two methods. We may base our procedure upon a simple interest rate i
and choose the latest date mentioned in the problem as the focal date, or
we may base our procedure upon a simple discount rate d and choose the
earliest date mentioned in the problem as the focal date. If the former
method is followed all sums will accumulate at i to the focal date whereas
if the latter method is adopted all sums will be discounted at d to the focal
date.
Example. A owes B the following debts: $200 due in 60 days, $400
due in 90 days, and $600 due in 120 days. Find the time when these debts
may be canceled by a single payment of their sum, $1,200.
Solution. We have the debts and the payment as shown by the line
diagram.
$1200
$200 $400
60 90 n 120 days
Let n days from now be the equated date.
We choose the focal date at the latest date, 120 days from now, and
assume an interest rate i.
The first debt, $200, will be at interest for 60 days and its value on the
focal date is
/ 60 \
200 (H 1).
\ 360 /
The second debt, $400, will be at interest for 30 days and its value on
the focal date is
400
/ 30 \
(1 + 1).
\ 360 /
The third debt, $600, due on the focal date, bears no interest and hence
its value then is
600 H i
360
The single payment, $1,200, will be at interest (120 n) days and
thus its value on the focal date is
120 -n
1 +
1,200 ( 1
\ 360
Expressing by an equation the fact that the value of the payment on
the focal date is equal to the sum of the maturity values of the debts on
that date, we have
which reduces to
Simple Interest and Discount 27
/ 120 - n \
1,200 ( 1 H -- i }
V 360 /
/ 60 \ / 30 \ / \
= 200 1 -t i } + 400 ( 1 + f 1 + 600 ( 1 + i }
\ 360 / V 360 / \ 360 /
to
/120 - n \ / 60 \ / 30 \ / \
1,200 ( - i = 200 I t ) + 400 ( i) + 600 ( i }
V 360 / \360 / \360 / \360 /
Note. The student should note that the last equation written above simply states
that the interest on the payment equals the sum of the interest increments on the debts,
all calculated from their due dates to the focal date.
Multiplying the last equation by 360 and dividing through by lOOi,
we get
12(120 - n) = 2(60) + 4(30)
1,440 - 12n = 120 + 120
- 12n = -1,200
n = 100.
Hence, the $1,200 may be paid 100 days from now and the equities be
the same as if the debts were paid as originally scheduled.
Note. The fact that the interest rate i divides out as a factor in solving the equa-
tion of value shows that the value of n is independent of i,
Exercise. Solve the preceding example by assuming a discount rate
d and choosing (a) the earliest date, 60 days, as the focal date, and (b) the
present or "now" as the focal date.
By following a line of reasoning similar to that used in solving the pre-
ceding example, we will solve the general problem.
Problem. Let Di, Z>2, , D k be k debts due in ni, n2, , n k years
respectively, and let their maturity values be Si, 82, ,$&. We wish to
find the equated time, that is, the time when the k debts may be settled by
a single payment of Si + 82 + + Sk.
Solution. We shall assume n\ < n<z < ns < < n kj and we shall take
the latest date, n*, to be the focal date. Also we let n years from now be
the equated time. The diagram gives us the picture.
n 2 n n 3 n k years
28 Financial Mathematics
Assuming an interest rate i, the accumulated values of Si, 82, etc., at
n k are Si[l + (nk ni)i\, 82(1 + (nk ^2)^'], etc., we then have the equa-
tion of value
[Si + S 2 + + S k ][l + (n k - n)i] =
Si[l + (n k - ni)i] + &[1 + (n k - n 2 )i] + + S*[l + (n k - n k ){\.
Subtracting Si + 82 + + Sk from both sides of the equation we
have
(Si + S 2 + ---- h Sk)(n k - n)i =
Si(nk n\)i + Szfak ri2)i + + S k (n k n k )i.
Note. This equation shows that the interest on the payment equals the sum of the
interest increments on the maturity values, all calculated from their due dates to the
focal date.
Solving for n we obtain
Si/i! + S 2 n 2 + S 3 n 3 + + S k n k
n = - Q j_ c _i_ <? _L - Z~Q --
Si + 6 2 + 6 3 + + iS/j
If Di, Dz, , D k are not interest-bearing debts, DI = Si, Z>2 = 82,
, Dk = Sk, and equation (12) becomes
If the debts involve short periods of time it is usually more convenient
to express n, ni, 712, etc., in terms of either months or days.
Exercise. Derive formula (12) by assuming a discount rate d and
choosing "now" as the focal date.
Exercise. The equated time has an interesting "teeterboard" prop-
erty in that it is the "center of balance" when the maturity values are
suspended as weights with lever arms measured from n. That is, let the
lever arms be fii = n\ n, n 2 = U2 n, etc., respectively. Then,
Sifii + S 2 n 2 + S 3 rl 3 + + S k n k = 0.
13. To find the equated date of an account. To find the equated date
of an account means we must find the date when the balance of the account
can be paid without loss to either the debtor or the creditor.
As hi Art. 12, we assume that the sum of the values, as of the focal
date, of all credits including the balance, is equal to the sum of the values
on that date of all debits. Obviously, we may select the focal date in
many ways. We may, for example, choose the earliest date mentioned in
the problem as the focal date and discount all credits and debts to this
Simple Interest and Discount 29
point. We shall illustrate this procedure in our discussion first by a spe-
cific example and then by the general problem.
Example. What is the equated date of the account?
1944 1944
May 1, Mdse., $1,500 May 11, Cash, $400
June 19, Mdse., $1,000 May 31, Cash, $900
Solution. The total of the debts is $2,500 and the total of the credits
is $1,300. Our problem is to find the date when the balance, $1,200, can
be paid without loss to either the debtor or the creditor. The line dia-
gram gives us the picture.
$400 $1,200 $900
$1,500 j j j $1,000
Mayl May 11 ( ) May 31 June 19
10 n 30 49 days
We let the earliest date, May 1, be the focal date. Let n days from
May 1 be the equated date. We assume a discount rate d and set up the
equation of value.
/ 10 \ / n \ / 30 \
400(1 d) + 1,200(1 d) + 900(1 d)
\ 360 / \ 360 / \ 360 /
/ \ / 49 \
= 1,500(1 d) + 1,000(1 d)-
\ 360 / V 360 /
Subtracting 2,500 from both sides of the equation and multiplying by
(-1), we get
/ 10 \ / ^ \ / 30 \
400 ( d } + 1,200 ( d } + 900 ( d }
\360 / V360 / \360 /
/ \ / 49 \
= 1,500 ( d } + 1,000 I <n-
\360 / V360 /
Note. This equation shows that the sum of the discounted values of the credits,
as of May 1, equals the sum of the discounted values of the debts as of the same date.
Further, since the last equation written above is divisible by d, the value of n is inde-
pendent of the discount rate.
Multiplying the last equation by 360 and dividing by lOOd, we have
40 + 12n + 270 - 490
12n = 180
n = 15 days.
Thus the equated date is 15 days after May 1, or May 16.
30 Financial Mathematics
Exercise. Solve the preceding example by assuming an interest rate
i and choosing the latest date, June 19, as the focal date.
By following a line of reasoning similar to that used in solving the pre-
ceding example, we will solve the general problem.
Problem. Let Di, Do, DS, , D k be k debts due in n\, 712, na, - , n k
years from now respectively, and let their maturity values be Si, 82, 83,
, Sk. Also, let Ci, C<2, Cx, , C m be m credits entered 01, 02, 03, ,
o m years from now respectively. We wish to find the equated date of the
account, that is, the date when the balance 5,
B = (Si +&+&++ S t ) - (Ci + C a + C 8 + - + C m ),
can be paid without loss to either debtor or creditor.
Solution. We shall assume n\ < ri2 < na < < n^ and 01 < 02 <
03 < <o m . For the sake of variety we shall take "now" to be the focal
date. We assume a discount rate d and let n equal the number of years
from now to the equated date. The line diagram gives us the picture.
Si
91
i
i
s 2 i
S 3 i
c 3 c
i 4
s k
n x Q! n 2 o 2 n 3 n 03 o m n k years
By equating the sum of the credits, including the balance, discounted
to the present, 0, and the sum of the debts as of the same date, we have
the equation of value
Subtracting Ci + 2 + 3 + + C m + B from both sides of this
equation, then multiplying by ( 1), we get
+ C 2 o 2 d + Czozd + + C m o m d + End
Note. This last equation shows that the sum of the discounted values of the pay-
ments equals the sum of the discounted values of the debts, all discounted to the focal
date, "now." Also, since every term of this equation contains the factor d, which may
be divided out, the equated date is independent of d.
Dividing out d and solving for n, we get, replacing B by its value,
(Sini+S 2 n a +S3n3+'" + S fe n fe )-(C 1 Q 1 + C 2 o a + C 3 Q3 + -- + C m oJ
(S 1 +S,+B+---+S jk )-(C 1 +C 1 +C,+ ...+C jn )
(13)
Simple Interest and Discount 31
If the debts are not interest-bearing, Si = Di, 82 = #2, e ^c., in which
case (13) becomes
(Z) 1 n 1 +D 2 n a +Z)3n3+ - - +D k n k )- (C 1 o 1 +C 2 Q 2 +C 3 O3+ +C m o m )
In practice we usually let the earliest date mentioned in the problem
be "now," then n\ = and the first term in the numerator vanishes.
When accounts involve short periods of time, we usually express n,
n\, 7i2 y 113, ' "> n k, 01, 02, 03, , o my in months or days.
Note. An account becomes interest-bearing on the equated date and the debtor should
pay interest on the balance of the account from the equated date until the balance is paid.
Exercise. Derive formula (13) by assuming an interest rate i and
choosing the latest date, n k , as the focal date.
Exercises
1. An obligation of $500 is due in 3 months and another obligation of $520 is due in
9 months. Assuming money worth 6% simple interest, compare the values of these
obligations (a) now, (b) 6 months from now, (c) 12 months from now.
2. Solve Exercise 1, assuming money worth 9% simple interest.
3. A note for $600 drawing 5% simple interest will be due in 5 months, and another
note for $600 drawing 6% interest will be due in 9 months. Assuming money worth
7% simple interest, compare the values of these obligations 7 months from now.
4. Solve Exercise 3, assuming money worth 8% simple interest.
6. A owes B $500 due in 3 months, $600 due in 5 months, and $700 due in 8 months.
Find the equal payments to be made at the end of 6 months and 12 months, respectively,
which will equitably discharge the three debts if money is worth 5%.
6. Assuming 6% simple interest, find the equal payments that could be made in 3
months, 6 months, and 9 months, respectively to equitably discharge obligations of
$500 due in 2 months and $800 due in 5 months.
7. Solve Exercise 5, assuming that the three debts draw 6% simple interest.
8. A owes B the following debts: $700 due in 5 months at 7% interest, $500 due in
6 months at 7% interest, and $600 due in 9 months at 5% interest. Assuming money
worth 6%, find the single payment that is necessary to equitably discharge the above
debts 8 months from now.
9. Find the time when the following items may be paid in a single sum of $3,000:
$1,500 due May 1, $500 due June 12, $800 due June 25, and $200 due July 20.
10. Find the time when the following items may be paid in a single sum of $2,300:
$500 due March 1, $300 due April 10, $800 due April 25, and $700 due June 1.
11. Find the time when obligations of $350 due in 2 months, $600 due in 3 months,
and $850 due in 6 months may be settled by a single payment of $1,800.
12. Find the time for settling in one payment of $1,600 the following debts: $200 due
in 3 months, $400 due in 5 months, $300 due in 6 months, and $700 due in 8 months.
32 Financial Mathematics
13. Find the date when the following items may be paid in a single sum of $2,000:
Sept. 1, Mdse., 30 days, $400*
Sept. 27, Mdse., 60 days, $500
Nov. 9, Mdse., 2 months, $1,100
Cheek the correctness of the date by assuming 6% simple interest and showing that
the interest on the past due items as of the equated date is the same as the interest
from the equated date to the due dates of the items not yet due.
Find the time when the following accounts may be paid in single amounts:
14. 1941 16. 1941
January 2, Mdse., 30 da., $800 July 1, Mdse., 60 da., $550
January 17, Mdse., 1 mo., $500 July 10, Mdse., 1 mo., $450
March 1, Mdse., 2 mo., $300 August 1, Mdse., 2 mo., $750
March 30, Mdse., net $400 Sept. 1, Mdse., net, $350
Sept. 10, Mdse., 30 da., $400
Find the time when the balance of the following accounts may be paid in single
amounts:
16. 1941 1941
April 1, Mdse., $700 April 20, Cash, $400
April 10, Mdse., $500 May 10, Cash, $300
July 1, Mdse., $800 May 31, Cash, $300
17. 1941 1941
July 1, Mdse., net, $575 July 10, Cash, $440
July 5, Mdse., 1 mo., $435 Aug. 1., Cash, $720
Aug. 1, Mdse., 60 da., $990
18. 1944 1944
January 1, Balance, $1,900 Jan. 15, Cash, $1,560
January 20, Mdse., 1 mo., $1,450 Jan. 30, Note, f 2 mo., $1,200
March 10, Mdse., Net, $1,325 Feb. 1, Note, f 90 da., with interest, $500
19. 1944 1944
May 1, Balance, $500 May 15, Cash, $700
May 10, Mdse., 2 mo., $1,000 June 20, Cash, $1,000
June 7, Mdse., 30 days, $2,000 July 10, Cash, $400
July 1, Mdse., $600
* When terms of credit are given on the different items, we must first find the due
date of each item.
f When a note is given without interest, the time is figured to the due date of the
note, but when the note bears interest the time is figured to the date that the note is given.
Simple Interest and Discount 33
Review Problems *
1. A man derives an income of $205 a year from some money invested at 4% and
some at 5%. If the amounts of the respective investments were interchanged, he would
receive $200. How much has ne in each investment?
2. A man has one sum invested at 4% and another invested at 5H%. His total
annual interest is $320. If both sums had been invested at 6%, the annual interest
would have been $390. Find the sums invested at each rate.
3. A man made three loans totaling $15,000, the first at 4%, the second at 5% and
the third at 6%, receiving for the whole $770 per year. The interest on the second part
is $70 less than on the sum of the first and third parts. How was the money divided?
4. A man has three sums invested at 4%, 6%, and 7% respectively, the total interest
received being $280. If the three sums had been invested at 6%, 7% and 4% respec-
tively, the total interest would have been $305. How much was invested at each rate,
if the sum invested at 4% was $500 more than the sum invested at 7%?
6. One half of a man's property is invested at 4%, one third at 5%, and the rest at
C%. How much property has he if his income is $560?
6. One man can do a piece of work in 10 days, another in 12 days, and a third in 15
days. How many days will it require all of them to do it when working together?
7. A certain tank can be filled by a supply pipe in 6 hours. It can be filled by another
pipe in 8 hours and a third pipe can empty it in 12 hours. If all three pipes are running
at the same time, how soon will it be filled?
8. How much cream that contains 32% butter fat should be added to 500 pounds of
milk that contains 3% butter fat to produce a milk with 4% butter fat?
9. A merchant desires to mix coffee selling at 24 cents a pound with 80 pounds selling
at 30 cents a pound and 60 pounds selling at 33 cents a pound to produce a mixture
which he can sell at 28 cents a pound. How many pounds of the 24 cent coffee must
he use?
10. How large a 6% interest-bearing note should be given April 1 to cancef a debt
of $1,200 due July 1?
11. What is the difference between the true and bank discount on a debt of $1,000
due in 4 months, the interest rate and the discount rate being 7J^%?
12. A note for $2,500, bearing 5% interest, dated June 1 was due November 10.
What should be paid for this note August 18, (a) if 6% simple interest is to be realized?
(b) if 6% discount is to be realized?
13. A note of $500, bearing 6% interest, is dated March 1. If it is due in 4 months,
what would be its value May 1 at 4}^%?
14. A merchant is offered a bill of goods invoiced at $748.25 on 4 months' credit.
As a settlement he gives his note with interest at 7^% for a sum which, at maturity,
will cancel the debt. Find the face of the note.
15. On March 5, a bill of merchandise valued at $3,000 was bought on 6 months'
credit. On May 8, $1,500 was paid on the account. On July 22 the present value of
the balance of the debt was paid. Assuming money worth 6%, find the amount of
the final payment.
* Many of these problems are review problems of algebra. For additional review
problems in interest and discount, see end of this book.
34 Financial Mathematics
16. A piece of property was offered for sale for $2,900 cash or for $3,000 due in 6
months without interest. If the cash offer was accepted, what rate of interest was
realized?
17. The cash price of a certain article is $90 and the price on 6 months' credit is
$95. How much better is the cash price for the purchaser, if money is worth 7%?
18. The present value at 5% of a debt due in 72 days is $396.04. What is the amount
of the debt?
19. Find the true discount on a debt of $3,600 when paid 6 months before maturity,
assuming 5% simple interest.
20. A father wishes to provide an educational fund of $2,000 for his daughter when
she reaches the age of 18. What sum should he invest at 4% simple interest on her
thirteenth birthday in order that his wishes may be realized?
21. What cash payment on July 1 will cancel a debt of $2,400 due December 8, if
money is worth 8%?
22. A merchant buys a bill of goods from a jobber for $1,500 on 4 months' credit.
If the jobber can realize 6% simple interest on his money, what cash payment should
he be willing to accept from the merchant?
23. A man borrows $10,000. He agrees to pay $1,000 at the end of each year for
10 years and 4% simple interest on all unpaid amounts. Find the total sum paid in
discharging the debt.
24. Find the sum: 1 + (1.06) + (1.06) 2 -f - + (1.06).
25. Find the sum: (1.03)- 10 + (1.03)- 9 + (1.03)- 8 + + (1.03) -.
26. Solve for n: (1.05) n = 6.325.
27. Solve for n: (1.045) ~ n = 0.753.
28. Find the rate of interest when, instead of paying $100 cash for an article, the
purchaser pays $10 down and 10 monthly installments of $10 each.
29. A man buys a bill of goods amounting to $50. Instead of paying cash, he pays
$5 down and 5 monthly installments of $10 each. Find the actual rate of interest paid.
30. On a cash bill for $150, $15 is paid down, followed by 10 monthly payments of
$15 each. Find the rate of interest paid.
31. The cash price of an article is C. Instead of paying cash the purchaser makes a
down payment D followed by monthly installments of R at the end of each month for
n months. Show that the interest rate i is given by the formula
24(nR -f D - C)
n(2C -2D-nR+R)
if all amounts are focalized at the time of the last payment.
32. (a) Using formula (12) show that R at the end of each month for n months is
equivalent to nR at (n + l)/2 months.
(b) Using the data of Exercise 31 and the conclusion of (a), focalizing all amounts at
(n + l)/2 months, show that
. 24(nfl -f D - C)
1 ~ (n + 1)(C - D) '
(c) Note that (nR -J- D C) is the total carrying charge and (C D) is the unpaid
balance.
CHAPTER II
COMPOUND INTEREST AND COMPOUND DISCOUNT
14. Compound interest. Simple interest is calculated on the original
principal only, and is proportional to the time. Its chief value is its
application to short-term loans and investments. Long-term financial
operations are usually performed under the assumption that the interest,
when due, is added to the principal and the interest for the next period of
time is calculated on the principal thus increased, and this process is con-
tinued with each succeeding accumulation of interest. Interest when so
computed is said to be compound. Interest may be compounded annually,
semi-annually, quarterly, or at some other regular interval. That is,
interest is converted into principal at these regular intervals. The time
elapsing between successive periods, when the interest is converted into
principal, is commonly defined as the conversion period. For example,
if the interest is converted into principal semi-annually, the conversion
period is six months. The rate of interest is nearly always expressed on
an annual basis and if nothing is specified as to the conversion period, it is
commonly assumed to be one year. The final amount at the end of the
time, after all of the interest has been converted into principal, is defined
as the compound amount. Consequently, the compound interest is equal
to the compound amount minus the original principal.
Example. Find the compound amount and compound interest on $600
for four years at 5%, the interest being converted annually.
Solution. The interest for the first (conversion period) year is
$600(0.05) = $30.00. When this is converted into principal, the amount
at the end of the first year becomes $630. The interest for the second year
is $630(0.05) = $31.50, and when this is converted the principal becomes
$661.50. Continuing this process until the end of the fourth year, we
find the compound amount to be $729.30; and the compound interest for
the given time is $129.30, the difference between $729.30 and $600.
35
36 Financial Mathematics
The solution of the above example can be written in the following form :
Interest for first year = $600(0.05)
Principal at end of first year = $600 + $600(0.05)
= $600(1 + 0.05) = $600(1.05)
Interest for second year = $600(1.05) (0.05)
Principal at end of second year = $600(1.05) + $600(1.05) (0.05)
= $600(1.05) (1.05)
= $600(1.05) 2
Interest for third year = $600(1.05) 2 (0.05)
Principal at end of third year = $600(1.05) 2 + $600(1.05) 2 (0.05)
= $600(1.05) 2 (1.05)
= $600(1.05) 3
Interest for fourth year = $600(1.05) 3 (0.05)
Principal at end of fourth year = $600(1.05) 3 + $600(1.05) 3 (0.05)
= $600(1.05) 3 (1,05)
= $600(1.05) 4
= $600(1.21550625)
= $729.30.
15. Compound interest formula. If we let P be the original principal,
i the yearly rate of interest and S the amount to which P will accumulate
in n years and reason as in the illustrated example of Art. 14, we will obtain
the compound interest formula.
The interest for the first year will be Pi and the principal at the end of
the first year will be p + pi = p ^ + ~
The interest for the second year will be Pi(l + i) and the principal at
the end of the second year will be
P(l + i] + Pi(l + i) = P(l + i) 2 .
By similar reasoning we find that the amount at the end of the third
year is
P(l + t) 2 + Pt(l + *) 2 = Pd + i) 3 ,
and in general the amount at the end of n years is P(l + i) n . Thus we
have the formula
s = p(i + n . (i)
This relation is easily visualized by the following line diagram:
P n years at i% S = P(l + i) n
I - . - 1
compound interest
Compound Interest and Compound Discount 37
Example 1. Find the compound amount and compound interest on
$500 for 8 years at 6%, the interest being converted annually.
Solution. Here, P = $500, i = 0.06, n = 8.
Substituting in (1) we haVe
8 = 500(1.06) 8 .
From Table III, (LOG) 8 = 1.59384807,
and S = 500(1.59384807) = $796.92.
The compound interest is
$796.92 - $500.00 = $296.92.
Example 2. Find the compound amount on $850 for 12 years at
, the interest being converted annually.
Solution. Here, P = $850, i = 0.0625, n = 12,
and S = 850(1.0625) 12 .
We do not find the rate, 6M%, in Table III, so we use logarithms to
compute S.
log 1.0625 = 0.02633
12 log 1.0625 = 0.31596
log 850 - 2.92942
log S = 3.24538
S = $1,759.50.
Using a table of seven place logarithms we find S = $1,759.41, which
is correct to six significant digits. When we use a table of five place
logarithms for computing, our results will be accurate to four and never
more than five significant digits.
When P, n, and i are given, the amount S computed by (1) is frequently
called the accumulated value of P at the end of n years. Hence, to accumu-
late P for n years at i% we find the amount S by using (1). The quantity
(1 + i) is called the accumulation factor.
Similarly, when S, n, and i are given, the principal P is called the dis-
counted value of S due at the end of n years. Hence, to discount S for
n years at i% we find the principal P by using (1). The principal P is also
called the present value of S.
38 Financial Mathematics
Exercises
1. Find the amount of $1,000 invested 15 years at 4%.
2. Find the amount of $1,000 invested 12 years at 6%.
3. Accumulate $500 for 15 years at 6%.
4. Discount $800 for 20 years at 3%.
6. Find the difference between the amount of $100 at simple interest and at com-
pound interest for 5 years at 5%.
6. At the birth of a son a father deposited $1,000 with a trust company that paid
4%, the fund accumulating until the son's twenty-first birthday. What amount did
the son receive?
7. In the following line diagram each section represents 1 year. The point denotes
any given time. Any point to the right of O denotes a later time and any point to the
left of denotes an earlier time. Consider $100 at 0. Based upon i = 4%, what is its
value at B1 at A?
Solution. At B the value is that of $100 accumulated for 5 years, or 100(1 + .04) 6 .
At A the value is that of $100 discounted for 4 years or 100(1 + .04) "~ 4 .
$100
I 1 1 | 1 j | 1 1 {
A B
8. In the following line diagram, based upon i = 5%, find the values at A, B, C t
and D of $100 at 0.
$100
I 1 1 1 1 1 1 1 1 1 , , 1
A :- ' B C D
1 year
16. Nominal and effective rates of interest. The effective rate of interest
is the actual interest earned on a principal of $1 in one year. When interest
is converted into principal more than once a year, the actual interest
earned (effective rate) is more than the quoted rate (nominal rate). Thus,
if we have a nominal rate of 6% and the interest is converted semi-annually,
the effective rate is by a method similar to that used in Art. 14,
(1.03) 2 - 1 = 0.0609 = 6.09%.
Then, on a principal of $10,000, a nominal rate of 6% convertible
semi-annually gives in one year $609.00 interest.
Similarly, if the rate is 6%, convertible quarterly, the effective rate is
(1.015) 4 - 1 = 0.06136 = 6.136%.
If we let i stand for effective rate, j for nominal rate, and m for the
J
number of conversions per year, then will be the interest on $1 for one
m
Compound Interest and Compound Discount 39
conversion period. Hence, the amount of $1 at the end of one year will
be given by
(2)
and the effective rate will be given by the equation
We may also write
H(I+Y
\ m/
be substituted for (1 + i) in (1), we obtain the equation
(5)
This equation gives the amount of a principal P at the end of n years at
rate j convertible m times per year. If m = 1, (5) reduces to (1). Hence
we say that (5) is the general compound interest formula and (1) is a special
case of (5).
From (4), we may easily find j in terms of m and i. Extracting the
mth root of each member and transposing, we find
j = m [(1 + z) 1/m - 1]. (40
Sometimes the nominal rate j is written with a subscript to show the
frequency of conversion in a year. Thus j m means that the nominal rate
is j with m conversion periods in a year. We also find it convenient at
times to use the symbol "j m at i" to mean "the nominal rate j which con-
verted m times a year yields the effective rate i." Values of j for given
values of m and i are found in Table IX.
Example 1. Find the effective rate corresponding to a nominal rate
of 5% when the interest is converted quarterly.
Solution. Here, j = 0.05 and m = 4.
Substituting in (3), we have
i = (1.0125) 4 - 1
= (1.05094534) - 1 = 0.050945
= 5.0945%.
40 Financial Mathematics
Example 2. Find the amount of $750 for 15 years at 5% converted
quarterly.
Solution. Here P = $750, j = 0.05, n = 15 and m = 4. Substituting
in (5) we have
S = 750(1,0125) 60 .
From Table III, (1.0125) 60 = 2.10718135,
and S = 750(2.10718135) = $1,580.39.
Example 3. Find the compound amount of $500 for 120 years at 3%.
Solution. Here P = $500, i = 0.03, n = 120. We find no value of
(1 + i) n in the table when n = 120, but we may apply the index law,
a x. a v _ a x+v t
Hence, (1.03) 120 = (1.03) 100 -(1,03) 20
*= (19.21863198) (1.80611123)
= 34.710987
and 8 = 500(34.710987) = $17,355.49.
This example illustrates a method by which the table can be used when the
time extends' beyond the table limit.
Example 4. To what sum does $5,000 amount in 7 years and 9 months
at 4% converted semi-annually.
Solution. The given time contains 15 whole conversion periods and
3 months. Now, the compound amount at the end of the 15th period is
S = 5,000(1.02) 15 = $6,729.34.
The simple interest on $6,729.34 for the remaining 3 months is
6,729.34 X ^2 X 0.04 = $67.29.
Hence, the amount at the end of 7 years and 9 months is
$6,729.34 + $67.29 = $6,796.63.
The solution of Example 4 illustrates a plan that is usually used for
finding the compound amount when the time is not a whole number of
conversion periods. We may state the plan as follows:
Compound Interest and Compound Discount 41
I. Find the compound amount for the whole number of conversion
periods, using (5).
II. Find the simple interest on the resulting amount at the given rate for
the remaining time.
III. Add the results of I and II.
Exercises
1. Find the amount of $800 invested for 8 years at 5%, convertible annually.
2. Solve Example 1, when the interest is converted (a) semi-annually, (b) quarterly.
Use formula (5).
3. Find the compound interest on $2,500 at 6H% for 8 years, if the interest is con-
verted semi-annually.
4. A man pays $1,000 for a 10 year bond that is to yield 5%, payable semi-annually.
What will be the amount of the original investment at the end of 10 years if the dividends
are immediately reinvested at 5%, payable semi-annually?
6. On January 1, 1928, $1,500 was placed on time deposit at a certain bank. For
10 years the bank allowed 4% interest converted annually. During the next 4 years
3%, converted quarterly, was allowed, and on January 1, 1942 the interest rate allowed
on such deposits was reduced to 2J^%, converted semi-annually. What was the accumu-
lated value of this original deposit as of January 1, 1945?
6. Find the effective rate equivalent to 6% nominal converted (a) semi-annually,
(b) quarterly, (c) monthly.
7. A savings bank paid 5% compound interest on a certain deposit for 6 years and
then 4% for the next 4 years. What single rate (equivalent rate) during the 10 years
would have produced the same effect?
Solution. Let i equal the equivalent rate.
Then (1 -f i) 10 - (1-05) 6 (1.04)*.
log 1.05 = 0.0211893
log 1.04 = 0.0170333
6 log 1.05 0.1271358
4 log 1.04 =0.0681332
10 log (1 -ft) =0.1952690
log (1 + i) = 0.0195269
(1 + i) = 1.04599
i = 0.04599 - 4.599%.
The value obtained for (1 + i) is correct to six significant digits. A seven place
table of logarithms was used here. When we use a table of seven place logarithms, we
can be sure that our results are accurate to six significant digits.
8. What is the effective rate for 20 years equivalent to 6%, converted annually for
the first 8 years; 5% converted semi-annually for the next 7 years; and 4%, converted
quarterly for the last 5 years?
42 Financial Mathematics
9. An individual has a sum of money to invest. He may buy saving certificates,
paying 5J^% convertible semi-annually, or deposit it in a building and loan association,
which pays 5% convertible monthly. Assuming that the degree of safety of the two is
the same, should he buy the certificates or deposit his money in the association?
10. Find the compound amount on $750 for 8 years 9 months at 5% converted
semi-annually.
11. Representing time along the horizontal axis and the computed values of S along
the vertical axis, make graphs of S = 100(1 -f 0.04ra) and S = 100(1.04) n . Take for n
the values 1, 5, 9, 13, 17, 21, 25 and use the same scale for both graphs.
12. Repeat Exercise 11, when the interest rate is 6%.
13. Accumulate $2,000 for 12 years if the interest rate is 5% compounded monthly.
14. A house is offered for sale. The terms are $4,000 cash, or $6,000 at tne end of
10 years without interest. If money is worth 4%, interest converted semi-annually,
which method of settlement is to the advantage of the purchaser?
15. Find the effective rate equivalent to 7% converted (a) monthly, (b) quarterly,
(c) semi-annually.
16. Find the nominal rate, converted quarterly, that will yield an effective rate of
(a) 4%; (b) 5%; (c) 6%.
17. Present value at compound interest. In Art. 6 we defined the
present value P of a sum S, due in n years, from the standpoint of simple
interest. The definition of present value will be the same here, except
that compound interest is used in the place of simple interest. From the
definition of present value, it follows that the present value P of a sum S
may be obtained by solving equation (1), Art, 15 for P. Solving this
equation for P, we have
where v =
The number v is called the discount factor.
If the rate of interest is j, converted m times a year, we have from (5)
Art. 16
S
Compound discount is commonly defined as the future value S minus the
present value P. If D stands for compound discount on S, we have
D = S - P. (8)
Compare the above formula with (8), Art. 7.
Compound Interest and Compound Discount 43
Since P is defined as the principal that will accumulate to S, at com-
pound interest, in n years, the difference S P also stands for the com-
pound interest on P. Therefore, we may say that the compound discount
on the accumulated value is the same as the compound interest on the
present value for the given time at the specified interest rate.
Example 1. Find the present value and compound discount of $4,000
due in 10 years at 5% converted annually.
Solution. Here, S = $4,000, i = 0.05, and n = 10.
Substituting in (6), we have
P = 4,000(1.05) ~ 10 .
From Table IV, (1.05) ~ 10 = 0.61391325
and P = 4,000(0.61391325) = $2,455.65,
Also, D = 4,000.00 - 2,455.65 = $1,544.35.
Example 2. Find the present value of $2,000 due in 8 years at 4%%
converted semi-annually.
Solution. Here, S = $2,000, j = 0.0475, m = 2, and n = 8.
Substituting in (7), we have
P = 2,000(1.02375) ~ 16 .
We do not find the rate, 2Ji%, in Table IV, so we use logarithms to
compute S.
log 1.02375 = 0.0101939
16 log 1.02375 = 0.1631024
log (1.02375) ~ 16 - 9.8368976 - 10
log 2,000 = 3.30103
logP = 3.13793
P = $1,373.81.
Example 3. Find the present value of $5,000 due in 7 years with interest
at 6% converted semi-annually, assuming money worth 5%,
Solution. We first find the maturity value of the debt and then find
the present value of this sum.
44 Financial Mathematics
Hence, fi = 5,000(1.03) 14
= 5,000(1.51258972)
= $7,562.95,
and P = S(1.05)- 7
= 7,562.95(1.05) ~ 7
= 7,562.95(0.71068133)
= $5,374.86.
$5,000
int.
at (j = .06, *
S = $7,562.95
p
1
2
3 4
int. at i=.05
5
6
7
$7,562.95
1
2
3 4
5
6
7
This example illustrates a method for finding the present value of an
interest-bearing^debt.
Problems
1. What is the present value of a note of $200 due in 6 years without interest, assum-
ing money worth 6%?
2. Find the present value of $3,000 due in 5 years, if the nominal rate is 5%, con-
vertible semi-annually.
3. What sum of money invested now will amount to $4,693.94 in 25 years if the
nominal rate is 5%%, convertible semi-annually?
4. A note of $3,750 is due in 4^ years with interest at 6% payable semi-annually.
Find its value 3 years before it is due, if at that time money is worth 5%.
6. What is the present value of a $1,000 note due in 5 years with interest at 8%
payable semi-annually, when money is worth 6%?
6. Compare the present values of non-interest-bearing debts of $400 due in 3 years
and $450 due in 5 years, assuming money worth 6% converted semi-annually. Compare
the values of these debts 2 years from now, assuming that money is still worth 6%
converted semi-annually.
7. An investment certificate matures in 3 years for $1,000. Its present cash value is
$860. If one desires his money to earn 5% annually, should he purchase the certificate?
8. A debt of $4,500 will be due in 10 years. What sum must one deposit now in a
trust fund, paying 4}^% converted semi-annually, in order to pay the debt when it
falls due?
9. What is the present value of $300, due in 4 years and 3 months without interest,
when money is worth 5%?
10. A father wishes, at the birth of his son, to set aside a sum that will accumulate
to $2,500 by the time the son is 21 years old. How much must be set aside, if it accumu-
lates at 3% converted semi-annually?
Compound Interest and Compound Discount 45
Of Sf
11. Draw graphs of P = - and P = . for integral values of n from
(1 + i) n 1 + m
to 10. For convenience, take S = 10 and i = 0.05. Take values of n along the
horizontal axis and corresponding values of P along the vertical axis, using the same
scale and set of axes for both graphs. Use Table IV for finding the values of P =
12. If $2,500 accumulates to $3,700.61 in a certain time at a given rate, what is the
present value of $2,500 for the same time and rate?
13. Find the present value of a debt of $250, due in 5 years 3 months and 15 days,
if money is worth 5%.
14. An investment certificate matures in 7 years for $500. If money is worth 4%
for the first 3 years and 3H% thereafter, what is the present value of the certificate?
16. A man desires to sell a house and receives two offers. One is for $2,500 cash
and $5,000 in 5 years. The other is for $3,000 cash and $4,000 to be paid in 3 years.
On a 5% basis, which is the better offer for the owner of the house and what is the dif-
ference between the two offers?
16. An insurance company allows 3}^% compound interest on all premiums paid
one year or more in advance. A policy holder desires to pay in advance three annual
premiums due in 1 year, 2 years, and 3 years respectively. How much must he pay the
company now if each annual premium is $21.97?
17. Making use of the binomial theorem (assuming n greater than 1) show that
(1 + i) n is greater than (1 + ni). Using Table III compare these values when n - 5
and i = 0.06.
18. Other problems solved by the compound interest formulas.
Formulas (1) and (5) each contain four letters (assuming m in (5) to be
fixed). Any one of these letters can be expressed in terms of the other
three. In Art. 16 we solved problems in which S was the unknown and
in Art. 17 we solved for P. We shall now solve some problems when the
value of n or j is required.
Example 1. In how many years will $742.33 amount to $1,000 if in-
vested at 6%, converted quarterly?
Solution. From (5), Art. 16, we have
1,000 = 742.33(1.015) 4 .
Taking logarithms of both members of the above equation, we get
log 1,000 = log(742.33) + 4n log(1.015).
46 Financial Mathematics
Solving for n,
= lo gQ> QQQ ) - log(742.33) _ 3.00000 - 2.87060
41og(1.015) ~ 4(0.00647)
0.12940
0.02588
= 5.
Hence, $742.33 will amount to $1,000 in 5 years, if the rate is 6%
converted quarterly.
Example 2, How long will it take $1,000 to amount to $1,500 at 5%
converted semi-annually?
Solution. Substituting in (5), Art. 16, we have
1,500 = l,000(1.025) 2w .
The above equation reduces to
(1.025) 2 - = 1.5.
From the 2^% column in Table III, we find that (1.025) 2 " = 1.4845 0562
when 2n = 16; and when 2n = 17, (1.025) 2 * = 1.5216 1826. The nearest
time, then, is 16 semi-annual periods or 8 years. That is, $1,000 amounts
to $1,484.51 in 8 years at 5% converted semi-annually. We now find the
time required for $1,484.51 to amount to $1,500 at 5% simple interest.
Here, P = $1,484.51, / = $15.49, and i = 0.05. We solve for n as in
illustrated Example 4, Art. 70.
_ 15.49 15.49
(1,484.51) (0.05) ~ 74.2255
= 0.209 year (approximately), or 2 months and
15 days.
Hence, we find that $1,000 will amount to $1,500 in 8 years 2 months
and 15 days at 5% converted semi-annually.
Examples 1 and 2 illustrate methods for finding n, when S, P, and i
are given.
Example 3. At what rate would $2,500 amount to $5,000 in 14 years
if interest were converted semi-annually?
Solution. From (5), Art. 16, we have
5,000 = 2,500 (l +
Compound Interest and Compound Discount 47
Taking logarithms of both members of the above equation, we get
/ A
log 5,000 = log 2,500 + 28 log ( 1 + J - ) ,
log
28
3.69897 - 3.39794 0.30103
28 28
= 0.01075.
J -} = 1.025
~ = 0.025
2
j = 0.05 = 5%.
That is, the rate is 5% nominal, convertible semi-annually. From (4)
Art. 16 we find the effective rate to be i = 5.0625%.
Example 4. At what rate would $1,500 amount to $2,500 in 9 years,
if the interest were converted annually?
Solution. From (1), Art. 15, we have
2,500 = 1,500(1 + i) 9 .
Dividing the above equation through by 1,500, we get
(1 + ;)Q = j 6667 ( to 4 decimal places).
In Table III we notice that when i = 0.055, (1 + z) 9 = 1.6191;
when i = 0.06, (1 + i) g = 1.6895. Hence, i is a rate between 5^% and
6%.
By interpolation, we find
i = 0.055 + (0.005) ( 47 %)4)
= 0.055 + 0.00338 = 0.05838.
Hence, the rate is 5.84% (approximately). The student should also
solve this example by logarithms.
Examples 3 and 4 illustrate methods for finding the rate when S, P,
and n are given.
48 Financial Mathematics
Exercises
1. In what time will $840 accumulate to $2,500 at 5%, converted annually?
2. If $1,000 is invested in securities and amounts to $2,500 in 15 years, what is the
average annual rate of increase?
3. At what rate must $10,000 be invested to become $35,000 in 25 years?
4. In how many years will $400 amount to $873.15 at 5% annually?
6. How long will it require any sum to double itself at effective rate i ?
6. How long will it require a principal to double itself at (a) 5%, (b) 6%?
7. How long will it take $1,500 to amount to $5,000 at 6% converted quarterly?
8. At what rate will $2,000 amount in 30 years to $10,184.50 if the interest is con-
verted semi-annually?
9. A will provides that $15,000 be left to a boy to be held in trust until it amounts to
$25,000. When will the boy receive the fund if invested at 4% converted semi-annually?
10. A man invested $1,500 in securities and re-invested the dividends from time to
time and at the end of 10 years he found that his investments had accumulated to
$2,700. What was his average rate of interest?
19. Equation of value. In Art. 11 the equation of value was defined
and used in connection with simple interest. The equation of value used
here will have the same meaning as in Art. 11. That is, it is the equation
that expresses the equivalence of two sets of obligations on a common date
(focal date). In Art. 11 we assumed, for convenience, that the equation
of value is true for any focal date. However, this assumption is only
approximately true, as was pointed out by a particular example. That is,
when simple interest is used the equivalence of two sets of obligations
actually depends upon the focal date selected. The equivalence of two
sets of sums, however, is independent of the focal date when the sums are
accumulated or discounted by compound interest. That is, if we have an
equation of value for a certain focal date, we may obtain an equation of
value for any other focal date by multiplying or dividing the first equation
through by some power of (1 + i) or of (1 + j/m).
Example 1. A owes B the following debts: $300 due in 3 years without
interest and $700 due in 8 years without interest. B agrees that A may
settle the two obligations by making a single payment at the end of 5 years.
If the two individuals agree upon 6% as a rate of interest, find the single
payment.
Solution. Let x stand for the single payment, and choose 5 years from
now as the focal date.
The $300 debt is due 2 years before the focal date and amounts to
300(1.06) 2 on the focal date.
Compound Interest and Compound Discount 49
The $700 debt is due 3 years after the focal date and has a value of
700(1.06) ~ 3 on the focal date.
The single payment x is to be made on the focal date and has a value
of x on that date.
x
i
- $300 I $700
012345678
Then, for the equation of value, we have
x = 300(1.06) 2 + 700(1.06) ~ 3
= 300(1.12360000) + 700(0.83961928)
= 337.08 + 587.73
= 924.81.
Hence, the two debts may be discharged by a single payment of $924.81
five years from now.
Had we assumed 8 years from now as focal date, our equation of value
would have been
z(1.06) 3 = 300(1.06) 5 + 700.
Dividing the above equation through by (1.06) 3 , we get
x = 300(1.06) 2 + 700(1.06) ~ 3 ,
which is the equation of value obtained when 5 years from now is taken
as the focal date. This is an illustration of the fact that an equation of
value does not depend upon our choice of a focal date.
The student will observe that in the construction of the line diagram
we place at the respective points the maturity values of the debts. Further,
it should be observed that the payment and the debts are placed at differ-
ent levels.
Example 2* Smith owes Jones $500 due in 4 years with interest at 5%
and $700 due in 10 years with interest at 4J^%. It is agreed that the two
debts be settled by paying $600 at the end of 3 years and the balance at
the end of 8 years. Find the amount of the final payment, assuming
an interest rate of
Solution. Let x stand for the final payment and choose 8 years from
now as the focal date.
50 Financial Mathematics
The maturity value of the $500 debt is 500(1.05) 4 and its value on
the focal date is 500(1.05) 4 (1.055) 4 .
The maturity value of the $700 debt is 700(1.045) 10 and its value on
the focal date is 700(1.045) 10 (1.055)- 2 .
The value of the $600 payment is 600(1.055) 5 on the focal date.
The value of the final payment is x on the focal date.
10
600
v 600 (1.05)*
X
| 700 (1.045)
| v v-
0123456
7 8 9 10
Expressing the fact that the value of the payments equals the value of
the debts (on the focal date), our equation of value becomes
600(1.055) 6 + x = 500(1.05) 4 (1.055) 4 + 700(1.045) 10 (1.055)- 2 .
Making use of Tables III and IV and performing the indicated multi-
plications, we have
784.176 + x = 752.900 + 976.688
and x = 945.41.
Hence, the payment to be made 8 years from now is $945.41.
20. Equated time. In Art. 12 equated time was discussed and a formula
(based upon simple interest) for finding this time was developed. Basing
our discussion on compound interest, we shall now solve a particular
example and then consider the general problem, thereby developing a
formula.
Example 1. Find the time when debts of $1,000 due in 3 years without
interest and $2,000 due in 5 years with interest at 5% may be settled by a
single payment of $3,000, assuming an interest rate of 6%.
Solution. Choose "now" as the focal date and let x stand for the time
in years, measured from the focal date ("now"), until the single payment
of $3,000 should be made. Our equation of value becomes
3,000(1.06) -*> 1,000(1.06)^ + 2,000(1.05) 5 (1.06)~ 5
3,000(1.06) = 1,000(0.83962) + 2,000(1.27628) (0.74726),
3,000(1.06) -* = 839.52 + 1,907.43 = 2,747.05,
Compound Interest and Compound Discount 51
2,747.05
(1.06) ~* =
3,000.00 '
x log 1.06 = log 3,000 - log 2,747.05,
log 3,000 - log 2,747.05
x =
log 1.06
3.47712 - 3.43886
0.02531
= 1.51.
Hence, the two debts may be settled by a single sum of $3,000 in 1 year,
6 months from "now."
Problem. Given that A owes B debts of DI, Z>2, Da, having ma-
turity values of Si, &, &, and due in m, 712, wa, years respectively.
Assuming an interest rate of i%, find the time when the debts may be
settled by making a single payment of S = Si + 82 + & +
Solution. Choose "now" as the focal date and let n stand for the time
in years, measured from the focal date (now), until the single payment
of S should be made.
Reasoning as in Example 1, the equation of value becomes
;)-"' + (9)
Solving the above equation for (1 + i) ~ n , we get
Q q Q
01 + 02 + 03 +
and
n -4- *> = Si + 82 + 83 +
( "" ~ ni
Taking logarithms of both sides of the above equation and solving for
n, we have
n =
i(l+0"" 1 ^
log (1 + i)
(10)
52 Financial Mathematics
Formula (10) gives the exact value for the equated time. However,
it is obviously very involved and is rather tedious to apply. We naturally
seek a satisfactory approximation formula. We shall now proceed to
find one.
If (1 + i)" n is expanded by the binomial theorem, we have
n<\-n i -,*(" + 1) ., n(n + 1) (n + 2) ., ,
(1 + i)~ n = 1 - ra + - - - 1 2 -- - z 3 + .
Neglecting all powers of i higher than the first gives (1 ra) as an
approximate value of (1 + i)~ n .
Applying the binomial theorem to (1 + i)~ ni , (1 + i)""* 1 *, and
dropping powers of i higher than the first, we obtain (1 nif), (1 n^i),
as approximate values of (1 + i)~ ni , (1 + i)~ n2 , respectively.
If in (9), (1 + i}~ n and (1 + i)-*, (1 + i)~ n2 , are replaced by
their approximate values, we get, on solving for n,
Now, if the original debts, DI, Z>2, DS, are non-interest-bearing, Si
2, $3, , may be replaced by DI, #2, 3, , respectively, and the
above equation becomes
^
n
We notice that (11) is essentially the same as (12), Art. 12. When the
periods of time involved are short and the debts, Z>i, Z>2, Ds, do not
draw interest, (11') gives us a close approximation of the equated time.
However, when the periods of time are short and the debts Di, Z>2, I>3,
draw interest (11) gives a good approximation to n.
Example 2, Find the equated time for paying in one sum debts of
$300 due in 3 years and $150 due in 5 years.
Solution. Choosing "now" as focal date and substituting in (ll')j we
have
(300)3 + (150)5
n== 300 + 150 -a- 8
Compound Interest and Compound Discount 53
Assuming an interest rate of 6% and applying (10), we find
log 450 - log [300(1.06) ~ 3 + 150(1.06) ~ 5 ]
ft =
log 1.06
2.65321 - 2.56118 _ 0.09203
0.02531 ~ 0.02531
= 3.64 years.
We notice that the results by the two methods differ by only 0.03 of a
year or about 11 days.
21. Compound discount at a discount rate. In Art. 17 we defined
the compound discount on the sum S as S P, the difference between S
and its present value P. The present value P has been found at the effec-
tive rate i% and at the nominal rate (7, m ) to be
P = 5(1 + i)- n = 5(1 + j/ro)-~*.
We may also find the present value P for a given discount rate. If the
discount rate is d convertible annually, we have from (10) Art. 9 that
d = i/(l + i) and 1 + i = 1/(1 d). Hence we have
/> = 5(1 + 1)-" = S(l-d) n (12)
as the present value of a sum S due in n years at the effective discount
rate d. The compound discount on S is
D = S ~ P = S - S(l - d) n = S[l ~ (1 - d) n ]. (13)
If the discount is converted m times a year at the nominal rate /, the
corresponding effective rate is the discount on $1 in 1 year. We shall find
the relation between d and /.
1
m 1 m periods
Consider $1 due at the end of 1 year (m conversion periods). Its
value at the end of the first discount period is 1 f/m. Its value at the
end of the second discount period is (1 //m) 2 , and at the end of the wth
discount period, that is at the beginning of the year, is (1 f/m) m . But
by Art. 7 its present value is 1 d. Therefore, we have
(14)
as the equation that expresses the relation between the nominal and
54 Financial Mathematics
effective rates of discount. This is similar to (4) Art. 16, which shows the
relation between the nominal and effective rates of interest.
Further, we have upon substituting in (12)
P = S(l - d) n = S(l - f/m) mn (15)
as the present value of a sum S due in n years discounted at a nominal
rate of discount / convertible m times a year. Immediately we have the
corresponding compound discount
D =: s - P = S[i - (1 - //m) mn ]. (16)
22. Summary of interest and discount. Let P be the principal and
P S
n i n years.
S be the accumulated value or amount of P at the end of n years. Then:
I. Simple interest and discount.
1. At simple interest rate i:
P = T ^~~. S = P(l + ro).
1 + ra
2. At simple discount rate d:
P - 5(1 - nd) S =
1 -nd
In each case
3. S P = simple interest on P for n years.
= simple discount on S for n years.
Combining 1 and 2 we obtain
d 7 *'
4. e = ; d =
1 nd 1 + ro ;
II. Compound interest and discount.
1. At effective rate of interest i:
2. At nominal rate of interest ( j,m) :
P = S(l + j/m)~ S = P(l
Compound Interest and Compound Discount 55
3. At effective rate of discount d:
P = S(l -d) n S =
4. At nominal rate of discount (/,m) :
p = s(l - f/m) mn S =
Combining 1 and 2 we obtain
5. 1 + t = (1+j/w)" 1 .
Combining 3 and 4 we obtain
6. l-d= (l-// m )*
In each case
7. S P = compound interest on P for n years.
= compound discount on S for n years.
Problems
1. A debt of $1 ,500 is due without interest in 5 years. Assuming an interest rate of
5%, find the value of the debt (a) now, (b) in 3 years, (c) in 6 years.
2. Solve Problem 1, assuming that the debt draws 6% interest convertible semi-
annually.
3. A debt of $500, drawing 6% interest will be due in 4 years. Another debt of $750,
without interest will be due in 7 years. Assuming money worth 5%, compare the debts
(a) now, (b) 4 years from now, (c) 6 years from now.
4. Set up the equation of value for Example 2, Art. 19, assuming now as the focal
date and show that the equation is equivalent to the one used in the solution of the
example.
6. A person is offered $2,500 cash and $1,500 at the end of each year for 2 years.
He has a second offer of $3,100 cash arid $800 at the end of each year for 3 years. Assum-
ing that money is worth 6% to him, which offer should he accept?
6. A owes B debts of $1,000 due at the end of each year for 3 years without interest.
A desires to settle with B in full now and B agrees to accept settlement under the assump-
tion that money is worth 5%. How much does A pay to B1
7. (a) In Problem 6 find the value of the debts 3 years from now, assuming 5%
interest, (b) Also, find the present value of this result, assuming money worth 5%.
(c) How does the result of (b) compare with the answer to Problem 6? Explain your
results.
8. Smith owes Jones $1,000 due in 2 years without interest. Smith desires to dis-
charge his obligation to Jones by making equal payments at the end of each year for
3 years. They agree on an interest rate of 6%. Find the amount of each payment.
9. A man owes $600 due in 4 years and $1,000 due in 5 years. He desires to settle
these debts by paying $850 at the end of 3 years and the balance at the end of 6 years.
Assuming money worth 6%, find the amount of the payment to be made at the end
of 6 years.
56 Financial Mathematics
10. Solve Problem 9, assuming that the debts draw 5% interest.
11. A man owes $2,000 due in 2 years and $3,000 due in 5 years, both debts with
interest at 5%. Find the time when the two obligations may be paid in a single sum
of $5,000, if money is worth 6%, converted semi-annually.
12. A owes B $200 due now, $300 due in 2 years without interest, and $500 due in
3 years with 4% interest. What sum will discharge the three obligations at the end of
lYi years if money is worth 6%, converted semi-annually?
13. There are three debts of $500, $1,000, and $2,000 due in 3 years, 5 years and 7
years respectively, without interest. Find the time when the three obligations could be
paid in a single sum of $3,500, money being worth 5%.
14. Solve Problem 13, making use of the approximate formula, (!!')
16. Money being worth 6%, find the equated time for paying in one sum the follow-
ing debts: $400 due in 2 years, $600 due in 3 years, $800 due in 4 years and $1,000 due
in 5 years. Choose 2 years from now as focal date and set up an equation of value as
in Example 1, Art. 12. Check the results by making use of the approximate formula.
16. Assuming money worth 5% show that $500 now is equivalent to $670.05 six years
from now. Compare these two values on a 6% interest basis.
17. Show that:
I J_
ft. T t b. = - (17)
m i-L m i+i
m m
18. Find the values of (j, 2) and (f, 2) that correspond to i = 0.06.^
19. A money lender charges 3% a month paid in advance for loans. What is tho
corresponding nominal rate of interest? What is the effective rate?
20. I purchase from the Jones Lumber Company building material amounting to
$1,000. Their terms are "net 60 days, or 2% off for cash." What is the highest rate of
interest I can afford to pay to borrow money so as to pay cash?
21. If a merchant's money invested in business yields him 2% a month, what dis-
count rate can he afford to grant for the immediate payment of a bill on which he quotes
"net 30 days"?
22. Find the nominal rate of interest convertible quarterly that is equivalent to
(j = .06, m = 2).
Hint. The two nominal rates arc equivalent if they produce the same effective rate.
Let i represent this common effective rate. Then 1 -f i - (1 -f .03) 2 = (1 -f- J/4) 4 .
23. Find the nominal rate of interest convertible semi-annually that is equivalent to
(j = .06, m = 4).
24. If $2,350 amounts to $3,500 in 4% years at the nominal rate (j, 4), find j. Solve
(a) by interpolation, and (b) by logarithms.
25. How long will it take a sum of money to double itself at (a) i = .06, (b) (j = .06,
m = 2), (c) (j - .04, m - 2)?
26. A man bought a house for $4,000 and sold it in 8 years for $7,000. What interest
rate did he earn on his investment?
CHAPTER III
ANNUITIES CERTAIN
23. Definitions. An annuity is a sequence of equal payments made at
equal intervals of time. Strictly speaking, the word "annuity" implies
yearly payments, but it is now understood to apply to all equal periodic
payments, whether made annually, semi-annually, quarterly, monthly,
weekly, or otherwise. Typical examples of annuities are : monthly rent on
property, monthly wage of an individual, premiums for life insurance,
dividends on bonds, and sinking funds.
An annuity certain is one whose payments extend over a fixed number
of years. A contingent annuity is one whose payments depend uponj/he
happening of some event whose occurrence cannot be accurately foretold.
The payments on a life insurance policy constitute a contingent annuity.
In this chapter we shall be concerned entirely with annuities certain, i
The time between successive payments is called the payment period.
The time from the beginning of the first payment period to the end of the
last payment period is called the term of the annuity.
Annuities certain may be classified into three groups: Ordinary annui-
ties, annuities due, and deferred annuities. An ordinary annuity is one
whose first payment is made at the end of the first payment period. If the
first payment is made at the beginning of the first payment period, the
annuity is called an annuity due. If the term of the annuity is not to begin
until some time in the future, the annuity is called a deferred annuity.
The periodic payment into an annuity is frequently called the periodic
rent. The sum of the payments of the annuity which occur in a year is
called the annual rent.
Illustration. A sequence of payments of $100 each, at the end of each
quarter for 3 years, constitutes an annuity whose payment period is one-
fourth of a year. The term begins immediately (one quarter before the
first payment) and ends at the close of three years. The periodic rent is
$100 and the annual rent is 4($100), or $400. This annuity is pictured in
the line diagram.
8888
4 -H T-4 i <
8
8 8
T-l r-l
8
888
8
7
1
2
3 years
J
I
''
57
1
58 Financial Mathematics
There are four general cases of ordinary annuities to which we shall give
especial consideration. They are briefly described by the outline:
A. Annuity payable annually.
I. Interest at effective rate i.
II. Interest at nominal rate 0, w).
B. Annuity payable p times a year.
I. Interest at effective rate i.
II. Interest at nominal rate (j, m).
A. ANNUITY PAYABLE ANNUALLY
24. Amount of an annuity. The sum to which the total number of pay-
ments of the annuity accumulate at the end of the term is called the amount, or
the accumulated value, of the annuity. We shall illustrate.
Example 1. $100 is deposited in a savings bank at the end of each
year for 4 years. If it accumulates at 5% converted annually, what is the
total amount on deposit at the end of 4 years?
Solution. Consider the line diagram.
100 100 100 100
i : 1 1 1 1
01234
It is evident that the first payment will accumulate for 3 years. Hence
its amount at the end of 4 years will be $100(1. 05) 3 .
The second payment will accumulate for two years, and its amount
will be $100(1.05) 2 , and so on.
Hence, the total amount at the end of 4 years will be given by
$100(1.05) 3 + $100(1.05) 2 + $100(1.05) + $100
or $100 + $100(1.05) + $100(1.05) 2 + $100(1.05) 3 . (1)
We may compute the above products by means of the compound interest
formula; their sum will be the amount on deposit at the end of 4 years.
However, we notice that (1) is a geometric progression, having 100 for the
first term, (1.05) for the ratio, and 4 for the number of terms.
m^ , A 100[(1.05) 4 - 1]
Therefore, Amount = ~ - (2)
.05
Annuities Certain
59
Evaluating (2) by means of Table III, we have
100[(1.05) 4 - 1] 100(1.2155062 - 1)
.05
.05
= 431.01.
Hence, the amount of the above annuity is $431.01.
The arithmetical solution of the above example may be tabulated as
follows :
End of
Year
Annual
Deposit
Interest
Total Increase
in Deposit
Total on
Deposit
1
$100 00
$100 00
$100 00
2
100.00
$ 5.00
105.00
205.00
3
100.00
10.25
110.25
315.25
4
100.00
15.76
115.76
431.01
Totals
$400.00
$31.01
$431.01
We shall now find the amount of an annuity of $1 per annum for n
years at an effective rate i. The symbol s^ is used to represent the amount
of an annuity of 1 per annum payable annually for n years at the effective
rate i. The first payment of 1 made at the end of the first year will be at
interest for n 1 years and will accumulate to (1 + i)"" 1 .
The second payment of 1 will be at interest for n 2 years and will
accumulate to (1 + i) n ~ 2 .
The third payment of 1 will be at interest for n 3 years and will
accumulate to (1 + t) n ~ 3 > an d so on.
The last payment will be a cash payment of 1. We have then
Sn\i = (1 + *') n ~ 1 + (1 + t) W " 2 + (1 + i) n ~ 3 + . . . + (1 + t) + 1
= 1 + (1 + i) + (1 + i) 2 + - + (1 + i)- 2 + (1 + t) 1 . (3)
This is a geometric progression of n terms, having 1 for first term and
(1 + i) for ratio. Finding the sum (Alg. : Com. Slot.,* Art. 60), we have f
If the annual rent is R and if S represents the amount, we have
(5)
* When it is not desired to emphasize the interest rate, this symbol is frequently
written s^.
f Sec page x of this text for a list of formulas from Alg.: Com. Stat.
60 Financial Mathematics
Example 2. Find the amount of an annuity of $200 per annum for 10
years at 5% converted annually.
Solution. Here, R = $200, n = 10, and i = 0.05. Substituting in
(5), we get
S = 200. % ,. 06 = 200^1^^.
In Table V we find the amount of an annuity of 1 per period for n
periods at rate i per period.
When n = 10 and i = 0.05, we find
= 12.57789254
and S = 200(12.57789254) = 2515.58.
Hence, the amount of the annuity is $2515.58.
Exercises
Find the amount of the following annuities:
1. $300 per year for 10 years at 4% interest converted annually.
2. $500 per year for 20 years at 5% converted annually.
3. $200 per year for 6 years at 3% converted annually. Make a schedule showing
the yearly increases and the amount of the annuity at the end of each year.
4. $150 per year for 10 years at 6% converted annually.
6. In order to provide for the college education of his son, a father deposited $100
at the end of each year for 18 years with a trust company that paid 4% effective. If
the first deposit was made when the son was one year old, what was the accumulated
value of all the deposits when the son was 18 years old?
6. A corporation sets aside $3,700 annually in a depreciation fund which accumulates
at 5%. What amount will be in the fund at the end of 15 years?
7. Write series (3) in the summation notation. (Alg.: Com. Slat., Art. 63.)
8. If $1,000 is deposited at the end of each year for 10 years in a fund which ia
accumulated at 4% effective, what is the amount in the fund 4 years after the last
deposit?
9. To create a fund of $5,000 at the end of 10 years, what must a man deposit at the
end of each year for the next 10 years if the deposits accumulate at 4% effective?
10. One man places $4,000 at interest for 10 years; another deposits $500 a year in
the same bank for 10 years. Which has the greater sum at the end of the term if interest
is at 4% effective?
Let us now find the amount of an annuity where the payments are
made annually but the interest is converted more than once a year. We
shall illustrate by an example.
Annuities Certain 61
Example 3. $100 is deposited in a savings bank at the end of each year
for 4 years. If it accumulates at 5% converted semi-annually, what is the
total amount on deposit at the end of 4 years?
Solution. Consider the line diagram.
100 100 100 100
I , 1 , 1 , 1 , 1
01234
It is evident that the first deposit will accumulate for 3 years and at
the end of 4 years, ((5), Art. 16), will amount to $100(1.025) 6 .
The second payment will amount to $100(1. 025) 4 , and so on.
Hence, the total amount at the end of 4 years will be given by
$100(1.025) 6 + $100(1.025) 4 + $100(1.025) 2 + $100
or $100 + $100(1.025) 2 + $100(1.025) 4 + $100(1.025). (6)
We notice that (6) is a geometrical progression, having 100 for the
first term, (1.025) 2 for ratio, and 4 for the number of terms. Substituting
in (8) Art. 60, Alg.: Com. Slot., we have
100[(1.025)8 - 1]
S- Amount= (U)25)a _ j - (7)
It is evident that Table V cannot be used here, but we may use Table III.
100(1.21840290 - 1)
Thus, S =
1.05062500 - 1
100(0.2184029)
0.050625
By writing (7) in the form
fl-ioo/ 1 - 025 ^- 1 ' 25
.025 (1.025) 2 - 1 '
we can identify the last two terms in the product as % 25 an d
Then
S = 100.
= 100(8.7361 1590) = 431.41
as was obtained by the first method.
Hence, the amount of the annuity is $431.41.
62
Financial Mathematics
The arithmetical solution of the above example may be tabulated as
follows:
End of
Year
Annual
Deposit
Interest
Total Increase
in Deposit
Total on
Deposit
H
1
$100 00
$100 00
$100 00
\y>
$2.50
2 50
102 50
2
2 1 A
100.00
2.56
5 13
102.56
5 13
205.06
210 19
3
$y>
100.00
5.25
7 89
105.25
7 89
315.44
323 33
4
100.00
8.08
108.08
431.41
We notice that the amount in Example 3 is 40 cents more than the
amount in Example 1. This is due to the fact that the interest is con-
verted semi-annually in Example 3 and only annually in Example 1.
If the interest is converted m times per year, we may substitute, [(4)
Art. 16], ( 1 + ^ ) for (1 + i) and ( 1 + ^ j - 1 for i in (5) and obtain
\ m/ \ m/
S = R
m
(8)
^
We can transform (8) into a form involving the annuity symbol s^ by
writing it in the form
fif-i
ml
= R
m
m
m
- 1
<8a)
Example 4. Find the amount of $200 per annum for 10 years at 5%
converted quarterly.
Annuities Certain 63
Solution. Here, R = $200, n = 10, j = 0.05, and m = 4. Sub-
stituting in (8a), we have
S = 200. 5^.0125
200(51.4895 5708)
4.075G 2695
= 2,526.71
Hence the amount is $2,526.71.
Why is the amount in Example 4 greater than the amount in Example 2?
Exercises
Find the amount of the following annuities:
1. $300 per year for 8 years at 6% interest, converted semi-annually.
2. $250 per year for 25 years at 5% converted quarterly.
3. $500 per year for 5 years 4% converted semi-annually. Make a schedule showing
the increases each six months and the amount of the annuity at the end of each six
months.
4. $600 per year for 30 years at 4^% converted semi-annually.
6. $750 per year for 15 years at 4.2% converted semi-annually. (Hint: Use loga-
rithms to evaluate (1.021) 30 .)
6. On the first birthday of his son a father deposits $100 in a savings bank paying
3H% interest, converted semi-annually. If he deposits a like amount on each birthday
until the son is 21 years old, how much will be on deposit at that time?
7. A man deposits $1,000 at the end of each year in a bank that pays 4% effective.
Another man deposits $1,000 at the end of each year in a bank that pays (j = .035,
m =2). At the end of 10 years how much more does the first man have than the second?
8. A man deposited $1,000 a year in a bank. At the end of 15 years he had $19,000.00
to his credit. What effective rate of interest did he receive? Solve by interpolation.
9. Solve r Exercise 1 with the interest converted quarterly.
10._Solve Exercise 1 with the interest converted monthly.
11. Set up the series for the amount of an annuity of R at the end of each year for
n years with interest at the nominal rate (j, m). Sum this scries by (9) Art. GO, Alg.:
Com. StaL, and thus obtain (8), Art. 24.
25. Present value of an annuity. The present value of an annuity is
commonly defined as the sum of the present values of all the future payments.
Suppose an individual is to receive R dollars each year as an ordinary
annuity and the payments are to last for n years. The individual may
64 Financial Mathematics
do any one of three things with this annuity: (a) He may spend the pay-
ments as they are received; (6) accumulate the payments until the end
of the last rent period (n years) ; (c) or sell the future payments to a bank
(or similar institution) at the beginning of the first rent period.
If the same rate of interest is used to accumulate the payments as is
used by the bank (or similar institution) in finding the present value of the
future payments, it is evident that the sum (present value) paid to the
individual by the bank at the beginning of the first rent period is equivalent
to the present value of the sum to which the future payments will accumu-
late by the end of the last rent period. Consequently, we may also define
the present value of an annuity as that sum, which, placed at interest at a
given rate at the beginning of the first rent period, will accumulate to the amount
of the annuity by the end of the last rent period. Thus, it is the discounted
value of S.
Example 1. It is provided by contract that a young man receive $500
one year from now and a like sum each year thereafter until 5 such pay-
ments in all have been received. Not wishing to wait to receive these pay-
ments as they come due, the young man sells the contract to a bank.
If the bank desires to invest its funds at 6% interest compounded annually,
how much does the young man receive now for his contract?
Solution.
A
S
i
500 500
i 1
500 500
, i. - i
500
1
The first payment is made one year from now and has a present value
of $500(1.06) - 1 .
The second payment is due two years from now and has a present value
of $500(1.06)~~ 2 , and so on until the last payment which has a present
value of $500(1.06) ~ 5 . Summing up, we have
Present value = $500(1.06) - 1 + $500(1.06) ~ 2 + . . . + $500(1.06) ~ 5 . (9)
We notice that (9) is a geometrical progression having 500(1. 06) ~ 1 for
the first term, (1.06)- 1 for ratio, and 5 for the number of terms. Sub-
stituting in (8), Art. 60, Alg.: Com. Stat. we find
A ^ , 500(1.06) -^(l.OB) ~ 5 - 1]
A = Present value =
(l.OoJ 1
Annuities Certain 65
Multiplying the numerator and denominator of the above expression
by (1.06),
500[(1.06)- 5 - 1]
1 - (1.06)
1 - (1.06)
A = Present value =
0.06
A = 500(4.21236379) [Table VI]
A = $2,106.182.
If the young man had waited to receive the payments as they became
due and immediately invested them at 6% converted annually, his invest-
ments at the end of 5 years would have amounted to
s = 500 -' = $2,818.546.
U.Uo
We notice that $2,818.546 is the amount of $2,106.182 for 5 years at
6%. For
$2,106.182(1.06) 5 = 2,106.182(1.33822558) = $2,818.546.
We shall now find the present value of an annuity of $1 per annum for
n years at the effective rate i. The symbol a^, or a^ is used to represent
the present value of this annuity. To find this value, we shall discount
each payment to the beginning of the term.
Ill 11
- 1 - 1 - 1 - 1 - 1
1 2 3 n-ln
The first payment of 1 made at the end of the first year when dis-
counted to the present, by Art. 17, has the present value of (1 + i)" 1 -
Similarly, the second payment when discounted to the present has a present
value of (1 + i)~ 2 . And so on for the other payments. We then have
a m = (1 + i)- 1 + (1 + i)- 2 + (1 + i)~ 3 ++ (1 + 0-" (10)
This is a geometric progression in which a = (1 + i)" 1 * r = (1 + i)" 1 ,
I = (i + ;)-*. Finding the sum (Alg.: ComStat., Art. 60), we obtain
1 - (1 + 0"*
The functions a^ and s m are the two most important annuity func-
tions. We frequently write them a^ and 8&.
66 Financial Mathematics
Formula (11) may be easily derived from (5) Art. 24. For a m is, by
definition, the discounted value of s^ t . That is,
z &
If the annual rent is jffi, payable at the end of each year for n years, and
if A represents the present value,
A = R-a^ = /?*"" (1 f +g) " (12)
If the interest is at the nominal rate (j, m), using the relation (4)
Art. 16,
we find
/ - \
1 - ( 1 + ~
, (13)
m
which is easily reduced to
A = fl-fl^y - (13a)
m
26. Relation between and .
We have a m (1 + i) n = s~ nl [Art, 25]
and (1 + z) n
Substituting for (1 + i) n in the equation
: = s^ii we have
Multipl^ng through by i and dividing through by s^, we find
1 1
or = -- i. (14)
Annuities Certain 67
Table VII gives values for According to (14), values for are
a ni SST
obtained by subtracting the rate i from the table values of Thus,
to find l/ssoi.04, we 1^ U P Table VII and obtain l/a^ M = 0.0735 8175
Using relation (14), we find
= 0.0735 8175 - .04 = 0.0335 8175.
S 2oj.04
27. Summary. Formulas of an ordinary annuity of annual rent R
payable annually for n years.
I. Interest at effective rate i.
2. A = R
II. Interest at nominal rate (j, m).
/ n \ ran
. S-R
1 -
2. A = R
where
J. ___ !_ _ .
28. Other derivations of a^, and s^. We have derived the formulas for
0% and SHI by setting up series and then finding their sums by the formula
for summing a geometric progression. It is of great value to derive the
68 Financial Mathematics
formulas by a method called "direct reasoning" by some authorities, or
"verbal interpretation" by other authorities.
Consider $1 at 0. Its value at the end of n years is (1 + i) n .
i d+0 n
i - 1 - 1 - j - 1
1 2 3 n
Also, from another point of view, $1 at will produce an annuity of i
at the end of each year for n years and leave the original principal intact
at the end of n years. For, at the end of the first year the amount is
(1 + i). Deposit the i into a separate account, and let the original prin-
cipal $1 again earn interest. It amounts to (1 + i) at the end of the
second year. We again deposit the i in the second account, and let the
principal $1 again earn interest. We continue this for n years. We thus
find that $1 at is equivalent to an annuity of i for n years plus the original
principal $1 at n. In other words,
1 at = [an annuity of i for n years] + 1 at n.
i i i t+1
1 - 1 - 1 - 1 - 1
1 2 3 .... n
Let us now focalize all sums at the end of n years. Then
(l + z) = z^ + l,
or, solving for s^,
If we focalize all sums at the present, 0, we have
or,
Exercises
1. An individual is to receive an inheritance of $1 ,000 at the end of each year for
15 years. If money is worth 5% effective, what is the present value of the inheritance?
2. Find the present value of an ordinary annuity of $1,000 a year for 12 years at
(j = .05, m = 2).
3. How much money, if deposited with a trust company paying (,;' = .04, m = 2), is
sufficient to pay a person $2,000 a year for 20 years, the first payment to be received 1
year from the date of deposit?
Annuities Certain 69
4. An article is listed for $2,000 cash. A buyer wishes to purchase it in four equal
annual installments, the first to be made 1 year from the date of purchase. If money is
worth 6%, what is the amount of each installment?
6. A house was purchased for $12,000, of which $3,000 was cash. The balance was
paid in 10 equal annual installments which began one year from the date of purchase.
If money is worth (j - .06, m = 2), find the amount of each installment.
6. A house is offered for sale on the following terms: $1,000 down, and $500 at the
end of each year for 10 years. If money is worth 6%, what is a fair cash price?
7. Prove: sn + s^\ -f sj] -f -f H| ~ r 2
8. Prove: ==^^.
10. Evidently $1 at is equivalent to an annuity of l/a%\ at the end of each year for
n years since the present value of the annuity is 1. Use this fact with Art. 28 to prove that
J_ _ 1
11. Show that a^FHl = ai| + (1 + *)~ m al = ail + (1 + O" 11 ^-
(a) by verbal interpretation. Draw line diagram.
(b) algebraically.
12. Find the value of ai2o) >0 4 by using the relation in Exercise 11.
13. Show that s^+Til = (1 + t) n m| + *j = (1 + i) m Hl + *rn|-
14. Find the value of si25|.o4 by using the relation in Exercise 13.
15. What do the formulas in Exercises 11 and 13 become if m = 1?
B. ANNUITY PAYABLE p TIMES A YEAR
29. Amount of an annuity, where the annual rent, /?, is payable in p
equal installments. In Art. 15, we derived the value of the compound
amount of $1 for n years, (1 + i) M , for integral values of n. We shall
assume this relation to hold for fractional as well as for integral values of
n. Consider
Example 1. $50 is deposited in a savings bank at the end of every six
months for 4 years. If it accumulates at 5% interest, converted annually,
what is the total amount on deposit at the end of 4 years?
Solution.
50 50
50 50
50 50
50 50
. i
, i
, i
1
2
3
4
70 Financial Mathematics
The first deposit of $50 is made at the end of six months and accumu-
lates for 3^A years. At the end of 4 years it will amount to $50(1.05)**.
The second deposit of $50 is made at the end of the first year and will
amount to $50(1. 05) 3 at the end of 4 years.
The third deposit of $50 will amount to $50(1.05)** at the end of 4 years,
and so on.
Next to the last deposit will be at interest six months and will amount
to $50(1.05)** at the end of 4 years and the last deposit will be made
at the end of 4 years and will draw no interest.
Hence, the total amount at the end of 4 years will be given by
$50(1.05)^ + $50(1.05) 3 + . . . + $50(1.05)* + $50;
or $50 + $50(1.05)* + $50(1.05) + . . . + $50(1. 05)* 4 (15)
We notice that (15) is a geometrical progression having 50 for first term,
(1.05)* for ratio, and 8 for the number of terms.
Substituting in (8), Art. 60, Alg.: Com. Stat., we have
* A . 50[(1.05) 4 - 1]
S = Amount^ (L05) * _ x '
Using Table III and Table VIII, we have
50(1.21550625 - 1)
1.02469508 - 1
S = Amount =
Hence, the amount of the above annuity is $436.34.
Let us now find the amount of an annuity of $1 per annum, payable in
p equal installments of l/p at the end of every pth part of a year for n
years at rate i t converted annually.
To assist him in following this discussion the student should draw a line
diagram.
The amount of an annuity of $1 per annum, payable in p equal install-
ments at equal intervals during the year, will be denoted by the symbol,
8$. If the interest is converted annually, and i is the rate, s$ can be
expressed in terms of n, i, and p as follows: At the end of the first pth
part of a year, l/p is paid. This sum will remain at interest for (n l/p)
years and will amount to l/p(l + i) n ~ 1/p .
The second installment of l/p will be paid at the end of the second
pth part of a year and will be at interest for (n 2/p) years, amounting to
Annuities Certain 71
+ i) n ~~ 2/p at the end of n years, and so on until np installments
are paid.
Next to the last installment will be at interest for one pth part of a
year and will amount to l/p(l + i) l/p .
The last installment will be paid at the end of n years and will draw no
interest. Adding all of these installments, beginning with the last one,
we have
s% = ~ + - (1 + i) l/p + - (1 + i) 2/p + ...+- (1 + i) n ~ l/p * (16)
P P P P
We notice that (16) is a geometrical progression having 1/p for first
term, (1 + t) 1/p for ratio, and np for the number of terms. Substituting
in (8), Art. 60, Alg.: Com. Stat., we have
If the annual rent is R, we have
For convenience in evaluating, (18) may be written, (4') Art. 16,
(1 _L fin _ I I
or S^R(s^-r)- (19a)
Table X gives values of
JP
Example 2. Find the amount of an annuity of $1,200 per year paid
in quarterly installments of $300 for 7 years if the interest rate is 5% con-
verted annually.
Solution. Here, R = $1,200, n = 7, p = 4, and i = 0.05. Substitut-
ing in (19), we have
a-VM ^- 1 ^-..--MM,'
0.05
Using Table V and Table X, we have
S - 1,200(8.14200845) (1.01855942) = 9,951.74.
Hence, the amount of the above annuity is $9,951.74.
72 Financial Mathematics
Example 3. Find the amount of an annuity of $200 per year paid in
semi-annual installments for 10 years, the interest rate being 4.3% con-
verted annually.
Solution. Here, R = $200, n = 10, p = 2, and i = 0.043. The rate,
4.3%, is not given in our tables. We will evaluate by means of logarithms,
using (18).
(1.043) l - 1
log 1.043 = 0.0182843 (Table II.)
10(log 1.043) = 0.1828430
(1.043) 10 = 1.5235 (Table I.)
^(log 1.043) = 0.0091422
(1.043) H = 1.021274 (Table II.)
200(1.5235 - 1) 100(0.5235) , M ^
* = 2(1.021274 - 1) = -002i27T = 2 - 460J5 -
Consequently, the amount of the above annuity is $2,460.75, and it is
accurate to five significant digits. That is, the exact value is between
$2,460.75 and $2,460.65.
Exercises
Find the amount of the following annuities:
1. $300 per year paid in semi-annual installments for 10 years at 4% interest con-
verted annually.
2. $500 per year paid in quarterly installments for 20 years at 5% converted annually.
3. $50 per month for 10 years at 4% interest converted annually.
4. $250 at the end of every six months for 15 years at 4H% converted annually.
Evaluate by logarithms, using (18), and then check the result by using Tables V and X.
6. $100 quarterly for 12 years at 3}% converted annually.
6. A young man saves $50 a month and deposits it each month in a savings bank
for 25 years. If the bank pays 3H% interest, converted annually, how much does he
have on deposit at the end of the 25 years?
7. Solve Exercise 1, if it were paid in quarterly installments. Is the answer more or
less than the answer of Exercise 1? Explain the difference.
8. Solve Exercise 2, if it were paid in semi-annual installments. Is the answer more
or less than the answer of Exercise 2? Explain the difference.
9. $100 is deposited in a savings bank at the end of every 3 months. If it accumu-
lates at 3% converted annually, how much is on deposit et the end of 4 years? Solve
fundamentally as a geometrical progression.
Annuities Certain 73
Let us now find the amount of an annuity paid in p equal installments
each year where the interest is converted more than once a year. We will
illustrate by an example.
Example 4. $25 is deposited in a savings bank at the end of every
three months for 4 years. If it accumulates at 5% interest, converted
semi-annually, what is the total amount on deposit at the end of 4 years?
Solution. The first deposit of $25 is made at the end of three months
and is at interest for 3% years. At the end of 4 years it will amount to
$25(1.025) l . [(5), Art. 16.]
The second deposit of $25 is made at the end of six months and is at
interest for 3J^ years. At the end of 4 years it will amount to
$25(1.025) 7 .
The third deposit of $25 is made at the end of nine months and is at
interest for 3J years. At the end of 4 years it will amount to
$25(1.025) I% , and so on.
Next to the last deposit of $25 will be at interest for % year and will
amount to $25(1.025)*.
The last deposit of $25 is made at the end of 4 years and draws no interest.
Hence, the total amount on deposit at the end of 4 years will be given by
$25(1.025) lf4 + $25(1.025) 7 + + $25(1.025)* + $25
or $25 + $25(1.025)* + $25(1.025) + . . . + $25(1.025)^. (20)
We notice that (20) is a geometrical progression having 25 for first term,
(1.025)* for ratio, and 16 for the number of terms. Substituting in (8),
Art. 60, Alg.: Com. Stat., we have
25{[(1.025)*P 6 ~1}
8 - Amount -
_ 25K1.025)* ~ 1]
S ~ (1.025)* - 1
Using Table III and Table VIII, we have
25(1.21840290 - 1)
S = Amount =
1.01242284 -
_ 25(0.21840290) _
0.01242284
~ } for (1 + i) in (18) and obtain
( \ mn
1 + ~ I ~"
7717
74 Financial Mathematics
Hence, the amount of the above annuity is $439.50.
If the interest is converted m times per year, we may substitute
(22)
Let us consider further equation (21). Here p = 4, m = 2, and hence
p/m is an integer. We may write (21) in the form
100 [(1.025) 8 - 1] .025
2 .025 " 2[(1.025)^ - 1] "
.025
J2
P
When is an integer, (22) becomes
m
s = * . ._,. . (22a)
m ~ j p at rate j
" m
When p/m is an integer, (22) can easily be reduced to (22a) in which
case we may apply Tables V and X. This transformation simplifies the
arithmetical computation since >S is expressed as a continued product.
Example 5. Find the amount of an annuity of $600 per year paid in
quarterly installments for 8 years, if the interest rate is 5% converted
semi-annually.
Solution. Here, R = $600, n = 8, p = 4, m = 2, and j = 0.05. Sub-
stituting in (22), we have
(1.025) 16 - 1
4[(1.025) ?/4 -
(1.025) 16 -
600
30 2[(1.025)*-1]
n n9.^i6 _
300
0.025 2K1.025)* - 1]
Using Table V and Table X, we find
S = 300(19.38022483) (1.00621142) = $5,850.18.
Annuities Certain 75
Example 6. Solve Example 5, if the interest is converted quarterly.
Solution. Here, R = $600, n = 8, m = p = 4, and j = 0.05. Sub-
stituting directly in (22),
= 150(39.05044069) (Table V)
= $5,857.57.
Example 7. Solve Example 5, if the payments are made semi-annually
and the interest is converted quarterly.
Solution. Here, R = $600, n = 8, p = 2, m = 4, and j = 0.05.
Substituting in (22), we have
(1.0125) 32 - 1
S = 600 ;
= 300
(1.U125J* 1
1 /t812O51 "H
(Table III)
2[(1.0125)* -
(1.0125) 32 - 1
(1.0125) 2 - 1
300(1.48813051 - 1
1.02515625 -
300(0.48813051)
0.02515625
- $5,821.18.
In this example, m/p is an integer. When this is true we can write S
as a product. Thus,
600 (1.0125) 32 - 1 .0125
S2\
in which Tables V and VII may be applied. In terms of annuity symbols,
it is of the form
<S = ~-<Wi/ ~ ' (226)
P m SmU
p\m
When m/p is an integer, (22) can easily be reduced to (22b).
Formula (22) is our most general formula for finding the amount of an
annuity. The other forms (5), (8), and (18) are special cases of (22).
Thus,
76 Financial Mathematics
If m = p = 1, (22) reduces to
S = R ~ (5)
If p = 1 and m > 1, (22) reduces to
+ m
- 1 )
m/
If m = 1 and p > 1, (22) reduces to
If w = p, (22) reduces to
\np
(
1
- 1
(23)
We observe that (23) is of the same form as (5), where n is replaced
by np, i by j/p, and R by #/p.
In solving annuity problems the student should confine himself to the
use of the fundamental formulas. Thus, if his problem requires that he
find the amount of an annuity, he should use (5), (8), (18), or (22), and then
effect the necessary transformation to reduce it to the annuity symbols that
will entail the least amount of labor in obtaining the numerical result.
Exercises
1. A man deposits $150 in a 4% savings bank at the end of every three months.
If the interest is converted semi-annually, what amount will be to his credit at the
end of 10 years?
2. Solve Exercise 1, with the interest converted (a) annually, (b) quarterly.
3. A man wishes to provide a fund for his retirement and begins at age 25 to deposit
$125 at the end of every three months with a trust company which allows 3% interest
converted semi-annually. What will be the amount of the fund at age 60?
4. Solve Exercise 3, with the interest converted quarterly.
Annuities Certain
77
6. Fill out the following table for the amount of an annuity of $300 per year for
12 years at 4%:
Annuity
Payable
Interest Convertible
Annually
Semi-annua!ly
Quarterly
Annually
Semi-annually
Quarterly
6. Solve Exercise 5, with the rate of interest 5%.
7. Solve Exercise 5, with the rate of interest 4J^ %.
8. Find the amount of an annuity of $400 a year for 7 years at 7% interest con-
verted semi-annually. Solve fundamentally as a geometrical progression using the
principle of compound interest.
9. Solve Exercise 8, with the interest converted annually.
10. Solve Exercise 8, with the annuity payable in quarterly installments and the
interest converted semi-annually.
11. $250 is deposited at the end of every six months for 10 years in a fund paying
4% converted semi-annually. Then, $150 is deposited at the end of every three months
for 10 years and the interest rate is reduced to 3% converted quarterly. Find the total
amount on deposit at the end of 20 years.
12. A man has $2,500 invested in Government bonds which will mature in 15 years.
These bonds bear 3% interest, payable January 1 and July 1. When these interest pay-
ments are received they are immediately deposited in a savings bank which allows
3V% interest converted semi-annually. To what amount will these interest payments
accumulate by the end of 15 years?
13. A man begins at the age of thirty to save $15 per month, and keeps all of his
savings invested at an average rate of 4% effective. How much will he have as a retire-
ment fund when he is sixty-five years old?
14. A man 25 years of age pays $41.85 at the beginning of each year for 20 years
for which he receives an insurance contract which will pay his estate $1,000 in case of
his death before 20 years and pay him $1,000 cash, if living, at the end of 20 years.
He also decides to deposit the same amount at the beginning of each year in a savings
bank paying 3% interest. Compare the value of the two investments at the end of 20
years. On the basis of 3% interest what would you say his insurance protection cost
for the 20 years?
16. A man deposits $150 in a savings bank on his twenty-fifth birthday and a like
amount every six months. If the bank pays 3% interest convertible semi-annually,
how much does he have on deposit on his sixtieth birthday?
16. Solve Exercise 15, with the interest converted quarterly.
17. A man, age 25, pays $24.03 a year in advance on a $1,000, 20-payment life policy.
If he should die at the end of 12 years, just before paying the 13th premium, how much
78 Financial Mathematics
would his estate be increased by having taken the insurance instead of having deposited
the $24.03 each year in a savings bank paying 4% effective?
18. Find the amount of an annuity of $200 a year payable in semi-annual install-
ments for 7 years at 4% converted annually. Solve fundamentally as a geometrical
progression.
19. Solve Exercise 18, with the interest converted quarterly.
20. Assume that R/p dollars is invested at the end of l/pth of a year, at nominal
rate j converted m times a year, and that a like amount is invested every pth part of a
year until np such investments are made ; sum up as a geometrical progression and there-
by derive the formula (22).
30. Present value of an annuity of annual rent, /?, payable in p equal
installments. In Art. 25 we considered the problem of finding the present
value of an ordinary annuity with annual payments. We are now ready
to consider the problem of finding the present value of an ordinary annuity
of annual rent, R, with p payments a year.
A S
I - 1 - 1 - 1 - 1 - 1
1 2 3 - - n - 1 n
We have found the amount S of such an annuity. When the interest
is at the effective rate i, the amount S is given by (18) ; when the interest
is at the nominal rate (j, m), the amount S is given by (22). If, as usual,
A designates the present value of the annuity, evidently
"' <>
if the interest is at the effective rate i, and
(i + L\ mn _
i \ mn \ m/
+ i) -S-fi-T7 ^r ~ (25)
if the interest is at the nominal rate (j, m).
Solving (24) and (25) respectively for A, we obtain
' - '*a"+V'-''ir <26 >
and
/ j \-rnn
l_h + L\
(27)
Annuities Certain 79
We shall leave it as an exercise for the reader to show that (26) can be
reduced to the form
A = R-a^-j"
JP
and that (27) can be reduced to the forms
. R m p . .
A = ' a '> 7' an mte s er ( 28fl )
" * fa 3 "i
j p at rate
R 1 m . x /rtrtl .
-4 = - -'055)2 > an integer. (286)
jT|wi
It is of great value to derive the fundamental formulas (26) and (27)
by discounting each payment R/p to the present and finding the sum of the
respective series. See p. 59. We shall set up the series and leave the details
of summation and simplification to the reader.
If the interest is at the effective rate i } the present value is
= - [
(1 + i)~ 1/p + (1 + i)~ 2/p + . . . + (1 + *)
which simplifies to the value given in (26).
If the interest is at the nominal rate (j, w), the present value is
7? f/ <j\-m/P / <f\-2m/j / -' \-mn~l
A = "1(1 + 1.) +(l + A) +... + ( 1 + A)
p L\ m/ V m/ \ m/ J
which simplifies to the value given in (27).
Again we would advise the student, when solving annuity problems, to
confine himself to the fundamental formulas. Thus, if his problem re-
quires that he find the present value of an annuity, he should use (12),
(13), (26), or (27), and then effect the necessary transformation to reduce
it to the annuity symbols that will entail the least amount of labor in
obtaining the numerical result.
31. Summary of ordinary annuity formulas.
S = The Amount of the Annuity.
A = The Present Value of the Annuity.
80 Financial Mathematics
A. Annuity of annual rent R payable annually for n years.
I. At the effective rate i.
1. S = R (1 + " ~ 1 = R sa (5)
i '
2. A = R l ~ (1 + f> " = R-Oin (12)
II. At the nominal rate (j, m).
mn
) -1
(8)
_!
m
2. A = R , , vn , -- (13)
JS. Annuity of annual rent R payable p times a year for n years.
I. At the effective rate i.
2 - ^ -
II. At the nominal rate (j, m).
m "
1
2- >i = *r7 - Tv - i- (27)
Annuities Certain 81
Example 1. Find the present value of an annuity of $600 per year
paid in quarterly installments for 8 years, if the interest rate is 5% con-
verted semi-annually.
Solution. Here, R = $600, n = 8, p 4, w = 2, and j = 0.05.
Substituting in (27), we have
JHL "~~
= 300
41(1.025)*- 1] 2[(1.025r - 1]
1 - (1.025) - 16 0.025
0.025 ' 2[(1.025) H - 1]
.025
=* 300-OJ6|.025 . , A0 _,
j2 at .o&o
= 300(13.05500266) (1.00621142) [Tables VI and X]
= $3,940.83.
Example 2. Solve Example 1, with the interest converted quarterly.
Solution. Here, R = $600, n = 8, m = p = 4, and j = 0.05. Sub-
stituting in (27),
1 - (1.0125) - 32
O.Q125
= 150(26.24127418) [Table VI]
= $3,936.19.
Example 3. Solve Example 1, if the payments are made semi-annually
and the interest is converted quarterly.
Solution. Here, R = $600, n = 8, p = 2, m = 4, and j = 0.05.
Substituting in (27), we have
2[(1.0125) H - 1]
300 \-^
1 ft fi71Q84.ft7^
[Tables III and IV]
0125) 2 - 1
300(1 - 0.67198407)
1.02515625 - 1
300(0.32801593)
0.02515625
= $3,911.74.
82
Financial Mathematics
We may also solve this example by writing A in the form
1 - (1.0125) ~ 32 .0125
A ~ ^0125 (1.0125) 2 - 1
and applying Tables VI and VII. We find
A = 300(26.2412 7418) (0.4968 9441)
= $3911.74.
Exercises
1. Find the present value of an annuity of $700 per year running for 15 years at 5%
converted annually.
2. Solve Exercise 1, assuming that the interest is converted semi-annually.
3. A piece of property is purchased by paying $1,000 cash and $500 at the end of
each year for 10 years without interest. What would be the equivalent price if it were
all paid in cash at the date of purchase, assuming money is worth 5J^%?
4. In order that his daughter may receive an income of $800 payable at the end of
each year for 5 years, a man buys such an annuity from an investment company. If
the investment company allows 4% interest, converted annually, what sum does the
man pay the company?
6. Solve Exercise 4, if the daughter is to receive $400 at the end of each six months.
6." The beneficiary of a policy of insurance is offered a cash payment of $10,000
or an annuity of $750 for 20 years, the first payment to be made one year hence. Allow-
ing interest at 3J^% converted annually, which is the better option?
7. A building is leased for a term of 10 years at an annual rental of $1,200 payable
annually at the end of the year. Assuming an interest rate of 5.2% what cash payment
would care for the lease for the entire term of 10 years?
8. Show that the results of Examples 5, 6, and 7 of Art. 29 arc the compound amounts
of the results of Examples 1, 2, and 3 respectively of Art. 31.
9. An individual made a contract with an insurance company to pay his family an
annual income of $4,000, payable in quarterly installments at the end of each quarter
for 25 years. He paid for the contract in full at the time of purchase. Assuming money
worth 4%, what did it cost?
10. Find the cost of the above annuity, with the interest converted quarterly.
11. Fill out the following table for the present value of an annuity of $100 per year
for 10 years, interest at 4%.
Annuity
Payable
Interest Convertible
Annually
Semi-annually
Quarterly
Annually
Semi-annually
Quarterly
Annuities Certain 83
12. Solve Exercise 11, with the rate 5%.
13. Derive formulas (22) and (27) from (18) and (26) respectively by using the rela-
tion (1 + t) = (1 +j/m) m .
14. How much money, if deposited with a trust company paying (j = .04, m = 2),
would be sufficient to provide a man with an income of $100 a month for 25 years?
16. A house is sold "like paying rent" for $50 a month for 12 years. What is the
cash equivalent if money is worth 6%?
16. A coal mine is estimated to yield $10,000 a year for the next 12 years. The mine
is for sale. What is the present value of the total yield of the mine on a 5% basis?
17. State problems for which the following would give the answers:
(a) 8 = 600.8isioi5
S61.015
.025
(b) A = 200. 025J.025
j 2 at .025
18. A widow is to receive from a life insurance policy $50 a month for 20 years.
If money is worth 3%, what is a fair cash settlement?
19. A building and loan association accumulates its deposits at (j = .06, m = 2).
If a man makes monthly deposits of $35 each for 10 years, what sum should he have to
his credit at the end of this time?
20. A man is offered a piece of property for $10,000. He wishes to make a cash pay-
ment and semi-annual payments of $500 for 10 years. What should be the cash payment
if the seller discounts future payments at (j .06, m = 2)?
32. Annuities due. In the previous sections of this chapter^ we have
been concerned with ordinary annuities, that is, annuities in which the
payments were made at the ends of the payment periods. An annuity due
is one in which the payments are made at the beginnings of the payment
periods.
The term of an annuity due extends from the beginning of the first
payment period to the end of the last payment period. That is, it extends
for one payment period after the last payment has been made. The
amount of the annuity due is the value of the annuity at the end of the
last payment period, that is, at the end of the term. The present value
of an annuity due is the value of the annuity at the beginning of the term,
or at the time of the initial payment. The present value includes the
initial payment.
To solve problems involving annuities due it is neither necessary nor
desirable that we invent a number of new formulas*. We can always
analyze an annuity due problem in terms of ordinary annuities. It is
important, however, that the student have a clear picture of the problem.
* The symbols, s^ and ajj), in black roman type are frequently used to represent the
amount and the present value of an annuity due of 1 per year for n years.
84 Financial Mathematics
We submit the following line diagrams to assist the student in clearly
understanding the similarities and the differences between ordinary annui-
ties and annuities due.
'I
f:
A
1st
pay.
2nd
pay.
3rd
pay.
S
last
pay.
A'
1st
pay.
1
2nd
pay.
2
3rd
pay.
3
S'
last
pay.
n - I
Example 1. $50 is deposited in a savings bank now and a like amount
every six months until 8 such deposits in all have been made. How much
is on deposit 4 years from now, if money accumulates at 5% converted
annually?
Solution. Consider the line diagram.
S S'
50 50 50 50 50 50 50 50
1 1 1 1 1
01234
First method. The amount of the annuity, S, just after the last
deposit (at 3H years) is that of an ordinary annuity with R = 100, n = 4,
p = 2, i = .05. Using J5I1, Art. 31, we find this amount to be
100 [(1.05) 4 - 1]
o =
.05
= 100(4.3101 2500) (1.0123 4754)
= 436.3344.
Now evidently S' is the value of S accumulated for % a year. Hence
S' - /S(1.05) H - 436.3344(1.0246 9508)
= $447.11.
Annuities Certain 85
Second method. If a deposit of $50 had been made at the end of 4 years,
the amount would have been that of an ordinary annuity with R = 100,
n = 4% p = 2, i = .05. Again using fill, Art. 31, we find this amount
to be
S" = 50
It is clear that the amount just before such a deposit was made, which
is the amount S' that we are seeking, is
,,. f
= $447.11.
Example 2. Find the amount of an annuity due of annual rent $400
payable in quarterly installments for 8 years at 6% converted annually.
We shall leave it as an exercise for the student to show that the first
method leads to the solution
- 400(1.01467385) (9.89746791) (1.02222688)
[Tables VIII, V, X]
= $4,106.36.
Example 3. Solve Example 2 with the interest converted quarterly.
We shall leave it as an exercise for the student to show that the second
method leads to the solution
= 100 [ (1 - 01 5 ^ 3 5 ~ * - i] - 100 [ moi5 - 1]
= 100(42.29861233 - 1) [Table V]
= $4,129.86.
86 Financial Mathematics
Example 4. Solve Example 2 with the interest converted semi-annually.
The application of the first method leads to the solution
(1.03) 16 - 1
S' = 400(1.03)*
4[(1.03
= 200(1.01488916) (20.1568813) (1.00744458)
[Tables VIII, V, X]
= $4,120.85.
Exercises
1. Set up a series for the accumulated values of the payments in Example 1 above,
find the sum of the resulting geometric progression, and thus find S'.
2. Do the same for Example 3 above.
3. A man deposits $150 in a savings bank on his twenty-fifth birthday and a like
amount every six months. If the bank pays 3% interest convertible semi-annually,
how much does he have on deposit on his sixtieth birthday?
: 4. Solve Exercise 3, with the interest converted quarterly.
6. A man, age 25, pays $24.03 a year in advance on a $1 ,000, 20-pay life policy. If
he should die at the end of 12 years, just before paying the 13th premium, how much
would his estate be increased by having taken the insurance instead of having deposited
the $24.03 each year in a savings bank paying 4% effective?
6. An insurance premium of $48 is payable at the beginning of each year for 20 years.
If the insurance company accumulates these payments at 5% converted semi-annually,
find the amount of the payments at the end of the 20th year.
7. Find the amount of an annuity due of $200 a year payable in semi-annual install-
ments for 7 years at 4% converted annually. Solve fundamentally as a geometrical
progression.
8. Solve Exercise 7, with the interest converted quarterly.
We have defined the present value of an annuity due to be the value of
the annuity at the time of the initial payment. Consider the examples:
Example 1. An individual is to receive $50 cash and a like sum every
six months until 8 such payments in all have been made. What is the
cash value of the payments, if money is worth 5% converted annually?
A A'
50
1
50
50
1
50
50
1
50
50
1
50
. i
Solution. We shall solve this example by two methods.
First method. The payments constitute an ordinary annuity whose
term begins 6 months before the present and ends at 3J^ years. Its term
Annuities Certain 87
is therefore 4 years. We have for this annuity R = $100, n = 4, p = 2,
i = .05. Using J5I2, Art. 31, we find
Evidently A', the required present value, is the value A accumulated
1 A year at 5%. Hence
A' =
A
P ~ (1 - 5) " 4 1 F _ **
L ^ J [ 2[(1 05) *
= 100(1.02469508) (3.54595050) (1.01234754)
[Tables VI, VIII, and X]
Second Method. If we disregard the first payment, the remaining
7 payments constitute an ordinary annuity whose term begins now. The
value at of this annuity is the present value of an ordinary annuity with
R = 100, n = &A, p = 2, i = .05. Using 512, Art. 31, the value at is
_ .
~ (1.05)* -1
Hence
= 100(1.02469508) (3.54595050) (1.01234754)
[Tables VI, VIII, and X]
Example 2. Find the present value of an annuity due of $600 per
year paid in quarterly installments for 8 years, if the interest rate is 5%
converted semi-annually.
We shall leave it an exercise for the reader to show that the first method
leads to the solution
A' eooao25)^ 1 ~ (1 - 025) " 16
A - 600(1.025) 4[(L025) *_ y
2[(1>025)>4 _ u
16 - 025
= 300(1.01242284) (13.05500266) (1.00621142)
= $3,989.78. [Tables VI, VIII, and X]
88 Financial Mathematics
Example 3. Solve Example 2 with the interest converted quarterly.
We leave it an exercise for the reader to show that the second method
leads to the solution
" 31 = 150[1 + am 12sl
= 150(1 + 25.56929010) [Table VI]
= 150(26.56929010) = $3,985.39.
Example 4. Find the present value of an annuity due of $600 per year
paid in semi-annual installments for 8 years if the interest rate is 5%
convertible quarterly.
An application of the first method leads to the solution
1 - (1.0125) - 32
A! = 300(1.0125) 2
(1.0125) 2 - 1
= 300(1.0125) 2 -a - at .0125
$2|
- $4,010.15. [Tables III, VI and VII]
Exercises
1. Set up a series for the present value of the payments in Example 1 above, find the
sum of the resulting series, and thus find A'.
2. Do the same for Example 3 above.
3. A man leases a building for 4 years at a rental of SI 00 a month payable in advance.
Find the equivalent cash payment, if money is worth 5%.
4. A man pays $500 cash and $500 annually thereafter until 10 payments have
been made on a house. Assuming money worth 6% converted semi-annually, what is
the equivalent cash price?
5. An insurance policy provides that at the death of the insured the beneficiary is to
receive $1,200 per year for 10 years, the first payment being made at once. Assuming
that money is worth 3H% what is the value of a policy that will provide such a settle-
ment?
6. Allowing interest at 5%, converted quarterly, what is the present cash value of
a rental of $2,000 per year, payable quarterly in advance for a period of 15 years?
7. Solve Exercise 6, with the payments made semi-annually.
8. A man deposits $30 at the beginning of each month in a bank which pays 3%
interest converted semi-annually. He makes these deposits for 120 months. What
amount does he have to his credit at the end of the time.
9. Prove that a^| = 1 + a^r\\ .
10. Prove that s^ = (1 -f i)$nl = *nTTl - 1-
Annuities Certain
89
33. Deferred annuities. A deferred annuity is one whose payments are
to begin at the end of an assigned number of years or periods. When we say
that an annuity is deferred m payment periods, we mean that the annuity
is "entered upon ;; at the end of m payment periods and that the first pay-
ment is made at the end of (m + 1) payment periods. The m periods con-
stitute the period of deferment.
The amount of a deferred annuity is the value of the annuity imme-
diately after the last payment. The present value of a deferred annuity is
the value of the annuity at the beginning of the period of deferment.
The following line diagram emphasizes the characteristics of a deferred
annuity that continues t payment periods after being deferred m payment
periods.
A' A S
1st 2nd las
pay. pay. pa
312?
Period of
n m + 1 m + 2 m -
Term of ordinary
deferment
y --..,_ TVn-m f^f Aatt-
annuity
To solve problems involving deferred annuities, it is neither necessary
nor desirable that we invent a number of new formulas.* Problems in-
volving deferred annuities can always be analyzed in terms of ordinary
annuities. We shall illustrate the methods of solution by a few examples.
Example 1. A young man is to receive $500 at the end of 6 years and
a like sum each year thereafter until he has received 10 payments in all.
Assuming money worth 4% converted annually what is the present value
of his future income?
We shall solve this problem by two methods.
Solution. First method. Consider the line diagram.
A' A
500 500
H-
500
H
15
The value of the annuity at the end of 5 years is evidently
0.04
* The symbols m \ s^l and m \ a^ are frequently used to represent the amount and
the present value of an annuity of 1 per year for n years deferred m years.
90 Financial Mathematics
The value at 0, which is the value we are seeking, is A discounted 5
years at 4%. Hence,
A' = (1.04) -M
= (1.04) - 5 [500 1 ^^ 4) "
= 500(0.82192711) (8.11089578) [Tables IV and VI]
= $3,333.28.
Solution. Second method.
Imagine $500 paid at the end of each year for the first five years.
These payments together with the 10 given payments constitute an ordi-
nary annuity of $500 a year for 15 years. Its value at is 500. a^o^.
Now, if we subtract the value at of the five imaginary payments, namely
500 ag.o*, we have
A' = 500 a I 5 1 . 04 500 a^ 04 = 500
= 500(11.1183 8743 - 4.4518 2233) [Table VI]
= $3,333.28.
The second method is much simpler from the standpoint of computa-
tion. The student, however, should become skilled in the use of both
methods.
Example 2. Find the present value of an annuity of $600 per year
paid in quarterly installments for 8 years but deferred 5 years, assuming
money worth 5% converted semi-annually.
Solution. We leave it as an exercise for the reader to show that the
first method leads to
- (1.025) ~ 16
A' = 600(1.025) ~ 10
= 300(1.025) ~ 10
4[(1.025)* /4 - 1]
1- (1.025) ~ 16 0.025
0.025 *2[(1.025) H - 1]
= 300(0.78119840) (13.05500266) (1.00621142)
[Tables IV, VI and X]
= $3,078.57.
Annuities Certain 91
Example 3. Solve Example 2 with the interest converted quarterly.
Solution. We shall leave it as an exercise for the reader to show that
the second method leads to
A' = 150 * ~ (L0125) " 52 l - (J- 012 *)"
0.0125 0.0125 J
= 150 L #52"|.oi25 ~~* #20|.0125 1
= 150(38.06773431 - 17.59931613) [Table VI]
= 150(20.46841818) = $3,070.26.
Example 4. Find the present value of an annuity of $600 a year paid
in semi-annual installments for 8 years but deferred 5 years, assuming
money is worth (j = ,05, m = 4).
Solution. We leave it as an exercise for the reader to show that the
first method leads to
= -<'.>^7^
_ 300(0.78000855) (1 - 0.67198407) [Tables III and IV]
1.02515625 - 1
300(0.78000855) (0.32801593)
0.02515625
= $3,051.19.
Remark 1. It will be noted that we have given no examples that
involve finding the amount of a deferred annuity. The amount of a de-
ferred annuity is obviously the amount of an ordinary annuity to which we
have already given much attention.
Remark 2. The second method for evaluating the present value of a
deferred annuity is preferable when m = p.
Exercises
1. If money is worth (j - .04, m - 2), find the present value of an annuity of
$1,000 a year, the first payment being due at the end of 8 years and the last at the end
of 17 years.
2. Find the present value of an annuity of $1,000 per year, payable in semi-annual
installments, for 9 years but deferred 5 years assuming money worth 4% converted
semi-annually. Solve by two methods.
92 Financial Mathematics
3. Solve Exercise 2, with the interest converted annually.
4. Find the present value of an annuity of $800 per annum paid in quarterly install-
ments for 14 years, deferred 6 years, if money is worth 5% converted annually.
6. Solve Exercise 4, with the interest converted (a) semi-annually, (b) quarterly.
6. A will provides that a son, aged 15 years, is to receive $1,000 when he reaches 25
and a like sum each year until he has received 15 payments in all. Assuming money
worth 4% converted annually, what would be the inheritance tax of 5% on the son's
share?
7. A geologist estimates that an oil well will produce a net annual income of $50,000
for 10 years. Due to litigations the first income will not be available until the end of 4
years, but will come in at the end of each year thereafter until 10 full payments have
been made. Assuming money worth 5H%, what is the present value of the well?
8. What sum should be set aside now to assure a person an income of $150 at the
end of each month for 20 years, if the income is deferred for 12 years, assuming money
worth 4^% converted semi-annually?
9. A man offers to sell his farm for $15,000 cash or $7,500 cash and $2,500 annually
for 4 years, the first annual payment to be made at the end of 5 years. Assuming money
worth 6%, what is the cash difference between the two offers?
10. What sum of money should a man set aside at the birth of his son in order to
provide $1,000 a year for 4 years to take care of the son's education, if the first install-
ment is to be paid in 18 years? Assume 4% interest.
11. Prove: m I on = (1 + i)~ m OnH .
12. Prove: m \ a^a
34. Finding the interest rate of an annuity. We may find the approxi-
mate interest rate of an annuity by the method of interpolation. This
method will be sufficiently accurate for all practical purposes.
Example 1. At what rate, converted quarterly, will an annuity of
$100 per quarter amount to $5, 100 in 10 years?
Solution. Here, R = $400, m = p = 4, n = 10, and S = $5,100.
Substituting in Bill, Art. 31, we have
A 40
+ 4) - 1
5,100 = 100 - : = lQQs m i.
4
Then s m t = 51.0000. We now turn to Table V and follow n = 40
until we come to a value just less than 51.0000 and one just greater than
51.0000. We find the value 50.1668 corresponding to 1^% and the
Annuities Certain 93
value 51.4896 corresponding to 1M%- Hence, the rate y/4 lies between
and lJi%. Interpolating, we have
.00125
" S 4oi.oi25 = 51.4896-
51.0000i
x
sioi.01125 = 50.1668
.8332
1.3228
X ' 8332 , = 0.00079
.00125 1.3228 '
7 = 0.01125 + 0.00079 = 0.01204.
4
And j = 0.04816 = 4.816% (approximately).
This result may be checked by logarithms. We have
log 1.01204 = 0.0051977 (Table II)
40 log 1.01204 = 0.2079080
(1.01204) 40 = 1.6140. (Table I)
(1.6140 - 1)
And S = 100
0.01204
61 ' 40 = $5,099.67.
0.01204
This result is only 33 cents less than the $5,100 and the rate 4.816% is
accurate enough. If 7 place logarithms had been used to find the anti-
logarithm of 0.2079080, our result would have been $5,099.80 which
differs from the $5,100.00 by only 20 cents.
Example 2. The present value of an annuity of $400 per annum for
20 years is $5,000. Find the interest rate.
Solution. Here, R = $400, m = p = 1, n = 20, and A = $5,000,
Substituting 412, Art. 31, we have
I _ n _1_ f)~20
5,000 = 400 ^ ; = 400 a^.
Then a^ = 12.5000.
We now turn to Table VI and follow n 20, until we come to a value just
greater than 12.5000 and one just less than 12.5000. We find the value
94 Financial Mathematics
13.0079 corresponding to 4J^% and 12.4622 corresponding to 5%. Hence,
the rate i lies between 4J^% and 5%.
5079
Therefore, i = 0.045 + f (0.005)
0.5457
= 0.045 + 0.00465
= 0.04965 = 4.97% (approximately).
Example 3. A house is priced at $2,500 cash or for $50 a month in
advance of 60 months. What is the effective rate of interest charged in
the installment plan?
Solution. We have here an annuity due of 60 periods. Let us assume
for convenience that the nominal rate is j converted 12 times a year. Then,
m = p = 12, R = $600, and A' = $2,500. Substituting in 112, Art. 31,
we find
l-M+i 1 " 69 '
2,500 = 50 1 1 +
12
= 50(1 + aoi).
Then a m = 49.0000.
Turning to Table VI, we find that when
-j%, <*,- 49.7988;
when = %, a m = 47.5347.
= 0.00583 + 0.00059 = 0.00642.
And j = 0.07704 = 7.704%. (Approximate nominal rate).
Checking by logarithms as in Example 1, we find
A' = $2,499.14,
which is 86 cents less than the $2,500.00.
Annuities Certain 95
To find the effective rate, we have
(1 + i) = (1.00642) 12 (4) Art. 16.
log 1.00642 = 0.0027792
12 log 1.00642 = 0.0333504
(1.00642) 12 = 1.07982 = (1 + i).
Therefore, i = 0.07982 = 7.982%.
Exercises
1. At what rate of interest will an annuity of $500 a year amount to $25,000 in
25 years?
2. A house is offered for sale for $6,000 cash or $1,000 at the end of each year for the
next 8 years. If the installment plan is used, what rate of interest is charged?
3. The cash price of an automobile is $1,150. A man is allowed $525 on his old car
as a down payment. To care for the balance he pays $57.20 at the end of each month
for 12 months. What rate of interest is charged? Use simple interest. [See p. 34.]
4. A man deposits $9,500 with a trust company now with the guarantee that he (or
his heirs) is to receive $1,000 each year for 25 years, the first $1,000 to be paid at the
end of 10 years. What effective rate of interest is the man allowed on his money?
35. The term of an annuity. We illustrate by examples the method
of finding the term of an annuity.
Example 1. In how many years will an annuity of $400 per year
amount to $9,500, if the interest rate is 3^% converted annually?
Solution. Here, K = $400, S = $9,500, and i = 0.035. Substi-
tuting in All, Art. 31, we have
And 8ft = 9,500 ^ 400 = 23.7500.
We now turn to Table V and follow down the 3J^% column. We notice
that when ^ = ^ , = 22 ?050;
n = 18, 8^ = 24.4997.
It is evident that 18 payments of $400 will amount to more than
$9,500. In fact, it will amount to $9,799.88, which is $299.88 more
than is needed. Hence, $400 per year for 17 years and
$100.12, ($400.00 - $299.88)
at the end of 18 years will amount to exactly $9,500.
96 Financial Mathematics
Example 2. An individual buys a house for $5,000 paying $1,000 in
cash. He agrees to pay the balance in installments of $500 at the end of
each year. How long will it take to pay the $4,000 and interest at 6%
converted annually?
Solution. Here, A = $4,000, R = $500, m = p = 1, and i = 0.06.
Substituting in AI2, Art. 31, we have
4,000 = 500 ^y^ = 500^.
And a^ = 8.0000.
We now turn to Table VI and follow down the 6% column. We notice
that when
n = 11, a$ = 7.8869
n = 12, a^j = 8.3838.
Hence, the present value of 11 payments is less than $4,000 and the
present value of 12 payments is more than $4,000. Then, it is evident
that the debtor must make 11 full payments of $500 each and a 12th
payment, at the end of 12 years, which is less than $500.
If no payments were made, the original principal of $4,000 would
accumulate in 11 years to
4,000(1.06) n = 4,000(1.89829856) = $7,593.19.
However, if payments of $500 are made regularly for 11 years, they will
accumulate to
11 - = 500(14 97164264) = $7^85.82
0.06
Hence, just after the llth payment, the balance on the principal is
$7,593.19 - $7,485.82 = $107.37. That is, the debt could be cancelled
by making an additional payment of $107.37 along with the llth regular
payment. However, if the balance is not to be paid until the end of the
12th year, the payment would be $107.37 plus interest on it for 1 year
at 6%, or $107.37 + $6.44 = $113.81. Then, 11 payments of $500 and
a partial payment of $113,81 made at the end of 12 years will settle the debt.
Annuities Certain 97
Exercises
1. In how many years will an annuity of $750 amount to $10,000 if interest is at
Solve by interpolation.
2. Solve formula All, Art. 31, for n.
_ log (iS + B) - log R
H log (1 +
3. Solve Exercise 1 by the formula given in Exercise 2.
4. A man borrows $3,000 and desires to repay principal and interest in installments
of $400 at the end of each year. Find the number of full payments necessary and the
size of the partial payment, if it is made 1 year after the last full payment is made,
assuming an interest rate of 5%.
6. A man deposits $15,000 in a trust fund with the agreement that he is to receive
$2,000 a year, beginning at the end of 10 years, until the fund is exhausted. If the trust
company allows him 4% interest on his deposits, how many full payments of $2,000 will
be paid and what will be the fractional payment paid at the end of the next year?
36. Finding the periodic payment. In the early sections of this chap-
ter we solved the problems of finding the amount and the present value
of an annuity under given conditions when the periodic payment was
known. Our results were summarized in Art. 31.
We are now about to attack the inverse problem, that of finding the
periodic payment under given conditions, when the amount or the present
value of the annuity is known. The solution requires no new formulas.
We must merely solve the equations of Art. 31 for R or R/p according as
the annuity is payable annually or p times a year. Consider the following
examples.
Example 1. A man buys a house for $6,000 and pays $1,000 in cash.
The remainder with interest is to be paid in 40 equal quarterly payments,
the first payment being due at the end of three months. Find the quar-
terly payment if the interest rate is 6% converted annually.
Solution. Here, A = $5,000, n = 10, p = 4, i = 0.06. Substituting
in Art. 31, #12, and solving for JR, we have
4K1.06)*-!]
R = 5,000
JL ^i.uo; * v
n nftftAQfta$n
[Tables IV and IX]
1- (1.06) - 10
5,000(0.05869538)
1 - 0.55839478
= $664.57, annual payment.
7?
Then, - = $166.14, quarterly payment.
98 Financial Mathematics
Example 2. Solve Example 1, with the interest converted semi-
annually.
Solution. Using #112, Art. 31, we have
4[(1.03)*-H
- (1.03) -20
10,000 2[(1.03) H - 1]
1 - (1.03) - 20
10,000(0.02977831)
1-0.55367575
10-000(0.02977831)
0.44632425
IV and IX]
=
T>
Then, -7 = $166.80, quarterly payment.
4
Example 3. Solve Example 1, with the interest converted quarterly.
Solution. Here, m = p = 4, and 5112, Art. 31 gives us
= 5,000(0.03342710) [Table VII]
= $167.14, quarterly payment.
Example 4. How much must be set aside semi-annually so as to have
$10,000 at the end of 10 years, interest being at the rate of 5% converted
annually?
Solution. Here, S = $10,000, n = 10, p = 2, i = 0.05. Substituting
in J3I1, Art. 31, we have
72 = 10,000-
^i.uo;^ i
nfUQ3Qm.fft
[Tables III and IX]
X -.J5) 10 j
10,000(0.04939015)
1.62889463 -
= $785.35, annual payment.
R
Then, ~ = $392.67, semi-annual payment.
Annuities Certain 99
Example 6. Solve Example 4, with the interest converted semi-
annually.
Solution. Here, m = p 2, the other conditions being the same as
in Example 4. Substituting in J5II1, Art. 31, we have
- = 10,000- ^- - = 10,000
' '
, ,
2 ' (1.025) 20 -! ' figa.025
= 10,000(0.06414713 - 0.025) [Table VII]
= 10,000(0.03914713) = $391.47.
Example 6. Solve Example 4, with the interest converted quarterly.
Solution. Here, m 4, the other conditions being the same as in
Example 4. Substituting in Bill, Art. 31, we have
= 20,000
(1.0125) 40 - 1
(1.0125) 2 - 1 0.0125
0.0125 (1.0125) 40 - 1
= 20,000(2.01250000) (0.01942141) [Tables V and VII]
= $781.71, annual rent.
r>
Then, = $390.86, semi-annual payment.
t
Exercises
1. A man buys a farm for $10,000. He pays $5,000 cash and arranges to pay the
balance with 5% interest converted semi-annually, by making equal payments at the
end of each six months for 14 years. How much is the semi-annual payment?
2. How much must be set aside annually to accumulate to $5,000 in 8 years, if
money is worth 4j/% converted semi-annually?
3. Solve Exercise 2, with the interest converted (a) annually, (b) quarterly.
4. In order to finance a school building costing $100,000 a city issues 20-year bonds
which pay 5% interest, payable semi-annually. How much must be deposited, at the
end of each six months, in a sinking fund which accumulates at 4J^% converted semi-
annually, if the bonds are to be redeemed in full at the end of 20 years? What total
semi-annual payment is necessary to pay the interest on the bonds and make the sinking
fund payment?
100 Financial Mathematics
6. Solve Exercise 4, with the interest on the sinking fund converted quarterly.
6. At the maturity of a $20,000 endowment policy, the policyholder may take the
full amount in cash or leave the full amount with the insurance company to be paid to
him in 40 equal quarterly payments, the first payment to be made at the end of three
months. If 4% interest, converted quarterly, is allowed on all money left with the com-
pany, how much is the quarterly payment?
7. A building is priced at $25,000 cash. The owner agrees to accept $5,000 cash and
the balance, principal and interest, in equal annual payments for 15 years. If the inter-
est rate is 7% effective, what is the annual payment?
8. Solve Exercise 7, with the interest converted quarterly.
37. Perpetuities and capitalized cost. An annuity whose payments
continue forever is defined as a perpetuity. It is evident that the amount of
such an annuity increases indefinitely, but the present value is definite.
The symbol A^ 9 will denote the present value of a perpetuity of R dollars
per annum, payable annually.
It is evident that the interest on A w for one year at nominal rate (j, m )
must equal R.
Hence, A (l + ^Y- A = R,
\ ui/
and A
R m Rm 1 , x
A. - 7-7 TV; = -y- r ( 29 )
^ * m lm
When w = 1, j = i, and (29) reduces to
A n = f (29')
Example 1. Find the present value of a perpetuity of $500 per annum,
if money is worth 4% converted annually.
Solution. Here, R = $500, i = 0.04.
500
Then, A M - - $12,500. [Formula (2901
Example 2. Solve Example 1, with the interest converted quarterly.
Annuities Certain 101
Solution. Here, R = $500, j = 0.04, and m = 4. Substituting in
(29),wehave 500 0.01 0.01 1
A * ~ ' ~ 5 ' 000
(1.01) 4 -
= 50,000(0.24628109) [Table VII]
= $12,314.05.
There are times when a perpetuity must provide for payments at
intervals longer than a conversion period. The symbol A 00t r , will denote
the present value of a perpetuity of C dollars payable every r years.
It is evident that the compound interest on ^loo.r, for r years at rate j
converted m times a year must equal C.
( J\ mr
Hence, <A>,rl 1 H ) -Aoo.r^C,
and -Aoo r =
2.
- m Cm 1
or 4 w , r = -- - - = .j. (30)
mr
m m
If m = 1, j = ij and
A.,, --? (300
$j\i
Example 3. What is the present value of a perpetuity of $2,000 pay-
able every 4 years, if money is worth 5% converted annually?
Solution. Here, C = $2,000, r = 4, i = 0.05. Substituting in (30'),
we have
2,000 0.05
= 40,000(0.23201183) [Table VII]
- $9,280.47.
Example 4. Solve Example 3, with the interest converted semi-
annually.
102 Financial Mathematics
Solution. Here, C = $2,000, j = 0.05, m = 2, and r = 4. We have
2,000 0.025
= 80 ' 000 (I5lP^-i = 80 ' 000
= 80,000(0.11446735) = $9,158.39. [Table VII]
/Example 5. A section of city pavement costs $50,000. Its life is 25
years. Find the amount of money required to build it now and to replace
it every 25 years, indefinitely, if money is worth 4% converted annually.
Solution. It is evident that the amount required to replace the pave-
ment indefinitely is the present value of a perpetuity of $50,000 payable
every 25 years at 4%.
= 1,250,000(0.02401196) [Table VII]
= $30,014.95.
Hence, the amount required to build the pavement plus the amount to
replace it indefinitely equals
$50,000 + $30,014.95 = $80,014.95.
This amount is called the capitalized cost. That is, the capitalized cost is
the first cost plus the present value of a perpetuity required to renew the project
indefinitely.
If we let K stand for the capitalized cost of an article whose first cost
is C, and which must be renewed every r years at the cost C, we have
m
m Cm 1
m m
(31)
Annuities Certain 103
If m = 1, j = t,
then K= -. (31')
1 a n/
Example 6. An automobile costs $1,000 and will last 7 years when it
must be replaced at the same cost. Another automobile, which would
serve the same purpose and would last 10 years, could be purchased. What
could one afford to pay for the second automobile if it is to be as economical
in the long run as the first, assuming money worth 5%?
Solution. When somebody says that a certain article is just as econom-
ical (cheap) in the long run as another article, he simply means that the
two articles have the same capitalized cost.
The first automobile has a capitalized cost of
1,000 0.05 rT1 . , , x ,
(31')]
If we let x stand for the cost of the second automobile, it will have a
capitalized cost of
Assuming that the two automobiles are equally economical, we have
JL 0.05 _ 1^000 0.05
0.05*1 - (1.05) - 10 ~~ 0.05 *1 - (1.05) ~ 7
-(1.05)- 10 0.05
0.05 l-(1.05)- 7
= 1,000(7.72173493) (0.17281982) [Tables VI, VII]
= $1,334.47.
That is, one can afford to pay $1,334.47 for the automobile that lasts
10 years, or $334.47 more, for the additional 3 years of service.
We shall now find the additional cost w required to increase the life of a
given article x years assuming money worth i%.
Let C = original cost of an article to last n years. Its capitalized
cost is
C i
il - (1 + '
104 Financial Mathematics
Let C + w = cost of an article to last n + x years. Its capitalized
cost is n .
C + w i
Equating capitalized costs, we have
C + w i C
The student may solve the above equation for w and get
w = C
(1 + i)* 1 i
= C^ (32)
If the interest rate is j converted m times per year, (32) may be written
m ' . "' (32')
m
Example 7. A cross tie costs $1.00 and will last 10 years. The life of
the tie can be extended to 18 years by treating with creosote. If money is
worth 5%, how much could one afford to spend for the treatment?
Solution. Here, C = $1.00, n = 10, x = 8, and i = 0.05. From
(32) we have 1
w = 1.00 -asi.05 -~-~
S 10|.05
= (0.07950458) (6.46321276) [Tables VI, VII]
= $0.51.
That is, 51j could profitably be spent to treat the tic, if the service life
would be extended 8 years.
Exercises
1. What amount would a railroad company be justified in expending per tie to
extend the life of cross ties costing $1.50 each from 12 to 20 years, money being worth
2. A hospital receives an annual income of $120,000 as a perpetuity from a trust
fund. What is the value of this perpetuity, money being worth 5% effective?
Annuities Certain 105
3. Solve Exercise 2, if the interest rate were 5% converted quarterly.
4. A railroad company has been paying a watchman $1,600 a year to guard a crossing.
The company decides to build an overhead crossing at a cost of $22,000. If the over-
head crossing must be rebuilt every 35 years at the same cost, how much does the com-
pany save by building it? Assume money worth 5%.
5. An office building is erected at a cost of $100,000. It requires a watchman at
an annual salary of $1,500, and $4,000 for repairs and renovation every 8 years. It
must be rebuilt every 80 years at the original cost. How much money is required now
to provide for its construction, maintenance, guarding and rebuilding, assuming money
worth 3%? (Hint: Every 80 years when the building is rebuilt, the $4,000 allowed for
repairs and renovation is not needed. This amount may be applied on the $100,000
for rebuilding, thereby reducing it to $96,000.)
6. A state highway commission has a certain road graded and ready for surfacing.
It may be graveled at a cost of $2,000 per mile, or paved at a cost of $10,000 per mile.
It will cost $200 per year to maintain the gravel road and it will need regraveling every
8 years at the original cost. The maintenance cost of the pavement is negligible and it
will need repaving only every 40 years at the original cost. If the cost for clearing the
road bed of the old paving is $1,000, which type of road is more economical, assuming
that the state can borrow money at 4%?
38. Increasing and decreasing annuities. A sequence of periodic
payments in which each payment exceeds by a fixed amount the preceding
payment is called an increasing annuity. If each payment is less by a fixed
amount than the preceding, the sequence is called a decreasing annuity.
Consider the following examples.
Example 1. Find the amount and the present value of a decreasing
annuity with payments of $250, $200, $150, $100, $50, at the ends of the
next five years if money is worth 4%.
Solution. Here is the picture.
250 200 150 100 60
I 1 1 1 1 1
012345
These payments are equivalent to the following five ordinary annuities,
superimposed: (1) $50 a year for 5 years; (2) $50 a year for 4 years; (3) $50
a year for 3 years; (4) $50 a year for 2 years; (5) $50 a year for 1 year.
These annuities are exhibited in the following diagram.
50
50
50
50
50
50
50
60
50
50
50
u I
50
1
60
1
50
1
60
106 Financial Mathematics
To find the amount of a decreasing annuity, we first find its present
value. The present value of the given decreasing annuity is
A = 50 an + 50 a^ + 50 a^ + SOa^ + 50 ag-,
= 50 I 5 "" as]04 1 Exercise 8, Art. 28
L .04 J
= 1250(5 - 4.4518 2233)
= $685.222.
The amount of this annuity is clearly
S = 4(1.04) 5 = 685.222(1.2166 5290)
- $833.68.
Example 2. Find the amount and the present value of an increasing
annuity with payments of $50, $100, $150, $200, $250 at the ends of the
next five years if money is worth 4%.
Solution. Here is the picture.
50 100 150 200 250
I-
These payments are equivalent to five ordinary annuities that we ex-
hibit in the following diagram.
50
50
50
50
50
50
50
50
50
50
50 50
i 1 1
50
1
50
50
1
0123
The amount of this increasing annuity is
8 = 50 !, + 50 s% + 50 % + 50 sj| + 50
50 Exercise 7, Art. 28
= 1250[(1.04) (5.4163 2256) - 5]
= $791.22.
Annuities Certain 107
The present value of this increasing annuity is clearly
A = S(1.04)~ 5 = 791.22(0.8219 2711)
= $650.33.
Exercises
1. If money is worth 5%, find the amount and the present value of the increasing
xnnuity pictured on the diagram.
100 200 300 400 500 600 700
1 1 1 1 1 1 1
0123456 7years
2. If money is worth 5%, find the amount and the present value of the decreasing
annuity pictured on the diagram.
700 600 500 400 300 200 100
1 ! 1 1 , 1 1
0123456 7 years
Problems
1. Show that formulas (12), (13), and (26) are special cases of formula (27). Follow
method on page 76.
2. In order to accumulate $20,000 in 14 years, how much must be deposited in a
savings bank at the end of each year, if the interest is converted annually at 4%?
3. An automobile is bought for $400 cash and $62 a month for 15 months. What
is the equivalent cash price if money is worth 7% converted monthly?
4. Find the rate of interest if an annuity of $700 a year amounts in 15 years to
$15,000.
6. The proceeds of a $5,000 insurance policy is to be paid in monthly installments
of $50 each. If money is worth 5% converted monthly, find the number of monthly
payments. The first payment is made at the end of the first month.
6. A man buys a house for $8,000, paying $2,000 cash. He arranges for the balance,
principal, and interest at 6%, to be paid in 60 monthly installments. Find the size
of each installment if the interest is converted monthly.
7. A son is to receive $1,000 a year for 12 years, the first payment being due 6 years
hence. Find the present value of the son's share assuming 5% interest converted^
annually.
8. The beneficiary of an insurance policy is offered $15,000 in cash or equal annual
payments for 12 years, the first payment being due at once. Find the size of the annual
payments if money is worth 4%.
9. A wooden bridge costs for construction $22,500, and requires rebuilding every
20 years. How much additional money can be profitably expended for the erection
of a concrete bridge instead, if money is worth 5% and the service life is extended to
40 years?
108 Financial Mathematics
10. A building costs $40,000 and has a life of 50 years. If it requires $2,000 every
5 years for upkeep, what endowment should be provided at the time it is built to con-
struct it, rebuild it every 50 years and provide for its upkeep? At the end of every
50 years the $2,000 allowed for upkeep may be applied towards the reconstruction cost.
Assume money worth 4%.
11. How much can a railroad company afford to pay to abolish a grade crossing
which is guarded at a cost of $1,000 per year, when money is worth 5% converted semi-
annually?
12. A certain machine costs $2,000 and must be replaced every 12 years at the same
cost. A certain device may be added to the machine which will double its output, but
the machine must then be replaced every 10 years. Assuming money worth 4%, what
is the value of the device?
13. $100 is deposited in a savings bank at the end of every six months for 10 years.
During the first 6 years 3%, converted semi-annually, was allowed but during the last
4 years the rate was reduced to 2J^%, converted semi-annually. Find the amount on
deposit at the end of 10 years.
14. Derive formula (32), Art. 37.
15. An income of $10,000 at the end of each year is equivalent to what income at
the ead of every 5 years, assuming money worth 5% converted semi-annually?
16. Solve Exercise 15, with the interest converted (a) annually, (b) quarterly.
17. A building has just been completed at a cost of $250,000. It is estimated that
$2,500 will be needed at the end of every two years for repairs, and that every 15 years
there must be renovation to the extent of $10,000, and that the building will have a
service life of 60 years with a salvage value of $20,000. Find what equal annual amount
should be set aside at 4% interest to cover repairs, renovations, and replacements.
How should the $2,500 repair fund and the $10,000 renovation fund be used at the
end of every 60 years?
13. A man borrowed $10,000 with the understanding that it be repaid by 20 equal
annual installments including principal and interest at 6% annually. Just after the
10th equal annual payment had been made the creditor agreed to reduce the principal
by $1,000 and reduce the rate to 4j^%. Find the annual payment for the first 10 years
and the annual payment for the last 10 years.
19. A mortgage for $5,000 was given with the understanding that it might be repaid,
principal and interest, by 15 equal annual payments. Find the annual payment if
the interest rate was 7% for the first 8 years and 5% for the last 7 years.
20. A person pays $12,500 into a trust fund now with the guarantee that he or his
heirs will receive equal annual payments for 30 years, the first payment to be made at
the end of 7 years. If the trust fund draws 4% interest, find the equal annual payment.
21. A perpetuity of $25,000 a year is divided between a man's daughter and a uni-
versity. The daughter receives the entire income until she has received as her share
one half the present value of the perpetuity. Find the number of full payments she
receives and the size of the last payment, if money is worth 5% converted annually.
22. A man pays $6,000 into a trust fund and receives $500 at the end of each year
for 20 years. What rate of interest converted annually did he*earn on his money?
23. At his son's birth a father set aside a sum sufficient to pay the boy $1,000 a
Annuities Certain 109
year for 7 years, the first $1,000 to be paid on his 18th birthday. What sum was set
aside, if money was worth 4% converted semi-annually?
24. The amount of an annuity of $800 per year is $20,000 and the present value is
$9,235. Find the rate of interest.
25. A man buys a piano for $300 and pays $50 cash. The balance is to be paid for
at $12.50 at the end of each month for 24 months. What effective rate of interest
does the purchaser pay? (Hint: Assume that the interest is converted monthly and
find the nominal rate. Then find the effective rate.)
26. Is it economical to replace a machine which costs $500 and lasts 8 years by one
that costs $650 and lasts 12 years? Assume that the annual running expense of each
machine is the same and that money is worth 5%. Also assume that the two machines
have the same output.
27. A person considers replacing a machine which costs $400 and lasts 6 years by a
machine which costs $750 and answers the same purpose as the other machine. If the
exchange is to be economical, how long should the new machine last? Assume that the
annual running expense is the same for each machine and that money is worth 4%.
28. Derive formula (29') by setting up a series and finding its sum. From (290
derive (29).
29. Derive formula (30') by setting up a series and finding its sum. From (30')
derive (30).
30. Derive formula (29') from (5) by showing that limit R a^ - R/i.
n oo
31. If money is worth (j = .04, rn = 2), find the present value of the decreasing
annuity: $5,000, $4,500, $500 payable semi-annually.
32. If money is worth (j - .04, ra = 2), find the present value of the increasing
annuity: $500, $1,000, $5,000 payable semi-annually.
33. State a problem for which the answer would be
MOO
1000
(1.01)' - 1
34. State a problem for which the answer would be
1- (1.025)-"
(1.025) 2 - I/,
35. State a problem for which the answer would be
36. In an increasing annuity, R is paid at the end of the first year, 2R at the end of
the second year, and so on for n years. Show that
A = [(l+r>Hi-n(l
110 Financial Mathematics
37. In a decreasing annuity nR is paid at the end of the first year, (n l)R at the
end of the second year, and so on for n years. Show that
S = I [n(l + - ssil,
n
A = - [n - aft].
Review Problems *
1. $1,000 Harrisburg, Pennsylvania,
July 12, HMf).
Four months after date I promise to pay Joe Brown, or order, one
thousand dollars with interest from date at 5%.
(Signed) JOHN JONES.
(a) Three months after date Brown sold the note to Bank B who discounted the note
at 6% discount rate. What did Brown receive for the note?
([>) Immediately after purchasing the above note, Bank B sold the note to a Federal
Reserve Bank at a re-discount rate of 4%. How much did Bank B gain on the trans-
action?
2. Same note as in Problem 1. Would it have been to Brown's advantage to have
sold the note to friend C, to whom money was worth 6%, rather than to Bank B?
3. I bought a bill of lumber from the Jones Lumber Company who quoted the terms
"net 60 days or 2% off for cash." What nominal rate of interest, je, could I afford to
pay to borrow money to take advantage of the discount? What effective rate?
4. A note for $1,000 with interest at (j = .06, m = 2), and another for $800 with
interest at (j = .05, m = 2), both due in 3 years, were purchased to net 7% effective.
How much was paid for them?
5. A bank pays 4% interest on time deposits and loans money at 6% discount rate.
What is the annual profit on time deposits amounting to $100,000?
6. The Jones Lumber Company estimates that they can earn 3% a month on their
money. If I buy a $1,000 bill of lumber from them, what amount of discount can they
afford to offer me to encourage immediate settlement in lieu of $1,000 at the end of the
month? What is the nominal rate of discount, fu, that they can afford to offer?
7. A son is now 10 years old. The father wishes to provide now for the college and
professional education of the son by depositing the proper amount with a trust company
that pays (j - .04, m = 2) on funds. It is estimated that the son will need $1,000 a
year for 7 years, the first payment to be made when the son is 18 years of age. Find the
amount of the deposit.
8. A man owes a $6,000 balance on a home. The balance is at (j = .06, m 2).
The man agrees to pay the balance with payments of $300 at the end of each half year.
After how many payments will the balance be paid in full? What is the amount of the
final partial payment?
9. A man at the age of 50 invests $20,000 in an annuity payable to him if living
(to his estate if he is dead) in equal monthly installments over a period of 15 years, the
first installment to be due at the end of the first month after he reaches 65. On a 3J^%
basis, what is the monthly installment that he receives?
10. A man bought a refrigerator for $250 paying $50 down and the balance in 12
monthly installments of $20 each. What rate of interest does the purchaser pay? [Use
simple interest.]
* For additional review nroblems. see end of this book.
CHAPTER IV
SINKING FUNDS AND AMORTIZATION
39. Sinking funds. When an obligation becomes due at some future
date, it is usually desirable to provide for its payment by accumulating a
fund with periodic contributions, together with interest earnings. Such
an accumulated fund is called a sinking fund.
Example. A debt of $6,000 is due in 5 years. A sinking fund is to be
accumulated at 5%. What sum must be deposited in the sinking fund at
the end of each year to care for the principal when due?
Solution. Here, S = $6,000, n = 5, and i = 0.05.
we have from All, Art. 31,
Since m
R = 6,000 -
0.05
= 6,000-
1
(1.05)* - 1 ' *r, 06
= 6,000 (0.18097480) [Table VII]
= $1,085.85.
The amount in the sinking fund at any particular time may be shown
by a schedule known as an accumulation schedule. The following is the
schedule for the above problem:
Years
Annual Deposit
Interest on Fund
Total Annual
Increase
Value of Fund at
End of Each Year
1
$1,085.85
$1,085.85
$1,085.85
2
1,085.85
$ 54.29
1,140.14
2,225.99
3
1,085.85
111.30
1,197.15
3,423.14
4
1,085.85
171.16
1,257.01
4,680.15
5
1,085.85
234.01
1,319.86
6,000.01
40. Amortization. Instead of leaving the entire principal of a debt
standing for the term to be cancelled by a sinking fund, we may consider
any payment over what is needed to pay interest on the principal to be
112
Financial Mathematics
applied at once toward liquidation of the debt. As the debt is being paid
off, a smaller amount goes towards the payment of interest, so that with
a uniform payment per year, a greater amount goes towards the payment
of principal. This method of extinguishing a debt is called the method
of amortization of principal.
Example. Consider a debt of $2,000 bearing 6% interest converted
annually. It is desired to repay this in 8 equal annual installments, includ-
ing interest. Find the annual installment.
Solution. Here, A = $2,000, n = 8, i = 0.06, m = p
tuting in -412, Art. 31, we have
1. Substi-
R = 2,000
0.06
= 2,000--
1
= 2,000 (0.16103594) [Table VII]
= $322.07.
The interest for the first year will be $120; hence $202.07 of the first
payment would be used for the reduction of principal, leaving $1,797.93
due on principal at the beginning of the second year. The interest on
this amount is $107.88; hence, the principal is reduced by $214.19, leaving
$1,583.74 due on principal at the beginning of the third year, and so on.
This process may be continued by means of the following schedule known
as an amortization schedule:
Year
Principal at
Beginning of Year
Annual Payment
Interest at 6%
Principal Repaid
1
$2,000.00
$322.07
$120.00
$202.07
2
1,797.93
322.07
107.88
214.19
3
1,583.74
322.07
95.02
227.05
4
1,356.69
322.07
81.40
240.67
5
1,116.02
322.07
66.96
255.11
6
860.91
322.07
51.65
270.42
7
590.49
322.07
35.43
286.64
8
303.85
322.07
18.23
303.84
$9,609.63
$576.57
$1,999.99
Such a schedule gives us the amount remaining due on the principal at
the beginning of any year during the amortization period. The principal
Sinking Funds and Amortization 113
at the beginning of the last year should equal the last principal repaid, and
the sum of the principals repaid should equal the original principal.
Exercises
1. Find the annual payment that will be necessary to amortize in 10 years a debt of
$2,500, bearing interest at 8% converted annually. Construct a schedule.
2. A mortgage of $5,000 is due in 8 years. A man wishes to take care of this prin-
cipal when due by depositing equal amounts at the end of each year in a sinking fund
which pays 5% interest. Find the annual deposit and check by an accumulation
schedule.
3. A man owes $10,000 and agrees to pay it in 10 equal annual installments. Find
the amount of each installment, allowing 6% for interest. Check by an amortization
schedule.
4. A farmer buys a farm for $10,000. He has $6,000 to pay down and secures a
federal farm loan for the balance to be amortized in 30 years at 5%. Find the annual
payment and build up a schedule for the first 10 years.
6. In order to construct a filtering plant a city votes bonds for $50,000 which bear
6% interest, payable semi-annually. A city ordinance requires that a sinking fund be
established to retire the bonds when they mature in 15 years. What semi-annual
deposit must be made into the sinking fund, if it accumulates at 4%, converted semi-
annually? What is the total semi-annual expense for the city?
6. A mortgage for $1,000 was given and it was agreed that it might be repaid, prin-
cipal and interest, by 5 equal annual payments. Build up an amortization schedule if
the interest rate is to be 5% for the first two years and 4% for the last three years.
41. Book value. The book value of an indebtedness at any time may
be defined as the difference between the original debt and the amount in
the sinking fund at that time. Thus, in the example of Art. 39, we see
that the book value of the debt at the end of the third year is $2,576.86,
($6,000 $3,423.14). If the debt is being amortized, then the book value
of the debt at the beginning of any year is the outstanding principal at that
time. Thus, in the example of Art. 40, we observe that the book value of
the debt at the beginning of the fourth year (at the end of the third year
just after the third payment has been made) is $1,356.69. The subject of
book value will be discussed further in connection with depreciation and
valuation of bonds.
42. Amount in the sinking fund at any time. To find the amount in
the sinking fund at the end of fc payment periods, k < up, we have only
to find the accumulated value of an annuity of annual rent R for k payment
periods by using the appropriate formula of Art. 31.
Example 1. Find the amount in the sinking fund of the Example of
Art. 39, at the end of 4 years.
114 Financial Mathematics
Solution. Here, R = $1,085.85, k = 4, m = p = 1, and i = 0.05.
Hence, using All, Art. 31, the amount is given by
Sj, = 1,085.85 'Q0 5 = 1,085.85-^.05
= 1,085.85 (4.31012500) [Table V]
= $4,680.15,
which checks with the amount given in the sinking fund schedule for the
fourth year.
Example 2. A debt of $3,000 is due in 12 years. A sinking fund is
created by making equal annual payments. If the interest rate is 5%
converted annually, find the annual payment and the amount in the sink-
ing fund just after the eighth annual payment has been made.
Solution. Here, S = $3,000, n = 12, i = 0.05, k = 8, p = m = 1.
R = 3 ' Q Q (1.06)" 1 " $188 - 48 '
n os") 8 _ i
and $q = 188.48 ^~- - = $1,799.82.
U.Uo
Hence, the amount in the sinking fund at the end of 8 years is $1,799.82.
43. Amount remaining due after the ftth payment has been made.
When loans are paid by the amortization process it is necessary at times
to know the amount of indebtedness (book value) after a certain number
of payments have been made. After k payments of R/p dollars have been
made there remain (np k) payments and these remaining payments
form an annuity whose present value is exactly the amount due on the debt
after the fcth payment has been made, and the debt could be cancelled
by paying this present value.
Example 1. Find the amount of unpaid principal just after making
the fifth payment in the Example of Art. 40.
Solution. Here, R/p = $322.07, n = 8, i = 0.06, m = p = 1, and
k = 5. We have three payments remaining. Hence
1 _ (i 06) ~ 3
An = 322.07 - ^ } [Formula AI2, Art. 31]
0.06
= 322.07 (2.67301195) [Table VI]
= $860.90.
Sinking Funds and Amortization 115
This checks with the value given in the amortization schedule for the
principal at the beginning of the 6th year (just after the fifth payment has
been made).
Example 2. A debt of $2,500 is to be amortized by 7 annual install-
ments with interest at 6%. Find the amount unpaid on the principal just
after making the fifth annual payment.
Solution. Here, A = $2,500, n = 7, k = 5, m = p = 1, and i = 0.06.
We have, using A 12, Art. 31,
R- 2,500 rr ^Lp- H17.84.
= 447.84 (1.83339267) = $821.06.
Hence, the amount unpaid on the principal at the end of the fifth year or
just at the beginning of the sixth year is $821.06.
Exercises
1. A man has been paying off a debt of $2,800 principal and interest in 20 equal
quarterly payments with interest at 5% converted quarterly. At the time of the
13th payment what amount is necessary to make the payment that will extinguish the
entire debt?
2. In order to pay a mortgage of $5,000 due in 7 years, a man pays into a sinking
fund equal amounts at the end of each month. If the sinking fund pays 6% interest
converted monthly, how much has he accumulated at the end of 5 years?
3. A man owes $4,000, which is to be paid, principal and interest, in 10 equal annual
payments, the first payment falling due at the end of the first year. If the interest rate
is 6%, find the balance due on the debt just after the 6th payment is made.
4. A building and loan association sells a house for $7,500, collecting $1,500 cash.
It is agreed that the balance with interest is to be paid by making equal payments at the
end of each month for 10 years. If the interest rate is 7%, converted monthly, find the
monthly payment. What equity does the purchaser have in the house just after making
the 50th payment? What is his equity after the 70th payment has been made?
6. A person owes a debt of $8,000, bearing 5% interest, which must be paid by the
end of 10 years but may be paid at the end of any year after the fourth. He pays into
a sinking fund equal amounts at the end of each year, which will accumulate to $8,000
at the end of 10 years. Just after making the 7thjpayment into the sinking fund, how
much additional money would be required to pay the debt in full, if the sinking fund
accumulates at 5%.
116 Financial Mathematics
44. The amortization and sinking fund methods compared. We shall
make this comparison by discussing a problem.
Problem. Let us consider a debt of principal A^ which is due in n
years and draws interest at rate r payable p times a year.
Discussion. This debt may be amortized by making np equal pay-
ments direct to the creditor, or it may be cared for by the sinking fund
method.
If the amortization method is used the periodic payment will be
(1)
Formula (1) gives us the total periodic expense, if the method of amor-
tization is used.
It is easily seen that, since = i -\ -- ,
a 'n\ -Vj
and (1) may be written
r
(2)
m ,
If the sinking fund method is used, the interest at rate r payable p
times a year is paid direct to the creditor and a fund to care for the principal
when it becomes due n years from now is created by depositing equal pay-
ments p times a year into a sinking fund which accumulates at rate j con-
verted p times a year. If this method is used, the total expense per period
will be the sum of the periodic interest and the periodic payment into the
sinking fund and is given by
Sinking Funds and Amortization 117
Now, if the sinking fund rate is the same as the interest rate on the
debt (j = r), then E of (3) is the same as R/p of (2). That is, when j = r,
the periodic expense is the same by either plan, and the amortization
method may be considered a special case of the sinking fund method where
the creditor has charge of the sinking fund money and allows the same
rate of interest on it that he charges on the debt.
If the sinking fund rate is less than the rate on the debt, that is, if
j < r, then - > - and E in (3) is greater than R/p in (2).
sm at 3/P **npi at r/p
That is, the sinking fund method is more expensive for the debtor than
the amortization method.
If j > r, then - < - - and E is less than R/p. That is,
*i*l at 3/P s w\ at r /P
the sinking fund method is less expensive for the debtor than the amortiza-
tion method.
Example 1. A debt of $10,000, with interest at 6%, payable semi-
annually, is due in 10 years. Find the semi-annual expense if it is to be
cared for by the amortization method.
Solution. Here, A# = $10,000, r = 0.06, p = 2, and n = 10. We
have
fi/2 = 10,000 ^^^o [Formula (1)]
= 10,000 (0.06721571) [Table VII]
= $672.16.
Example 2. Find the semi-annual expense in Example 1, if a sinking
fund is accumulated at (j = .05, p = 2).
Solution. Here, j = 0.05 and the other conditions are the same. We
have
E = 10,000(0.03) + 10,000 ' [Formula (3)]
\L.\
= 300.00 + 10,000(0.03914713) [Table VII]
= 300.00 + 391.47 = $691.47.
Example 3. Find the semi-annual expense in Example 1, if the sinking
fund is accumulated at (j = .06, p = 2).
118 Financial Mathematics
Solution. Here, j = 0.06 and the other conditions are the same. We
have 003
E = 10,000(0.03) + 10,000 (1 Q3)2Q _ 1
= 300.00 + 10,000(0.03721571)
= 300.00 + 372.16 = $672.16.
Example 4. Find the semi-annual expense in Example 1, if the sinking
fund is accumulated at (j = .07, p = 2).
Solution. Here, j = 0.07 and the other conditions are the same. We
have
E = 10,000(0.03) + 10,000
= 300.00 + 10,000(0.03536108)
= 300.00 + 353.61 = $653.61.
Compare the answers of Examples 1, 2, 3, and 4. Are the results con-
sistent with the conclusions that we have already drawn?
Exercises
1. A man secures a $15,000 loan with interest at 6M>%> payable annually. He may
take care of the loan (a) by paying the interest as it is due and paying the principal in
full at the end of 10 years; or (b) by paying principal and interest in 10 equal annual
installments. If a sinking fund can be accumulated at 5%, converted annually, which
is the more economical method and by how much?
2. A debt of $8,000 bears interest at 7%, payable semi-annually, and is due in
7 years. How much should be provided every six months to pay the interest and retire
the debt when it is due, if deposits can be accumulated at 6%, converted semi-annually?
3. What would be the semi-annual expense in Exercise 2, if the debt could be retired
by paying principal and interest in 14 equal semi-annual installments?
4. A debt of $20,000 which bears interest at 5%, payable semi-annually, is to be paid
in full in 20 years. The debtor has the privilege of paying the principal and interest in
40 equal semi-annual payments, or paying the interest semi-annually and paying the
principal in full at the end of 20 years. Compare the two methods if a sinking fund may
be created by making semi-annual payments which accumulate at (a) 4%, converted
semi-annually; (b) 5%, converted semi-annually; (c) 6%, converted semi-annually.
46. Retirement of a bonded debt. In the retirement of a debt which
has been contracted by issuing bonds of given denominations, the periodic
payments cannot be the same, because the payment on principal at the
end of each period must be a multiple of the denomination (face value or
Sinking Funds and Amortization
119
par value) of the bonds or their redemption value* (if not redeemed at
par). By varying the number of bonds retired each time the payments can
be made to differ from each other by an amount not greater than the
redemption value of one bond. An example will make the method clear.
Example. Construct a schedule for the retirement, in 8 years, of a
$30,000 debt, consisting of bonds of $100 face value, bearing interest at
6% payable annually, by making annual payments as nearly equal as
possible.
Solution. If the annual payments were all equal, we would have
0.06
R = 30,000
- - $1,831.08.
- (1.06) -*
The interest for the first year is $1,800. Subtracting this amount from
$4,831.08 leaves $3,031.08 available for the retirement of bonds. This will
retire 30 bonds, for $3,000 is the multiple of $100 which is nearest to
$3,031.08. This makes a total payment (for interest and bonds retired) of
$4,800 for the first year. Subtracting the $3,000 which has been paid on
the principal from $30,000 leaves $27,000 as the principal at the beginning
of the second year. The interest on this amount is $1,620, which when
subtracted from $4,831.08 leaves $3,211.08 to be used for retiring bonds
the second year. This will retire 32 bonds, because $3,200 is the multiple
of $100 which is nearest to $3,211.08. Continuing this process, we obtain
the following schedule :
Unpaid
Number
Value
Year
Principal at
Beginning of
Interest Due
at End of Year
of Bonds
Retired
of Bonds
Retired
Annual
Payment
Year
1
$30,000.00
$1,800.00
30
$3,000.00
$4,800.00
2
27,000.00
1,620.00
32
3,200.00
4,820.00
3
23,800.00
1,428.00
34
3,400.00
4,828.00
4
20,400.00
1,224.00
36
3,600.00
4,824.00
5
16,800.00
1,008.00
38
3,800.00
4,808.00
6
13,000.00
780.00
41
4,100.00
4,880.00
7
8,900.00
534.00
43
4,300.00
4,834.00
8
4,600.00
276.00
46
4,600.00
4,876.00
Totals
$144,500.00
$8,670.00
300
$30,000.00
$38,670.00
* See Art. 54 for definitions.
120 Financial Mathematics
As a check on the work of the schedule the interest on the total of the
unpaid principals should equal the total of the interest due; and the sum
of the totals in the third and fifth columns should equal the total in the
sixth column.
We notice that the annual payment each year varies from the com-
puted payment, $4,831.08, by an amount less than $50 (one-half the face
of one bond).
Exercises
1. Solve the illustrative Example when the bonds have a $500 face value.
2. A city borrows $100,000 to erect a school building. The debt is in the form of
bonds of face value $1,000 bearing interest at 5% converted annually. The bonds are
to be retired by 10 annual installments as nearly equal as possible. Set up a schedule
showing the number of bonds retired each year.
Problems
1. Construct the amortization schedule for the repayment of a loan of $10,000,
principal and interest at 5% nominal, payable semi-annually, in ten semi-annual
payments.
2. Construct an accumulation schedule for the accumulation of $10,000 in 10 equal
semi-annual installments at 6% interest, converted semi-annually.
3. A man deposits in a sinking fund equal quarterly payments sufficient to accumu-
late to $5,000 in 5 years at 6% converted quarterly. What is the amount in the sinking
fund just after the 9th quarterly payment has been made?
4. A debt of $8,000 bearing 5% interest, converted quarterly, is arranged to be paid
principal and interest in 30 equal quarterly payments. How much remains unpaid on
the principal just after the 17th payment is made?
5. The cash price of a house is $7,000. $2,000 cash is paid and it is arranged to pay
the balance by 70 equal monthly payments, including interest at 6%, converted monthly.
Just after the 50th payment is made, what is the balance due on the principal?
6. A mortgage for $7,500, bearing 6% interest payable semi-annually, is due in 12
years. A fund to care for the principal when it becomes due is established by making
semi-annual payments into a sinking fund, (a) Find the semi-annual expense of the
mortgage if the sinking fund accumulates at 5% semi-annually. (b) Find the semi-
annual expense of the mortgage if it is amortized by equal semi-annual payments.
7. What is the book value of the debt in Problem 6 at the end of 7 years, (a) if the
sinking fund method is used, (b) if the amortization method is used?
8. A man buys a house for $5,500, paying $1,500 cash. The balance with interest at
6% is to be cared for by paying $700 at the end of each year as long as such a payment
is necessary and then making a smaller payment at the end of the last year. Find the
number of full payments and the amount of the final payment. What amount remains
due just after making the 5th payment?
Sinking Funds and Amortization 121
9. A city borrows $100,000 at 5%. The debt is to be retired in 10 years by the
accumulation of a sinking fund that is invested at 4% effective. What is the total annual
expense to the city?
10. A county borrows $50,000 to build a bridge. The debt is to be paid by amortiza-
tion of the principal in 15 years at 5%. At the end of the tenth year what principal
remains outstanding?
11. A fraternity chapter borrows $60,000 at 6% to build a house. The debt is to be
amortized in 25 years. What is the annual payment?
12. A fraternity chapter borrows $60,000 at 6% to build a house. A sinking fund
can be built up at 5%. What amount must be raised annually to pay this debt if the
payments are to extend over 30 years?
Review Problems *
1. A well-known finance company requires payments of $7.27 a month for 18 months
for a loan of $100. What rate of interest does the borrower pay?
2. The cash price of an automobile is $995. An advertisement of a dealer stated,
"If you want to buy on terms, pay a little more for the convenience, $329 down and $63
a month for 12 months." What rate of interest does one pay who purchases the car on
the installment plan?
3. An automobile, cash price $1,300, was purchased on the terms, $507 down and
$57.50 a month for 18 months. What rate of interest was paid?
4. Solve A = Ras{i(l 4- i)~ m (a) for m\ (b) for n.
5. If C is the first cost and D is the renewal cost of an article whose life is r years,
show that the capitalized cost, K, at the rate i is given by
6. A machine costs $2,500 new and must be replaced at the end of each 10 years.
Find the capitalized cost if money is worth 5% and if the old machine has a salvage
value of $500.
7. A debt of $10,000 with interest at (j = .06, m 12) is to be amortized by pay-
ments of $100 a month. After how many payments will the debt be paid in full? What
is the final partial payment?
8. A $10,000 bequest invested at 4% is to provide a scholarship of R at the end of
each year for 25 years at which time the bequest is to be exhausted. Find R.
9. The Empire State Building was erected at a cost of $52,000,000. If its estimated
useful life is 100 years and its salvage value is to pay for its demolishing, what net annual
income for 100 years would yield 5% on the investment?
10. If interest is at 5% for the first 10 years and 4% thereafter, what equal annual
payments for 15 years will repay a $10,000 loan?
11. Show (a) by verbal interpretation and (b) algebraically that
when r ^ n.
* For additional review problems, see end of this book.
CHAPTER V
DEPRECIATION
46. Definitions. A building, a machine or any article of value into
which capital has been invested will be referred to as an asset. These
assets decrease in value due to use, action of the elements, lack of care, old
age, and other causes. A part of this decrease in value may be taken
care of by proper repairs, but repairs will not cause an asset to retain
its original value. In fact, some assets will decrease in value whether
they are used or not. This may be due to new inventions or decreases
in the [market prices or a combination of these and other causes. For
example, an automobile will decrease in value even though it does not
leave the floor of the showroom. (Why?) That part of the decrease in
value of an asset which can not be cared for by repairs is commonly known
as depreciation.
Good business principles demand that capital invested in an asset or a
business consisting of several assets, should not be impaired. Hence, from
the revenues of the asset or the business there should be set aside, periodi-
cally, certain sums, such that the accumulation of these sums at any time
plus the value of the asset at that time shall equal its original value. The
fluid into which these periodical sums are set aside is known as a depreciation
reserve. This depreciation reserve is usually retained in the business but is
carried as a separate item on the books of the business. The object of the
accounting for depreciation and the setting aside of a depreciation reserve
is to recover only the capital originally invested in the asset. The account-
ant is not concerned with the replacement of the asset, whether lower or
higher than the original cost. His chief concern is that the original capital
be not impaired, for this is a fund that must be considered as belonging to
the holders of the stock in the business.
These assets may never be replaced at any price for the company may
go out of business. Then this accumulated value would be used to retire
the capital stock. If the assets are replaced at a lower cost, then only a
part of this accumulated value may be considered as used for the replace-
ment. If the assets have to be replaced at a higher cost, then the differ-
122
Depreciation 123
ence between this cost and the accumulated value reserve must be met
by increasing the original capital. Regardless of the way that depreciation
is considered by the accountant, the mathematical principles involved in
the treatment of the subject remain the same.
Although an asset may become obsolete or useless for the purpose for
which it was intended originally, it may be of value for some other purpose.
This value is commonly known as the scrap value or trade in value of the
asset and the time it was in use up to the date it was replaced or discarded
is known as its useful life. The original value minus the scrap value is
defined as the wearing value or the total depreciation of the asset. At any
time during the life of an asset its book value may be defined as the original
value (or value when it became a part of the business) minus the value of
the depreciation reserve. The amount by which the depreciation reserve
increases any year is known as the annual depreciation charge.
47. Methods of treating depreciation. There arc many methods of
treating depreciation. We shall treat four of the most common methods:
(a) The straight line method.
(b) The sinking fund method.
(c) The fixed percentage on decreasing value method.
(d) The unit cost method.
Some of the other methods used are the compound interest method,
the service output method, the maintenance method, and so on.
48. The straight line method. By this method the total depreciation
(wearing value) is distributed equally over the life of the asset and the
amounts in the depreciation reserve do not earn interest. If we let C stand
for the original value (cost) of the asset, /S stand for its scrap value, n
stand for its useful (probable) life, W stand for its wearing value, and D
stand for the annual depreciation charge to be made, it follows from the
above definition of the straight line method that
D== f' (1)
where W = C - 8.
Example. A certain asset costs $2,250. It is assumed that with proper
care it will have a scrap value of $170 after a useful life of 8 years. Using
the straight line method, show by schedule and graph the value of the
depreciation reserve and the book value of the asset at any time.
124 Financial Mathematics
Solution. We have, C = $2,250, 8 = $170, n = 8, and W = $2,080.
Therefore, D
8
The value of the depreciation reserve at the end of the first year will
be $260 and this will increase each year by the constant amount, D $260,
until at the end of 8 years it will contain $2,080. The book value of the
asset will decrease each year by the constant amount, D = $260, until
at the end of 8 years it will be $170 (the scrap value).
The following schedule shows the book value of the asset and the
amount in the depreciation reserve at any time.
SCHEDULE OF BOOK VALUE AND DEPRECIATION
STRAIGHT LINE METHOD
Age in Years
Book Value
Depreciation Charge
Total in
Depreciation Reserve
$2,250.00
1
1,990.00
$260.00
$260.00
2
1,730.00
260.00
520.00
3
1,470.00
260.00
780.00
4
1,210.00
260.00
1,040.00
5
950.00
260.00
1,300.00
6
690.00
260.00
1,560.00
7
430.00
260.00
1,820.00
8
170.00
260.00
2,080.00
Observing the above schedule, we notice that the book value at the
end of any year plus the total in the depreciation reserve at that time equals
the original cost of the asset.
The changes in the book value and depreciation reserve may also be
shown by graphs. [See Fig. 1J
Observing the graphs for depreciation and book value, we notice that
the ordinate for depreciation at any time plus the ordinate for book value
at the same time equals the original value of the asset. We also observe
that the graphs which represent the book value and depreciation reserve
are straight lines. This suggests why this method is known as the straight
line method.
Depreciation
125
$2,250
2,000
1,750
1,500
1,250
1,000
750
500
250
\
v
\
y
/
^
v
>
/
^
/
k
A
Y
\
<f
*/
\
/
\
/
8
Years
Fig. 1. Graphical Representation of Book Value and Depreciation
Straight Line Method.
49. Fixed-percentage-on-decreasing-value method. This method de-
rives its name from the fact that the book value at the end of any year is
obtained by decreasing the book value at the end of the preceding year by
a fixed percentage. It is assumed that the book value is reduced from the
original cost C to the scrap value S at the end of n years, and the amounts
in the depreciation reserve do not earn interest.
Let C stand for the original cost of an asset and let x be the fixed per-
centage by which the book value is decreased each year.
During the first year the decrease in book value is Cx and consequently,
the book value at the end of the first year is
C l = C - Cx = C(l - x).
The book value at the end of the second year is
C 2 = Ci(l - x) = C(l - z)(l - x) = C(l - x) 2 .
The book value at the end of the third year is
C 3 = C 2 (J - *) = C(l - z) 2 (l - x) = C(l - x}*.
126 Financial Mathematics
Continuing our reasoning we find the book value at the end of n years to be
C n = C(l - X)\
But the book value of the asset at the end of its useful life, n years, equals
its scrap value S. Hence, we have*
C (1 - x) n = S (2)
logd-,)^ 1 ^^ 10 ^. (3)
n
Using (3), the fixed percentage may be computed for any particular case.
If we let Ck represent the book value of the asset at the end of k years,
we observe thnt
C k = C(l ~ x)*, (4)
and log C A = log C + k log (1 - x) (5)
We further observe that by using (3) and (5) and allowing fc to assume
all consecutive integers from 1 to n inclusive, we may compute, entirely
by the use of logarithms, the successive book values of the asset. An
example will illustrate the method.
Example. Find by the fixed percentage method the book values at the
end of each year for a machine costing $800, and having an estimated life
of 8 years and a scrap ^value of $80. Construct a schedule showing the
book values and amount in the depreciation reserve at the end of each
year.
Solution. Here, C = $800, S = $80, n = 8.
Using (3), we get
log (l-.) = ^ 9.87500 -10.
8
Then using (5), we have
log Ck = log 800 + (9.87500 - 10).
= 2.90309 + &(9.87500 - 10).
* It will be observed from (2) that, when S = 0, we have x - 1 for any assigned
value of n. That is, the book value is reduced to zero at the end of 1 year, no matter
what is the estimated value of n. This means that the method is impractical when S is
zero. Even if the ratio of S to C is small, the depreciation charge is likely to be unreason-
ably large during the first years of operation.
Depreciation
127
Giving k all values from 1 to 8, we get
log Ci = 2.77809, Ci = $599.91.
log C 2 = 2.65309, C 2 = 449.87.
log C 3 = 2.52809, C 3 = 337.35.
log C 4 = 2.40309, C 4 = 252.98.
log C 5 = 2.27809, C 5 = 189.71.
logC 6 = 2.15309, C 6 = 142.26.
log C 7 = 2.02809, CV = 106.68.
logC 8 = 1.90309, Cs = 80.00.
The student will observe that the actual value of x (fixed percentage)
was not needed in the above computations. Should we desire the value
of x, we find that 1 x is the antilogarithm of 9.87500 10, or 0.7499.
Hence, x - 0.2501 = 25.01%.
Since the book value at the end of the first year is $599.91, the deprecia-
tion charge for that year is
$800.00 - $599.91 = $200.09.
The depreciation charge for the second year is
$599.91 - $449.87 = $150.04
and the total in the depreciation reserve at the end of two years is
$200.09 + $150.04 = $350.13.
The following schedule shows the book values and the amount in the
depreciation reserve at the end of each year.
SCHEDULE OF BOOK VALUE AND DEPRECIATION
FIXED PERCENTAGE METHOD
Age in Years
Annual Depreciation
Total in
Depreciation Reserve
Book Value
o
$800.00
1
$200.09
$200.09
599.91
2
150.04
350.13
449.87
3
112.52
462.65
337.35
4
84.37
547.02
252.98
5
63.27
610.29
189.71
6
47.45
657.74
142.26
7
35.58
693.32
106.68
8
26.68
720.00
80.00
128
Financial Mathematics
The changes in the book value and depreciation reserve may also be
shown by graphs.
>BUU
700
600
500
400
300
200
100
\
IT """'
\
^
^>
^
k
^>
A
^
\
y
S
/
K
<\
/
^
/
"^
^
z_
12345678 Years
Fig. 2. Graphical Representation of Book Value and Depreciation
Fixed Percentage Method.
50. The sinking fund method. In the sinking fund method the total
depreciation (wearing value) of the asset is provided for by accumulating
a sinking fund at a given rate of compound interest. The annual payment
into the sinking fund is the payment on an annuity which will have an
amount equal to the total depreciation (wearing value) of the asset at the
end of its useful life.
If C is the cost, S the scrap value, W the wearing value, and n the esti-
mated useful life of the asset, we find, using All, Art. 31, the annual pay-
ment into the sinking fund to be
i W
R = W 1 = , (6)
where W = C - S.
By this method the depreciation charge for the first year is R and the
amount in the depreciation reserve at the end of the first year is R. How-
ever, the depreciation charge increases each year and for any subsequent
year it is R plus the interest on the amount in the depreciation reserve
during that year.
Depreciation
129
Example. Assuming money worth 4j%, apply the sinking fund
method to the Example discussed in Art. 49.
Solution. Here, C = $800, S = 80, n = 8, i = 0.045, and W =
C - S = $720.
Using (6), we get
The depreciation charge for the first year is R - $76.76. Consequently,
the amount in the depreciation reserve at the end of the first year is $76.76
and the book value of the asset at that time is $800.00 less $76.76 or $723.24.
The depreciation charge for the second year is K, ($76.76), plus the interest
on $76.76 (the amount in the depreciation reserve during the second year)
at 4%. Thus, the depreciation charge for the second year is $76.76 +
$3.45 = $80.21. Then, the amount in the depreciation reserve at the end
of two years is $76.76 plus $80.21 or $156.97 and the book value of the asset
at that time is $643.03. Values for subsequent years are found in a similar
manner.
The following schedulelwill show the values for each year.
SCHEDULE OF BOOK VALUE AND DEPRECIATION
SINKING FUND METHOD
Age in
Years
Annual
Payment
Interest
on Fund
Annual
Depreciation
Charge
Amount in
Depreciation
Reserve
f
Book Value
of Asset
o
$SOO.OO
1
$76.76
$0.00
$76.76
$76.76
723.24
2
76.76
3.45
80.21
156.97
643.03
3
76.76
7.06
83.82
240.79
559.21
4
76.76
10.84
87.60
328.39
471.61
5
76.76
14.78
91.54
419.93
380.07
6
76.76
18.90
95.66
515.59
284.41
7
76.76
23.20
99.96
615.55
184.45
8
76.76
27.70
104.46
720.01
79.99
The above information is shown by means of graphs in Fig. 3.
130
Financial Mathematics
$800
700
600
500
400
300
200
100
n
\
V
N,
<%
s
/
s
^
>
/
N^
\
/
A
/ s
^
'/
>
\
/
/
>
\
x
>
12345678 Years
Fig. 3. Graphical Representation of Book Value and Depreciation-
Sinking Fund Method.
51. The unit cost method. None of the three methods of depreciation
already discussed takes into consideration the question of improvements
in machinery. The unit cost method is based upon the principle that the
value of the old machine should be decreased from year to year to such an
extent that the net cost of a unit of output of the machine should be the
same as the net cost of a unit of output of a new machine with which it
could be replaced. The old machine should be so valued that its unit
cost of production, after taking into account all charges for depreciation,
repairs, interest, and operating expenses, is the same as that of a new
machine. Let us illustrate by an example.
Example 1. Consider the replacement of a machine which costs $300
a year to operate, costs $100 a year for repairs, turns out 25 units of work
per year and has a probable life of 5 years. A new machine costs $2,500,
costs $400 a year to operate, costs $100 a year for repairs, turns out 40
units of work per year, and has a probable life of 9 years. Find the value
of the old machine, assuming money worth 4%.
Solution. Let x be the value of the old machine. The cost of repairs
and operation on the old machine is $400. 0.04# is the interest on the
investment, and Q Q ,
X (1.04) 5 - 1
Depreciation 131
is the annual payment required to accumulate the value of the old machine
in 5 years.
- 04 * + * (IMP- 1 = - 22462711 *-
Hence, the unit cost of production for the old machine is
400 + 0.2246271 Ix
25
- 16 + 0.0089851z.
Reasoning the same as above, we find the yearly cost for operating
the new machine to be
400 + 100 + 2,500(0.04) + 2,500 - ^ - = 836.232475.
(1.04) J 1
Hence, the unit cost of production for the new machine is
= 20.905812.
40
According to the principle of the unit cost method, we have
16 + 0.0089851* = 20.905812,
4.905812
and
Hence, assuming money worth 4%, the value of the old machine as com-
pared with the value of the new is $546.00.
We shall now derive a formula for determining the value of the old
machine as compared with the new machine. Let
C = the original cost of the new machine,
N = the estimated lifetime of the new machine,
= the annual operating expense of the new machine not including
repairs,
R = the annual cost of repairs for the new machine,
K =i the annual rent of an annuity required to accumulate C in N
years,
U = the number of units of output per year.
Let the corresponding letters o, r, k, and u denote the corresponding
quantities for the old machine. Let c be the value of the old machine at
132
Financial Mathematics
the time of making the comparison, and n the remaining lifetime of the
old machine. Let i be the rate of interest.
The unit cost for the new machine is
+ R + K + Ci
U
and the unit cost of the old machine is
o + r + k + ci
u
According to the principle of the unit cost method, we have
+ R + K + Ci o + r + k + ci
U
u
(7)
Since,
K = and k = ,
and
Then (7) becomes
K + Ci = C\i + ) = [(14), Art. 26]
( 1\ c
k -f ci = ci i H ) =
\ Ssi/ 051
+ R + o+ r +
U
u
Solving (8) for c, we have
+ R +
^
(8)
(9)
e u j
If the number of units of output of the old and new machines are the
same, U = u, (9) reduces to
(10)
(U)
+ R + -o-rl-
OJTI J
If O o, along with U u, (10) reduces to
c = aJ/2 + r)-
\ a^ /
Depreciation 133
If + R = o + r, then (10) becomes
c-^S- (12)
a JT\
Example 2. A machine having a remaining service life of 6 years turns
out 30 units of work per year. Its operation costs $300 per year, and
repairs cost $225 per year. A new machine, that turns out 40 units of
work, costs $1,000. It has a probable life of 10 years and will cost $350
a year for operation and $250 a year for repairs. Assuming money worth
5%, find the value of the old machine.
Solution. Here, C = $1,000, N = 10, = $350, R = $250, U = 40,
n = 6, o = $300, r = $225, and u = 30.
-L = _L = 0.12950458,
a N\ a lQ\
a^ = agj = 5.07569206.
Substituting in (9), we have
[350 + 250 + 1,000(0.12950458) 300 + 225
c - 30(5.07569206)
= 30(5.07569206) [18.23761 - 17.50000]
152.2708(0.7376) = $112.31.
Exercises
1. A fanner pays $235 for a binder. The best estimates show that it will have a life
of 8 years and a scrap value of $15. Find the annual depreciation charge by the straight
line method and construct a schedule of depreciation.
2. A tractor costs $1,200. It is estimated that with proper care it will have a life
of 8 years with a scrap value of $50 at the end of this time. Construct a depreciation
schedule, using the sinking fund method and assuming 4% interest.
8. An automobile, costing $950, has an estimated life of 5 years and a scrap value
of $50. Prepare a depreciation schedule using the fixed percentage method.
4, A machine costs $5,000. The best estimates show that after 10 years of use its
scrap value will be $1,000. (a) Making use of the fixed percentage method, find the
134 Financial Mathematics
book value of the machine at the ends of 7 and 8 years, respectively, (b) What is the
depreciation charge for the 8th year?
6. Solve Exercise 4, making use of the sinking fund method and assuming an interest
rate of 5%.
6. Solve Example 2 of Art. 51, if the new machine could turn out 45 units of work
per year. Interpret the results.
7. How many units of work must be turned out by the new machine of Example 2,
Art. 51, so that the old machine would not have any value?
8. From formula (9) derive a formula for the number of units a new machine should
turn out in order to make the old machine worthless.
9. A machine having a probable life of 18 years has been in use for 8 years
and turns out 200 units of work each year. The cost for operating is $600 per year and
repairs are $400 per year. A new machine costs $3,000 and has a probable life of 20
years and will turn out 200 units of work per year. It would cost $500 per year to
operate this machine and repairs would cost $300 per year. Neither machine is sup-
posed to have any salvage value. What is the value of the old machine on a 6% interest
basis?
10. What output for the new machine in Exercise 9 would render the value of the
old machine zero?
11. An asset costs $1,000. It is estimated that with proper care it can be used for
8 years at which time it will have a value of $50. Using the sinking fund method and
assuming 4% interest, find the wearing value that remains at the end of 5 years. [Hint:
The wearing value that remains at the end of any year equals the total wearing value
minus the amount in the depreciation reserve at that time. Observing the schedule for
the Example of Art. 50, we see that the wearing value that remains after 5 years of use
is ($720.00 - $419.93) = $300.07].
12. Solve Exercise 11, making use of the fixed percentage method.
13. Solve Exercise 11, making use of the straight line method.
62. Depreciation of mining property. Investment in mines, oil wells,
and timber tracts should yield not only interest on the investment, but
additional income to provide for the restoration of the original capital
when the asset is exhausted. The mining engineer can estimate the net
annual return on the mine and the number of years before the mineral
will be exhausted. From this net annual return, interest on the capital
invested must be taken and also an annual payment to a depreciation
reserve which shall accumulate to the original cost of the mine, less the
salvage value, by the time it is exhausted.
An important problem in connection with mining property is, having
given the net annual yield and the number of years this yield will con-
tinue, to determine the price that should be paid for the mines so that this
net annual yield will provide a sufficient rate of interest on the investment
and an annual payment to the depreciation reserve.
Depreciation 135
Assume that R is the net annual return and that this yield will con-
tinue for n years. Also assume that the rate of yield on the invested cap-
ital is to be r and the depreciation reserve is to be accumulated at rate i.
If we let P stand for the purchase price of the property, then the an-
nual return on the capital invested would be Pr. Hence, the amount
left from the net annual return, for the annual contribution to the deprecia-
tion reserve, would be (R Pr), and this must accumulate to P S in
n years at rate i, where $ is the salvage value.
Therefore, we have
P - S = (R - Pr) (1 + ?* ~ 1 = (R - Pr)s m . (12')
When S = 0,
*
Example. A mining engineer estimates that a copper mine will yield
a net annual income of $50,000 for the next 20 years. What price should
be paid for the mine, if the depreciation reserve is to accumulate at 5%,
if 10% is to be realized on the capital invested, and if S = 0?
Solution. We have, R = $50,000, n = 20, r = 10%, and i = 5%.
Making use of (13), we get
50,000 50,000
0.10 + .' . 0.10
(1.05) 20 ^ 1 ' 5201-05
50,000.00 50,000.00
0.10 + (0.03024259) 0.13024259
= $383,899, purchase price.
This would give a return of $38,389.90 on the invested capital and leave
$50,000 - $38,389.90 = $11,610.10 for the annual payment into the de-
preciation reserve. This annuity in 20 years at 5% will amount to
$383,899.
Exercises
1. An oil well which is yielding a net annual income of $30,000 is for sale. The
geologist estimates that this annual income will continue 10 years longer. What should
be paid for the well, if the depreciation reserve is to accumulate at 4}^%, and 8% is to
be realized on the invested capital?
136 Financial Mathematics
2. A gold mine is yielding a net annual income of $100,000. Careful estimates show
that the mine will continue to yield this net annual income for 25 years longer, at which
time it will be exhausted. Find its value, if a return of 9% on the invested capital is
desired and the depreciation reserve accumulates at 5%.
3. A 1,000 acre tract of timber land is for sale. It is estimated that the net annual
income from the timber will be $125,000 for the next 5 years, at which time the land
will be worth $25 per acre. How much per acre should be paid for the land, if the
purchaser desires 10% on his investment and the depreciation reserve can be accumu-
lated at 5%?
4. $750,000 is paid for a mine which will be exhausted at the end of 25 years. What
net annual income is required from the mine, if 8% is to be realized on the investment
after the annual payments have been made into the depreciation reserve which accumu-
lates at 4%?
53. Composite life of a plant. We will consider that a manufactur-
ing plant consists of several parts, each having a different probable life.
By the composite life of a plant we mean a sort of average lifetime of the
several parts, and we may define it more precisely as the time required for
the total of the equal annual payments to the depreciation reserves of the several
parts to accumulate to the total wearing value of the plant.
Let Wi, W2, Ws, , W r be the wearing values of the several parts,
with probable lives of m, n?, ft.3, , n t respectively, and let W = W\ +
W<2 + Wz + + W r be the wearing value of the entire plant. Also
let DI, Z>2, D.J, -,D r be the annual payments to the depreciation
reserves for the several parts and let D = DI + Z>2 + DZ + + D r
be the depreciation for the whole plant.
Then by the straight line method, we have
= W = W, + W z + W 3 + + W r
71 D D! + D 2 + D 3 + + D, '
or
*Ti + W 2 + W 3 + + W r
* ** r
--- 1 --- j --- l_ . . . _J --
HI na n 3 n r
( )
Example 1. A plant consists of parts A, B, and C, having the following
values, scrap values, and probable lives, respectively:
A $25,000 $5,000 20 years
B 20,000 2,000 18 years
C 8,000 1,000 14 years
Find its composite life.
Depreciation 137
Solution. Here, Wi = $20,000, W 2 = $18,000, TF 3 = $7,000, m = 20,
tt2 = 18, fta = 14. Using (14), we get
__ 20,000 + 18,000 + 7,000
H ~ 20,000 18,000 7,000
20 18 14
2,500
Hence, the composite life is 18 years.
If the sinking fund method is used, we have
(Di + D 2 + + D r ) s^ {i = (Wi + W 2 + - - + TFr),
or D*n = W, (140
where DI = W\ , and so on.
^Hi
Solving (14') for n by the use of logarithms, we get
n = , ' ' (15)
log (1 + i)
The value for n obtained from (15) gives us the composite life. We
may also express (14') in the form
(i + Q" - i _w
$n\i - . - D
and read the approximate value for n from Table V.
Example 2. Solve Example 1, using the sinking fund method and 5%
interest.
Solution. Here, Wi = $20,000, TF 2 = $18,000, Ws = $7,000, m 20,
n 2 = 18, n z = 14, i = 0.05.
Whence, DI = 20,000 = $604.85,
D 2 = 18,000 -i- = $639.83,
Z> 3 = 7,000-^- = $357.17,
D = $1,601.85 and W = $45,000.
138
Financial Mathematics
Using (16), we get
5 ].05 ~
(1.05) n - 1 45,000
0.05
1,601.85
= 28.0925.
From Table V, we notice that the nearest value of n is 18. In fact,
when n = 17, the table value is 25.8404, and when n = 18, the table
value is 28.1324. Hence, n is a, little less than 18 and we say the com-
posite life is approximately 18 years.
Using (15), we have
log (3,851.85) - log (1,601.85)
H ~ log (1.05)
3.58567 - 3.20462 0.38105
0.02119
0.02119
= 17.98, or approximately 18.
Exercises
1. Allowing interest at 5%, find the composite life of the plant consisting of the
following parts.
Parts
Original Cost
Scrap Value
Life
Building
$150,000
$40,000
25 years
Machinery
Patterns
Tools
75,000
15,000
25,000
25,000
5,000
25 years
10 years
12 years
2. Solve Exercise 1, using the straight line method.
3. Allowing interest at 4%, find the composite life of the plant consisting of the
following parts.
Parts
Cost
Scrap Value
Life
A
$200,000
$30,000
50 years
B
150,000
20,000
40 years
C
50,000
10,000
35 years
D
30,000
5,000
20 years
E
25,000
5,000
25 years
4. Solve Exercise 3 by the straight line method.
Depreciation
139
Problems
1. A church with a probable life of 75 years has just been completed at a cost of
$125,000. It is free of debt. For its replacement at the end of its probable life the
congregation plans to make annual payments from their current funds into a sinking
fund that will earn 4% effective. What is the annual payment?
2. The value of a machine decreases at a constant annual rate from the cost of $1,200
to the scrap value of $300 in 6 years. Find the annual rate of decrease, and the value
of the machine at the ends of one, two, and three years.
3. The United States gross imports of crude rubber increased from 252,922 long
tons in 1920 to 563,812 long tons in 1929. Find the annual rate of increase during this
period, assuming that the annual rate of increase was constant.
4. A dormitory is planned at a cost of $250,000. Its probable life is estimated to
be 50 years at the end of which time its scrap value will be zero. To reconstruct the
building at the end of its probable life, a sinking fund, into which semi-annual payments
will be made, is to be created, the fund earning interest at (j = .04, m = 2). What is
the semi-annual payment?
6. It is estimated that a quarry will yield $15,000 per year for 8 years, at the end
of which time it will be worthless. If a probable purchaser desires 8% on his investment
and is able to accumulate a redemption fund at 4%, what should he pay for the quarry?
6. On a 3% basis find the annual charge for replacement of a plant, and its composite
life, if the several parts are described by the table:
Part
Life in years
Cost
Scrap Value
A
40
$200,000
$10,000
B
25
50,000
3,000
C
15
20,000
1,000
D
10
10,000
1,000
7. A philanthropist wishes to donate a building to cost $200,000 and to provide for
its rebuilding every 50 years at the same cost. He also wishes to provide for its complete
renovation every 10 years at a cost of $20,000 and for annual repairs at a cost of $2,000.
What amount should he donate, if the sums can be invested at 4%?
8. In starting a transfer business it is planned to purchase 10 cabs annually for 5
years at a cost of $1,000 per cab. On a 4% basis, what is the present value of these
purchases if the first allotment is purchased immediately?
It is estimated that 5 years is the service life of these cabs. It is also planned to
replace the worn out cabs by making annual payments at the end of each year into a
sinking fund that earns 4% effective, R at the end of the first year, 2R at the end of the
second year, 3R at the end of the third year, 4# at the end of the fourth year, 5R at the
end of the fifth and later years. What is the annual payment into the sinking fund at
the end of the first year? at the end of the second year? at the end of the fifth year?
What is the amount in the sinking fund just after the first allotment for replacements?
(See Art. 38.)
140 Financial Mathematics
9. In starting a transfer business it is planned to purchase 10 cabs immediately, 8
cabs at the beginning of the second year, 6 at the beginning of the third year, 4 at the
beginning of the fourth year and 2 at the beginning of the fifth year. On a 4% basis,
what is the present value of these purchases if each cab costs $1,000? (See Art. 38.)
10. Find the present value of the output of an oil well on the assumption that it will
produce a net return of $25,000 the first year, diminishing each year by $5,000 until it is
exhausted at the end of the fifth year. Use interest at 8% effective.
11. Show that the unit cost plan of appraisal of value gives the same result as the
sinking fund method when the new and the old machines have the same output and the
same annual expense charge for operation and upkeep.
Review Problems *
1. A quarry has sufficient stone to yield an income of $20,000 a year for 5 years at
the end of which time it will be exhausted. Find the value of the quarry if the invest-
ment is to yield 8% and the redemption fund is accumulated at 4%.
2. Telephone poles set in soil last 12 years, in concrete 20 years. If a telephone pole
set in soil costs $6, what can the company afford to pay to set the pole in concrete if
money can be invested at 4%?
3. In computing the annual return at rate i on the capitalized cost, K, of an article,
show that the return would be equivalent to allowing interest on the original investment,
C, and allowing for depreciation by (6) Art. 50. (See Problem 5, page 121.)
4. A city incurs a debt of $200,000 in constructing a high-school building. Which
would be better: to pay the debt, principal and interest at 63^% in 20 annual install-
ments, or to pay 6% interest each year on the debt and pay a fixed amount annually for
20 years into a sinking fund which accumulates at 4%?
6. A county borrows $75,000 to build a bridge. The debt is to be paid by the amor-
tization of the principal in 15 years at 6%. At the end of the tenth year what part of
the debt is unpaid?
6. A man pays $1,000 a year for 4 years and $2,000 a year for four years on a debt
of $10,000 bearing interest at 6%. What part of the debt is unpaid at the end of 8
years?
7. A machine costing $5,000 has an estimated life of 10 years and a scrap value of
$500. Find the constant rate at which it depreciates. What is its value at the end of
the second year?
8. If W r is the wearing value of a machine at the end of r years by the sinking fund
method, show that
W r - W
* For additional review problems, see end of this book.
CHAPTER VI
VALUATION OF BONDS
54. Definitions. A bond may be defined as a certificate of ownership
in a portion of a debt due from a city, corporation, government, or an
individual. It is a promise to pay a stipulated sum on a given date, and
to pay interest or dividends at a specified dividend rate and at definite inter-
vals. The interval between dividend payments is usually a year, a half
year, or a quarter year. The amount named in the bond is called the
face value or par value. When the sum due is repaid as specified in the
bond, the bond is surrendered to the debtor and it is said to be redeemed.
The price at which a bond is redeemed is called the redemption price.
It may be redeemed at par, below par or above par. When the redemption
price of a bond is the same as the face value, it is said to be redeemed
at par] if it is more than its face value it is said to be redeemed at a pre-
mium] and if it is less than its face value it is said to be redeemed at a
discount.
66. Purchase price. Bonds are usually bought to yield the purchaser
a certain rate of interest on his investment. This rate may be very differ-
ent from the rate of interest specified in the bond. To avoid confusion,
we shall designate the rate of interest specified in the bond as the dividend
rate and the rate of interest received by the purchaser, on his investment,
as the investment rate. When an individual buys a bond he expects to
receive the periodic dividends as they fall due from the date of purchase
to the redemption date and also receive the redemption price when due.
It is clear then that the purchase price is really equal to the present value of
the"*redemption price plus the present value of the annuity made from the
periodic dividends, both figured at the investment rate.
Example 1. Find the purchase price of a $1,000, 4}^% bond, dividends
payable annually, to be redeemed at par in 18 years when the investment
rate is to be 6% annually.
142 Financial Mathematics
Solution. Here, the redemption price is $1,000, the dividend is $42.50
annually. Denoting the purchase price by P, we get
P = 1,000(1.06)-" + 42.50 l ~ () ~ l *
= 1,000(0.3503438) + 42.50(10.8276035)
= 350.34 + 460.17 = $810.51.
Example 2. Find the purchase price of the above bond if it is to be
redeemed at $950.
1 (1 06) ~ 18
Solution. P = 950(1.06) ~ 18 + 42.50 - ^-'-~
O.Oo
= 332.83 + 460.17 = $793.00
If we let C = the redemption price,
(jt ) nominal investment rate,
n = number of years before redemption,
R the annual rent of the dividends,
p = the number of dividend payments each year,
and P = the purchase price,
we may write down the following general formula which will give the pur-
chase price under all conditions.
-1
Now, if m = p (that is, if the interest is converted at the same time
that the dividends are paid), the above formula reduces to
- np
P
In most cases formula (2) will apply.
Valuation of Bonds 143
When P is greater than (7, the bond is bought at a premium. The differ-
ence, (P C), is the premium. Similarly, when P is less than C, the bond
is bought at a discount. The difference, (C P), is the discount. When P
equals C the bond is bought at par. The bond in Example 1 was bought
at a discount of ($1,000 - $810.51), or $189.49.
Example 3. Find the purchase price of a $500, 6% bond, dividends
payable semi-annually, to be redeemed at par in 20 years, when the invest-
ment rate is to be 5^% converted semi-annually.
Solution. Here, C = $500, n = 20, j = 5J^%, R = $30, m = p = 2.
Using formula (2), we have
i _ /i 027^ ~ 40
P = 500(1.0275) ~ 40 + 15
= 500(0.33785222) + 15(24.07810106)
= 168.926 + 361.172 = $530.10
Premium = $530.10 - $500
= $30.10.
Example 4. A $500, 5% bond, dividends payable semi-annually, is
to be redeemed in 15 years at 104 (at 104% of the face). What should its
purchase price be, if the investment rate is to be 6% converted semi-
annually?
Solution. Since the bond is to be redeemed at 104, we have C = $520.
n = 15, j = 6%, R = $25, m = p = 2.
Making use of (2), we find
1 (1 03) - 30
P = 520(1.03) ~ 30 + 12.50 - - ^p
- 520(0.41198676) + 12.50(19.60044135)
= 214.233 + 245.006 = $459.24.
Discount = $520 - $459.24 == $60.76
144 Financial Mathematics
If we let K equal the present value of the redemption price =
/ A-"*
C [ 1 H j , and g equal the ratio of the annual rent of the dividends to
\ P/ R
the redemption price = , formula (2) reduces to
C
p = K + 9 - (C - K). (3)
The student will notice that (3) does not require an annuity table for
its evaluation. It was first established by Makeham, an English actuary.
CAUTION. Formula (2) was derived under the assumption m = p.
Formula (3) was derived from (2). Therefore, (3) may be used only
when m = p.
Exercises
Find the purchase price of each of the following:
1. A $500, 6% bond, dividends payable semi-annually, redeemable in 10 years at
par, the investment rate to be 5% convertible semi-annually.
2. A $1,000, 5% bond, dividends payable semi-annually, redeemable in 12 years at
105, the investment rate to be 6% convertible semi-annually.
3. A $10,000, 4% bond, dividends payable quarterly, redeemable in 20 years at 110,
the investment rate to be 5% convertible quarterly.
4. A $5,000, 7% bond, dividends payable annually, redeemable in 18 years at par,
the investment rate to be 6% convertible annually.
5. A $500, 5J^% bond, dividends payable semi-annually, redeemable in 14 years
at 102, the investment rate to be 6% convertible semi-annually.
6. Establish formula (3).
7. Use formula (3) to solve Example 3.
8. A $2,000, 5% bond, dividends payable semi-annually, will be redeemed at 105
at the end of 10 years. Find the purchase price to yield 7% converted semi-annually.
9. Solve Exercise 8, with the yield rate (investment rate) 7% converted annually.
10. Should an investor, who wishes to make 6% (converted semi-annually) or more
on his money, buy bonds at 88 which are to be redeemed in 10 years and bear 5%
dividends payable semi-annually?
11. A $5,000, 6% bond, dividends payable semi-annually, is to be redeemed in 16
years at 106. What should be paid for the bond if 5% (convertible annually) is to be
realized on the investment?
56. Premium and discount. If we subtract C from both members of
formula (3) we will obtain the excess of purchase price over the redemption
price. This result may be positive, negative, or zero. That is, the pur-
chase price may be greater than the redemption price, less than the redemp-
tion price, or equal to the redemption price.
Valuation of Bonds 145
We have, if E is the excess,
j
- g -^ (C - K)
,0- J
If we let k equal the excess of purchase price per unit of redemption
price, it follows from the above equation that
(4)
P
E = P - C = Cfc, and P = C + Cft. (5)
Example 1. A $1,000, 6% semi-annual bond is to be redeemed in 10
years at $1,050. Find the purchase price if the investment is to yield 5%
scmi-annually.
Solution. Here, C = $1,050, n = 10, j = 0.05, m = p = 2, and
r/\
(7 = = 0.057143. Substituting in (4), we have
0.057143 - 0.05 1 - (1.025) ~ 20
2 ' 0.025
= (0.003571) (15.58916229)
= 0.055669.
And from (5), we get
E = P - C = 1,050(0.055669) = $58.45.
Hence, the purchase price is $58.45 more than the redemption price and
P = $1,050 + $58.45 = $1,108.45.
146 Financial Mathematics
In actual practice bonds are usually redeemed at par. Then C becomes
the face value and g = becomes the actual dividend rate. Also, the
L/
value, fc, obtained from (4) is the excess of purchase price per unit of face
value, and the value, P C, obtained from (5) is the premium or discount
at which the bond is purchased. In fact, k is the premium or discount
per unit of face value. It is evident that & is a premium when
9 > r,
is a discount when
< JJ
is at par when
9 = ]
Example 2. Solve Example 1, if the bond is to be redeemed at par.
Solution. Here, C = $1,000, n = 10, m = p = 2, j = 0.05, and
g = 0.06. We have
. 0.06 - 0.05 I - (1.025) -*> rT? . ,.,.
k = - - --- - [Formula (4)]
= (0.005) (15.58916229)
= 0.0779458.
And E = P - C = 1,000(0.0779458) = $77.95
= the premium.
Hence, P = $1,000 + $77.95 = $1,077.95.
Example 3. A $500, 5% semi-annual bond is to be redeemed in 15
years at par. Find the purchase price if the investment is to yield 5J^%
semi-annually.
Solution. Here, C = $500, n = 15, m = p = 2, j = 0.055, and
gr = 0.05. We have
7 0.05 - 0.055 1 - . ^ f /JVI
* - - - --- - [Formula (4)]
= - (0.0025) (20.24930130) =- 0.0506233.
And E = P - C = 500(- 0.0506233) = - $25.31
Hence, P = $500 - $25.31 = $474.69.
That is, the discount is $25.31 and the purchase price is $474.69.
Valuation of Bonds 147
Exercises
Use formulas (4) and (5) in the solution of the following:
1. Find the purchase price of a $1,000, 5% bond, dividends payable annually,
redeemable in 20 years at par, if the investment rate is to be 5^ % convertible annually.
2. Find the*pur chase price of a $5,000, 4J^% bond, dividends payable semi-annually,
redeemable in 15 years at 102, if the investment rate is to be 4% convertible semi-
annually.
3. What should be the purchase price of a $10,000, 3J^% bond, dividends payable
gemi-annually, redeemable in 35 years at par, if 4% (convertible semi-annually) is to
be realized on the investment?
4. Find the purchase price of a $500, 4j/% bond, dividends ^payable quarterly, to
be redeemed in 18 years at par, if the investment rate is to be 5% convertible quarterly.
6. What is the purchase price of a $10,000, 6% bond, dividends pay able '"semi-
annually, redeemable in 30 years at 105, the investment rate to be 4J^% convertible
semi-annually? i
6. What should be the purchase price of a $1,000, 5% bond, dividends payable
annually, to be redeemed in 10 years at 110, if the investment rate is to be 6% converti-
ble annually?
T.^Establish formula (4).
8. Use formulas (4) and (5) to solve Exercises 3 and 5, Art. 55.
9. Use formulas (4) and (5) to solve Exercise 10, Art. 55.
67. Amortization of premium and accumulation of discount When
a bond is bought for more than the redemption value, provision should be
made for restoring any excess of the original capital invested over the
redemption price. The excess of interest on the bond over the interest
required at the investment rate can and should be used for the gradual
extinction of the excess book value* over the redemption price. TheJ>ook
value of a bond bought above redemption price thus diminishes at each
interval until the redemption date, at which time its book value is equal
to .the redemption price. This amortization of the excess of purchase
price<over redemption price is called amortization of the premium.
When a bond is bought for less than the redemption price, we may
think of it as having a periodically increasing book value, approaching the
redemption price at maturity. The accumulation of the excess of redemp-
tion price over the purchase price is called accumulation of the discount.
We shall illustrate by examples.
* The book value of a bond on a dividend date is the price P at which the bond would
sell at a given investment rate.
148
Financial Mathematics
Example 1. A $1,000, 6% bond, dividends payable annually, redeem-
able in 6 years is bought to yield 5% annually. Find the purchase price
and construct a schedule showing the amortization of the premium.
Solution. Here, C = $1,000, n = 6, j = 0.05, m
Hence, k = 0.0507569.
Premium = P - C = $50.76.
p = 1, and g = 0.06.
And
P = $1,050.76.
Now the book value of the bond at the date of purchase is $1,050.76.
At the end of the first year a $60 dividend is paid on the bond. However,
5% on the book value for the first year is only $52.54. This would leave
a difference of $60 $52.54 = $7.46 for the amortization of premium for
the first year. This would reduce the book value to $1,043.30 for the
second year. The interest on this amount at 5% is $52.17. This leaves
$60 $52.17 = $7.83 for the amortization of premium for the second
year, and so on.
The following schedule shows the amount of amortization each year
and the successive book values.
SCHEDULE OF AMORTIZATION SCIENTIFIC METHOD
At End
of Period
Dividend
on Bond
Interest Earned
on Book Value
Amortization
of Premium
Book Value
$1,050 76
1
$60.00
$52.54
$7.46
1,043.30
2
60.00
52.17
7.83
1,035.47
3
60.00
51.77
8.23
1,027.24
4
60.00
51.36
8.64
1,018.60
5
60.00
60.93
9.07
1,009.53
6
60.00
60.48
9.52
1,000.01
Total
$50.75
The amortization of the premium may also be cared for by the straight
line method. By this method the premium is divided by the number of
periods and the book value is decreased each period by this quotient. Thus,
in the present problem we would have $60.76 -r 6 = $8.46. The following
schedule illustrates the method.
Valuation of Bonds
SCHEDULE OF AMORTIZATION STRAIGHT LINE METHOD
149
At End of Period
Dividend on Bond
Amortization
Book Value
o
$1,060.76
1
$60.00
$8.46
1,042.30
2
60.00
8.46
1,033.84
3
60.00
8.46
1,025.38
4
60.00
8.46
1,016.92
5
60.00
8.46
1,008.46
6
60.00
8.46
1,000.00
Example 2. A $10,000, 4% bond, dividends payable semi-annually,
redeemable in 4 years, is bought to yield 5% semi-annually. Find the
purchase price and construct a schedule showing accumulation of the
discount.
Solution. Here, C = $10,000, n = 4, j = 0.05, m = p = 2, and
g = 0.04. Hence,
k = - 0.0358506,
Discount = P - C = - $358.51.
And P = $9,641.49.
The following schedule shows the accumulation of discount for each
period and the book value for each period.
SCHEDULE OF ACCUMULATION SCIENTIFIC METHOD
At End
of Period
Dividend
on Bond
Interest Earned
on Book Value
Accumulation
of Discount
Book Value
$9,641.49
1
$200.00
$241.04
$41.04
9,682.53
2
200.00
242.06
42.06
9,724.59
3
200.00
243.11
43.11
9,767.70
4
200.00
244.19
44.19
9,811.89
5
200.00
245.28
45.28
9,857.17
6
200.00
246.43
46.43
9,903.60
7
200.00
247.59
47.59
9,951.19
8
200.00
248.78
48.78
9,999.97
Total
$358.48
150 Financial Mathematics
Exercises
1. A $1,000, 5% bond, dividends payable semi-annually, redeemable in 7 years at
par, is bought to yield 6% semi-annually. Construct an accumulation schedule.
2. A $1,000, 5% bond, dividends payable annually, redeemable in 10 years, is bought
to yield 4J^% annually. Construct an amortization schedule.
3. Construct a schedule for the amortization of the premium of the bond in Exer-
cise 1, Art. 55.
4. Construct an accumulation schedule for the bond of Exercise 6, Art. 56.
5. A $500, 5% bond, pays dividends semi-annually and will be redeemed at 105 on
January 1, 1946. It is bought on July 1, 1942, to yield 6% converted semi-annually.
Find the purchase price and form a schedule showing the accumulation of the discount.
6. A $5,000, 6% bond, paying semi-annual dividends will be redeemed at 110 on
September 15, 1947. Find the price on September 15, 1942, to yield 5% converted
semi-annually, and form a schedule showing the amortization of the premium.
68. Bonds purchased between dividend dates. We shall consider two
cases.
(a) When the bond is bought at a certain quoted price and accrued
interest with no apparent regard for yield.
(b) When the bond is bought on a strictly yield basis.
By accrued interest in case (a) is meant accrued simple interest on the
face value at the rate named in the bond. In other words, we mean the
accrued dividend. We shall illustrate by an example.
Example 1. A bond of $1,000 dated July 1, 1940, bearing 6% interest
payable semi-annually, was purchased March 1, 1941, at 98.5 and accrued
interest. What was paid for the bond?
Solution. The dividend dates are July 1, and Jan. 1. The price quoted
on this bond is evidently $985.00. Hence, the price paid on March 1
is $985.00 plus the interest on $1,000 from Jan. 1 to March 1 at 6%, or
$985.00 + $10.00 - $995.00, purchase price.
The student should observe that the purchase price is equal to the quoted
price plus the dividend accrued from the last dividend date to the time of pur-
chase.
When the bond is bought at a price to yield a given rate of interest
on the investment, the purchase price is equal to the value (purchase price) of
the bond at the last dividend date (the one just before the date of purchase) plus
the interest, at the investment rate, on this value, from the last dividend date to
the date of purchase. In practice, ordinary simple interest is used.
Valuation of Bonds 15 1
If Po stands for the purchase price at the last dividend date and d is the
number of days from the last dividend date to the date of purchase, the
purchase price may be defined by the formula
Example 2. A bond of $500 issued March 1, 1930, at 4% payable
semi-annually arid to be redeemed March 1, 1947, was purchased May 10,
1938, to realize 5% (converted semi-annually) on the investment. What
should have been paid for the bond? Find the quoted price.
Solution. The time from March 1, 1938 (the last dividend date) to
March 1, 1947 (the redemption date), is 9 years, and the purchase price
as of the last dividend date is
1 - (1.025) ~ 18
Po = 500(1.025) ~ 18 + 10 -- - - - - = $464.12.
0.025
The time from March 1, 1938 (the last dividend date), to May 10, 1938
(the date of purchase), is 70 days.
Podj (404. 12) (70) (0.05)
Hence, - = -- = $4.51
360 360
and P = 464.12 + 4.51 = $468.63, the purchase price on May 10, 1938.
Now, the quoted price as of May 10, 1938, is the purchase price as of
that date minus the dividend accrued from March 1, 1938, to May 10, 1938.
The accrued dividend is the ordinary simple interest on $500 for 70 days
at 4%, or $3.89.
Hence, the quoted price is
$468.63 - $3.89 = $464.74.
The student should observe the difference between purchase price and
quoted price. Bonds are usually quoted on the market at a certain price
plus accrued interest (at the dividend rate), guaranteed to yield a certain
rate of interest on the investment. In the case of the above bond the
quoted price as of May 10, 1938, would have been $464.74 (or 92.95% of
face) and accrued interest to yield 5% semi-annually on the investment
if held to the date of redemption.
152 Financial Mathematics
Exercises
1. A $1,000, 6% bond, dividends payable semi-annually, dated January 1, 1942,
was purchased September 10, 1944, at 97.5 and accrued interest. What was paid for
the bond?
2. The bond described in Exercise 1 is to mature January 1, 1949. What should
have been paid for it September 10, 1944, if purchased to yield 7% semi-annually?
3. At what price should a $500, 6% semi-annual bond, dated April 1, 1939, and
maturing April 1, 1946, be bought July 10, 1940, to yield 5J^%, semi-annually, on the
investment? Find the quoted price.
4. Should an investor, who wished to make 5% nominal, converted semi-annually,
on his investment, have bought government bonds quoted at 89 on February 1, 1920?
These bonds were redeemable November 15, 1942, and bore 4J4% interest, payable
semi-annually.
5. On July 20, 1935, a man bought 5% semi-annual bonds, due October 1, 1945, on
a 6% semi-annual basis. The interest dates were April 1 and October 1. What price
did he pay? Find the quoted price for that date.
6. A $1,000, 6% bond, dividends payable March 15 and September 15, is redeemable
March 15, 1950. It was bought January 1, 1944, to yield 5^% converted semi-annually.
Find the purchase price and the quoted price.
7. Find the quoted price for the bond of Exercise 6, as of July 5, 1947.
69. Annuity bonds. An annuity bond is an interest-bearing bond,
payable, principal and interest, in equal periodic payments or installments.
It is evident that these equal periodic payments constitute an annuity
whose present value is the face of the bond. The periodic payment can
be found by using Art. 31. The purchase price at any date is the present
value (figured at the investment rate) of the annuity composed of the
periodic payments yet due. Let us illustrate by an example.
Example. At what price should a 4% annuity bond for $5,000, payable
in 8 equal annual payments, be purchased at the end of 3 years (just after
the third payment has been made), if 5% (converted annually) is to be
realized on the investment?
Solution. Using Art. 31, we find the periodic payment to be
0.04
The purchase price at the end of 3 years is equal to the present value of
an annuity of $742.64 for 5 years at 5% converted annually.
1 - (1.05) ~ 5
Hence, P = 742.64 - - - i_ $3,215.24.
0.05
Valuation of Bonds 153
60. Serial bonds. When selling a set of bonds, a corporation may
wish to redeem them in installments instead of redeeming all of the bonds
on one date. When a bond issue is to be redeemed in several installments
instead of all the bonds being redeemed on one date, the issue is known as a
serial issue and the bonds of the issue are known as serial bonds. Evidently,
the purchase price at any date is equal to the sum of the purchase prices of
the installments yet to be redeemed.
Example. A city issues $40,000 worth of 4% bonds, dividends payable
semi-annually, to be redeemed by installments of $4,000 in 2 years, $6,000
in 4 years, $8,000 in 6 years, $10,000 in 8 years and $12,000 in 10 years.
An insurance company buys the entire issue on the date of issue so as to
realize 5% (converted semi-annually) on the investment. What price
was paid for the entire issue?
Solution. The purchase price of the entire issue is equal to the sum
of the purchase prices of the five installments to be redeemed. Using (5),
Art. 56, we have
1 - (1.025) ~ 4
4,000 - 4,000(0.005) - ^ ' = $ 3,924.76
6,000 - 6,000(0.005) ~ ( ' ' = $ 5,784.90
0.025
8,000 - 8,000(0.005) ~ ( ' - - $ 7,589.69
0.0^5
10,000 - 10,000(0.005) * ~ 1 o 25) = $ 9.347.25
12,000 - 12,000(0.005) ) - $11,064.65
and P^= $37,711.25.
Hence, the purchase price of the issue is $37,711.25.
Exercises
1. At what price should a 5% (payable semi-annually) annuity bond for $10,000,
payable in 26 equal semi-annual payments, be purchased at the end of 6 years, if I
(converted semi-annually) is to be realized on the investment?
154
Financial Mathematics
2. A $25,000 serial issue of 6% bonds, with semi-annual dividends, is to be redeemed
by payments of $5,000 at the end of 3, 4, 5, 6, and 7 years respectively. Find the pur-
chase price of the entire issue, if bought now to realize 5% (converted semi-annually)
on the investment. [Use (4) and (5) Art. 56.]
3. What is the purchase price of a bond of $20,000 payable $5,000 in 4 years, $8,000
in 6 years, $5,000 in 7 years, and $2,000 in 9 years, with dividends at 5% semi-annually,
if the purchaser is to receive 6%, converted semi-annually, on his investment?
4. Find the purchase price of a 10-year annuity bond for $25,000, to be paid in
semi-annual installments with interest at 6% converted semi-annually, if purchased at
the end of 4 years to yield 5% converted semi-annually.
5. Find the purchase price on the date of issue of a $2,000 bond bearing 4%, the
principal and interest to be paid in 6 equal annual installments, if the purchaser is to
realize 5% (convertible semi-annually) on his investment.
61. Use of bond tables. Tables are available which give the purchase
prices of bonds corresponding to given dividend rates, investment rates
and times to maturity. These tables may be made as comprehensive as
their purpose demands. The dividend rates may range from as low as 2
per cent to 8-or 9 per cent by intervals of H per cent. The investment rates
may have about the same range, but with smaller intervals. The times
to maturity may range from J4, y% or 1 year to 50 or 100 years by intervals
of M> 1 A or 1 year depending on whether or not the dividends are payable
quarterly, semi-annually or annually. These tables may be arranged in
various forms. The following is a brief portion of a bond table :
TABLE SHOWING PURCHASE PRICES OF A 4% BOND FOR $1,000 WITH
DIVIDENDS PAYABLE SEMI-ANNUALLY
Investment Rate
Time to Maturity
Converted
Semi-annually
5 Years
10 Years
15 Years
20 Years
2.00
$1,094.71
$1,180.46
$1,258.08
$1,328.35
2.50
1,070.09
1,131.99
1,186.67
1,234.95
3.00
1,046.11
1,085.84
1,120.08
1,149.58
3.50
1,022.75
1,041.88
1,057.97
1,071.49
4.00
1,000.00
1,000.00
1,000.00
1,000.00
4.50
977.83
960.09
945.89
934.52
5.00
956.24
920.05
895.35
874.49
5.50
935.20
885.71
848.14
819.41
6.00
915.70
851.23
804.00
768.85
Valuation of Bonds 155
Example. A $500, 4% bond, dividends payable semi-annually,
redeemable in 15 years at par, is bought to yield 5 1 A% convertible semi-
annually. Find its purchase price.
Solution. Observing the above table, we find the purchase price of a
$1,000 bond corresponding to the given dividend rate, investment rate and
time to maturity is $848.14. But we are considering a $500 bond. Con-
sequently, its purchase price is $424.07.
Exercises
1. Consider a $500 bond due in 20 years, and bearing semi-annual dividend coupons
at 4% per annum. Find by the use of the above table the purchase price if the invest-
ment rate is to be 4H%. Check the result by calculations independent of the table.
2. Solve Exercise 1, if the investment rate is to be (a) 3%; (6) 3M%J (c) 5%; (d) 6%.
3. Consider a $500, 4% bond, dividends payable semi-annually, which matures in
10 years. Using the above table and the method of interpolation find the approximate
purchase price when the investment rate is to be (a) 3%% (6) 5J^%. Check (a) by
using formula (5), Art. 124 and logarithms.
4. Solve Exercise 3, if the investment rate is to be (a) 3J%; (&)
62. Determining the investment rate when the purchase price of a
bond is given. At times the price of a bond is quoted on the market,
guaranteed to yield a certain rate of interest on the investment, provided
the bond is held until the date of maturity. At other times the price is
quoted, but no investment rate is given. Before purchasing a bond at a
certain price, the prospective buyer would naturally want to know (approxi-
mately at least) the rate of interest that would be realized by such an
investment. Therefore, it is very important that we have a method of
finding the investment rate when the purchase price is given. We shall
discuss two methods: (a) when bond and annuity tables are available;
(b) when no tables are available.
(a) When either bond or annuity tables are given the approximate
investment rate may be found by the method of interpolation. We shall
illustrate by examples.
Example 1. Find the rate of income realized on a 6% bond purchased
for $105, 10 years before maturity.
Solution. Since the bond is bought at a premium the investment rate
will be less than the dividend rate. Let us try 5%.
156
Financial Mathematics
Then,
P- 100(1.05)-
0.05
= 61.39 + 46.33 - $107.72.
Evidently the investment rate is greater than 5%. Let us now try
Then,
P = 100(1.055) -> + 6
= 58.54 + 45.23 = $103.77.
We observe that the investment rate must lie between 5% and
Arranging the results thus obtained, we have
Cost
107.72
105.00
103.77
Investment
Rate
5%
x%
Interpolating, we have
107.72 - 105.00
107.72 - 103.77 =
2.72 _x- 5
3^95 ~ fc '
3.95* = 21.11,
x = 5.344%.
Example 2. Find the rate of income realized on a 4% semi-annual
bond, purchased for $94.50, 10 years before maturity.
Valuation of Bonds
Solutim. Try 4H%. Then,
157
P = 100(1.0225) -*> + 2
1 - (1.0225) ~ 80
0.0225
- $96.01
The rate is evidently greater than 4>%. We shall now try 5%.
1 (1.025) - 20
P = 100(1.025) ~ 20 + 2
0.025
= $92.20.
We observe that the rate lies between 4}^% and 5%.
Cost
96.01
94.50
92.20
Investment
Rate
Interpolating, we have
96.01 - 94.50
96.01 - 92.20 4J - 5 '
1.51 x - &A
1.51 = 7.62o: - 34.29,
7.62* = 35.80,
x - 4.7%.
The student will observe that we find a rate that gives a purchase price
ailittle larger than the given purchase price and then a rate which gives a
purchase price a little smaller than the given purchase price. We then
find the approximate rate by interpolation.
158 Financial Mathematics
Example 3. A $1,000, 4% bond, dividends payable semi-annually, was
bought 20 years before maturity at $850.25. Using the above bond table,
Art. 61, find the approximate investment rate.
Solution.
When j = 0.050, P = $874.49.
When j = 0.055, P = $819.41.
874.49 - 850.25
Then, j = 0.0500 H (0.055 - 0.050)
874.49 - 819.41
24.24
= 0.0500 + (0.005)
55.08
= 0.0500 + 0.0022 = 0.0522 = 5.22%.
Exercises
1. Find the rate of income realized on a 5% semi-annual bond maturing in 18% years
when bought at $103.35.
2. A $1,000, 5% bond with semi-annual dividends, is redeemable at par at the end
of 12 years. If it is quoted at $1,075.60, what is the investment rate?
3. Find the effective rate realized by investing in 5% bonds with semi-annual divi-
dends, redeemable at par, which are quoted at 84.2, 10 years before redemption.
4. A state bond bearing 5% interest, payable semi-annually, and redeemable in
8 years at par, was sold at 95. Find the yield rate.
6. On November 15, 1930, a certain United States Government bond sold at 90.
If this bond is redeemable November 15, 1952, and bears 4% interest, payable semi-
annually, find the yield rate on November 15, 1930.
6. A 4% bond, dividends payable semi-annually, was bought 15 years before matur-
ity at 92.5. Using the bond table, find the approximate investment rate.
7. Using the bond table, find the approximate investment rate, when a 4% bond,
dividends payable semi-annually, is bought 10 years before maturity at 106.3.
(b) When tables are not available the approximate investment rate
may be found by solving formula (4), Art. 56 for j. This formula may
be written
(7)
(A -n
1+ i)
Valuation of Bonds 159
(j\-np
1 + - ) by the binomial theorem and neglecting all
P'
terms that involve j 3 and higher powers of j, we get
(
l
-"p _ np(np + 1) f_
HJ 2 V
, y-j
and r~
. n(np + 1) n(np + l)j
f n-
2p J 2p
Multiplying the above equation through by n and dividing out the
right-hand member, we obtain
n(g j) up -f- 1
= 1 H -- ~ j (approximately),
/c Zp
Solving for j, we have
- *>
which will give the approximate investment rate.
Example 4. Let us now apply formula (8) to Example 1, of Art. 62.
Solution. Here, & = 0.05, n = 10, p = 1, and g = 0.06.
>
and the approximate investment rate is 5.353%.
We notice that the result obtained by using formula (8) is approx-
imately the same as that obtained by using annuity tables.
Ordinarily, (8) will give a result which is accurate enough. At least,
it is accurate enough for the layman who might be interested in the pur-
chasing of bonds. Naturally, bond houses and individuals dealing in
bonds and quoting bond prices, to yield a certain rate of interest on the
investment, would require a more accurate method. However, these
people would have comprehensive bond and annuity tables available, by
which the investment rate could be found to the required degree of accuracy.
160 Financial Mathematics
Exercises
1. Apply formula (8) to Examples 2 and 3 of Art. 62.
2. Apply formula (8) to Exercises 1, 3, and 5 of Art. 62(a), page 158.
3. Apply formula (8) to Exercises 2, 4, and 6 of Art. 62 (a), page 158.
4. A person bought a $1,000, 5% bond, dividends payable semi-annually, 18 years
before maturity for $975. Find the investment rate by using annuity tables and then
check the result by using formula (8).
6. A $500, 3J% Government bond, dividends payable June 15 and December 15,
was bought June 15, 1945, for $530. If this bond is to be redeemed December 15, 1956,
find the investment rate as of June 15, 1945.
Problems
1. A $1,000, 5% bond, dividends payable April 15 and October 15, maturing Octo-
ber 15, 1946, was bought April 15, 1943, to yield (j = .06, m = 2). Construct a schedule
showing the accumulation of the discount.
2. A $1,000, 6% bond, dividends payable semi-annually, maturing in 4 years, was
bought to yield (j = .05, m = 2). Construct a schedule showing the amortization of the
premium.
3. A $300,000 issue of highway bonds bearing 4% interest, payable semi-annually,
dated January 1, 1944, matures $100,000 January 1, 1945, 1946 and 1947. What price
should be paid for the issue to realize (j = .03, m = 2)?
4. A $1,000 bond paying 5% semi-annually, redeemable at $1,040 in 10 years, has
been purchased for $970. Find the investment rate.
6. A 4%, J. and J.,* bond is redeemable at par on January 1, 1952. Find the yield
if it is purchased July 1, 1939, at 89.32.
6. A $1,000, 6%, J. and J., bond is redeemable at par on July 1, 1950. Find the
price to yield (j = .05, m = 2) on August 16, 1940.
7. Find the purchase price of a $100, 5% bond, dividends payable semi-annually and
redeemable at par in 10 years, to yield 6% effective.
8. Find the purchase price of a $100, 4% bond, dividends payable semi-annually
and redeemable in 20 years at 120, to yield 5% effective.
* That is, th dividends are payable January 1 and July 1.
CHAPTER VII
PROBABILITY AND ITS APPLICATION IN LIFE INSURANCE
63. The history of probabilities Aristotle (384-322 B.C.), the
Greek philosopher, is credited with the first attempt to define the measure
of a probability of an event. Aristotle says an event is probable when the
majority, or at least the majority of the most intellectual persons, deem it
likely to happen.
But the first real mathematical treatment of probability originated
as isolated problems coming from games of chance. Cardan (1501-1576)
and Galileo, two Italian mathematicians, solved many problems relating to
the game of dice. Aside from his regular occupation as a mathematician,
Cardan was also a professional gambler. As such he had evidently noticed
that there was always more or less cheating going on in the gambling
houses. This led him to write a little treatise on gambling in which he
discussed some mathematical questions involved in the games of dice then
played in the Italian gambling houses. The aim of this little book was
to fortify the gamester against such cheating practices. Galileo was not
a gambler, but was often consulted by a certain Italian nobleman on
problems relating to the game of dice. As a result of these consultations
and his investigations he has left a short memoir. Pascal (1623-1662) and
Fermat (1601-1665), two great French mathematicians, were also con-
sulted by professional gamblers and this led them to make their contribu-
tions to the subject of chance.
The Dutch physicist, Huyghens (1629-1695), and the German mathe-
matician, Leibnitz (1646-1716), also wrote on chance. However, the first
extensive treatise on the subject of chance was written by Jacob Bernoulli
(1654-1705). In this treatment of the subject which was published in
1713, the author shows many applications of the new science to practical
problems.
The first English treatise on probabilities was written by Abraham de
Moivre (1667-1754). This was a remarkable treatment and may yet be
read with profit. This book was translated into German by the Austrian
mathematician, E. Czuber.
162 Financial Mathematics
It was left for La Place (1749-1827), that great French mathematician,
to leave the one really famous treatise on the theory of chance, "Th6orie
Analytique des Probabilities." Since the time of La Place many books
and articles on the theory have been written by mathematicians in all lands.
The subject of probability has become so widespread in its applica-
tions that the best minds of the world have undertaken its further develop-
ment. Today, the physicist, the chemist, the biologist, the statistician, the
actuary, depend upon the results of the theory of probability for the
development of their respective fields.
Probably the earliest writer on the application of the theory of proba-
bility to social phenomena was John Graunt (1620-1674) who, in 1662,
published his " Observations on the London Bills of Mortality." The
astronomer, Edmund Halley, published his Mortality Tables in 1693.
Adolphe Quetelet (1796-1874) devoted his life to the applications of proba-
bilities to scientific research, particularly to the study of populations.
Following the work of these investigators, life insurance organizations
began to function. With the organization of the Equitable Society of
London in 1762, life insurance was successfully placed on a scientific basis.
The company employed the mathematician, Dr. Richard Price, to be the
actuary to determine the premiums which should be charged. He drew
up the Northampton Table of Mortality in 1783, and from this event
insurance as a science may be said to date.
64. Meaning of a priori probability. A box contains three white and
four black balls. One ball is drawn at random and then replaced and this
process is continued indefinitely. What proportion of the balls drawn will
be black? Here there are seven balls to be drawn or we may say there are
seven possibilities, and either of the seven balls is equally likely to be drawn
or any one of the seven possibilities is equally likely to happen. Of the
seven possibilities, any one of three would result in drawing a white ball
and any one of four would result in drawing a black ball. We would say
then that three possibilities of the seven are favorable to drawing a white
ball and the other four possibilities are favorable to drawing a black ball.
We put the above statement in another way by saying that in a single
draw the probability of drawing a white ball is % and the probability of
drawing a black ball is %. This does not mean that out of only seven
draws, exactly three would be white and four black. But it does mean
that, if a single ball were drawn at random and were replaced and this
process continued indefinitely, % of the balls drawn would be white and
ff would be black. Or the ratio of the number of white balls drawn to
the number of black balls drawn would be as 3 to 4.
Probability and Its Application in Life Insurance 163
Reasoning similarly to the above led La Place to formulate the follow-
ing a priori definition of probability:
If h is the number of possible ways that an event will happen and/ is the
number of possible ways that it will fail and all of the possibilities are
equally likely, the probability that the event will happen is p =
/
and the probability that it will fail is q =
h +f
It is evident, then, that the sum of the probability that an event will
happen and the probability that it will fail is 1, the symbol for certainty.
In analyzing a number of possibilities we must be sure that each of
them is equally likely to happen before we attempt to apply the above
definition of probability.
Example: What is the probability that a man aged 25 and in good
health will die before age 30? In this case we might reason thus: The
event can happen in only one way and fail in only one way, and conse-
quently the probability that he will die before age 30 is 3/. But this reason-
ing is false for we are assuming that living five years and dying within five
years are equally likely for a man now 25 years old. But this is not the
actual experience. This example will be discussed in Art. 65.
Exercises
1. A bag contains 7 white and 5 black balls, and a ball is drawn at random. What
is the probability (a) that the ball is white? (b) that the ball is black?
2. A deck of 52 cards contains 4 aces. If a card is drawn at random, what is the
probability that it will be an ace?
3. A coin is tossed. What is the probability that it will fall head up?
4. If the probability of winning a game is %, what is the probability of losing?
6. If the probability of a man living 10 years is 0.6, what is the probability of his
dying within 10 years?
6. If a cubical die is tossed, what is the probability that it will fall with 6 up?
7. Two coins are tossed at random. What is the probability of obtaining (a) two
heads? (b) one head and one tail?
8. Two cubical dice are tossed at random. Find the probability that the sum of
the numbers is 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12.
9. A box contains 45 tickets numbered from 1 to 45. If a ticket is drawn at ran-
dom, what is the probability that the number on it is (a) odd? (b) even? (c) divisible
by 5? (d) larger than 35?
10. A coin and a cubical die are tossed simultaneously. Find the probability that
they will fall with the coin head up and with a face on the die numbered less than 5.
11. Three coins are tossed. What is the probability of exactly two heads?
164 Financial Mathematics
12. Which is the more likely to happen, a throw of 4 with one die or a throw of 8
with two dice?
13. A and B each throw two dice. If A throws 8, find the probability that B will
throw a larger number.
66. Relative frequency. Empirical probability. In the example and
the? exercises of Art. 64 the probabilities are derived in each case by an
a priori determination of all the equally likely ways in which the event in
question can happen. There are many classes of events in which the
notion of probability is important although it is impossible to make an
a priori determination of all the equally likely ways an event can happen
or fail. In such cases we determine an approximate probability empiri-
cally by moans of a large number of observations. Such determinations
are necessary in the establishment of life insurance, pension systems, fire
insurance, casualty insurance, and statistics.
If we have observed that an event has happened h times out of n
possible ways, we call h/n the relative frequency of the event. When n is a
large number, h/n may be considered a fair estimate of the probability
derived from observation. Our confidence in the estimate increases as
the number n of observed cases increases. If, as n increases indefinitely,
the ratio h/n approaches a limiting value, this limiting value is the prob-
ability of the happening of the event. That is
limit h
n co 7^
In statistical applications the limit of h/n cannot in general be determined,
but satisfactory approximations to the limit may be found for many
practical purposes.
We are now ready to solve the problem which was stated in Art. 64.
The American Experience Table of Mortality shows that out of 89,032 men
living at age 25, the number living at age 30 will be 85,441. Then the
number dying before age 30 is 89,032 85,441 or 3,591. Hence the proba-
bility that a man aged 25 will die before age 30 is -~ = .0403. In this
problem, n equals 89,032 and h equals 3,591. 89,032
We have previously stated that the value h/n is only an estimate, but it
is accurate enough (when n is a large number) for many practical purposes.
Life insurance companies use the American Experience Table of Mortality
as a basis to determine the proper premiums to charge their policy holders.
Probability and Its Application in Life Insurance 165
Exercises
1. Among 10,000 people aged 30, 85 deaths occurred in a year. What was the rela-
tive frequency of deaths for this group?
2. Out of 10,000 children born in a city in a given year, 5,140 were boys and 4,860
were girls. What was the relative frequency of boy babies in the city that year?
3. A group of 10,000 college men was measured as to height. Of these, 1,800 were
between 68 and 69 inches high. Estimate the relative frequency of height of college
men between 68 and 69 inches.
66. Permutations. Number of permutations of things all different.
Each of the different ways that a number of things may be arranged
is known as a permutation of those things. For example the different
arrangements of the letters abc are : a6c, acb y 6ac, bca, cab, cba. Here there
are 3 different ways of selecting the first letter and after it lias been selected
in one of these ways there remain 2 ways of selecting the second letter.
Then the first two letters may be selected in 3-2 or 6 ways. It is clear
that we have no choice in the selection of the third letter and consequently
the total number of permutations (or arrangements) of the three letters is 6.
This example illustrates the following:
Fundamental Principle : // one thing may be done in p ways, and after
it has been done in one of these ways, another thing may be done in q ways,
then the two things together may be done in the order named in pq ways.
It is evident that for each of the p ways of doing the first thing there
are q ways of doing the second thing and the total number of ways of doing
the two in succession is pq.
The above principle may be extended to three or more things.
Exercises
1. If 2 coins are tossed, in how many ways can they fall?
2. If 3 coins are tossed, in how many ways can they fall?
3. If 2 dice are thrown, in how many ways can they fall?
4. If 2 dice and 3 coins are tossed, in how many ways can they fall?
5. How many signals can be made by hoisting 3 flags if there are 9 different flags
from which to choose?
6. In how many different ways can 3 positions be filled by selections from 16 differ-
ent people?
7. How many four-digit numbers can be formed from the numbers 1, 2, 3, 4, 5, 6,
7, 8, 9?
Now suppose there are n things all different and we wish to find the
number of permutations of these things taken r at a time, n & r.
166 Financial Mathematics
Since only r of the n things are to be used at a time, there are only r
places to be filled. The first place may be filled by any one of the n things
and the second place by any one of the n 1 remaining things. Then
the first and second places together may be filled in n(n 1) ways. The
third place may be filled by any one of the n 2 remaining things. Hence
the first three places may be filled in n(n l)(n 2) ways. Reasoning
in a similar way we see that after r 1 places have been filled, there
remain n (r 1) things from which to fill the rth place. Applying the
fundamental principle stated above we have
V, = n(n - l)(n - 2) -- (n - r + 1). (1)
When r = n, (1) becomes,
n P n = n(n - l)(n - 2) 3-2-1 = fn!. (2)
Exercises
1. A man has two suits of clothes, four shirts and three hats. In how many ways
may he dress by changing suits, shirts and hats?
2. How many arrangements of the letters in the word "Mexico" can be made, using
in each arrangement (a) 4 letters? (b) all the letters?
3. Four persons enter a street car in which there are 7 vacant seats. In how many
ways may they be seated?
4. Three different positions in an office are to be filled and there are 15 applicants,
each one being qualified to fill any one of the positions. In how many ways may the
three positions be filled?
5. How many signals could be made from 5 different flags?
6. Find the number of permutations, P, of the letters aabbb taken 5 at a time.
Hint:P-2!-3! -5!.
7. If P represents the number of distinct permutations of n things, taken all at a
time, when, of the n things, there are n\ alike, n% others alike, n$ others alike, etc., then:
8. How many distinct permutations can be made of the letters of the word attention
taken all at a time?
9. How many distinct permutations of the letters of the word Mississippi can be
formed taking the letters all at a time?
* The symbol n P r is used to denote the number of permutations of n things taken r
at a time.
t n! is a symbol which stands for the product of all the integers from 1 up to ana
including n, and is read "factorial n."
Probability and Its Application in Life Insurance 167
10. How many ways can ten balls be arranged in a line if 3 are white, 5 are red, and
2 are blue?
11. How many six-place numbers can be formed from the digits 1, 2, 3, 4, 5, 6, if
3 and 4 are always to occupy the middle two places?
12. In how many ways can 3 different algebras and 4 different geometries be arranged
on a shelf so that the algebras are always together?
13. In how many ways can 10 boys stand in a row when:
(a) a given boy is at a given end?
(b) a given boy is at an end?
(c) two given boys are always together?
(d) two given boys are never together?
67. Combinations. Number of combinations of things all different.
By a combination we mean a group of things without any regard for order of
arrangement of the individuals within the group. For example abc, acb,
bac, bca, cab, cba are the same combination of the letters abc, but each
arrangement is a different permutation.
By the number of combinations of n things taken r at a time is meant
the number of different groups that may be formed from n individuals
when r individuals are placed in each group. For example ab, ac, and be
are the different combinations of the letters abc when two letters are used
at a time.
The symbol n C r is universally used to stand for the number of combina-
tions of n things taken r at a time. We will now derive an expression for
n C r . For each one of the n C r combinations there are r\ different permuta-
tions. And for all of the n C r combinations there are n C r -r\ permutations,
which is the number of permutations of n things taken r at a time. Hence,
and
n C T = =. (3)
Since
we have
n(n- l)(n-
- W
(4')
r/(n-r)/ ^ '
168 Financial Mathematics
Exercises
1. Find the number of combinations of 10 things taken 7 at a time:
Solution. Here, n - 10 and r = 7.
10-9-8-7-6-6-4
loC7 - 7.6.5.4.3.2 - 12 '
2. How many committees of 5 can be selected from a group of 9 men?
3. Out o/ 8 Englishmen and 5 Americans how many committees of 3 Englishmen
and 2 Americans can be chosen?
4. How many different sums can be made up from a cent, a nickel, a dime, a quarter,
arid a dollar?
6. An urn contains 5 white and 7 black balls. If 4 balls are drawn at random what
is the probability that (a) all are black, (b) 2 are white and 2 are black?
Solution, (a) The total number of ways that 4 balls may be drawn from 12 balls
is i2C4 or 495 ways. And the number of ways that 4 black balls may be drawn is 704
or 35 ways. Hence the probability of drawing 4 black balls is 3 %gr> or % 9 .
(b) Two white balls may be drawn in sC-2 or 10 ways. And for each one of these
10 ways of drawing two white balls, two black balls may be drawn in ?C2 or 21 ways.
Then two white balls and two black balls may be drawn together in 10 X 21 or 210 ways
(Fundamental Principle, Art. 66). Hence, the probability of drawing 2 white and
2 black balls is 21 % 9 5 or 1^3-
6. A bag contains 4 white, 6 black, and 7 red balls. If 4 balls are drawn at random,
what is the probability that (a) all are black, (b) 2 black and 2 red, (c) 1 white, 1 black,
and 2 red?
7. Prove that n C r = n C n -r*
8. Prove that the expansion of the binomial (a + 6) n may be written
(a + b) n = a n + n Cia n ~ l b + n C 2 a n ~V + + n C r a n - r b r + - + b n
r-n
/ j n
r =
n n ~ r h r
" O ,
if we define M Co to be 1.
9. How many straight lines are determined from 10 points, no 3 of which are in the
same straight line?
10. How many different sums can be made from a cent, a nickel, a dime, a quarter,
a half-dollar, and a dollar?
11. From 10 books, in how many ways can a selection of 6 be made: (a) when a
specified book is always included? (b) when a specified book is always excluded?
12. Prove that n C r + n C r ~i = n+ iC r .
13. Out of 6 different consonants and 4 different vowels, how many linear arrange-
ments of letters, each containing 4 consonants and 3 vowels, can be formed?
14. A lodge has 50 members of whom 6 are physicians. In how many ways can a
committee of 10 be chosen so as to contain at least 3 physicians?
Probability and Its Application in Life Insurance 169
15. In the equation of Exercise 8, make a = 6 = 1, and chow that
nCl + C 2 + + nC n - 2* - 1.
16. Solve Exercise 4 above, using Exercise 15.
17. In how many ways can 7 men stand in line so that 2 particular men will not
be together?
18. A committee of 7 is to be chosen from 8 Englishmen and 5 Americans. In how
many ways can a committee be chosen if it is to contain: (a) just 4 Englishmen? (b) at
least 4 Englishmen?
19. Prove: n-^CV+i = n CV+i + 2- n C r + nCr-i-
20. If n P r = 110 and n C r = 55, find n and r.
21. If rA = nCi find n '
22. If ,A = 10/21 GA), find n.
23. If 2nC n -i = 91/24( 2n _ 2 C n ), find n.
24. Prove: B Ci + 2- n C 2 + 3- n C 3 + + n- n C n = n(2) n ~ 1 .
26. How many line-ups are possible in choosing a baseball nine of 5 seniors and 4
juniors from a squad of 8 seniors and 7 juniors, if any man can be used in any position?
68, Some elementary theorems in probability. Sometimes it is con-
venient to consider an event as made up of simpler events. The given
event is then said to be compound. Thus, the compound event may be
made of simpler mutually exclusive events, simpler independent events, or
simpler dependent events.
A. Mutually Exclusive Events. Two or more events are said to be
mutually exclusive when the occurrence of any one of them excludes
the occurrence of any other. Thus, in the toss of a coin the appearance
of heads and the appearance of tails are mutually exclusive. Also, if a
bag contains white and black balls and a ball is drawn, the drawing of a
white ball and the drawing of a black ball are mutually exclusive events.
Theorem. Mutually exclusive events. If pi, p2 y . . ., p r are the separate
probabilities of r mutually exclusive events, the probability that one of these
events will happen on a particular occasion when all of them are in question is
P=A+A+A + ...+A, (5)
the sum of the separate probabilities.
This theorem follows from the definition of mutually exclusive events.
For if a\, a%, as,**-, a r , indicate the number of ways the separate
events can happen, then the number of ways favorable to some event
170 Financial Mathematics
is ai + a% + as + +0r- If m represents the total number of possibili-
ties, favorable and unfavorable, then
H
r = 1 , & , 5& , _____ , ^
m m m m
When two mutually exclusive events are in question, the probabilities
are frequently called either or probabilities. Thus, if a die is thrown, the
probability of either an ace or a deuce is K + H r 1 A-
B. Independent Events. Two or more events are dependent or
independent according as the occurrence of any one of them does or does
not affect the occurrence of the others. Thus, if A tosses a coin and B
throws a die, the tossing of heads by A and the throwing of a deuce by B
are independent events. However, if a bag contains a mixture of white
and black balls and a ball is drawn and not returned to the bag, the prob-
abilities in a second drawing will be dependent upon the first event.
Theorem. Independent events. If pi, p2,. . ., p r are the separate prob-
abilities of r independent events, the probability that all of these events will
happen together at a given trial is the product of their separate probabilities.
Let pi = oi/wi, p2 = fl2/W2,. . . 9 p r = a r /m r be the simple probabilities ;
where ai, a2,. . . , a r are the ways favorable to the happening of the separate
events; and mi, mz,. . ., m r are the possible ways in which the separate
events may happen or fail. By the Fundamental Principle, Art. 66, the
number of ways favorable to the happening together of the r events is
aia2...a r . And by applying the same principle we get m\m2. . .m r as
the number of possible ways that the r events might happen or fail. Con-
sequently,
7711/722 771 r
= AA'--Jri (6)
and the theorem is proved.
Corollary. If Pi, Pi,. . ., p r are the separate probabilities of r inde-
pendent events, the probability that they mil all fail on a given occasion is
(l-p 1 )(l-p 2 )...(l-p r ), (7)
and the probability that the first k events mil succeed and the remainder fail is
(1 - Pr). (8)
Probability and Its Application in Life Insurance 171
C. Dependent events. The following theorem for dependent events
may be proved by a similar method to that used for independent events.
Theorem. Dependent events. Let pi be the probability of a first event]
let P2 be the probability of a second event after the first has happened] let p%
be the probability of a third event after the first two have happened; and so on.
Then the probability that all of these events will occur in order is
..p r . (9)
Exercises
1. The probability that A will live 20 years is J^, the probability that B will live
20 years is J^, and the probability that C will live 20 years is J. What is the prob-
ability that all three will be living in 20 years?
Solution. We have here three independent events, where
Pi = Mi P2 = 34 and p 3 = J.
Hence, P = (H) (K) (H) = Jiio-
2. Find the probability of drawing 2 white balls in succession from a bag containing
4 white and 7 black balls, if the first ball drawn is not replaced before the second drawing
is made.
Solution. We have here two dependent events. The probability that the first
4 4
draw will be white is = -- ; the probability that the second draw will be white is
_3_ = 3 4 + 711
3 + 7 ~ 10*
Hence, pi = fi, P2 = ?fo,
and P = (!) ( ) Ms-
3. A and B, with others, are competitors in a race. The probability that A will win
is Y and the probability that B will win is */. What is the probability that either
A or B will win?
Solution. We have here two mutually exclusive events.
Hence, P = M + M = 7 /i2-
4. Four coins are tossed at once. What is the probability that all will be heads?
5. A bag contains 3 white balls, 4 black balls and 5 red balls. One ball is drawn and
not replaced, then a second ball is drawn and not replaced and then a third ball is drawn.
What is the probability (a) that a ball of each color will be drawn, (b) that 2 blacks and
1 red will be drawn, (c) that all will be red?
6. Suppose that in Exercise 5 the balls are replaced after each draw. Then answer
(a), (b) and (c).
7. Three men ages 28, 30 and 33 respectively form a partnership. What is the
probability (a) that all three will be living at the end of 10 years, (b) that the first two
172 Financial Mathematics
will be living, (c) that one only of the three will be living? Use the American Experi-
ence Table of Mortality, Table XI.
8. A man and wife arc 29 and 25 years of age when they marry. What is the proba-
bility that they will both live to celebrate their golden wedding?
9. A, B, and C go bird-hunting. A has a record of 1 bird out of 2, B gets 2 out of 3,
and C gets 3 out of 4. What is the probability that they will kill a bird at which all
shoot simultaneously?
10. If the probability that A will die within a year is ${Q and the probability that B
will die within a year is y\ o, what is the probability that (a) both A and B will die
within a year? (b) both A and B will live a year? (c) one life will fail within a year?
(d) at least one life will fail within a year?
11. The probability that A will solve a problem is J/s and that B will solve it is %.
What is the probability that if A and B try the problem wiil be solved?
12. From a group of 6 men and 5 women, a committee of 5 is chosen by lot. What
is the probability that it will consist of (a) all women? (b) all men? (c) 3 men and 2
women?
13. A committee of 7 is chosen from a group of 8 Englishmen and 5 Americans.
What is the probability that it will contain (a) exactly 4 Englishmen? (b) at least 4
Englishmen?
14. From a lottery of 30 tickets marked 1,2, ... ,30, four tickets are drawn. What
is the probability that the numbers 1 and 15 are among them?
15. From a pack of 52 cards, 3 cards are drawn at random. What is the prob-
ability that they are all clubs?
69. Mathematical expectation. The expected number of occurrences of
an event in n trials is defined to be up where p is the probability of occur-
rence of the event in a single trial.
Illustrations. If 100 coins are thrown or if one coin is thrown 100
times, theoretically, we " expect " 50 heads and 50 tails, for n = 100 and
p = K.
If a die is rolled 36 times we " expect " an ace to turn up 6 times, for
n = 36 and p = Jtf .
If 0.008 is the probability of death within a year of a man aged 30, the
" expected " number of deaths within a year among 10,000 men of this
age would be 80, for n = 10,000 and p = 0.008.
If p is the probability of obtaining a sum of money, k, then pk represents
the mathematical expectation.
Illustration. Suppose that 1,000 men, all aged 30, contribute to a
fund with the understanding that each survivor will receive $1,000 at age
60. The mortality tables show that approximately 678 will be alive.
Hence, the expectation of each would be $678. The fund must contain
$678,000 in order that each survivor receive $1,000. Hence, neglecting
interest, each of the 1,000 men will have to contribute $678 to the fund.
Probability and Its Application in Life Insurance 173
70. Repeated trials. When the probability that an event will happen
in a single trial is known, it becomes a question of importance to determine
the probability that the event will happen a specified number of times in a
given number of trials.
To familiarize us with the method of proof of the general theorem of
repeated trials, let us consider the
Example. What is the probability of throwing 2 aces in 4 throws of
a die?
The conditions of the problem are met if in the first 2 throws we obtain
aces and in the next 2 throws not-aces; or if in the first throw we get ace,
the second throw not-ace, the third throw ace, and the fourth throw not-
ace; and so on. We shall illustrate the possibilities symbolically as
follows:
Considering the first case, the probability of throwing an ace on any
throw is M- The probability of not throwing an ace on any throw is %.
Hence the probability of throwing an ace on the first and second throws
and not throwing an ace on the two remaining throws is (%) 2 (%) 2 .
In the second case, the probability of events occurring as the symbol
above indicates is (X)0*)(H)(%) = (K) 2 ON0 2 .
The remaining cases may be treated in a similar manner, and in each
instance the result for any specified set is (K) 2 (%) 2 . Now it is evident that
the 2 aces can be selected from the 4 possible aces in 4^2 = 6 ways. Since
the 6 cases are mutually exclusive, the chance that one or the other of the
specified cases occurs is 6(K) 2 (!M0 2 = 15 %296.
Let us now consider the important
Theorem of Repeated Trials. If p is the probability of the success of an
event in a single trial and q is the probability of its failure, (p + q = 1),
then the probability P r that the event will succeed exactly r times in n trials is*
Pr = nC,ff-'. (10)
For the probability that the event will succeed in each of r specified
trials and will fail in the remaining (n r) trials is, by (6), p r q n ~ r . Further,
it is possible for the r successes to occur out of n trials in n C r different ways.
These ways being mutually exclusive, by (5) the probability in question is
Pr =
* It will be noted that (10) is the (n r + l)th term of the expansion (p +
and the (r -f l)th term of the expansion (q + p) n .
174 Financial Mathematics
The various probabilities are indicated in the following table:
VALUES OF P r FOR VARIOUS VALUES OF r
r
Tfo Probability That in n
^ r Trials There Will Be
n
n - 1
n -2
p n n successes
n Cip n ~ l q n -I
n^2p n ~ 2 3 2 n 2 "
failures
1 "
2 "
n r
n C r p n ~ r q r n r successes
r failures
r
2
1
.Crff r
n -r '
n-2 '
n - 1 '
n
n C>2p 2 q n ~ 2 2
n Cipq n ~ l 1
q n
Total. . . .
(p + q) n = 1
From the above table we have at once the following:
Corollary. The probability that an event will succeed at least r times
in n trials is P r + Pr+i H ----- h Pn, that is:
n C 2 p"- V
(11)
It will be noted that (11) consists of the first (n r + 1) terms of
the expansion (p + q) n .
Example 1. An urn contains 12 white and 24 black balls. What is
the probability that, in 10 drawings with replacements, exactly 6 white
balls are drawn?
Solution. We have:
P = 12 /3* = 1 A, q = 2 %G = %,
n = 10, r = 6, n - r = 4.
Hence,
3360
310 '
Probability and Its Application in Life Insurance 175
Example 2. The American Experience Mortality Table states that for
an individual aged 25 the probability of survival a year is p = 0.992 and
the probability of death within a year is q = 0.008. Out of a group of
1,000 individuals aged 25, how many are expected to survive a year? What
are some conclusions that may be drawn from the terms of the binomial
expansion (.992 + .008) i- 000 ?
Solution. We have n = 1,000, p = 0.992, q = 0.008. By Art. 69,
we expect np = 1,000(0.992) = 992 to survive the year, and nq = 1,000
(0.008) = 8 to die within a year.
The terms of the expansion
(.992 + .008) 1 - 000 = (.992) 1 ' 000 + 1,000(.992) 999 (.008)
+ i,oooC 2 (.992) 9 ^(.008) 2 + . . . +(.008) 1 ' 000
give, by equation (10), the following probabilities:
(.992) 1 - 000 gives the probability that 1,000 will survive a year;
1,000(.992) 999 (.008) 1 gives the probability that 999 will live a year and
1 will die within a year, and so on.
Problems
1. If there are five routes from London to Cambridge, and three routes from Cam-
bridge to Lincoln, how many ways are there of going from London to Lincoln going by
the way of Cambridge?
2. Out of 20 boys and 25 girls, in how many ways can a couple be selected?
3. A committee of 5 is to be chosen from 15 Englishmen and 18 Americans. If the
committee is to contain exactly 3 Americans and 2 Englishmen, in how many ways may
it be chosen?
4. From 10 Democrats and 8 Republicans a committee of 3 is to be selected by lot.
Find the probability that it will consist (a) of 2 Democrats and 1 Republican, (b) of 2
Republicans and 1 Democrat, (c) of 3 Democrats, (d) of 3 Republicans. What is the
sum of the four answers?
6. Out of a party of 12 ladies and 15 gentlemen, in how many ways can 4 ladies and
4 gentlemen be selected for a dance?
6. In how many ways can 3 men choose hotels in a town where there are 6 hotels?
7. In how many ways can A, B, and C choose hotels in a town where there are 6
hotels, if (a) A and B refuse to stay at the same hotel, (b) they all stay at different hotels,
(c) they all stay at the same hotel?
8. In how many ways can 7 books be arranged on a shelf, if 3 particular books are
to be together?
9. How many signals can be made with 7 flags of different colors by arranging them
on a mast (a) all together, (b) 4 at a time, (c) at least 1 at a time?
10. If the probability that A will die in 10 years is 0.2, that B will die in 10 years is
0.3, and that C will die in 10 years is 0.25, what is the probability that at the end of
176 Financial Mathematics
10 years (a) all will be dead, (b) all will be living, (c) only two will be living, (d) at
least two will be living?
11. If two dice are thrown, what is the probability of obtaining an odd number for
the sum?
12. In tossing 10 coins, what is the probability of obtaining at least 8 heads?
13. A man whose batting average is !o will bat 4 times in a game. What is the
probability that he will get 4 hits? 3 hits? 2 hits? at least 2 hits?
14. A machinist works 300 days in a year. If the probability of his meeting with
an accident on any particular day is Hooo> what is the probability that he will entirely
escape an accident for a year?
16. If it is known that 2 out of every 1,000 dwelling houses worth $5,000 burn
annually, what is the risk assumed in insuring such a house for one year?
16. According to the American Experience Mortality Table out of 100,000 persons
living at age 10 years, 91,914 are living at the age of 21 years. Each of 100 boys is
now 10 years old. What is the probability that exactly 60 of them will live to be 21?
71. Meaning of mortality table. If it were possible to trace a large
number of persons, say 100,000, living at age 10 until the death of each
occurred, and a record kept of the number living at each age x and the
number dying between the ages x and x + 1, we would have a mortality
table.
However, mortality tables are not constructed by observing a large
number of individuals living at a certain age until the death of each, for
it is evident that this method would not be practicable, but would be next
to impossible, if not impossible. Mathematical methods have been devised
for the construction of such tables, but the scope of this text does not per-
mit the discussion of these methods.
Table XI is known as the American Experience Table of Mortality and
is based upon the records of the Mutual Life Insurance Company of New
York. It was first published in 1868 and is used for most life insurance
written in the United States. It will be used in this book as a basis for all
computations dealing with mortality statistics. It consists of five columns
as follows: The first giving the ages running from 10 to 95, the different
ages being denoted by x; the second giving the number living at the begin-
ning of each age x and is denoted by l x ; the third giving the number dying
between ages x and x + 1 and is denoted by d x ] the fourth giving the proba-
bility of dying in the year from age x to x + 1 and is denoted by q x \ and
the fifth giving the probability of living a year from age x to age x + 1
and is denoted by p x .
The American Experience Table, now 77 years old, is not expected to
represent present-day experience. It is conservative in its estimates for
insurance and thereby contributes a factor of safety to policies. What-
ever added profit comes from its use is generally passed on to policy-
Probability and Its Application in Life Insurance 177
holders as dividends. It is now generally prescribed in the state laws as
the standard for insurance evaluations.
While the American Experience Table furnishes a safe basis for insur-
ance valuations, it is not at all suitable for the valuation of annuities.
Annuities are paid to individuals during the years that they live, and com-
putations based upon a table with mortality rates lower than the actual
might easily cause a company to lose money. For the valuation of annui-
ties, the American Experience Table is not legally prescribed so that the
companies have been free to employ tables that more accurately represent
the mortality they experience. The American Annuitants' Table is widely
used for the valuation of annuities.
The American Experience Table and the American Annuitants' Table
are " select " tables inasmuch as they show the mortality rates after the
selection caused by medical examination. In 1915 the larger insurance
companies of the United States cooperated in developing the American
Men Mortality Table. It too is a " select " table.
Many mortality tables have been based upon the experience of the
general population. Such a table includes many in poor health and others
engaged in hazardous or unhealthy occupations. Since the rates of
mortality in a table constructed from population records are higher than
the rates of mortality of the select tables, such a table is unsuitable for
life insurance valuations.
The United States Life Tables* shows the rates of mortality among
the general population in certain parts of the United States. For purposes
of comparison, these tables are very enlightening, though they are inappli-
cable for insurance and annuity evaluations.
The following table shows the rates of mortality per 1,000 for a few
ages according to the mortality tables that we have mentioned.
RATES or MORTALITY PER 1,000
American Annuitants'
US T ifp
Age
Experience
Men
Table, 1910
Male
Female
30
8.43
4.46
6.51
4.99
4.52
35
8.95
4.78
8.04
6.00
5.27
40
9.79
5.84
9.39
7.51
6.39
45
11.16
7.94
11.52
9.78
8.07
50
13.78
11.58
14.37
13.15
10.56
* United States Life Tables, J. W. Glover, published by the Bureau of Census, Wash-
ington, D. C.
178 Financial Mathematics
Exercises
1. What is the probability that a man aged 30 will live to be 65? What is the
probability that the same man will die before reaching 65? What is the sum of the two
probabilities?
2. Find the probability that a man aged 70 will live 10 years.
3. Suppose 100,000 lives age 10 were insured for one year by a company for $1,000
each, what would be the cost to each individual, neglecting the interest?
4. What would be the cost of $1,000 insurance for one year of an individual 30 years
old, neglecting the interest, if based upon (a) the American Experience Table? (b) the
American Men Table? (c) the United States Life Table?
5. Solve Exercise 4 for an individual aged 50?
72. Probabilities of life. In Art. 71 we discussed the meaning of the
mortality table and gave something concerning its history. We now
derive some useful formulas based upon this table. We notice certain
relations existing among the elements l x , d x , p x and q x of the table.
Since l x +i denotes the number of people living at age x + 1 and 1 9
denotes the number living at age x, the probability, p x , that a person age x
will live one year is given by
lx + l , .
p x = - (12)
Since d x stands for the number of people dying between the ages x and
x + 1, the probability, q xj that a person age x will die within a year is
given by
* - - (13)
Since it is certain that a person age x will either live one year or die
Yvithin the year, we have
P* + q* = 1. (14)
From (12) and (13), we get
x ~r Qx =
/* -I yx
/ 7 /
lx vx *>x
Hence,
and
d x = l x - Z+i. (15)
Probability and Its Application in Life Insurance 179
The number of deaths between the ages x and x + n is given by
lx lx+n = d x + d x + 1 + -{- d x + n i. (16)
When (x + n) exceeds the oldest age in the table,
L+n = 0, and (16) becomes
lx = d x + d x +i H to end of table. (17)
The probability that a person aged x will live n years is denoted by the
symbol n p x . Thus 15^10 means the probability that a person aged 10 will
live 15 years and is 89,032 -f- 100,000 or 0.89032.
In general,
nP* = l ~"- (18)
The probability that a person aged x will die within n years is denoted
by | n <fr. Since a person aged x will either live n years or die within that
time, we have
npx + |n3* = 1, or
(19)
'
U = j ~~ n - (20)
*JC
The probability that a person aged # will die in the year after he reaches
age x + n is denoted by n | q x . This may be regarded as the compound
event that consists of a person aged x living n years and one aged x + n
dying within that year. Thus we have
n 1 9* = n2V2*+n (Art. 68)
Since
ta; tx '*
and
(22)
180 Financial Mathematics
We observe from (22) that the probability that a person aged x will die
in the year after reaching age (x + n) is equal to the probability that a
person aged x will live n years minus the probability that a person aged x
will live n + 1 years.
The probability that a person aged x will live n years, and one aged y
will die within that period is
i Art - 68]. (23)
1. Verify from the table that
Exercises
2. Verify that #15 = - Does p\$ -f- #15 = 1?
lib
3. Verify that 15 /is = ^15 + <^i6 + dn.
4. Verify that lw = ^90 4- ^91 ~f- - to end of table.
6. What is the probability that a person aged 20 will live 30 years and die within the
next year?
6. Find the probability that a person aged 30 will live to be 65.
7. What is the probability that a person aged 25 will die within 10 years? What is
the probability that he will die in the year after he reaches 35?
Problems
1. Find the probability that a man aged 40 will live to be 70.
2. What is the probability that three persons, each age 40, will all reach the age of
50? What is the probability that none will reach that age?
3. A boy 15 years old is to receive $20,000 on attaining the age of 21. Neglecting
interest, what is the value of the boy's expectation?
4. Show that the probability that at least one of two lives aged x and t/, respec-
tively, will survive n years is given by the expression n p x -f- n py npx'nPv Hint: We
have here three mutually exclusive events.
5. A father is 40 years old and his son is 15. What is the probability that both will
live 10 years? What is the probability that at least one will live 10 years?
6. What is the probability that a person aged 40 will die in the year just after
reaching 60?
7. If we assume that out of 10,000 automobiles of a certain class there are 70 thefts
during the year, what would it cost an insurance company to insure 1,000 such cars
against theft at $700 each? What would be the premium on one such car? In this
problem running expenses and interest on money are neglected.
8. Show that the probability that at least one of three lives x, y, z, respectively, will
survive n years is given by the expression:
nPx'nPy'nP* (nPx'nPy + nPv'nP* + nPx'npg) + nPx + nPy + nPz-
Probability and Its Application in Life Insurance 181
9. A man 35 years of age and his wife 33 years of age are to receive $10,000 at the
end of 10 years if both are then living to receive it. Neglecting interest, what is the
value of their expectation?
10. Two persons, A and B, are 42 and 45 years of age respectively. Find the prob-
ability (a) that both will survive 10 years, (b) that both will die within 10 years, (c) that
A will survive 10 years and B will die during the time, (d) that B will survive 10 years
and A will not survive. What is the sum of the four answers?
11. A man 50 years old will receive $5,000 at the end of 10 years if he is alive. At
4 % interest, find the present value of his expectation.
12. What is the probability that a man aged 50 will live 20 years longer?
13. Given two persons of ages x and y, express the probability that:
(a) both will live n years,
(b) both will die within n years,
(c) exactly one will live n years,
(d) exactly one will die within n years.
14. To what events do the following probabilities refer?
(a) 1 - npx'nPv
(b) (1 - npx)(l - nPv).
(c) 1 - \ n qx'\nq y .
(d) nPx'Px+n-
15. Each of 7 boys is now 10 years old. What is the probability that (a) all seven
will live to be 21 years old? (b) at least five of them will live to be 21?
16. Given 1 ,000 persons aged x, write expressions in terms of p x and q x for the fol-
lowing probabilities:
(a) that exactly 10 will die within a year.
(b) that not more than 10 will die within a year.
17. Prove: m+npx = mPx'np x \m = npx'mpx+n-
18. Prove: 5 p x = p x -p x+ rp x ^p x ^'p x ^.
19. Translate the symbolic statement of Problem 18 into words.
20. Prove: n p x = p X 'Px+i'Px+2 . . . Px+n-i.
CHAPTER VIII
LIFE ANNUITIES
73. Pure endowments. A pure endowment is a sum of money payable
to a person whose present age is x, at a specified future date f provided the per-
son survives until that date. We now find the cost of a pure endowment of
$1 to be paid at the end of n years to a person whose present age is x. The
symbol, n E x , will represent the cost of such an endowment.
Suppose l x individuals, all of age x, agree to contribute equally to a fund
that will assure the payment of $1 to each of the survivors at the end of n
years. From the mortality table we see that out of the l x individuals
entering this agreement, l x+n of them would be living at the end of n years.
Consequently, the fund must contain l x + n dollars at that time in order that
each of the survivors receives $1. The present value of this sum is
7 ,n 7
V 'Ix+ny
where
.
The present value of the money contributed to the fund by the l x individ-
uals is
If we equate the present value of the money contributed to the fund and
the present value of the money received from the fund by the survivors,
we have
l x - n E x = v./ x+li
and
A - 3*- (1)
*JC
The preceding method of derivation is known as " the mutual fund "
method. The formula may also be derived by using the notion of mathe-
matical expectation.
182
Life Annuities 183
It is clear that n E x will be the present value of the mathematical expecta-
tion, which is the present value of $1 due in n years multiplied by the
probability that a person aged x will live n years. Consequently
^x = ^- n p x = ^~^,
tx
which is the same as (1).
It should be emphasized that n E x , the present value of $1 payable in n
years to a person aged x if he lives to receive it, is dependent upon the rate
of interest i and the probability that (x) will live n years.* Since these
two fundamental factors v n and n p x are generally each less than unity,
n E x is generally less than unity. Further, considering both interest and
survivorship, the quantity n E x may be looked upon as a discount factor
being the discounted value of 1 due in n years to (x). Similarly, the
quantity \/ n E x may be looked upon as an accumulation factor, being the
accumulated value at the end of n years of 1 due now to (x). The line
diagram shows the equivalent values.
nEx n years 1
It is obvious that the present value A, of R payable in n years to (x),
is given by
A = R- n E x . (10
If the numerator and the denominator of (1) be multiplied by v x , we get
v*l x '
and if we agree that the product v x l x shall be denoted by the symbol D x ,
(1) becomes
A - +- (2)
D x is one of four symbols, called commutation symbols, that are used
to facilitate insurance computations (see Table XII). This table is based
on the American Experience Table of Mortality and a % 1 A% interest rate
is used. There are other commutation tables based upon different tables
of mortality and different rates of interest.
* We shall frequently use the symbol (x* to mean "a person aged x" or "a life aged x."
184 Financial Mathematics
It will be observed as the theory develops that we rarely use the values
given in the mortality table except to compute the values of the commu-
tation symbols.
Unless otherwise specified, all computations in the numerical exercises
will be based upon the American Experience Table of Mortality with 3^
per cent per annum as the interest rate.
Exercises
1. Find the present value (cost) of a pure endowment of $5,000, due in 20 years and
purchased at age 30, interest at 3J^%.
Solution. Here, x = 30, n = 20, and
0.410587. [Formula (2) and Table XII]
DSQ 30440.8
Hence, A = (5,000.00) 20^30 = 5,000 (0.410587)
= $2,052.94.
2. Solve Exercise 1, with the rate of interest 3%.
3. An heir, aged 14, is to receive $30,000 when he becomes 21. What is the present
value of his expectation on a 4% basis?
4. Find the cost of a pure endowment of $2,000, due in 10 years and purchased at
age 35, interest at 3J^%.
6. What pure endowment due at the end of 20 years could a person aged 45 purchase
for $5,000? Assume 3 1 A % interest.
6. Solve Exercise 5, assuming 4% interest.
7. A boy aged 12 is to receive $10,000 upon attaining age 21. Find the present value
of the inheritance on a 4% basis.
8. A man aged 30 has $10,000 that he wishes to invest with an insurance company
that operates on a 3J^% basis. He wishes the endowment to be payable to him when
he attains the age of 50 years. What would be the amount of the investment at that
time if he agrees to forfeit all rights in the event of death before he reaches age 50?
9. Two payments of $5,000 each are to be received at the ends of 5 and 10 years
respectively. Find the present value at 3}^%
(a) if they are certain to be received;
(b) if they are to be received only if (25) is alive to receive them.
10. What pure endowment payable at age 65 could a man age 25 purchase with
$1,000 cash?
11. To what formula would the formula for nE x reduce if (x) were sure to survive n
years? To what would it reduce if money were unproductive?
12. Show that
(a) m+nEx mEz'nEx+m't
(b) n E x - lE x -iE x +i- A+2 .". . lEx+n-L
Life Annuities 185
74. Whole life annuity. A whole life annuity is a succession of equal
periodic payments which continue during the entire life of the individual
concerned. It is evident that the cost of such an annuity depends upon
the probability of living as well as upon the rate of interest.
The terms payment interval, annual rent, term, ordinary, due, deferred,
have similar meanings in life annuities that they have in annuities certain.
Unless otherwise specified, the words life annuity will be taken to mean
whole life annuity.
75. Present value (cost) of a life annuity. We now propose to find
the present value of an ordinary life annuity of $1 per annum payable to
an individual, now aged x. The symbol, a x , is used to denote the cost of
such an annuity. We see that the present value of this annuity is merely
the sum of the present values of pure endowments, payable at the ends of
one, two, three, and so on, years. Consequently,
a x = iE x + 2&X + 3#* + ... to end of table
D x+ l D x +2 Ac+3 , , i r , t i
a- __r_ _| _ 1 _ 1- ... to end of table
D x D x D x
-P*+i + Dx+2 + Ac+3 + ... to end of table
D,
a, = ^~ (3)
where
N x = A, + Ar+i + D x +2 + ... to end of table (4)
[See Table XII]
The symbol N s (called " double bar W ") as defined above is that gen-
erally adopted in America. In actuarial parlance, it is frequently called
the American N. The English textbooks use the single bar N which is
defined by the equation
N x = Ac+i + A*+2 + Ac +3 + ... to end of table.
In this book we shall use the " double bar " American N.
Exercises
1. What is the cost of a life annuity of $600 per annum for a person aged 30, interest
at
186 Financial Mathematics
Solution. From (3), Art. 75, we have
tTabiexni
Hence, the annuity has a cost (present value) of
600(18.60538) = $11,163.23.
2. Find the present value of a life annuity to a person aged 60, the annual payment
to be $1,200.
3. What annual life income could a person aged 50 purchase with $10,000.
4. Derive the formula for a x by the mutual fund method.
6. Show that a x = vp x (l -f o*+i)
(a) algebraically,
(b) by verbal interpretation or direct reasoning using the following line diagram :
Ages x+l x+2
1 1 year of death
1 - 1 - 1
6. A man aged 60 is promised a pension of $600 at the end of each year as long as
he lives. What is the present value of the pension?
7. The beneficiary, age 50, of a life insurance policy may receive $25,000 cash or an
ordinary life annuity of annual rent R. If she chooses the annuity, find R.
76. Life annuity due. When the first payment under an annuity is
made immediately, we have what is called an annuity due. The present
value of an annuity due of $1 per annum to a person aged x is denoted by
a x . An annuity due differs from an ordinary annuity (Art. 75) only by
an immediate payment. Consequently, we have*
a, = 1 + a x (5)
N x+ i = D x + W x+ i
+ D x D x
D x + D x +i + D x + 2 + ... to end of table
D x
a,-f ; (6)
77. Deferred life annuity. When the first payment under an annuity
is not made until some specified future date, and then only in case the
individual, now aged x, is still living, we have what is called a deferred
annuity. Since the first payment under an ordinary annuity is made
at the end of one year, an annuity providing for first payment at the end
* Values of a x and a z may be found in Table XII.
Life Annuities 187
of n years is said to be deferred n 1 years. Then in an annuity deferred
n years the first payment would not be made until the end of n + 1 years.
These payments are illustrated by the diagram
\< - n years - >! Ill 1 (x) dies
j - 1 - 1 - 1 - : - 1 - 1 - 1 - 1 - 1
Age x x-\-l x+n
The present value of an annuity of $1 per annum, deferred n years,
payable to an individual now aged x, if he is then living is denoted by the
symbol, n | a x . It is evident that the present value of such an annuity is
merely the sum of the present values of pure endowments payable at the
end of n + 1, n + 2, n + 3, and so on, years so long as the individual
survives.
Consequently,
n | a x = n+iEx + n +2#* + n+3#z + . . . to end of table
Dx+n+1 . -Dx+n+2
Let n | a* denote the present value of a deferred whole life annuity due,
that is, a succession of $1 payments to be made at the ends of n years,
n + 1 years, and so on as long as (x) survives. These payments are illus-
trated by the following line diagram:
n years >l 1 1 1 (x) dies
: - 1 - 1 - 1 - 1 - 1 - 1 - h
Age a; x+l x-\-n
The value at age x + n of these payments is a^+n, and the value at
age x, the present value, is a*+ n -n^*. Consequently
* ^ X+n Dx+n X+n
78. Temporary life annuity. When the payments under a life annuity
stop after a certain time although the individual be still living, we have
what is called a temporary annuity. Such an annuity of $1 per annum
which ceases after n years is denoted by the symbol, a x ^. It is clear that
the present value of a temporary annuity is equal to the sum of present
188 Financial Mathematics
values of pure endowments of $1 payable at the ends of 1, 2, 3, . . ., n years.
Thus,
a x -ft = \E X + 2E X + . + nE x
D x
D x +i + D x +2 + ... to end of table
= ~D X
D I+n +i + D x+n +a + to end of table
N x+i - N x+n+ i
a t -a\ = - - -- (9)
**X
If the first of the n payments be made immediately and the last pay-
ment be made at the end of n 1 years, we then have a temporary annuity
due. Letting a^ represent the present value of such an annuity we get
a^ == 1 + a^^rri
I Dx+l + D *+
= D* + D x+ l + D x + 2 + . . . + Ar+n-l
D,
_ _ N, - NI+*
xn| - D -- ^ ^
Exercises
1. An insurance company accepts from a man, aged 30, $85.89 per annum in advance
for 10 years if living as payment for insurance. What would be the equivalent single
premium based upon the American Experience Table of Mortality and 3J^% interest?
2. A will provides that a son is to receive a life annuity of $1,500 a year, the first
payment to be made when the son attains the age of 60. What is the value of the son's
share when he is 40 years old?
3. A man aged 50 pays $10,000 for a life annuity whose first payment is to be made
when he is 60 years old. W T hat will be his annual income beginning at age 60?
4. A will provides that a son who is now 25 years old is to receive $1,200 at the end
of one year, and a like amount at the end of each year until 10 payments in all have
been made. If each payment is contingent upon the son being alive, what is the value
of his estate at age 25?
Life Annuities 189
6. Make n = in formula (7) and show that it reduces to formula (3). What does
this mean?
6. Show that a x = a x ^ + n \a x
(a) algebraically,
(b) by direct reasoning with the aid of an appropriate line diagram.
7. Derive formulas (7) and (9) by the mutual fund method.
8. Derive formula (8) by finding the sum of appropriate pure endowments.
9. Draw line diagrams to illustrate the meaning of the following symbols:
10
10. Prove & X m+ni = a^^I + mEx'&x+mnl
(a) algebraically,
(b) by direct reasoning.
11 Prove the following identities:
(a) SLxft = 1 + Oafn=TI,
(b) a z = a, x | + n \ a z .
12. A beneficiary, age 50, of a life insurance policy may receive $25,000 cash or a
temporary life annuity due for 15 years. If she chooses the annuity, find its amount.
79. Forborne temporary life annuity due. An individual aged x may
be entitled to a life annuity due of $1 per annum, but forbears to draw it.
Instead he requests that the unpaid installments be allowed to accumulate
as pure endowments until he is aged x + n. Such an annuity is known
as a forborne temporary life annuity due.
The problem here is to find the value of such an annuity, taken at age x,
to the person at age x + n if he is still alive. This value is equal to the
n-year pure endowment that the present value of a temporary life annuity
due of $1 per annum will buy. The present value of a temporary life an-
nuity due of $1 per annum is
D x
[(10) Art. 78]
D x+n
Since [(2) Art. 73] will buy an n-year pure endowment of $1, $1
L/x
will buy an n-year pure endowment of , and consequently
>D x + n D x
will buy an n-year pure endowment of *
N x - N x + a D x W x -
nU * * 7) ' D = D
V x V x+n D x
* The symbol n u x is customarily used to stand for the amount at age x -f n of the
forborne temporary life annuity due of $1 per annum. It is one of the most useful
functions for the actuary.
190 Financial Mathematics
It follows that R per annum payable in advance for n years as a tem-
porary life annuity will buy an n-year pure endowment of
S = R. nUl = R N *~ N *+ n - (12)
Mc+n
N
Since is the cost of a life annuity due of $1 per annum for an
D x +n
individual aged x + n, $1 at age x + n will buy a life annuity due of *
N x +n
per annum, and at age x + n will buy a life annuity due of
i/r+n
D x+n N x+n N x+n
Hence, it follows that with $1 per annum payable in advance by an individ-
flT _ fir
ual now aged x, a life annuity due of ~- per annum, beginning at
^vs-l-n
age x + n, may be bought.
Then R dollars per annum payable in advance as a temporary life
annuity by an individual now aged x, will buy a life annuity due of
(13)
beginning at age x + n.
It may be shown that
per annum payable in advance for n years by an individual now aged a*,
will buy him a life annuity due of K dollars per annum beginning when
he is aged x + n. Here, an individual aged x is buying a regular life
annuity of K dollars per annum, deferred n 1 years, by paying
K ^ dollars annually in advance.
-tV* IVx+n
80. Summary of formulas of life annuities. Examples.
R = the annual payment,
(x) = the person of age x.
Life Annuities 191
Pure Endowment: A =
Whole life annuity: A = R(a x ) = Rf~* ( 3/ )
Whole life annuity due: A = R(a x ) = J? -~- (6')
Deferred life annuity: A = /?( | a,) = R~^~^. (?')
Deferred life annuity due: 4 = J?( n | a x ) = jR ^hn. (8')
Temporary life annuity: A = ^(a^,^) = I? 1 * 1 - 1 *+ >l + 1 , (9')
Temporary life annuity due: A = /^(a^^) = fl x Jf+M (10')
Forborne temporary life 7^ _ TV-
annuity due: S = /?(!/* ) = .R - ^^ (12)
Example 1. A man aged 30 pays an insurance company $1,000 annually,
in advance, for 20 years for the purchase of a pure endowment. What
will be the amount of the endowment if he lives to claim it?
Solution. The annual payments constitute a forborne temporary life
annuity due in which x = 30, n = 20, R = 1,000. Using (12), we find
596,804 -181,663
12,498.6
= $33,215.00.
Example 2. A man aged 30 pays an insurance company $100 annually,
in advance, 'for 35 years to purchase a life annuity, the first payment to
be made when the annuitant reaches age 65. What is the annual rent of
his annuity?
Solution. Consider the line diagram.
R R ......
100 100 100 ...... 100 I !
I - 1 - 1 - ! - 1 - , -
Age 30 31 32 ...... 64 65 66 ......
192 Financial Mathematics
We shall choose age 65 as the focal time.
The value at age 65 of the payments is that of a forborne temporary
life annuity due with x = 30, n = 35, R = 100. Using (12) we find
The value of the benefit is that of a life annuity due on a life aged 65.
Using (6'), the value of the benefit is
w.^
A =,
Therefore,
A)5 '
and
596,804 -48,616.4
NCS 48,616.4
R = $1,127.58.
Exercises
1. In the settlement of an estate a man, aged 30, is to receive $1,000 and a like
amount at the end of each year. However, he requests that this annuity be forborne
until he reaches the age of 60. What will be the amount of these forborne payments at
that time on a 3J^% interest basis?
2. A young man, aged 25, pays $300 per annum in advance to accumulate as a pure
endowment until age 60. What will be the amount of his endowment at age 60 on a
3/^% basis? Suppose that at age 60 he does not take the amount of his endowment in
cash, but instead purchases a life annuity due. What would be his annual income on a
3^% basis?
3. An individual now aged 30 desires to make provisions for his retirement at age 60.
How much per annum, in advance, must he pay for the next 30 years to guarantee a
life annuity due of $3,000 per annum beginning at age 60?
4. A person whose present age is 25 desires to have a life income of $1,500 beginning
at age 60. How much must he invest annually in advance for the next 35 years to
guarantee his desired income?
6. A man aged 50 pays an insurance company $20,000 for a contract to pay him
a life annuity with the first payment to be made at age 65. Find the annual payment
of the annuity.
6. A corporation has promised to pay an employee, now aged 50, a pension of $1,000
at the end of each year, starting with a payment on his 65th birthday. What is the
.present value of this expectancy?
Life Annuities 193
7. A certain insurance policy on a life aged 30 calls for premiums of $100 at the
beginning of each year as long as he lives. Find the present value of the premiums.
8. A certain insurance policy on a life aged 30 calls for premiums of $100 at the
beginning of each year for 20 years. Find the present value of the premiums.
9. A man aged 30 wishes a life annuity of $1,000 a year, the first payment to be
made when he is 65 years old. To provide for this, he will pay R per year in advance
for the next 20 years. Find R.
10. A man aged 55 is to receive a life annuity of $1,000 a year, the first payment
to be made immediately. He wishes to postpone the annuity so that the first payment
will occur on his 65th birthday. What will be his annual income?
11. A certain life insurance policy matures when the policy-holder is aged 50 and gives
him an option of $10,000 in cash or a succession of equal payments for 10 years certain
and as long thereafter as he may live. Should he die during the first ten years, the
payments are to be continued to his heirs until a total of ten have been made. Find
the annual payment under the optional plan.
Hint. The equation of value is R(&IQ\ + 10 |a 60 ) = 10,000.
12. Show by direct reasoning that the annual premium for n years, beginning at
age x, for an annuity of 1 per year, beginning at age x + n, is given by as+n/nW*.
13. A boy of age 15 is left an estate of $50,000 which is invested at 4% effective.
He is to receive the income annually, if living, and at age 25 he is to receive the principal,
if living. Find the present value of the inheritance.
14. How much must an individual now aged x invest at the beginning of each year
for n years, if living, to secure an annuity of R dollars per annum payable for t years
certain and as long thereafter as he may live?
Hint. Focalize at age x -f- n. Let y be the annual payment. The equation of
value is
15. A person whose present age is 25 desires to have an income of $1,000 a year for
10 years certain and as long thereafter as he may live, first payment at age 60. How
much must he invest annually in advance for the next 35 years to guarantee this income?
81. Annuities payable m times a year. Optional provisions are usu-
ally made in annuity contracts so that the periodical payments may be
made m times a year. The symbol a x (m) is used to denote the present
value of a life annuity of $l/m payable m times a year, and & x (m) is used
to denote the present value of a life annuity due of $l/m payable m times
a year. Theoretically, it follows from Art. 73, that
1#* + &+ ] (15)
m m
It is apparent that (15) would involve considerable computation
and besides the mortality table does not take into consideration fractional
194 Financial Mathematics
parts of years. However, we may derive an approximate formula for a m>
which is accurate enough for most purposes.
The deferred annuity due may be written
| a* = (1 + a x ) -
and
i|a, (l + o*)- 1.
By simple proportion,
1 | a* = (1 + a z ) -- = a x -\ --
mm
and, in general,
k m ~ k
a* =
m
Assume that we have m such annuities, where the first payments are
to be madeatthecnds of , , ,, of a year, respectively. Together
m m m m
they will provide $1 at the end of each th of a year. Hence,
m
/ /w, _ 1\ /
mal m)
/ m-l\ ( m-2\ ( m-k\
= \ a* H -- ) + \ a ^ H -- j H ----- h I a x H -- 1
\ m / \ m/ \ m/
\
)
/
m m
The right-hand side of the above equation is the sum of an arithmetical
progression with a common difference of --- Consequently
m
(TO) m(m -
ma x ma x H
and
If the first payment is made at once, we have
< m \ 1 , ^ 1 , m 1
Life Annuities 195
The student should observe the difference between (16) and (17).
If we let n | a x (m) stand for the present value of an annuity of $1 deferred
n years and payable in m installments a year, and reason as in Art. 73,
we get
D x
Also, if we let a^\ stand for the present value of a temporary life
annuity of $1 payable in m installments a year and consider that a life
annuity is made up of a temporary annuity and a deferred annuity, we get
() _ () I I nW*
&X = a* ni + n I a*
and
~(m) (m) I (m) /IA\
0n| = a * ~~ n I a x (19)
Exercises
1. What is the present value of a life annuity of $100 payable at the end of every
month to a person aged 30?
Solution. Here, x 30 and m = 12. From (16), Art. 81, we have
30 = 30 ~t~ 1 M4'
and (1,200) a& 2> <1,200 (18.6054 + 0.4583) = $22,876.44.
2. Solve Exercise 1, with the annuity payable quarterly.
3. Find the cost of a temporary life annuity of $600 per annum, payable in 12 monthly
installments for 20 years, first payment due one month hence. Assume age 30.
4. Solve Exercise 3, with the annuity paid at the rate of $300 at the end of every six
months.
6. Find the cost of a life annuity due of $1,000 per annum, payable in quarterly
installments, for a person aged 40.
196 Financial Mathematics
Problems
1. Show that the present value of an annuity of $1, payable for n years certain and
so long thereafter as the individual, now aged x, survives (first payment due one year
hence) is given by
| + n | <**.
Also show that the present value of an annuity due of $1, payable for n years certain
and so long as an individual, now aged x, may live, is given by
2. What is the value of an annuity of $1,000 per annum payable at the end of each
year for 10 years certain and so long thereafter as an individual, now aged 60, survives?
3. According to the terms of a will a person aged 30 is to receive a life income of
$6,000 per annum, first payment at once. An inheritance tax of 4% on the present
value of the income must be paid at once. Find the present value of the income and the
amount of the tax.
4. What would be the present value of the income of Problem 3 if payments of
$500 a month were made at the beginning of each month?
5. What would be the value of the annuity in Problem 2, if the payments were made
at the end of each year for 20 years certain and for life thereafter?
6. A man carrying a $20,000 life insurance policy arranges it so that the proceeds at
his death shall be payable to his wife in annual installments for 10 years certain, first
payment upon due proof of death. What would be the annual installment, assuming
3}^% interest?
7. What would be the amount of the annual installments of Problem 6, if payable
for 10 years certain and so long thereafter as the beneficiary shall survive, assuming that
the beneficiary was 55 years of age at the death of the insured?
8. What would be the amount of the annual installments in the above problem, if
payments were to be made throughout the life of the beneficiary?
9. What would be the amount of the annual installments in Problem 8, if payable
for 10 years, each payment contingent upon the beneficiary being alive?
10. Assume that the proceeds in Problem 9 are to be paid monthly. What would
be the monthly installment?
11. Show that
n
J'
*- 2m D, *+n 2m
where a^ stands for the present value of a temporary annuity due of $1 payable in m
installments per annum.
12. A suit for damages due to the accidental death of a railroad employee 42 years
old and earning $175 a month was settled on the basis of three-fourths of the present
value of the expected wages of $175 a month during his after lifetime. What was the
amount of the damages?
13. By the terms of a will, a son is bequeathed an estate of $100,000 with the pro-
vision that he must pay his mother who is 60 years of age $200 monthly as long as she
lives? What is the value of the son's inheritance?
Life Annuities 197
14. Prove: m + n u x ~ m u x -- h n u x + m .
n&x+m
16. Prove:
(a) a^+i = iu x (a x - 1),
(b) a x = ( n U x + SL x +n)nE Xt
(c) 4|
/j\ (w) /i w \
W a i n\ = a *^I o (! ~ n#x).
2m
16. A woman aged 30 offers $20,000 to a benevolent organization if it will pay her
5% interest thereon at the beginning of each year for the remainder of her life. If the
institution can purchase the desired annuity for her from an insurance company which
operates on a 3^2% basis, will it pay to accept the offer?
17. A man aged 65 is to receive a life annuity of $1,000 a year, the first payment being
due immediately. He desires to postpone the annuity so that the first annual payment
will occur when he is aged 70. What will be the annual income from the new annuity?
18. A man aged 55 is entitled to a life annuity of $1,000. He agrees to use it to
purchase a 10-year pure endowment. What is the amount of the pure endowment?
19. A young man aged 25 is to receive as an inheritance a life income of $100 a
month, first payment immediately. An inheritance tax of 5% on the present value is
levied. Find the amount of the tax.
20. A man aged 60 is granted a pension of $1,000 a year for 10 years, first payment at
once, and $500 a year thereafter for the rest of his life. If all payments are contingent
on his survival, find their present value.
21. Show that the present value of a perpetuity of $1 per year, the first payment to
be made at the end of the year in which (a?) dies, is 1/i a x . See Art. 37.
CHAPTER IX
LIFE INSURANCE, NET PREMIUMS (SINGLE AND ANNUAL)
82. Definitions. A thorough mathematical treatment of life insur-
ance involves many very complex problems. However, there are a few
principles that are fundamental and it is these with which we wish to deal
in this chapter. Life insurance is fundamentally sound only when a large
group of individuals is considered. Each person contributes to a general
fund from which the losses sustained by individuals of the group are paid.
The organization that takes care of this fund and settles the claims for all
losses is known as an insurance company. The deposit made to this fund
by the individuals is called a premium. Since the payment of this premium
by the individual insures a certain sum or benefit at his death,* he is spoken
of as the insured and the person to whom the benefit is paid at the death
of the insured is called the beneficiary. The agreement made between the
insured and the company is called a policy and the insured is sometimes
spoken of as the policy-holder.
The fundamental problem of life insurance is the determination of the
premium that is to be charged the policy-holder in return for the benefits
promised him by the policy. It is clear that the premium will depend
upon the probability of dying and also upon the rate of interest on funds
left with the company. That is, the premium requires a mortality table
and an assumed rate of interest. The premium based upon these two
items is called the net premium.
The American Experience Table of Mortality is the standard, in the
United States, for the calculation of net premiums and for the valuation
of policies. We shall in all our problems on life insurance assume this
table and interest at 3J^%. In computing the net premiums, we shall also
assume that the benefits under the policy are paid at the ends of the years
in which they fall due.
The insurance company has many expenses, in connection with the
securing of policy-holders, such as advertising, commissions, salaries, office
* Certain insurance agreements specify the payment of an indemnity to the indi-
vidual himself in case he is disabled by either accident or sickness. This is known as
accident and health insurance, but we shall not attempt to treat it in this book.
198
Life Insurance, Net Premiums (Single and Annual) 199
supplies, et cetera, and consequently, must make a charge in addition to
the net premium. The net premium plus this additional charge is called
the gross or office premium. In this chapter we shall discuss only net
premiums and leave gross premiums for another chapter. The premium
may be single, or it may be paid annually, and this annual premium may
sometimes be paid in semi-annual, quarterly or even monthly installments.
All premiums are paid in advance.
83. Whole life policy. A whole life policy is one wherein the benefit
is payable at death and at death only. The net single premium on a whole
life policy is the present value of this benefit. The symbol A x will stand
for the net single premium of a benefit of $1 on (#).
Let us assume that each of l x persons all of age x, buys a whole life
policy of $1. During the first year there will be d x deaths, and conse-
quently, at the end * of the first year the company will pay d x dollars in
benefits, and the present value of these benefits will be vd x . There will
be d x+ i deaths during the second year and the present value of the death
benefits paid will be v*d x +i, and so on. The sum of the present values of
all future benefits will be given by the expression
vd x + v*d x +i + v*d x+ z H ---- to end of table.
Since l x persons buy benefits of $1 each, the present value of the total
premiums paid to the company is l x -A x .
Equating the present value of the total premiums paid and the present
value of all future benefits, we have
k'A x = vd x + v 2 d x +i + v z d x +2 H ---- to end of table.
Solving the above equation for A x , we get
vd x + v*d x +i + v*d x +2 -\ -to end of table
A x = - ; -- (1)
l>x
If both the numerator and the denominator of (1) be multiplied by
v*, we get
v*+ l d x + v*+%k+i H ---- to end of table
C x + C I+ i + C x+ 2 -\ -to end of table
_
* We assume that the death benefit is paid at the end of the year of death.
200 Financial Mathematics
where
Cx = v*+ l d x , C x +i = v x +*d x +i, and so on,
and
M x = C x + C x+ i + C x+ 2 + to end of table.
The expressions C x and M x are two new commutation symbols that are
needed in this chapter. They are tabulated in Table XII.
If in (1) d x be replaced by its equal l x l x +i, and so on, we get
I,
= / rf, + l + P'k.h2+---\ _ A4
\ I* / \
lx
A,, = v(l + a,) - a*. Art. 75. (3)
If (x) agrees to pay for the insurance of $1 on his life in one installment
in advance, the amount he must pay is A x . Most people do not desire, or
cannot afford, to purchase their insurance by a single payment, but prefer
to distribute the cost throughout life or for a limited period. For the con-
venience of the insured, the policies commonly issued provide for the
payment of premiums in equal annual payments. The corresponding net
premiums are called net level annual premiums.
A common plan is to pay a level premium throughout the life of the
insured. When this is the case the policy is called an ordinary life insurance
policy.
We will denote the net annual premium of an ordinary life policy of $1
by the symbol P x . The payment of P x , at the beginning of each year, for
life forms a life annuity due and the present value of this annuity must be
equivalent to the net single premium. Thus we have,
JfVa, = 4.. (4)
Solving for P x , we get
since
. M,
Life Insurance, Net Premiums (Single and Annual) 201
and
a, = 7~ [(6) Art. 76]
L/x
Another common plan probably the plan that occurs most fre-
quently is to pay for the insurance by paying the level premium for a
limited number of years. When this is the case, the policy is called a
limited payment life policy. The standard forms of limited payment policies
are usually for ten, fifteen, twenty, or thirty annual payments, but other
forms may be written.
Let us consider the n-payment life policy.
It is evident that the n annual premiums on the limited payment life
policy form a temporary life annuity due. It is also clear that the present
value of this annuity is equivalent to the net single premium A x . Hence,
if the net annual premium for a benefit of $1 be denoted by n P x , we may
write
Solving for n P x and substituting for a^ and A x , we have
Exercises
1. Use (1) Art. 83 to find the net single premium for a whole life policy to insure
a person aged 91 for $2,000.
2. Find the net single premium for a whole life policy of $10,000 on a life aged 30.
3. Find the annual premium for an ordinary life policy of $10,000 on a life aged 30.
4. Find the net annual premium on a 20-payment life policy of $5,000 for a person
aged 30.
6. Assuming that each of l x persons, all of age x, buys an ordinary life policy of $1,
show from fundamental principles that
Pxdx + Vlx+l + Mx + 2 +)= (Vd x + V*d x +l + t^+2 + )
and thereby derive (5) Art. 83.
6. Show that M x - vN x - N X +I.
7. Compare annual premiums on ordinary life policies of $10,000 for ages 20 and 21
with those for ages 60 and 51. Note the annual change in cost for the two periods of
life.
8. Find the net annual premium for a fifteen payment life policy of $10,000 issued
at age 50.
9. Find the net annual premium for a ten payment life policy of $25,000 issued at
age 55.
202 Financial Mathematics
10. Find the net annual premium on a twenty payment life policy of $5,000 for your
age at nearest birthday.*
11. Compare annual premiums on twenty payment life policies of $10,000 for ages
25 and 26 with those for ages 50 and 51. Note the annual change in cost for the two
periods of life.
12. Find the net annual premium for a twenty-five payment life policy of $10,000,
issued at age 35.
13. Find the net annual premium for a thirty payment life policy of $10,000, issued
at age 35.
14. Using (10) Art. 9 with n 1, and (3) Art. 83, show that A x 1 - da*.
16. Show that P x = -- d.
&x
16. Give a verbal interpretation of the formula A x = v(l 4- a x ) Q>* = va, x o>x-
17. Prove that A x = v da x .
18. Let r \ A x denote the net single premium for an insurance of $1 on (x) deferred r
years (that is, the benefit is paid only if the insured dies after age x -\- r). Show that
I A Mx + r
84. Term insurance. Term insurance is temporary insurance as it
provides for the payment of the benefit only in case death occurs within a cer-
tain period of n years. After n years the policy becomes void. The stated
period may be any number of years, but usually term policies are for five
years, ten years, fifteen years, and twenty years. The symbol A 1 ^ is
usually used to denote the net single premium on a term policy of benefit
$1 for n years taken at age x.
If we assume that each of l x persons, all of age x, buys a term policy
of benefit $1 for n years, the present value of the payments made by the
company will be given by
vd x + v*d x + i + v*d x +2 H ----- h v n d x+n -i.
Since each of l x persons buys a benefit of $1, the present value of the
premiums paid to the insurance company is l x -A\-^.
Equating the present value of the premiums paid to the company and
the present value of the benefits paid by the company, we have
fc- Al-% = vd x + v*d x+ i H ----- |- v n d x+n ~i
and
A i - _ vd x + v*d x+ i H ----- h v n d x + n -i
Acifl ;
* Your insurance age is that of your nearest birthday.
(?)
Life Insurance, Net Premiums (Single and Annual) 203
If both the numerator and the denominator of (7) be multiplied by
v x , we get
v x+l d x + v x + 2 d x+ i H to end of table
v x l x
v x+n+l d x+n -\ to end of table
v x l x
And
- M x - M x + n
A X n\ j: (o)
When the term insurance is for one year only the net premium is
called the natural premium. It is given by making n = 1 in (8). Thus,
A ^M x -M x+ _ C x
xl{ D x D x
The net annual premium for a term policy of $1 for n years will be
denoted by the symbol P\n\- It is evident that the annual premiums
for a term policy constitute a temporary annuity due. This annuity
is equivalent to the net single premium. Thus,
Solving for P\^ and substituting for a^ and A^, we get
[(10) Art. 78] and (8) above.
Exercises
1. Find the net single premium for a term insurance of $5,000 for 15 years for a man
aged 30.
Solution. Here, n = 15 and x = 30. Using (8) Art. 84, we have
M 30 - M 46 10,259 - 7,192.81 3,066.19
and 5,OOOAib is! = 5,000(0.10072) = $503.60.
2. Find the net single premium for a term insurance of $25,000 for 10 years for a
man aged 40.
3. What are the natural premiums for ages 20, 25, 30, 35 and 40 for an insurance of
$1,000.
204 Financial Mathematics
4. Find the net annual premium for a 20-year term policy of $10,000 taken at age 35.
6. Show that the net annual premium on a fc-payment n-year term policy of benefit
$1 (k < n) taken at age x is given by the expression
P i M *~ M *+. (11)
* *"' N x -N x + k
6. What is the net annual premium on a 20-payment 40-year term policy of $1,000
for a man aged 20?
7. A person aged 25 buys a $20,000 term policy which will terminate at age 65,
Find the net annual premium.
8. Find the net annual premium on a 7-year term policy of $5,000 taken at age 27.
85. Endowment insurance. In an endowment policy the company
agrees to pay a certain sum in event of the death of the insured within a speci-
fied period, known as the endowment period, and also agrees to pay this sum
at the end of the endowment period, provided the insured be living to receive
the sum. From the above definition it is evident that an endowment
insurance of $1 for n years may be considered as a term insurance of $1
for n years plus an n-year pure endowment of $1. (See Art. 73 and Art.
84.)
Thus, if we let the symbol A x -^ stand for the net single premium for
an endowment of $1 for n years, we have
D x D x
M x - M x+n + D x+n
D x
since,
(12)
[(8) Art. 84]
and
JE X = - [(2) Art. 73]
We shall now find the net annual premium for an endowment of $1 for
n years, the premiums to be payable for k years. The symbol k P x ^\ will
stand for the annual premium of such an endowment. It is clear that
Life Insurance, Net Premiums (Single and Annual) 205
these premiums constitute a temporary annuity due that is equivalent
to the net single premium. Hence,
Solving for kP X n\ and substituting for a^, and A^ we get
x
N x - N x + k
If the number of annual payments is equal to the number of years in
the endowment period, then k n, and (13) becomes
Exercises
1. Find the net annual premium on a $5,000 20-payment, 30-year endowment policy
taken at age 25.
Solution. Here, x = 25, n = 30 and k 20. Using (13), we have
Af 2 6 - M B 5 + #55
20^25 30] = - - - - -
iV25 -^45
11,631.1 - 5,510.54 + 9,733.40
~~ 770,113 - 253,745
516,368
and (5,000) 20 P2s 301 = 5,000(0.0307028) = $153.51.
2. Find the net annual premium for a $10,000 twenty payment endowment policy
maturing at age 65, taken out at age 21.
3. Find the net annual premium on a $25,000 15-year endowment policy, taken at
age 55.
4. A person aged 22 buys a $10,000 policy which endows at age 60. Find the net
annual premium. The premiums are to be paid until age 60.
6. Find the net single premium on a $10,000 10-year endowment policy, taken at
age 50.
86. Annual premium payable by m equal installments. In Art. 82
we mentioned the fact that the annual premium may be paid in semi-
annual, quarterly or monthly installments.
206 Financial Mathematics
We shall now find the total annual premium on an ordinary life insurance
of $1, when the premium is payable by m equal installments. The symbol
Pi m) will represent this total premium. It is evident that the premiums
constitute an annuity due of P m) per annum, payable in m equal in-
stallments of P^/m each, and the present value of this annuity must equal
the net single premium for an insurance of $1. Hence, we have
p(m) < (m) _ ^
Since,
aT = a x + '^~ [(17) Art. 81]
2m
we have
~frn\ A r
-* m + 1
a * + ~2^~
Example. Find the quarterly premium on an ordinary life policy of
$1,000 taken at age 30.
Solution. Here, x 30 and m = 4. From (15), we have
p(4) _ ^80
#30 + /^
0.33702
18.6054 + 0.6250
0.33702
[Table XII]
19.2304
and
1,000 -P$ = 1,000(0.01752) = $17.52.
The quarterly premium is therefore y ($17.52) = $4.38.
Making m 1, 2, and 4 in (15), we get
p _ A* p(2) _
JL x ~~~ * X ~~"
l + o, 1
respectively, which shows that twice the semi-annual premium is larger
than the annual and four times the quarterly premium is larger than twice
the semi-annual. This addition in premium takes account of two things
Life Insurance, Net Premiums (Single and Annual) 207
only: (1) the possibility that a part of the annual premium may be lost
in the year of death; and (2) loss of interest on part of annual premium
unpaid. On an annual basis the premium would be paid in full at the
beginning of the year of death, while on a semi-annual or quarterly basis
a part of the premium might remain unpaid at date of death, and the
interest on that part of the premium that is not paid at the beginning
of the year is lost annually.
However, in practice there is at least another element which is not
provided for in this theoretical increase and that is the additional expense
incurred in collecting premiums twice or four times a year instead of once.
And then, too, it is the observation of most companies that the percentage
of lapsed policies is greater when written on the semi-annual and quarterly
basis than when written on the annual basis.
It is evident, then, that this theoretical increase is not sufficient to
take care of the additional expenses incurred. To obtain the semi-annual
premium many companies add 4% to the annual rate and then divide by
2 and to obtain the quarterly premium they add 6% to the annual rate
and divide by 4.
We might derive formulas for the annual premiums on other types of
policies, but, as indicated above, these formulas are not really used in
practice.
Exercises
1. Find the total annual premium on an ordinary life policy $1,000 taken at age 50,
if the premiums are to be paid (a) semi-anrmally; (b) quarterly. Use formula (15) and
then use the method that is used in practice by most companies and compare results.
2. Show that (15), Art. 86 can be written
Make m = 1 and compare with (5), Art. 83.
3. Find the annual premium on an ordinary life policy of $5,000 taken at age 25,
if the premiums are to be paid (a) quarterly; (b) monthly.
87. Summary of formulas of life insurance premiums. In this chap-
ter we have discussed the "standard" policies and have derived the for-
mulas for computing the net single and the net annual premiums under
them. We summarize this information in the following table.
208 Financial Mathematics
x = the age of the insured; F = the face of the policy.
Name of Policy
Policy Benefits
Premiums
Paid
Single Premiums
Annual Premiums
Ordinary life
Whole life insurance
For life
f z
D*
M*
*N X
Whole life insurance
For n years
F M*
F D t
F M *
N x - N x+n
n-year term insurance
For n years
Sf x - M z+n
M x M x n
F D X
N x - N x+n
n-year 1 endow-
fc-payment / meat
(a) n-year pure endowment
(b) -year term insurance
For k years,
k < n
r, MX M x+n ~f~ Dx+n
M x -M x + n +Dx4*
r D X
N x -Nx +k
88. Combined insurance and annuity policies. The principles sum-
marized in Art. 87 enable us to compute the premiums on the well-
known standard policies. Today, combined insurance and annuity
policies are frequently written, and we shall now illustrate the methods of
computing the premiums for them. We shall merely need to apply the
equation of value:
Present value of payments = Present value of benefits.
(16)
Example 1. An insurance-annuity contract token out by a life aged
25 provides for the following benefits:
(a) 10-year term insurance for $5,000,
(b) a pure endowment of $10,000 at the end of 10 years.
It is desired to pay for these benefits in 10 equal annual premiums in
advance. What is the annual premium?
Solution. Let P be the required annual premium.
Present value of benefit (a) is 5,000 (^25 isf) ^ 5 >
M25 M35
[(8) Art. 84]
Present value of benefit (b) is 10,000 (10^25) = 10,000
Present value of the payments is P(a 2 5ioi) =
#25
[(!') Art. 80]
[(100 Art. 80]
Life Insurance 9 Net Premiums (Single and Annual) 209
Hence, using (16), we have
= 5,000 + 10,000
, ,
#25 #25 #25
5000(M 25 - M 35 ) + 10,000 #35
#25 #35
P = $855.98. Table XII.
Example 2. An insurance-annuity contract taken out by a life aged
40 provides for the following benefits:
(a) a $10,000 pure endowment payable at age 65,
(b) a $10,000 20-payment life insurance,
(c) a life annuity of $2,000 annually with the first payment at age 65.
If the premiums are to be paid annually in advance for 20 years, find
the annual premium P. Set up in commutation symbols.
Solution.
Dc*>
Present value of benefit (a) is 10,000 (25^40) = 10,000
#40
[(!') Art. 80]
M40
Present value of benefit (b) is 10,000 (Aw) = 10,000
#40
[(2) Art. 83]
#65
Present value of benefit (c) is 2,000 (25 | a 4 o) = 2,000 ^
#40
[(8') Art. 80]
Present value of the payments is P(a 40 2of) = P --
#40
[(100 Art. 80]
Hence, applying (16), we have
10,000 (#65 + M 40 ) + 2,000 #65
#40 #60
210 Financial Mathematics
Problems
1. Find the net annual premium for an endowment policy for $5,000 to mature at
age 85 and taken at age 40.
2. For purposes of valuation, a policy for $15,000 taken at age 35 provides that the
insurance of the first year is term insurance, and that of subsequent years is a 14 pay-
ment life insurance on a life aged 36, so that the insurance is paid up in 15 payments in
all. What is the first year premium and that of any subsequent year?
3. An insurance contract provides for the payment of $1,000 at the death of the
insured, and $1,000 at the end of each year thereafter until 10 installments certain are
paid. What is the net annual premium on such a contract for a person aged 40, if the
policy is to become paid up in 20 payments?
4. What would be the net annual premium in Problem 3, if it were written on the
ordinary life basis?
6. Assume that each of l x persons, all of age x, buys an n-payment life policy of $1 ;
equate the present value of all premiums paid and all benefits received; and derive (6),
Art. 83.
6. Reasoning as in Problem 5, derive (10), Art. 84.
7. Reasoning as in Problem 5, derive (14), Art. 85.
8. Prove that:
(aM, =--., <b)>,
Px + d
9. Prove that:
(a) A\T\ = v - ~ , (b)
10. Show that
and interpret this formula verbally.
11. A 20-payment life insurance policy for $1,000 issued to a life aged 30, for pur-
poses of valuation, is treated as a one-year term policy at age 30 plus a 19-payment
life policy at age 31. What is the net premium for the first year and the net level
annual premium for the subsequent 19 payments?
12. For purposes of valuation, an ordinary life policy of $1,000 issued to a life aged
30 is considered as a one-year term policy at age 30 and an ordinary life policy at age 31.
What is the first net annual premium and the subsequent annual net level premiums?
13. A person aged 45 takes out a policy which promises $10,000 if death occurs
before age 65. If the insured is living at age 65, he is to receive $1,000 annually as long
as he lives, the first $1,000 being paid when age 65 is reached. What is the net level
annual premium if the policy is issued on a 20-payment basis?
14. A life insurance policy issued on a life aged 30 provides for the following benefits:
In the event of death of the insured during the first 30 years the policy pays $1,000, with
a $5,000 cash payment if the insured survives to age 60. If the policy is issued on a
20-payment net level basis, find the net premium.
Life Insurance, Net Premiums (Single and Annual) 211
Find the net periodic premium for each of the following policies.
Problem
Benefits of Policy
Age of
Insured
Number of
Annual
Premiums
15.
(a) 10-year term insurance for $10,000,
(b) a pure endowment of $20,000 at end of 20
years.
45
10
16.
(a) Whole life insurance of $10,000,
(b) a pure endowment at age 60 of $10,000,
(c) a life annuity of $1,000 annually with first
payment at age 65.
30
20
17.
(a) $30,000 to beneficiary if death of insured
occurs between ages 30 and 40,
(b) $25,000 to beneficiary if death of insured
occurs between ages 40 and 50,
(c) $15,000 to beneficiary if death of insured
occurs between ages 50 and 60.
30
20
Hint. The benefits under the policy in Problem 17 are the same as those under a
policy providing for $5,000 10-year term insurance, $10,000 20-year term insurance,
$15,000 30-year term insurance, all issued to a life aged 30.
CHAPTER X
VALUATION OF POLICIES. RESERVES
89. Meaning of reserves. Except at very low ages, the probability
of dying in any year increases with increasing age. Consequently, the
cost of insurance provided by the given policy, as indicated by the natural
premium, increases with increasing age. The net level annual premium
for the policy is larger than the natural premium during the early years of
the policy and is therefore more than sufficient to cover the insurance, but,
in the later years of the policy the net level premium is smaller than the
natural premium and is therefore insufficient to cover the cost of the
insurance.
To illustrate the above remarks, let us consider a numerical example.
A man aged 35 takes out a $1,000 ordinary life policy. The ixet level
annual premium under the American Experience 3H% Table is $19.91.
The cost of insurance (natural premium) for the first policy year is $8.64,
leaving a difference ($19.91 - $8.64) = $11.27. During the second year
of the policy the cost of insurance (natural premium on a life aged 36) is
$8.78, and thus the insured pays ($19.91 - $8.78) = $11.13 more than the
expense due to mortality. This situation continues to age 57 when, and
for later years, the net level premium $19.91 is insufficient to meet the cost
of insurance, for, at age 57 the natural premium is $20.61. The following
table compares the net level premium $19.91 with the increasing cost of
insurance for an ordinary life policy of $1,000 on a life aged 35.
Attained
Acre
Natural
Premium
Excess
N.L.P.-N.P.
Attained
Ace
Natural
Premium
Excess
N.L.P.-N.P.
35
$8.64
$11.27
65
$38.77
-$18.86
40
9.46
10.45
70
59.90
- 39.99
45
10.79
9.12
75
91.18
- 71.27
50
13.32
6.59
80
139.58
-119,67
55
17.94
1.97
85
227.59
-207.68
60
25.79
-5.88
90
439.17
-419.26
212
Valuation of Policies. Reserves
213
It is evident that if an insurance company is to operate upon a solvent
basis, it must accumulate a fund during the early policy years to meet the
increased cost in the later policy years. These excesses of the net level
premium over the natural premiums that appear in the early policy years
are improved at interest and held by the company to meet the increased
cost during the later policy years. The accumulation of these excesses
results in a fund that is called the reserve or the value of the policy*
90. Computing reserves, Numerical illustration. A glance at the
American Experience Table of Mortality shows that of 100,000 persons
alive at age 10 there remain 81,822 alive at age 35.
Let us assume that each of 81,822 persons, all aged 35, buys an ordinary
life policy of $1,000. The total of the net annual premiums amounts to
$1,629,076.02. This amount accumulates to $1,686,093.68 by the end
of the first year. According to the table of mortality the death losses
to be paid at the end of the first year amount to $732,000.00, leaving
$954,093.68 in the reserve. This leaves a terminal reserve of $11.77
to each of the 81,090 survivors. The premiums received at the begin-
ning of the second year amount to $1,614,501.90, which when added to
$954,093.68 makes a total of $2,568,595.58, and so on. The following table
is self explanatory.
TABLE SHOWING TERMINAL RESERVES ON AN ORDINARY LIFE POLICY FOR $1,000
ON THE LlFE OF AN INDIVIDUAL AGED 35 YEARS
Funds on
Amount to
Policy
Year
Hand at
Beginning
Accumulated
Death
Losses
Funds at
End of Year
Credit of Each
Survivor,
of Year
Reserve
1
$1,629,076.02
$1,686,093.68
$732,000
$ 954,093.68
$11.77
2
2,568,595.58
2,658,496.43
737,000
1,921,496.43
23.91
3
3,521,324.66
3,644,571.02
742,000
2,902,571.02
36.46
4
4,487,625.03
4,644,692.94
749,000
3,895,692.94
49.40
5
5,465,835.36
5,657,139.60
756,000
4,901,139.60
62.75
This illustrates what is known as the retrospective method of computing
reserves because the reserve at the end of any policy year was determined
exclusively from facts that belong to the past history of the policy.
* The reserve on any one policy at the end of any policy year is known as the terminal
reserve for that year, or the policy value.
214
Financial Mathematics
Exercises
1. The premium on a 5-year endowment insurance for $1000 taken out at age 25 is
$183.56. Complete the following table and show that at the end of 5 years the fund
is just sufficient to pay each survivor $1000.00.
Funds on
Amount to
Policy
Year
Hand at
Beginning
of Year
Funds
Accumulated
at ZY 2 %
Death
Losses
Funds at
End of Year
Credit of Each
Survivor,
Reserve
1
$16,342,713.92
$16,914,708.91
$718,000
$16,196,708.91
$183.40
2
32,407,626.75
33,541,893.69
718,000
32,823,893.69
374.72
3
48,903,015.45
50,614,620.99
718,000
49,896,620.99
574.33
4
718,000
5
r.- 1 ' 1
719,000
2. The annual premium on a 10-payment life policy on a life aged 30 is $40.6078.
Prepare a table similar to that in Exercise 1 and thus compute the reserve on the policy
at the end of each policy year.
91. Fackler's accumulation formula. We will now develop a formula
which expresses the terminal reserve of any policy year in terms of the
reserve of the previous year. We will designate by r V x the terminal reserve
of the rth year on an insurance of $1, and let P x stand for the net annual
premium. The reserve then at the beginning of the (r + l)th year will
be rVx + Px- This is called the initial reserve of the (r + l)th year.
The aggregate reserve at the beginning of the (r + l)th year, for the
h+r individuals insured, will be
l,+r(rV, + P x ).
This last amount will accumulate, by the end of the year, to
Out of this amount the company will have to pay d x + r as death claims
for the year, leaving
as the total reserve to the l x+r +i surviving policy holders at the end of the
(r + l)th year.
Valuation of Policies. Reserves 21S
The terminal reserve then for the (r + l)th year is
lx+r(rV x +P x )(l+l) -4+r
r-fl
If we now define the valuation factors (see Table XIII)
(l + i)l* , . d*
u x = - - and k x = - ,
lx+\ lx+l
we have
= U x +,(rK, + />,) - *,+ r . (2)
This formula is known as Fackler's accumulation formula. It will evi-
dently work for any policy, for the factors u x + r and k x+r in no way depend
upon the form of the policy. This formula is used very extensively
by actuaries in preparing complete tables of terminal reserves. The valua-
tion functions u x and k x are based upon the American Experience Table of
Mortality and 3J^% interest and are given in Table XIII.
To find the terminal reserve for the first policy year we make r = 0,
and (2) becomes
iV, = u^P, - ft,, (3)
for it is evident that oV x = 0.
Exercises
D C
1. Show that u x == and k x = and verify the tabular values of u x and
Dx+l Dx+l
k x for the ages 20, 25, and 30 by making use of the C x and D x functions.
2. Making use of formulas (3) and (2) Art. 91, verify the reserves in the problem of
Art. 90.
Solution. From (3) we have
iF 3B = ^36/^5 - &36, and P 36 = 0.01991.
Hence iV 3 5 = 1.044343 (.01991) - 0.009027
= 0.011766.
Then, 1,000 iF 3 6 = $11-77.
Also, 2 F 35 = use (iF 35 + PSB) - fee, [(2) Art. 91]
= 1.044493 (0.011766 + 0.01991) - 0.009172
- 0.023913.
216 Financial Mathematics
Hence, 1,000 2 V 3B = $23.91.
3. Find the terminal reserve for each of the first five policy years on a ten payment
life policy for $5,000 taken at age 25.
4. The terminal reserve at the end of the fifteenth policy year on a twenty-year
endowment policy for $1,000 taken at age 25 is $665.59. Calculate the terminal reserves
for the succeeding policy years until the policy matures.
5. The terminal reserve at the end of the tenth policy year on a fifteen payment
life policy for $1,000 taken at age 30 is $272.96. Find the terminal reserve for the
eleventh and twelfth years.
6. The terminal reserve at the end of the twenty-fifth policy year on an ordinary
life policy for $1,000 taken at age 29 is $333.81. Find the terminal reserve for the
twenty-sixth year.
92. Prospective method of valuation, We now consider another
method of valuation and derive a formula for determining the terminal
reserve for any policy year independent of the reserve for the previous
year. At the end of the ri\\ policy year the sum of the terminal reserve and
the present value of the future premiums to be paid must equal the net single
premium for a new policy on the life of the insuredj who is now aged x + r.
If we consider an ordinary life policy the present value of the future
premiums to be paid would be F x & x + r and the net single premium
for a policy on the insured, now aged x + r, would be A X + T . Again denot-
ing the terminal reserve for the rth year by r V x , we obtain the relation,
r V, + P^ x+r = A,+ r , [(5) Art. 76]
and
T V X = A x+r - P x z x+r . (4)
We see from equation (4) that the rth year terminal reserve is equal to the
net single premium for the attained age x + r minus the present value of all
future net annual premiums. This definition of reserve will evidently
hold for all forms of policies.
The value of T V X may be expressed in terms of the commutation columns
by remembering that
4"* -*' [(2) Art - 83 l
M
~, [(5) Art. 83]
2V.
Valuation of Policies. Reserves 217
and
a,+ r = ^ r [(6) Art. 76]
l-'x+r
Then
Replacing A x + r by its equivalent P x+r (si x +r), equation (4) becomes
or
r V x = <P,+ r - P,)(a x+r ). (5)
P x +r is the net annual premium for an individual now aged x + r, but
since he took his insurance at age x instead of waiting until age x + r, his
annual saving in premium is (P x + r Px) and the present value of these
annual savings is (P x + r P*)(a;e+ r ) which is the policy reserve at the
end of the rth year. Hence we have a verbal interpretation of the for-
mula (5).
We will now derive an expression for the terminal reserve for the rth
year on an n-payment life insurance of $1. The symbol, r:n V XJ will denote
the rth year reserve for this policy. Immediately following equation (4),
Art. 92, we defined reserve and said this definition would hold for all forms
of policies. Here the net single premium for the attained age x + r
would be A x+r and the present value of all future premiums would be given
by
as they would constitute a temporary life annuity due, for n r years.
Consequently, we may write
finYx = A x + r ~ nP x 'B, x + r j[I^ (6)
Denoting the rth year terminal reserve on a fc-payment n-year endow-
ment insurance of $1 by r:t V,^ and following the same line of reasoning
used in obtaining (6), we get,
(7)
When r is equal to or greater than k formula (7) becomes
rijJVnl ^*+rrF,. (8)
218 Financial Mathematics
When the annual premiums are payable for the entire endowment
period, k =* n, and (7) reduces to
r'x~n\ == A x + r n-r\ * x~n\ '^Jt+r ii-r|- W/
Exercises
1. Find the 20th year reserve on an ordinary life policy for $5,000 taken at age 30.
Solution. Here, r = 20, x = 30. Then from (4) Art. 92, we have *
20^30 = ^ so ~ -Pao
But
Hence, 20^30 = 0.50849 - 0.01719(14.5346)
= 0.25864,
and 5,000-20^30 = $1,293.20.
2. Find the terminal reserve of the 15th policy year on a 15-payment life policy of
$5,000 taken at age 35. Explain why this result equals the net single premium on a
life policy taken at age 50.
3. Find the 20th year terminal reserve on a $10,000 policy which is to mature as an
endowment at age 65, if the policy was taken at age 30.
4. Find the 10th year reserve on a $20,000, 20-year endowment policy taken at
age 40.
6. Find the terminal reserve of the seventh policy year on a twenty payment life
policy of $2,500 taken at age 32.
6. Find the terminal reserve of the ninth policy year on an ordinary life policy of
$5,000 taken at age 40.
7. Verify the result for the third terminal reserve in Exercise 1, Art. 90.
8. Verify the result for the fifth terminal reserve of the illustrative problem in Art.
90.
9. Reduce formula (6) Art. 92 to commutation symbols.
93. Retrospective method of valuation. In preceding sections we have
alluded to the retrospective method of computing reserves. Fackler's
accumulation formula, Art. 91, was developed from facts that pertain to
the past history of the policy. It expresses the reserve of any policy year
in terms of the reserve of the previous year, and is therefore very useful
in preparing complete tables of terminal reserves. It cannot be used,
however, for computing the reserve on a given policy for a specified policy
year.
The problem of finding the reserve on a given policy for a specified policy
Valuation of Policies. Reserves 219
year was solved in Art. 92 by the prospective method. The thoughtful
student will naturally enquire: "Can we develop formulas by the retro-
spective method for computing the reserves on given policies for specified
policy years, and are the results consistent with those of Art. 92?"
We answer both questions in the affirmative.
From the retrospective point of view, the rth terminal reserve for a given
policy issued at age x is the accumulated value at age x + r of the past premiums
kss the accumulated value at age x + r of the past insurance benefits. The
past insurance benefits are those of an r-year term insurance on (x). That
is,
xt. rr . ,\ / Value at age \ / Value at age \
rth Termman / ill . 1
) = [ x + r I I x + r I
reserve / I - , . \ ,. ~ I
' \of past premiums/ \of past benefits/
Consider an ordinary life policy of $1 on (x).
P x the net annual premium, and r V x = the rth terminal reserve.
M x N x - W x+r
D x+r
[(5) Art. 83] [(12) Art. 79J
/Value at age x + A = p __
\of past premiums/
/Value at age x + r
\ = ^n = 4
I 77?
/ rJLjf
/ ' *
- M x+r D x M x -
\ of past benefits / T E X D x D x + r _.-, D x+r
[(8) Art. 84] [(2) Art. 73]
Hence,
v - ^* ^ x ~~ ^ x + r ^ x ~~ ^ x + r
r x AT 71 n
Jiv x J^x+r J-Sx+r
T. =-
N x D x+r
which is the same as (4') Art. 92.
Problems
1. How much does a person save by buying a $10,000 ordinary life policy at age 25
instead of waiting until age 30? See formula (5), Art. 92.
2. Show that when r - n, the right-hand member of (6) Art. 92, reduces to A x + n
and explain the meaning of this result.
3. Derive formula (7), Art. 92.
4. To what does the right member of (9) reduce when r = n?
5. Express formula (6) in terms of the commutation symbols.
220 Financial Mathematics
6. Express formula (9) in terms of the commutation symbols.
7. Making use of (3) and (5) Art. 83, show that
a x - a x + r 1 + a*+r a g +
8. Use formula (10) to find the twelfth year terminal reserve on a $2,000 ordinary
life policy taken at age 37.
9. (a) Show that r:nV x (n-rPx+r nPx) (bx+rn^) and interpret the result.
(b) Derive a similar expression for the r&-year endowment policy.
10. Build up a table of terminal reserves for the first 10 years on a 20-payment life
policy of $1,000 taken at age 30. Use (3) and (2), Art. 91 and check every 5 years by
using (6), Art. 92.
11. Build up a table of terminal reserves for the first 10 years on an ordinary life
policy of $1,000 taken at age 33. Use (3) and (2) Art. 91 and check for the fifth and
tenth years by using formula (10), Problem 7.
12. Build up a table of terminal reserves for the first 5 years on a 10 year endowment
of $1,000 taken at age 30. Use Fackler's formula and check the fifth year by using
formula (9) Art. 92.
13. Solve Exercise 10, with the policy taken at age 40.
14. Solve Exercise 11, with the policy taken at age 38,
15. Develop a formula similar to (9), Art. 92, but for term insurance for a term of
n years. Find the fifth year terminal reserve on a ten year term policy of $1,000 issued
at age 30.
16. Find the seventh year terminal reserve on a $1,000, 15 year term policy issued at
age 40.
CHAPTER XI
GROSS PREMIUMS, OTHER METHODS OF VALUATION, POLICY OPTIONS
AND PROVISIONS, SURPLUS AND DIVIDENDS
94. Gross Premiums. In Chapter IX a net premium was defined
and we found the net premiums for a number of the standard policies. We
saw that this net premium was large enough to take care of the yearly
death claims and to build up a reserve sufficient to care for all future
claims, but was not adequate to pay the running expenses of the company
and provide against unforeseen contingencies.* Hence to care for these
extra expenses a charge in addition to the net premium must be made.
This additional charge is sometimes spoken of as a loading, and the net pre-
mium plus this loading is called the gross premium.
In Chapter IX we enumerated some of the expenses of the insurance
company. To these we may add taxes imposed by state legislatures,
medical expenses for the examination of new risks, expenses for collecting
premiums, and many other minor ones.
We shall now discuss some of the methods used in arriving at a suffi-
cient gross premium. At first thought it might seem reasonable to add a
fixed amount to the net premium on each $1,000 insured regardless of age
or kind of policy. This would give the same amount for expenses on an
ordinary life policy for a young man, aged 25 say, as on a 20-year endow-
ment policy for the same amount and age. The percentage of loading on
the ordinary life policy would be about three times as large as that on the
endowment policy, while as a matter of fact the expenses of each policy
would be about the same percentage of the respective premium, for com-
missions are usually paid as a percentage of the premium, and taxes are
charged in a like manner. Hence, we see that a constant amount added to
a premium does not make adequate provisions and it is seldom used now
without modification.
Sometimes loadings are effected by adding a fixed percentage of the
net premium. Let us assume for the time being that this is 30%. Then
the loading at age 25 on an ordinary life policy would be $4.53 and on a
ten year endowment at age 65 it would be $32.75. ^ It is evident that this
method makes the loading very high for the older ages and thereby causes
the premium to be unattractive to the applicant. As a matter of fact the
* The influenza epidemic of 1918 is an example of this.
221
222 Financial Mathematics
$32.75 is more than is actually required to care for the expenses of the
10-year endowment taken at age 65. This method has its objections as
well as the first method described.
Often a constant amount plus a fixed percentage of the net premium
is added. This is a combination of the two methods described above. The
constant gives an adequate amount for administration expenses as this
depends more on the volume of insurance in force than on the amount of
premiums, and the percentage provides for those expenses that are a certain
percentage of the net premium.
If we add a constant $4 for each $1,000 of insurance and 15% of the net
premium we get a premium that is very satisfactory. For example the net
premium on an ordinary life policy of $1,000 at age 35 is $19.91. Adding
$4.00 and 15%, we get $26.89 as our office premium.
Another plan is a modification of the percentage method. If 33J^%
be the percentage, y$ of the net premium is added to obtain the office
premium on ordinary life. On limited payment life and endowment policies
K of the net premium for the particular policy is added and then J^ of
the net premium on an ordinary life for the same age. To illustrate:
Ordinary life, net rate, age 35 ................... $19 . 91
H of net rate ............................... 6.64
Gross premium ......................... $26 . 55
20-year endowment, net rate, age 35 ............. $40. 11
K of $40.11 ................................ 6.68
H of ordinary life rate ....................... 3 . 32
Gross premium ......................... $50 . 01
If we let PS stand for the gross premium of an ordinary life policy of $1,
and let r denote the rate of the percentage charge, and c the constant
charge per $1,000 of insurance, we may express by the formula,
the ideas mentioned above. If the loading is a constant charge, r will be
zero but if it is considered a percentage charge only, c will be zero.
Formula (1) may be modified to apply to the different forms of policies.
Nearly every company has its individual method of calculating gross
premiums but all companies get about the same results.
95. Surplus and dividends. The gross premium is divided into three
parts. The first part is an amount sufficient to pay the death claims for
Gross Premiums, Other Methods of Valuation 223
the year, where the number of deaths is based upon the American Experi-
ence Table of Mortality. The second part goes to build up the reserve.
The third part is set aside to meet the expenses of the company.
As all new policy holders are selected by medical examination it is
reasonable to expect that, under normal conditions, the actual number of
deaths will be much smaller than the expected. Hence, a portion of the
first part of the premium is not used for the current death claims, and
is placed in a separate fund known as the surplus.
The reserve is figured on a 3^% interest basis, but the average interest
earned by the funds of the company is usually considerably more than this.
This additional interest is also added to the surplus.
After an insurance company has become well organized and its terri-
tory has been thoroughly developed its annual expenses are usually much
less than the expected. Hence a portion of the third part of the premium
is saved and added to the surplus.
Since the surplus comes from savings on the premiums, a part of it is
refunded to the policy holders at the end of each year. These refunds are
called dividends, but they are not dividends in the same sense as the
interest on a bond. Most of these dividends come from savings on pre-
miums and only a small amount comes from a larger interest earning on the
reserve and other invested funds.
A large portion of this surplus must be held by the company for it is
as essential for an insurance company to have an adequate surplus as it
is for a trust company, a bank, or any other corporation. The surplus
represents the difference between the assets and the liabilities, and a
relatively large surplus is an indication of solvency.
96. Policy options. In any standard life-insurance policy there is a
nonforfeiture table giving the surrender or loan value, automatic extended
insurance, and paid-up insurance at the end of each policy year beginning
with the third.* In case the insured desires to quit paying any time after
three annual payments have been made, he may surrender his policy and
receive the cash value indicated in the table, or a paid-up policy for the
amount indicated in the table. Or he may keep his policy and remain
insured for the full face amount of the policy for the time stated in the
table.
97. Surrender or loan value. The surrender or loan value of a policy
at the end of any policy year is the terminal reserve for that year less what-
ever charge (known as a surrender charge) the company makes for a sur-
render. This charge is a per cent of the terminal reserve and decreases
* Some companies begin the non-forfeiture table at the end of the second year.
224 Financial Mathematics
each year. After 10 or 15 years there is usually no charge made upon
surrender. The surrender value at the end of the tenth year on an ordi-
nary life $1,000 policy, issued to a person age 25, is $89.43 less the surrender
charge. Insurance laws allow companies to make a surrender charge.
The companies, however, usually make a smaller charge than is allowed
them by law.
We give a few reasons for this charge: First, the company is at an
expense to secure a new policy holder in place of the one surrendered;
Second, life insurance companies claim that the greatest number of lapses
come from people who are in excellent health rather than from those in
poor health. This would tend to increase the percentage of mortality and
thereby decrease the surplus and dividends to policy holders. Third,
if policy values were not subjected to a surrender charge, it is the belief
that a large number of policy holders would either surrender their insurance
or take the full loan value during hard times and thus cause financial loss
to the company.*
98. Extended insurance. Whenever the insured fails to pay his
annual premium the company automatically extends his insurance for the
full face of the policy unless he surrenders his policy and requests the sur-
render value or paid-up insurance. The length of time that the company
can carry the insurance for the full amount, without further premiums,
depends upon the surrender value of the policy at that time.
In order to find the time of extension we must solve the equation
M *+'- M *++< = rVx [(8) Art. 84] (2)
for t. An example will show how this is done.
Example. The value at the end of the tenth year, of an 'ordinary life
policy of $1,000, taken at age 25, is $89.43. Find the time of the automatic
insurance.
Solution. Here, x = 25, r = 10, 10^25 = 0.08943, and
= 0.08943
>35
or
- (0.08943)Z) 35
= 9,094.96 - (0.08943) (24,544.7)
= 6,899.93.
* For a more complete discussion of surrender values see " Notes on Life Insurance "
by Fackler.
Gross Premiums, Other Methods of Valuation 225
This value of M^+t lies between M and Mi. By interpolation we find
that 35 + t 46 years 9 months, approximately, or t = 11 years 9 months.
Hence, the value $89.43 is enough to buy a term policy of $1,000 for 11
years and 9 months.
99. Paid-up insurance. If at any time the insured surrenders his
policy he may take a paid-up policy for the amount that his surrender
value at that time will purchase for him at his attained age. For example,
the value at the end of the tenth year, of an ordinary life policy of $1,000
taken at age 25 is $89.43. Find the paid-up insurance for that year. The
insured is now age 35 and an insurance of $1 will cost him
^35 = 0.37055.
Hence, he may buy for $89.43 as much insurance as .37055 is contained
in $89.43, or approximately $241.00.
The following is a non-forfeiture table for the first 10 years on an
ordinary life policy for $1,000 taken at age 25:
NON-FORFEITURE TABLE - $1,000, ORDINARY LlFE, AGE 25
Automatic
Extension
At End of
Cash or Surrender Value
Years
Months
Paid-up Insurance
3rd Year
$23.70
3
1
$73.00
4th
32.16
4
2
97.00
5th
40.91
5
5
121.00
6th
49.98
6
7
146.00
7th
59.35
7
10
170.00
8th
69.04
9
2
194.00
9th
79.07
10
5
2J8.00
10th
89.43
11
9
241.00
In the above table the values are all based upon the full level net
premium terminal reserves. In a standard policy these values would all
be some smaller due to the surrender charge. Usually, only even dollars
are published in non-forfeiture tables. If the preliminary term method
or modified preliminary term methods of valuation are used,* all the values
will be made somewhat smaller for the first few policy years.
' These methods are discussed in later sections.
226
Financial Mathematics
We shall now outline a method for determining the surrender values,
automatic extended insurance, and paid-up insurance for an endowment
policy. The surrender values will be determined just as terminal reserves
are determined (the surrender value is the terminal reserve less the sur-
render charge). The time for automatic extension must at no time extend
beyond the date of maturity. Hence, only such a part of the surrender
value will be used as is necessary to extend the insurance to the maturity
date. The balance of the surrender value for that year will go to buy
a pure endowment which will mature at the end of the endowment period.
Let us consider a $1,000, 20-year endowment for an individual aged 30.
The reserve (full level net premium method) for the fifth year is $177.83.
The cost of a 15-year paid-up term policy of $1,000 for the attained age,
35, is $111.61. This leaves (177.83 - 111.61) = $66.22 with which to
purchase a 15-year pure endowment. A pure endowment of $1 will cost
istfss = 0.50922. [(2) Art. 73]
Hence, $66.22 will buy as much pure endowment as 0.50922 is contained
in 66.22, or $130.00 (nearest dollar).
We now find the amount of the 15-year paid-up endowment that
$177.83 will buy. The cost of a $1, 15-year paid-up endowment for age 35
is $0.62083. Hence, $177.83 will buy a paid-up endowment of
177 83
Q 62Q83 = $286.00 (approximately).
The following is a non-forfeiture table for the first 10 years on a 20-year
endowment of $1,000 taken at age 30:
NON-FORFEITURE TABLE SI, 000, 20- YEAR ENDOWMENT, AGE 30
Automatic
Extension
Cash or
Pure
Paid-up
At end of
Surrender Value
Endowment
Endowment
Years
Months
3rd Year
$102 35
14
4
$175 00
4th
139.32
16
no
$47.00
231.00
5th
177.83
15
130.00
286.00
6th
217.95
14
208.00
341.00
7th
259.74
13
282.00
394.00
8th
303.29
12
353.00
447.00
9th
348.67
11
421.00
498.00
10th
395.98
10
491.00
554.00
Gross Premiums, Other Methods of Valuation 227
In the above table the values are all based upon the full level net
premium terminal reserves. However, these values would all be somewhat
smaller due to the surrender charge.
In the event the policy holder paid only five premiums and then lapsed
his policy, he could accept any one of the following options at the end of
five years: Receive $177.83 (less surrender charge) in cash, receive a paid-
up 15-year term policy for $1,000 and $130 in cash at age 50, if living, or
receive a paid-up endowment for $286.00.
Exercises
1. Make a non-forfeiture table for the first 10 years of a $1,000 ordinary life policy
taken at age 40.
2. Make a non-forfeiture table for the first 10 years of a $1,000 20-payment life
policy taken at age 40.
3. Make a non-forfeiture table for the first five years of a $1,000 20-year endowment
policy taken at age 40.
4. Make a non-forfeiture table for the first 10 years of a $1,000 policy taken at age 26,
which is to endow at age 60.
6. A man who has attained the age of 35 surrenders his policy and chooses to elect
the option which grants him extended insurance to the amount of $5,000 for eight years.
Find his surrender value.
6. A man who has attained the age of 35 surrenders his policy and elects the option
of paid-up insurance. If his surrender value is $5,000, find the amount of insurance he
should receive.
7. A man aged 25 took out a convertible $10,000 10-year term policy. At the end
of 5 years he converted it into an ordinary life policy as of his attained age. How much
ordinary life insurance did he obtain if all his reserve was used for that purpose?
8. A man aged 30 takes out an ordinary life policy for $10,000. When he is 55 years
of age, the company decides to go out of business. What sum is due him?
100. Preliminary term valuation. In Chapter X we considered what
is known as the full level premium method of valuation. By this method
the difference between the net annual premium and the natural premium for
the first year is placed into the reserve. It is clear that this leaves none of
the net annual premium to care for the first year's expenses of the policy.
The initial expenses of a policy are the greatest for they include an agent's
commission, medical examiner's fee, taxes, etc. To illustrate the above
remarks let us consider an ordinary life policy of $1,000 taken at age 35.
The net annual premium on this policy is $19.91 and the office premium is
$26.55, leaving only $6.64 to go towards initial expenses. The balance of
the first year's expenses must come from the surplus. But this seems
228 Financial Mathematics
unfair to the old policy holders as their contributions in the way of pre-
miums have built up this surplus. It is perhaps fair that they should
bear a small portion of the expenses of securing new business, but they
should not pay so much as is required under the full level premium method
of valuation. It is also evident that under this method it would be almost
impossible for a new company to build up an adequate surplus.
A method known as a preliminary term system has been devised to meet
the objections mentioned above, and we will now describe it. Under this
method all the first year premium is available for current mortality and
expenses. The first year's insurance then is term insurance and the policy
provides that it may be renewed at the end of the first year as a life or endow-
ment policy at the same office premium. The net premium for the first year is
the natural premium for the age when the policy was issued and the balance of
the gross premium is considered as first year loading and is available for
initial expenses. The net premium for the second and subsequent years is
the net premium at an age one year older than when the policy was issued.
Let us again consider the ordinary life policy of $1,000 taken at age 35.
Here the office premium is $26.55 and since the natural premium for the
first year is $8.65 there would be a first year loading of $17.90. The net
annual premium for subsequent years would be $20.55 * which would leave
$6.00 as a renewal loading. Had the policy been issued under the full
level net premium system there would have been a uniform loading of $6.64.
A 20-payment life policy taken at age 35 would have a gross premium
of $35.70. The first year natural premium would be $8.65, thus leaving
a loading of $27.05 for initial expenses, and the net premium for the sub-
sequent nineteen years would be the net premium on a 19-payment life
policy as of age 36. This would be $28.89, thus resulting in a renewal
loading of $6.81. Had this policy been issued under the full level net
premium system there would have been a uniform loading of $8.31.
The preliminary term method when applied to ordinary life policies
and limited payment life and endowment policies with long premium pay-
ing periods is sound in principle and is recognized by the best authorities.
However, the system has some objections when it is applied to limited
payment life and endowment policies of short premium paying periods.
These objections will be discussed in Art. 101 and a remedy will be devised.
It is evident that, since the whole of the first year's gross premium is
available for current mortality and expenses, there can be no terminal
reserve set up until the end of the second year. It is also clear that this
* That is, the premium on a $1,000 ordinary life policy as of age 36,
Gross Premiums, Other Methods of Valuation 229
reserve from year to year will be a little smaller than the full level net
premium reserve until the policy matures.
Example 1. For an ordinary life policy of $1,000 taken at age 30, find
the terminal reserve for the first three policy years under the preliminary
term system of valuation. Also find reserve for the twentieth year.
Solution. The insurance for the first year is term insurance and there
is no first year reserve. To get the terminal reserve for the second year
we make use of (3) Art. 91, letting x = 31. Then
= 1.043884 (0.01768) - 0.008583
= 0.00987,
and 1,000 i7 3 i = 1,000 (0.00987) = $9.87 (2nd year reserve).
AISO, 2^31 = ^32(l^3i + Psi) &32
= 1.043986 (0.00987 + 0.01768) - 0.008682
= 0.020179,
and 1,000 2^31 = 1,000 (0.020179) = $20.18 (3rd year reserve).
The reserve for the 20th year will be the 19th year reserve for age 31.
From (4), Art. 92, we get
= 0.50849 - 0.01768 (14.5346)
= 0.25151,
and 1,000 19^31 = 1,000 (0.25151)
= $251.51 (20th year reserve).
According to the full level premium method, the reserve for the third
year would have been $29.33 and that for the twentieth year would have
been $258.64. The student will observe that the difference between the
reserves, for any particular year, according to the two methods decreases
as the age of the policy increases. In fact, the reserves for the fortieth
year differ by only $3.64.
230 Financial Mathematics
Example 2. For a 20-payment life policy of $1,000 taken at age 30,
find the terminal reserve for the first three policy years under the prelimi-
nary term system. Also find the reserve for the twentieth year.
Solution. The insurance for the first year is term insurance and there
is no first year reserve. To get the terminal reserve for the second year
we make use of (3) Art. 91, letting x = 31. Then
= 1.043884 (0.02601) - 0.008583
= 0.018568,
and 1,000 iF 3 i = 1,000 (0.018568)
= $18.57 (2nd year reserve).
= 1.043986 (0.018568 + 0.02601) - 0.008682
= 0.037859,
and 1,000 2 F 3 i = 1,000 (0.037859)
= $37.86 (3rd year reserve).
The reserve for the 20th year will be the 19th year reserve on a
19-payment life taken at age 31. From (6), Art. 92, we get
19:19^31 = Aw = 0.50849,
and 1,000 19:19^31 = 1,000 (0.50849)
= $508.49 (20th year reserve).
According to the full level premium method, the reserve for the third
year would have been $53.94 and that for the twentieth year would have
been $508.49. We observe that the difference in reserve by the two
'methods is $16.08 at the end of the third year. However, at the end of
20 years there is no difference.
Gross Premiums, Other Methods of Valuation 231
101. Modified preliminary term valuation. In Art. 100 we mentioned
the fact that the preliminary term method of valuation is objectionable
when applied to limited payment life and endowment policies with short
premium paying periods. This can best be illustrated by an example.
Suppose we apply this method of valuation to a fifteen-payment endow-
ment policy for $1,000 taken at age 35. The office premium is $67.92
and since the natural premium for the first year is $8.65 there would be a
first year loading of $59.27. This is entirely too much for first year
expenses. It is evident then that the preliminary term system should be
modified when applied to short premium paying periods.
We found that in the case of the ordinary life policy taken at age 35
there was, according to the preliminary term system, a first year loading
of $17.90 and this was adequate for initial expenses. Hence, if this amount
is sufficient in the one case, it seems reasonable that the same amount, or
but little more, should be adequate for limited payment and endowment
policies of short premium paying periods. This then suggests a modifica-
tion. The ordinary life premium at any age forms the basis of the amount
which can be used for first year expenses for limited payment and endowment
policies taken at the same age.
Another method of modification is that provided by the laws of Illinois,
usually known as the " Illinois Standard. " Under the Illinois plan, twenty
payment life policies and all other policies having premiums smaller than that
of the twenty payment life policy for that age are valued on the preliminary
term plan without any modification. * Then the twenty payment life premium
forms the basis of the amount which can be used for first year expenses on all
policies whose premiums are greater than that of the twenty payment life.
The principles underlying the two methods of modification were recog-
nized by the " Committee of Fifteen," composed of Insurance Commis-
sioners and Governors, in 1906, and since that time the laws of many
states have been amended so as to adopt the recommendations of this
committee.
Some other states have other ways of modifying the preliminary term
system, but the two modifications that we have here described will be
sufficient for this discussion. We will now illustrate each of the above
methods with an example.
Example 1. Find the terminal reserves for the first three years on a
fifteen-year endowment policy of $1,000, issued at age 25, valued according
* This is spoken of as the full preliminary term plan to distinguish it from any one of
the modified plans.
232 Financial Mathematics
to the modified preliminary term system with the ordinary life as a basis
of modification.
Solution. We shall base all our computations on an insurance of $1
and then multiply by 1,000. The net premium for the first year is the
natural premium plus a certain excess, e. The subsequent net annual
premiums are the net ordinary life premiums for age 26, plus the same
excess, e, required to mature the policy.
Neglecting e each year the value of the policy at the end of 15 years
would be the full level net premium terminal reserve of the 14th policy
year on an ordinary life policy of $1 issued at age 26, or 14^25. However,
at the end of 15 years the policy must have a value of $1. Hence, the
excess payment of e each year must provide at maturity a pure endowment
of
(1 - 14^20).
This excess, e, is the annual payment on a forborne temporary annuity
due at age 25 (Art. 79), that will accumulate in 15 years to
(1 - 14^26).
Hence, e{ 25 ) = (1 14^20),
\ JL>40 /
and e = (1 14^20)
Y25
From (4), Art. 92, we get
14^26 = ^40 ~ ^26 840
= 0.41003 - 0.01548 (17.4461)
= 0.13997,
since, P 26 = 0.01548.
Then, e = (1 - 0.13997)
770,113 - 344,167
= 0.03983.
The terminal reserve for the first year is
iF 25 = 1/25(6 + Aim) - &25 1(2) Art. 91]
= u 2 !>-e = 1.043415(0.03983)
- 0.04156,
Gross Premiums, Other Methods of Valuation 233
since, Uzs'Alm = 25. [(9) Art. 84 and Exercise 1, Art. 91]
Then, 1,000 iF 2 5 = 1,000(0.04156) = $41.56 (1st year reserve).
2^25 = U26(lV 2 5 + P26 + e) - fe>6
= 1.043415(0.04156 + 0.01548 + 0.03983)
- 0.008197 = 0.09288.
Then, 1,000 2 F 25 = 1,000(0.09288) = $92.88 (2nd year reserve).
3^25 = ^27(2^25 + /V. + e) k<27
= 1.043554(0.09288 + 0.01548 + 0.03983)
- 0.008264 = 0.14638.
Then, 1,000 3 F 25 = 1,000(0.14638) = $146.38 (3rd year reserve).
According to the full level premium method, the reserve for the first
three years would be $48.87, $99.81, and $152.90, respectively. We
notice that the difference between the two methods for the first year is
$7.31 and for the third year the difference is $6.52. There would be no
difference for the fifteenth year.
Example 2. Find the terminal reserves for the first three years on a
ten-year endowment policy of $1,000, issued at age 25, valued according
to the Illinois standard.
Solution. The net premium for the first year is the natural premium,
^25 T\> plus an excess e. The subsequent net annual premiums are the net
premiums on a nineteen-payment life taken at age 26, plus the same excess e.
Neglecting e each year the value of the policy at the end of 10 years
would be the full level net premium terminal reserve of the 9th policy year
on a nineteen-payment life policy of $1, issued at age 26, or 9:19^26.
However, at the end of 10 years the policy must have a value of $1.
Hence, the excess payment of e each year must provide at maturity
a pure endowment of (1 9:19^26).
Therefore, e = (1 - 9:19^20)
and e = (1 9:19^20)
- Was
234
Financial Mathematics
From (6), Art. 92
since
and
Then,
Hence,
and
and
and
9:19^26 = Az5
= 0.17458,
A 35 = 0.37055,
19^26 = 0.02368, [(6) Art, 83]
a 35 To! = 8.27575. [(10) Art. 76]
e - (1 - 0.17458)
770,113 - 456,871
= 0.06468.
= U25 . e = 1.043415(0.06468)
= 0.06749,
1,000 iV 25 = 1,000(0.06749) = $67.49 (1st year reserve).
2^25 = ^2G(l^25 + 19^26 + e) fed
= 1.043484(0.06749 + 0.02368 + 0.06468)
- 0.008197 = 0.15443,
1,000 2 F 2 5 = 1,000(0.15443) = $154.43 (2nd year reserve).
3^25 = ^27(2^25 + 19^26 + e) k 2 7
= 1.043554(0.15443 + 0.02368 + 0.06468)
- 0.008264 = 0.24510,
1,000 3 F 25 = 1,000(0.24510) = $245.10 (3rd year reserve).
According to the full level premium method, the reserve for the first
year would be $82.08, for the second $167.66, and for the third $256.92.
The difference between the two methods for the first year is $14.59 and
the difference for the third year is $11.82.
Note. " It should be noted that a modification of premiums and reserves is employed
solely for the purpose of providing for large preliminary expenses in the first policy year,
and does not in any way affect the yearly amount of gross premium actually paid to the
Gross Premiums, Other Methods of Valuation 235
company by the policyholder. The modification is purely an internal transaction of
the life insurance company, which releases a larger part of the gross premium for expenses
in the first year and defers to a later date the setting up of a part of the reserve." *
102. Concluding remarks. Before completing this elementary treat-
ment of life insurance, we wish to emphasize the fact that we have
attempted to give a mere introduction into a broad field. There are many
topics that we have not touched. For the student who is interested in a
further study of this important field, we suggest the following books:
Moir, Henry, Life Assurance Primer, The Spectator Company, New York
City.
Menge, W. O., and Glover, J. W., An Introduction to the Mathematics of
Life Insurance, The Macmillan Company, New York City.
Knight, Charles K., Advanced Life Insurance, John Wiley and Sons, New
York City.
Spurgeon, E. F., Life Contingencies, The Macmillan Company, New York
City.
Exercises
1. For a twenty payment life policy of $1,000, taken at age 25, find the terminal
reserve for the 15th policy year both under the level net premium system and under the
preliminary term system of valuation.
2. Find the terminal reserve for the first three years on a 20-year endowment policy
of SI, 000, issued at age 40, valued according to the modified preliminary term system
with the ordinary life as a basis of modification.
3. Solve Exercise 2, using the Illinois Standard.
4. If the gross premium of a limited payment life policy of $1 on (x) is found by
increasing the net premium by a certain percentage r and adding to this a certain per-
centage s of the net ordinary life premium and further increasing this by a constant c,
per $1,000 insurance, show that the gross premium may be expressed by the formula
/>; =/Vs + ,A(l + r) + ~ (3)
6. Making use of formula (3) find the office premium on a fifteen payment life
policy of $1,000 for the ages 20, 25, 30 and 35, where r = 16%%, s = 16^%, and
c = 50 cents.
6. Making use of (1) find the office premiums on an ordinary life policy of $1,000
for the ages 20, 25, 30 and 35, where r = 33 Ji% and c = 50 cents.
* Menge, W. 0. and Glover, J. W., An Introduction to the Mathematics of Life Insur-
ance, 1935, p. 108,
236 Financial Mathematics
7. The formula
P'xnl - /V* + /Wl + + ^ (4)
gives the gross premium for an n-year endowment policy of $1 on (x). Interpret the
formula.
8. Making use of (4) find the office premium of a fifteen year endowment policy
of $1,000 for the ages 20, 25, 30 and 35, where r = s = 16^% and c - 50 cents.
Problems
1. By the terms of a will the income at 5% annually of a $20,000 estate goes to a
widow aged 50 during her lifetime. Find the value of her inheritance.
2. The will in Problem 1 requires that the residue of the estate shall go to a hospital
when the widow dies. Find the value of this residue at the time the inheritance comes
to the widow.
3. By the terms of a will the income at 5% annually of a $20,000 estate goes to a son
aged 25 for 10 years, or so long as he lives during the 10 years, after which the residue of
the estate goes to a university. Find the present value of each legacy.
4. A widow aged 55 is to receive a life income of $25,000 a year from her husband's
estate. The inheritance tax law requires that the bequest be valued on a 3K% basis.
The law grants the widow an exemption of $5,000, and on the remainder of the cash value
of her inheritance a tax of 3% must be paid of the first $50,000 over the exemption
value, and 5% on the next $50,000, then 10% on the cash value in excess of $100,000.
Find the inheritance tax on this bequest.
5. Under the Illinois Standard, the terminal reserve at the end of 25 years of a $1,000,
15-payment life policy issued at age 35 is $626.92. If the full amount of this reserve is
allowed as cash surrender value, how much paid-up insurance will it purchase?
6. Under the full preliminary term valuation, the terminal reserve at the end of
25 years on a $1,000 ordinary life policy issued at age 35 is $400.25. If the full amount
of this reserve is used to purchase extended insurance, how long is the extension?
7. Find the net first year and renewal premiums for an ordinary life policy of $1,OOC
issued at age 25 according to the full preliminary term method.
8. Same as Problem 7 but for a 20-payment life policy.
9. Same as Problem 7 but for a 20-payment 20 year endowment policy.
REVIEW PROBLEMS
Percentage
1. A building worth $15,000 is insured for $12,000. For what per cent of its value
is it insured?
2. A merchant fails, having liabilities of $30,000, and resources of $18,000. What
per cent of his debts can he pay? He owes Joe Brown $6,500. How much will Brown
receive?
3. A manufacturer sells to a wholesaler at a profit of 20%. The wholesaler sells to
the retailer at a 25% profit. The retailer sells to the consumer at a profit of 60%. If
the consumer pays $28.80, what is the cost to the manufacturer? To the wholesaler?
To the retailer?
4. Which is better for the purchaser, a series of discounts of 30%, 20%, and 10%,
or a single discount of 50%? What would be the difference on a bill of $1,000?
6. A coat listed at $100 is bought subject to discounts of 20%, 10%, and 8K%.
(a) Find the net cost rate factor. (6) Find the net cost, (c) What single discount
rate is equivalent to the given series of discounts? [Alg.: Com. Stat., p. 98.]
6. A coat cost a dealer $66. He marked the coat so that he could "drop" the marked
price 20% and still sell it so as to make a profit of 10% on the cost. What was the selling
price? The marked price?
7. I can buy a living room suite for $150, less 33K% and 20%. From another
dealer I can get the same suite for $125, less 25% and 12JHj%. The terms in each case
are "net 30 days or 2% off for cash." What is the least amount of cash for which I can
purchase the suite?
8. A bill of goods is purchased subject to discounts of r\ and r* Show that an equiv-
alent single discount is their sum less their product.
9. Goods are bought subject to discounts of 25% and 20%. Find the marked price
per dollar list if the goods are to be marked to realize a profit of 33J^%.
10. At what price should goods costing $432 be marked to make a profit of 25% of
the cost after allowing a discount of 20%?
Simple Interest and Discount
11. A note for $1,200 bearing interest at 5% and due in 8 months is sold to an inves-
tor to whom money is worth 6%. What does the investor pay for the note?
12. I purchased $400 worth of lumber from a dealer who will allow me credit for
60 days. If I desire to pay immediately, what should he be willing to accept if he esti-
mates that he earns 6% on his money?
13. A real estate dealer received two offers for a piece of property. Jones offered
$3,000 cash and $5,000 in 6 months; Smith offered $5,000 cash and $3,000 in 1 year.
Which was the better offer on a 6% basis?
237
238 Review Problems
14. The cash price of a washing machine is $75. It is bought for $10 down and
$10 a month for 7 months. What rate of interest is paid?
15. I borrow $500 for six months from a bank that charges 6% in advance. For
what amount do I make the note?
16. I owe $500 due in 3 months and $600 due in 12 months. I desire to pay these
debts by making equal payments at the ends of six and nine months. On a 6% basis,
find the equal payments. Choose 12 months as a focal date.
17. I owe William Brown $500 due in 3 months with interest at 8% and $800 due
in 12 months without interest. We agree that I may liquidate these debts with equal
payments at the ends of six and nine months on a 6% basis. Find the equal payments
by focalizing at 12 months.
18. When could I liquidate the debts in Problem 16 by a single payment of $1,100,
the equities remaining the same?
19. When could I liquidate the debts in Problem 17 by a single payment of $1,310,
the equities remaining the same? Solve by setting up an equation of value with focal
date at 12 months.
20. $1,000 LOUISVILLE, KENTUCKY
February 12, 1945
Nine months after date I promise to pay Robert Brown, or order, one
thousand dollars with interest at 7% from date.
Signed, GEORGE SANDERS.
(a) Five months after date, Brown sold the note to Bank B which operates on a
6% discount basis. What did Brown receive for the note?
(6) Bank B held the note for 1 month and then sold it to a Federal Reserve
Bank which operates on a 4% discount basis. What did Bank B gain on
the transaction?
Compound Interest and Discount
21. A man buys a house for $6,000, pays $2,000 cash, and gives a mortgage note at
6% for the balance. If he pays $1,000 at the end of two years and $1,000 at the end of
4 years, what will be the balance due at the end of 5 years?
22. I owe $1 ,500. I arrange to pay $R at the end of 1 year, $2/2 at the end of 2 years
and $3/2 at the end of 3 years. If money is worth 5% find R.
23. If (j = .08, m = 12), find i.
24. If a finance company charges 1% a month on loans, what is their effective
earning?
25. I owe two sums: $700 due in 6 months without interest and $1,500 due in
18 months with interest at (J = .06, m = 2). On a (j = .05, m = 2) basis what
amount will liquidate these debts at the end of 1 year?
26. A lot is priced at $2,000 cash. A buyer purchased it with equal payments now
and at the end of one year. On a 6% basis, what was the amount of the payments?
27. What sum payable in 2 years will discharge two debts, $1,500 due in 3 years
with interest at 5%, and $2,000 due in four years with interest at 6%, money being
worth 4%?
Review Problems 230
28. A merchant sells goods on the terms "net 90 days or 2% off for cash. 1 ' Find
the highest nominal rate of interest, j*4, at which a customer should borrow money in
order to pay cash. Find the effective rate.
29. If i = .06, find d, j*, and/ 4 .
30. If d = .06, find i, / 4 , and j 4 .
31. If / 4 = .06. find t, d, and j' 4 .
32. If j 4 = .06, find i, d, and/ 4 .
33. The Jones Lumber Co. estimates that money put into their business yields
/i 2
1/4% a month. Find the highest discount rate, , they can afford to offer to
1 &
encourage payment of a bill due in one month.
34. State a problem for which the answer would be the value of x determined by
the equation:
7,860 = z(1.03)~ 2 + z(1.03)- 4 + x.
36. State a problem for which the answer would be the value of x determined by
the equation:
a?(1.04) + 3(1.04) + 3 = 3,000(1.025) + 2,000(1.04) ~i.
36. I can buy a piece of property for $9,800 cash or for $6,000 cash and payments of
$2,000 at the ends of 1 year and 2 years. Should I pay cash if I can invest money at 6%?
Annuities
37. A purchaser of a farm agreed to pay $1,000 at the end of each year for 10 years,
(a) What is the equivalent cash price if money is worth 5%? (b) At the end of 5 years,
what must the purchaser pay if he desires to completely discharge his remaining liability
on that date?
38. I owe $6,000 due immediately. If money is worth (j .04, m = 4), what equal
quarterly payments will discharge the debt if the first payment occurs at the end of
3 years and the last at the end of 10 years?
39. A man buys a home for which the cash price is $10,000. He pays $1,200 down
and agrees to pay the balance with interest at (j = .05, m = 2) by payments of $1,200
at the end of each half-year as long as necessary with a final partial payment at the end
of the last payment period. How many full payments are necessary? What is the
final partial payment?
40. In Problem 39, find the principal outstanding just after the fifth payment of
$1,200.
41. Prove that (1 -f i)s^i -f 1 = s--Fi|
(a) by verbal interpretation;
(6) algebraically.
42. A man buys a house of cash value $25,000. He pays $5,000 down and agrees
to pay the balance with payments of $1,000 at the beginning of each half-year for
14 years. Find the nominal rate j> and the effective rate i that the purchaser pays.
240 Review Problems
43. An annuity of $100 a year amounts to $3,492.58 in 20 years. Find i.
44. A man purchased a property paying $3,000 down and $600 at the end of each
half-year for 10 years. If money was worth (j = .07, m *= 2), what was the equivalent
cash price?
45. A debt of $10,000 is being amortized, principal and interest, by payments of
$1,000 at the end of each half-year. If interest is at (j = .04, m 2), what is the final
payment?
46. The sum of $500 was paid annually into a fund for five years, and then $800
a year was paid. If the funds accumulated at 4%, when did the total amount to $12,000?
Obtain the final payment.
47. The sum of $100 was deposited at the end of each month for 8 years in a bank
that paid 4% effective. What was the value of the account two years after the last
deposit if no withdrawals were made?
48. A man deposited $200 at the end of every quarter in a savings bank that paid
3H% effective. When did the account total $10,000? What was the final partial
payment?
49. A machine costs $2,000 new and must be replaced at the end of 15 years at a cost
of $1,900. Find the capitalized cost if money can be invested at 4%.
50. Is it more profitable for a city to pay $2 per square yard for paving that lasts
five years than to pay $3 per square yard for paving that lasts 8 years, money being
worth 5%?
51. A lawn mower costs $10 and will last 3 years. How much can one afford to pay
for a better grade of mower that will last 5 years, money worth 4%?
Sinking Funds and Amortization
52. Find the annual payment necessary to amortize in 5 years a debt of $1,000
which bears interest at 7%. Construct a schedule.
63. A corporation issues $1,000,000, 6% bonds, dividends payable semi-annually.
The dividends are paid as they fall due and the corporation makes semi-annual deposits
into a sinking fund that will accumulate at j* = .04 to their face value in 15 years.
Find the sinking fund deposit. Find the total semi-annual expense to the corporation.
54. A debt of $100,000 bearing interest at 5% effective will be retired by a sinking
fund at the end of 10 years that earns 4% effective. Find the total annual expense.
At what rate of interest could the debtor just as well have agreed to amortize the debt?
55. Which will be better, to repay a debt of $25,000, principal and interest at 5%,
in 10 equal annual payments, or to pay 6% interest on the debt each year and accumu-
late a sinking fund of $25,000 in 10 years at 4%?
56. A man purchases a house for $12,000 paying one-half down. He arranges to
pay $1,500 per year principal and interest on the remaining amount until the debt is
paid. How many payments of $1,500 are made and what is the final payment at the
end of the year of settlement if the debt bears interest at 6%?
57. At the end of two years what was the purchaser's equity in the house in
Problem 56?
Review Problems 241
Depreciation
68. A dynamo costing $5,000 has an estimated life of 10 years and a scrap value of
$200. Find the constant rate of depreciation. What is the book value of the machine
at the end of 5 years?
69. What is the annual payment into the depreciation fund of the machine in
Problem 58 if the fund increases at 4%? What is the book value of the machine at the
end of 5 years?
60. A plant consists of three parts described by the table. Find the total annual
depreciation charge on a 3% basis:
Part Est. Life Cost Scrap Value
A 40 $20,000 $1,000
B 20 8,000 200
C 15 10,000 2,000
61. A Diesel engine costs $50,000, lasts 20 years and has a salvage value of $5,000.
(a) Find the amount that should be in the sinking fund at the end of 10 years at 4J^%.
(6) What is the amount of depreciation during the eleventh year?
62. An old machine turns out annually 1,200 units at a cost of $3,000 for operation
and maintenance. It is estimated that at the end of 12 years it will have a salvage value
of $500. To replace the old machine by a new one would cost $15,000, but 1,500 units
could be turned out annually at an average annual cost of $3,500 and this could be
maintained for 25 years with a salvage value of $1,000. On a 6% basis what is the
value of the old machine?
63. What number of units output annually of the new equipment in Problem 62
would reduce the value of the old machine to $4,000, all other data remaining the same?
64. What number of units output of the new machine in Problem 62 would render
the old machine worthless?
Valuation of Bonds
65. Find the cost of a $1,000, 5% J. and J. bond, redeemable at par in 10 years, if
bought to yield (j = .06, m = 2).
66. Find the cost of a $1,000 bond, redeemable in 8 years at 106, paying 6% con-
vertible quarterly if bought to yield 8% effective.
67. Find the cost of the bond described in Problem 66 if bought to yield (j - .08,
m = 4).
68. A $10,000, 4% J. and J. bond, redeemable at par January 1, 1940, was bought
July 1, 1936, to yield (j = .06, m = 2). Construct a schedule for the accumulation of
the discount.
69. What was a fair price for the bond described in Problem 68 if bought on August
13, 1936?
70. A $10,000, 7% J. and J. bond, was sold on June 1 at 102 H and accrued interest.
What was the selling price?
71. A $1,000, 5% J. and J. bond, redeemable at par in 10 years was purchased for
$970. Find the yield rate, j 2 .
242 Review Problems
Miscellaneous
72. If $100 invested at 5% simple interest accumulates to the same amount as $100
invested at 4% simple discount, find the time the investment runs.
73. Show that it takes three times as long for a principal P to quadruple itself at
i% as it does to double itself.
74. Jones considers two offers for a piece of property. A offers $3,000 cash and $5,000
in 6 months. B offers $5,000 cash and $3,000 in 1 year. On a 5% simple interest basis,
which is the better offer? Find the difference in the present values of the two offers.
76. If Do and D e denote ordinary and exact simple discounts on an amount S for
n years at d%, show that D e = D A>/73. [Compare (6), page 4.]
76. How long will it take a principal P to double itself at the compound discount
rate, d%?
77. Prove: = -- \- d.
78. Prove: a^ =
a
79. If R r denotes the amount in the depreciation fund at the end of r years under
the S.F. plan, prove that R r = Rs^\i.
80. If D r denotes the depreciation charge during the rth year under the S.F. plan,
prove that D r = R(l + if" 1 . [See Exercise 79 above.]
81. If ami = x and s^i; y, prove that i = (y x)/xy.
82. A debt D bearing interest at i% is being amortized by equal annual
payments 72. Show that the indebtedness remaining unpaid at the end of r years
is D - (R - Di)8w
83. Let C $, and show that (2), page 142, can be reduced to form (12'), page 135.
Explain how this can be true.
84. An alumnus, 50 years of age, proposes to give his college $50,000 provided the
college will pay him $2,500 a year as long as he lives. If the college can borrow money
at 4%, should it accept the proposition?
85. A note for $3,000 with interest (compound) at 5%, due in 5 years, is discounted
at the end of 2 years at discount rate of 4% compounded semi-annually. Find the pro-
ceeds and the discount.
86. A teacher provided for retirement by depositing $300 a year with a trust com-
pany that granted him (j = .04, m 2) interest rate. At the end of 25 years he retired
and withdrew $1,000 a year. For how many years could he enjoy this annuity?
87. It is estimated that a copper mine will produce $30,000 a year for 18 years. If
the investor desires to earn 12% on the investment and can earn 4% on the sinking
fund, what can he afford to pay for the mine?
88. A timber tract is priced at $1,000,000. It is estimated the tract will yield a net
annual income of $200,000 for 10 years and that the cleared land will be worth $20,000.
The lumber company wishes to earn 10% on the investment and can earn 4% on redemp-
tion funds. Is the tract a good buy?
89. Find the constant per cent by which the value of a machine is decreased if its
cost is $12,000, its scrap value $2,000, and its estimated life 15 years.
Review Problems 243
90. Expand (1 -j- j/m) m by the binomial theorem, let m become infinite, and show
that
(j\ m f f
l+) .i +J - + i + 2-+....
m) 21 31
The series on the right is the infinite series expansion of e j , where e * 2.71828 -f
and is called the base of the natural or Napierian logarithms. The series converges for
all values of j. Thus, as m becomes infinite, (1 -f i) approaches e j . (See page 139.)
When m becomes infinite, it is customary to replace j by d. Thus, for continuous
conversion we have
1 + i = e*
* i /i -\ logio (1 4- 1) logio (1 + i)
5 = log, (1 + i) - -- -
.43429
The quantity 6 is called the force of interest.
91. If 5 = .06, find i.
92. If i .06, find 5.
93. Show that if the interest is converted continuously for n years, the accumulated
value of S is
.S' = Pe nS .
94. The population of Jacksonville increased continuously from 130,000 in 1930 to
173,000 in 1940. Find the continuous rate of increase. (Use results of Exercise 93
above.)
96. Proceed as in Exercise 90 and show that
lirn
m
It is customary for continuous conversion of discount to replace/ by 5'. Then we have
1 - d = e -*' t
The quantity 5' is called force of discount.
96. Show that if the discount is converted continuously for n years, the discounted
value of S is
D ... Op nd
97. Find the amount of $1,000 for 10 years at 4% nominal, converted continuously.
98. A machine depreciated continuously from a value of $50,000 to a salvage value
of $10,000 in 20 years. Find the continuous rate of depreciation.
99. Jones bought a truck for $2,000. Its estimated life was 5 years and its salvage
value was $500. Jones estimated the truck earned $500 a year net. What did he earn
on his investment if deposits for replacement earned 3%? (See page 135.)
100. A college invests $400,000 in a dormitory. It is estimated that the college will
derive $25,000 net a year for 50 years at the end of which time the building will have a
salvage value of $100,000. What will the college earn on its investment if deposits for
replacement earn 3%? (See page 135.)
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
3
3
4
5
6
7
8
PP 1
100
00000
043
087
130
173
217
260
303
340
389
01
02
03
432
860
01284
475
903
326
518
945
368
561
988
410
604
*030
452
647
*072
494
689
*115
536
732
*157
678
775
*199
620
817
*242
662
44 43 42
04
05
06
07
08
09
703
02119
531
938
03342
743
745
160
572
979
383
782
787
202
612
*019
423
822
828
243
653
*060
463
862
870
284
694
*100
503
902
912
325
735
*141
543
941
953
3GS
776
*181
683
981
996
407
816
*222
623
*021
*036
449
857
*262
663
*060
*078
490
898
*302
703
*100
2
6
I
4.4 4.3 42
8.8 8.6 8.4
13.2 12.9 12.6
17.5 17.2 10.8
22.0 21.5 21.0
26.4 25.8 25.21
30.8 30.1 29.41
35.2 31.4 33.61
39.6 38.7 37.81
110
04139
179
218
258
297
336
376
415
454
493
1
11
12
13
632
922
05308
571
961
346
610
999
385
660
*038
423
689
*077
461
727
*115
600
766
*154
538
805
*192
576
844
*231
614
883
*269
652
41 40 99
14
15
16
17
18
19
690
06070
446
819
07188
555
729
108
483
856
225
591
767
145
521
893
2G2
628
805
183
558
930
298
684
843
221
695
967
335
700
881
258
633
*004
372
737
918
296
670
*041
408
773
956
333
707
*078
445
809
994
371
744
*115
482
846
*032
408
781
*151
518
882
2
3
|
1
4.1 4.0 3.9
8.2 8.0 7.8
12.3 12.0 11.7
16.4 16.0 15.6
20.5 20.0 19.5
24.0 24.0 23.4
28.7 28.0 27.3
32.8 32.0 31.2
36.9 36.0 35.1
120
918
954
990
*027
*063
*099
*135
*171
*207
*243
21
22
23
08279
636
991
314
672
*026
350
707
*061
386
743
*096
422
778
*132
458
814
*167
493
849
*202
529
884
*237
565
920
*272
600
955
*307
38 37 80
24
25
26
27
28
29
09342
691
10037
380
721
11059
377
726
072
415
755
093
412
760
106
449
789
126
447
795
140
483
823
100
482
830
175
517
857
193
517
864
209
551
890
227
552
899
243
685
924
261
587
934
278
619
958
294
621
968
312
653
992
327
656
*003
346
687
*025
361
1
5
6
8
9
3.8 3.7 3.8
7.0 7.4 7.2
11.4 11.1 10.8
15.2 14.8 14.4
19.0 18.5 18.0
22.8 22.2 21.6
26.6 25.9 25.2
30.4 29.6 28.8
34.2 33.3 32.4
130
394
428
461
494
528
561
594
628
661
694
3}
32
33
727
12057
385
760
090
418
793
123
450
826
156
483
860
189
516
893
222
548
926
254
681
959
287
613
992
320
646
*024
352
678
35 84 S3
34
35
36
37
38
39
710
13033
354
672
988
14301
743
066
386
704
*019
333
775
098
418
735
*051
364
808
130
450
767
*082
395
840
162
481
799
*114
426
872
194
613
830
*145
457
905
226
545
862
*176
489
937
258
577
893
*208
520
969
290
609
925
*239
551
*001
822
640
956
*270
582
!
1
I
10'.5 10!2 9.9
14.0 13.6 13.2
17.5 17.0 16.5
21.0 20.4 19.8
24.5 23.8 23.1
28.O 27.2 26.4
31.5 30.0 29.7
140
613
644
675
706
737
768
799
829
860
891
41
42
43
922
15229
534
953
259
564
983
290
694
*014
320
625
*045
351
655
*076
381
685
*106
412
715
*137
442
746
*168
473
776
*198
503
806
32 81 30
44
45
46
47
48
49
836
16137
435
732
17026
319
866
167
465
761
056
348
897
197
495
791
085
377
927
227
524
820
114
406
957
256
554
$50
143
435
987
286
584
879
173
464
*017
316
613
909
202
493
*047
346
643
938
231
522
*077
376
673
967
260
551
*107
406
702
997
289
580
i
3
S
6
3.2 3.1 3.0
6.4 6.2 6.0
9.6 9.3 0.0
12.8 12.4 12.0
16.0 15.6 15.0
19.2 18.6 18.0
22.4 21.7 21.0
25.6 24.8 24.0
28.8 27.9 27.0
150
609
638
667
696
725
754
782
811
840
869
N
1
8
4
5
6
7
8
9
PP
T I 1
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
!8
3
4
5
6
7
8
9
J
PP
150
17609
638
667
696
725
764
782
811
840
869
51
62
53
898
18184
469
926
213
498
955
241
526
984
270
564
*013
298
583
*041
327
611
*070
355
639
*099
384
667
*127
412
696
*156
441
724
29 28
54
55
56
57
58
59
752
19033
312
690
866
20 HO
780
061
340
618
893
167
808
089
368
645
921
194
837
117
396
673
948
222
865
145
424
700
976
249
893
173
451
728
*003
276
921
201
479
756
*030
303
949
229
507
783
*058
330
977
257
535
811
*085
358
*005
285
562
838
*112
385
2
I
5
6
8
9
:
1]
i<
17
2C
2i
2C
5.9 2.8
i.8 5.6
$.7 8.4
.6 11.2
1.5 14.0
.4 16.8
>.3 19.6
.2 22.4
1.1 ,25.2
160
412
439
466
493
520
648
575
602
629
656
61
62
63
683
952
21219
710
978
245
737
*005
272
763
*032
299
790
*059
325
817
*085
362
844
*112
378
871
*139
405
898
*165
431
925
*192
458
2
7 26
64
65
66
67
68
69
484
748
22011
272
531
789
511
775
037
298
557
814
637
801
063
324
583
840
564
827
089
350
608
866
690
854
115
376
634
891
617
880
141
401
660
917
643
906
167
427
686
943
669
932
194
453
712
968
696
958
220
479
737
994
722
985
246
505
763
*019
2
3
6
6
I
9
1
8
II
10
18
21
24
.7 2.6
.4 5.2
.1 7.8
.8 10.4
.5 13.0
.2 15.6
.9 18.2
.6 20.8
.3 23.4
170
23045
070
096
121
147
172
198
223
249
274
71
72
73
74
75
76
77
78
79
300
553
805
24055
304
551
797
25042
285
325
578
830
080
329
676
822
066
310
350
603
855
105
353
601
846
091
334
376
629
880
130
378
625
871
115
358
401
654
905
155
403
650
895
139
382
426
679
930
180
428
674
920
164
406
452
704
955
204
452
699
944
IBS
431
477
729
980
229
477
724
969
212
455
602
754
*005
254
602
748
993
237
479
528
779
*030
279
527
773
*018
261
503
2
3
5
6
1
25
2.5
5.0
7,5
10.0
12.5
15.O
17.5
20.O
22,5
180
527
551
675
600
624
648
672
696
720
744
81
82
83
768
26007
245
792
031
269
816
055
293
840
079
316
864
102
340
888
128
364
912
150
387
935
174
411
959
198
435
983
221
458
2
4 23
84
85
86
87
88
89
482
717
951
27184
416
646
505
741
975
207
439
669
529
764
998
231
462
692
653
788
*021
254
485
715
576
811
*045
277
508
738
600
834
*068
300
531
761
623
858
*091
323
554
784
647
881
*114
346
577
807
670
905
*138
370
600
830
694
928
*161
393
623
852
2
3
5
6
8
9
2
A
(
12
14
16
18
21
.4 2.3
.8 4.6
.2 6.9
.6 9.2
.0 11.5
.4 13.8
.8 16.1
.2 18.4
.6 20.7
190
875
808
921
944
967
989
*012
*035
*058
*081
91
92
93
28103
330
556
126
353
678
149
375
601
171
398
623
194
421
64G
217
443
663
240
466
691
262
488
713
285
511
735
307
533
758
2
V 21
94
95
96
97
98
99
780
29003
226
447
667
885
803
026
248
469
688
907
825
048
270
491
710
929
847
070
292
513
732
951
870
092
314
535
754
973
892
115
336
557
776
994
914
137
358
579
798
*016
937
159
3SO
601
820
*038
959
181
403
623
842
*060
981
203
425
645
863
*081
2
3
5
6
8
9
2
4
i!
1?
It
11
li
>.2 2.1
1.4 4.2
.6 6.3
5.8 8.4
.0 10.5
(.2 12.6
>.4 14.7
'.6 16.8
>.8 18.9
200
30 103
125
146
168
190
211
233
255
276
298
1
PP 1
T I 2
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
O
1
2
3
4
5
6
7
8
9
PP
200
30103
125
146
168
190
211
233
255
276
298
01
02
03
04
05
06
07
08
09
320
535
750
963
31 175
387
597
806
32015
341
557
771
984
197
408
618
827
035
363
578
792
*006
218
429
639
848
056
384
600
814
*027
239
450
660
869
077
406
621
835
*048
260
471
681
890
098
428
643
856
*069
281
492
702
911
118
449
664
878
*091
302
513
723
931
139
471
685
899
*112
323
534
744
952
160
492
707
920
*133
345
555
765
973
181
514
728
942
*154
366
576
785
994
201
<
i~i
3 6
4 8
5 11
6 12
7 15
8 17
9 19
J2 21
72 2.1"
.4 4.2
.6 6.3
.8 8.4
.0 10-5
.2 12.6
.4 14.7
.0 10.8
.8 18.9
210
222
243
263
284
305
325
346
366
387
408
11
12
13
428
634
838
449
654
858
469
675
879
490
095
899
510
715
919
531
736
940
552
756
960
572
777
980
593
797
*001
613
818
*021
20
14
15
16
17
18
19
33041
244
445
616
8i6
34 044
062
264
465
666
866
064
082
284
486
686
885
084
102
301
506
706
905
104
122
325
526
726
925
124
143
345
546
746
945
143
163
365
566
766
905
163
183
385
586
786
985
183
203
405
606
806
*005
203
224
425
626
826
*025
223
2
3
5
6
8
9
2.0
4.0
6.0
8.0
10.0
12.0
14.0
10.0
18.0
220
242
262
282
301
321
341
361
380
400
420
21
22
23
439
635
830
459
655
850
479^
674
869
498
694
889
518
713
908
537
733
928
557
753
947
577
772
967
596
792
986
616
811
*005
19
24
25
26
27
28
29
35025
218
411
603
793
984
044
238
430
622
813
*003
064
257
449
641
832
*021
083
276
468
660
851
*040
102
295
488
679
870
*059
122
315
507
698
889
*078
141
334
526
717
908
*097
160
353
545
736
927
*116
180
372
564
755
946
*135
199
392
583
774
965
*154
1
2
3
4
5
6
8
9
1.9
3.8
5.7
70
95
11.4
13.3
15.2
17.1
230
36 173
192
211
229
248
267
286
305
324
342
31
32
33
34
35
36
37
38
39
361
549
736
922
37 107
291
475
658
840
380
568
754
940
125
310
493
676
808
399
586
773
959
144
328
511
694
876
418
605
791
977
162
346
530
712
894
436
624
810
996
181
365
548
731
912
455
642
829
*014
11)9
383
500
749
931
474
661
847
*033
218
401
585
707
949
493
G80
866
*051
236
420
603
785
967
511
698
884
*070
254
438
621
803
985
530
717
903
*088
273
457
639
822
*003
2
3
4
5
6
8
9
18
1.8
3.0
5.4
7.2
9.0
10.8
12.0
14.4
16.2
240
38021
039
057
075
093
112
130
148
166
184
41
42
43
44
45
46
47
48
49
202
382
561
739
917
39094
270
445
620
220
399
578
757
934
111
287
463
637
238
417
596
775
952
129
305
480
655
256
435
614
702
970
146
322
498
672
274
4.53
632
810
987
164
340
515
690
202
471
OoO
828
*005
182
358
533
707
310
489
668
846
*023
199
375
550
724
328
507
686
863
*041
217
393
568
742
346
525
703
881
*058
235
410
585
7.09
364
543
721
809
*076
252
428
602
777
2
3
4
6
6
8
9
17
3.4
5 1
0.8
8.5
10.2
11 9
13.6
15.3
250
794
811
829
840
SG3
881
898
915
933
950
N
1
2
5
4
5
6
7
8
9
PP
T I 3
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
o
P I*
250
39 794
811
829
846
863
881
898
915
933
950
51
52
53
967
40 140
312
985
157
329
*002
175
3-16
*019
192
364
*037
209
381
*054
226
398
*071
243
415
*088
261
432
*106
278
449
*123
295
466
13
54
55
56
57
58
59
483
654
824
993
41 162
330
500
671
841
*010
179
347
518
688
853
*027
1'JG
303
535
705
875
*044
212
330
552
722
892
*OG1
229
397
. r >69
739
909
*078
246
414
f>86
756
926
*095
203
430
603
773
943
*111
280
447
620
790
960
*128
29G
464
637
807
976
*145
313
481
2
3
5
6
1
1.8
3.6
6.4
7.2
9.0
10.8
12.6
14.4
16.2 ,
260
497
514
531
547
5G4
5S1
597
614
631
647
61
62
63
004
830
9i)0
681
817
*012
697
803
*029
714
880
*013
731
896
*062
7-17
913
*078
761
.929
*0'J5
780
916
*iu
797
963
*127
814
979
*144
17
64
65
66
67
68
69
42 160
325
488
651
813
975
177
341
504
667
830
991
193
357
521-
684
846
*008
210
374
537
700
862
*024
226
390
553
716
878
*040
243
406
570
732
894
*05G
259
423
580
749
911
*072
275
439
602
705
927
*088
292
455
G19
781
943
*104
308
472
635
797
959
*120
1
2
3
6
6
7
8
9
1.7
3.4
5.1
6.8
8.5
10.2
11.9
13.6
15.3
270
43136
152
169
185
201
217
233
249
265
281
71
72
73
297
457
616
313
473
632
329
489
G48
345
505
GG4
361
521
680
377
537
696
393
553
712
409
569
727
425
584
743
441
600
759
1C
74
75
76
77
78
79
775
933
44091
248
404
560
791
949
107
264
420
576
807
905
122
279
436
592
823
981
138
205
451
607
838
996
154
311
467
G23
854
*012
170
326
483
638
870
*028
185
342
498
654
886
*044
201
358
514
6G9
902
*059
217
373
529
685
917
*075
232
389
545
700
2
I
5
6
8
9
1.6
3.2
4.8
6.4
8.0
9.6
11.2
12.a
1-1.4
280
716
731
747
762
778
793
809
824
840
855
81
82
83
84
85
86
87
88
89
871
45025
179
332
484
637
788
939
46090
886
040
194
347
500
652
803
954
105
902
056
209
362
515
667
818
969
120
917
071
225
378
530
682
834
9S4
135
932
086
240
393
545
697
8-10
-"000
150
948
102
255
408
561
712
864
*015
165
963
117
271
423
576
728
879
*030
ISO
979
133
286
439
591
743
894
*045
195
994
148
301
454
606
758
909
*060
210
*010
163
317
469
621
773
924
*075
225
2
3
5
?
8
9
15
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
290
240
255
270
285
300
315
330
345
359
374
91
92
93
94
95
96
97
93
99
389
538
687
835
982
47 129
276
422
567
404
553
702
850
997
144
290
436
582
419
568
716
864
*012
159
305
451
59G
434
583
731
879
*02G
173
319
465
611
449
598
746
894
*041
188
334
480
625
464
613
761
909
*056
202
349
494
C40
479
627
776
923
*070
217
363
509
654
494
642
790
938
*085
232
378
524
669
509
657
805
953
*100
246
392
538
683
523
672
820
967
*114
261
407
553
698
2
3
5
6
8
9
14
1.4
2.8
4.2
5.6
7.0
8 4
9.8
11 2
12.6
300
712
727
741
756
770
784
799
813
828
842
N
t
2
S
4
5
6
7
8
9
]
P
T i 4
TABLE I. COMMON LOGARITHMS OP NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
1
PF
300
47712
727
741
756
770
784
799
813
828
842
01
02
03
04
05
857
48001
144
287
430
871
015
159
302
444
885
029
173
316
458
900
044
187
330
473
914
058
202
344
487
929
073
216
359
601
943
087
230
373
515
958
101
244
387
530
972
116
259
401
544
986
130
273
416
558
15
06
07
08
09
572
714
855
996
586
728
869
*010
601
742
883
*024
615
756
897
*038
629
770
911
*052
643
785
926
"066
657
799
940
*080
671
813
954
*094
686
827
968
*108
700
841
982
*122
2
5
6
1.5
3.0
4.5
?:?
0.0
310
49136
150
164
178
192
206
220
234
248
262
g
10.5
12.0
11
12
13
14
15
16
276
415
554
693
831
969
290
429
56$
707
845
982
304
443
582
721
859
996
318
457
596
734
872
*010
332
471
610
748
886
*024
346
485
624
762
900
*037
300
499
638
776
914
*051
374
513
651
790
927
*065
388
527
665
803
941
*079
402
541
679
817
955
*092
9
13.5
14
17
18
19
50 106
243
379
120
256
393
133
270
406
147
284
420
161
297
433
174
311
447
188
325
461
202
338
474
215
352
488
229
365
501
2
I
1.4
2.8
5.6
330
515
529
542
558
569
583
596
610
623
637
8
7.0
8 4
21
22
23
24
25
26
27
28
OQ
651
786
920
51 055
188
322
455
587
7OA
664
799
934
068
202
335
468
601
1700
678
813
947
081
215
348
481
614
74 R
691
826
961
095
228
362
495
627
7CQ
705
840
974
108
242
375
508
640
779
718
853
987
121
255
383
521
654
786
732
866
*001
135
268
402
534
667
799
745
880
*014
148
282
415
548
680
812
759
893
*028
162
295
428
561
693
825
772
907
*041
175
308
441
574
706
838
8
9
0.8
11.2
12.6
13
1
1.3
330
851
865
878
891
904
917
930
943
957
970
I!
2.6
3.0
31
32
33
34
35
36
37
38
39
983
52 114
244
375
504
634
763
892
53020
996
127
257
388
517
647
776
905
033
*009
140
270
401
530
660
789
917
046
*022
153
284
414
543
673
802
930
058
*035
166
297
427
556
686
815
943
071
*048
179
310
440
569
699
827
956
084
*061
192
323
453
582
711
840
9G9
097
*075
205
336
466
595
724
853
982
110
*088
218
349
479
608
737
866
994
122
*101
231
362
492
621
750
879
*007
135
&
6
I
9
6.2
?i
0.1
10.4
11.7
340
148
161
173
186
199
212
224
237
250
263
13
41
42
43
44
45
46
47
48
49
275
403
529
656
782
908
54033
158
283
288
415
542
668
794
920
045
170
295
301
428
555
681
807
933
058
183
307
314
441
567
694
820
945
070
195
320
326
453
580
706
832
958
083
208
332
339
466
593
719
845
970
095
220
345
352
479
605
732
857
983
108
233
357
364
491
618
744
870
995
120
245
370
377
504
631
757
882
*008
133
258
382
390
17
643
769
895
*020
145
270
394
1
2
3
5
6
1.2
2.4
3.6
4.8
?i
8.4
9.6
10.8
350
407
419
432
444
456
469
481
494
606
518
g
]
P P
T 15
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
P P
350
54407
419
432
444
456
469
481
494
506
518
51
531
543
555
5GS
580
593
605
617
630
642
52
654
667
679
691
704
716
728
741
753
705
53
777
790
802
814
827
839
851
864
876
888
54
900
913
925
937
949
9G2
974
986
998
*011
55
55023
035
047
OGO
072
084
090
108
121
133
13
50
145
157
169
182
194
206
218
230
242
255
T~
i.s
2
2.6
57
267
279
291
303
315
328
340
352
364
376
3
3.9
58
388
400
413
425
437
449
461
473
485
497
4
52
59
509
522
534
546
558
570
582
594
606
618
5
6
G.f>
7.8
360
630~
642
654
6G6
678
691
703
715
727
739
8
9.1
10.4
61
751
7G3
775
787
799
811
823
835
847
859
9
11.7
62
871
883
895
907
919
931
943
955
967
979
63
991
*OQ3
*015
*027
*038
*050
*062
*074
*086
*098
64
56110
122
134
146
158
170
182
194
205
217
65
229
241
253
2G5
277
289
301
312
324
336
66
348
360
372
384
396
407
419
431
443
455
12
67
467
478
490
502
514
520
538
549
501
573
" j
1~2~
68
585
597
G08
020
G32
G44
656
6G7
679
691
2
\A
69
703
714
726
738
750
7G1
773
785
797
803
3
3.6
4 8
370
820
832
814
855
SG7
879
891
902
914
92G
5
6^0
71
937
019
901
972
084
99G
*C08
*019
*031
*043
6
7
7.2
8.4
72
57054
OGG
078
OS!)
101
113
124
130
148
159
8
9.6
73
171
183
194
20G
217
229
241
252
264
270
9
10.8
74
287
299
310
322
334
345
3-57
368
380
392
75
403
415
420
438
449
4(>1
473
484
490
507
76
519
530
542
553
565
576
588
600
611
623
77
631
646
657
6G9
680
692
703
715
726
738
78
749
7G1
772
784
795
807
818
830
841
852
H
79
864
875
8S7
898
910
921
933
944
955
967
~I~
~TT
380
97S~
990
*001
*013
*024
*035
*047
*058
*070
*081
2
3
2.2
3 3
81
580"92T
104
115
~~lW
138
149
161
172
184
195
4
4.4
82
206
218
229
240
252
2G3
274
286
297
309
5
5.5
A A
83
320
331
343
354
3G5
377
388
399
410
422
6
7
O.O
7.7
8
8.8
84
433
444
456
4G7
478
490
501
512
524
535
9
9.9
85
546
557
569
580
591
602
614
625
636
647
86
659
670
681
692
704
715
726
737
749
760
87
771
782
794
805
816
827
838
850
SGI
872
88
883
894
906
917
928
939
950
901
973
984
89
995
*OOG
*017
*028
*040
*0ol
*062
*073
*084
*095
390
59106
118
129
140
151
1G2
173
184
195
207
10
91
218
229
240
251
262
273
281
295
306
318
1
2
1.0
2
92
329
340
351
362
373
384
395
406
417
428
3
3.0
93
439
450
461
472
483
494
506
517
528
539
4
4.0
6.0
94
550
561
672
583
594
605
610
627
038
649
6.0
i n
95
660
671
682
693
704
715
720
737
748
759
7
g
/ .u
8
96
770
780
791
802
813
824
835
846
857
808
9
9.0
97
879
890
901
912
923
934
945
956
906
977
98
988
999
*010
*021
*032
*043
*054
*065
*07G
*080
99
60097
108
119
130
141
152
103
173
184
195
400
206
217
223
239
249
260
271
282
293
304
N
1
3
3
4
6
7
8
9
P P
T 16
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
F P
400
60206
217
228
239
249
260
271
282
293
304
Ot
314~
325
336
347
358
369
379
390
401
412
02
423
433
444
455
466
477
487
498
509
520
03
531
541
552
563
574
584
595
606
617
627
04
638
649
660
670
681
692
703
713
724
735
05
746
756
767
778
788
799
810
821
831
842
06
853
863
874
885
895
906
917
927
938
949
07
959
970
981
991
*002
*013
*023
*034
*045
*055
08
61066
077
087
098
109
119
130
140
151
1C2
11
09
172
183
194
204
215
225
236
247
257
268
T
1.1
410
278
289
300
310
321
331
342
352
363
374
2
3
2.2
3.3
11
384
395
405
416
426
437
448
458
469
479
4
5
4.4
5.5
12
490
500
511
521
532
542
553
563
574
584
6
6.6
13
595
606
616
627
637
648
658
669
679
690
8
7.7
8.8
14
700
711
721
731
742
752
763
773
784
794
9
0.9
15
805
815
826
836
847
857
808
878
888
899
16
909
920
930
941
951
962
972
982
993
*003
17
62014
024
034
045
055
066
076
086
097
107
18
118
128
138
149
159
170
180
190
201
211
19
221
232
242
252
263
273
284
294
304
315
430
325
335
346
356
366
377
387
397
408
418
21
428
439
449
459
469
480
490
500
511
521
22
531
542
552
562
572
583
593
603
613
624
4 A
23
634
644
655
665
675
685
696
706
716
726
10
1
10
2<
737
747
757
767
778
788
798
808
818
829
2
2.0
ft
25
839
849
859
870
880
890
900
910
921
931
4
o.U
4 o
26
941
951
961
972
982
992
*002
*012
*022
*033
5
5!o
6
6.0
27
63043
053
063
073
083
094
104
114
124
134
7
7.0
28
144
155
165
175
185
195
205
215
225
236
8
8.0
O ft
29
246
256
266
276
286
296
306
317
327
337
V .U
430
347
357
367
377
387
397
407
417
428
438
31
448
458
468
478
488
498
508
518
528
538
32
548
558
568
579
589
599
609
619
629
639
33
649
659
6G9
679
689
699
709
719
729
739
34
749
759
769
779
789
799
809
819
829
839
35
849
859
869
879
889
899
909
919
929
939
36
949
959
969
979
988
998
*008
*018
*028
*038
37
64048
058
068
078
088
098
108
118
128
137
9
38
147
157
167
177
187
197
207
217
227
237
T"
0.9
39
246
256
266
276
280
296
306
316
320
336
|
1.8
2.7
440
345*
355
365
375
385
395
404
414
424
434
5
3.6
4.5
41
444
454
464
473
483
493
503
513
523
532
6
5.4
42
542
552
662
572
582
591
601
611
621
631
7
6.3
f O
43
640
650
660
670
680
689
699
709
719
729
U
44
738
748
758
768
777
787
797
807
816
826
45
836
846
856
865
875
885
895
904
914
924
46
933
943
953
963
972
982
992
*002
*011
*021
47
65031
040
050
060
070
079
089
099
108
118
48
128
137
147
157
167
176
186
196
205
215
49
225
234
244
254
263
273
283
292
302
312
450
32T
331
341
360
360
369
379
389
398
408
N
1
3
4
5
6
7
8
9
P P
T 1-7
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
3
3
4
7
8
9
P
P
460
65321
331
341
350
360
369
379
3S9
398
408
61
52
53
54
55
56
57
58
59
418
514
610
706
801
896
992
66087
181
427
523
619
715
811
906
*001
096
191
437
533
629
725
820
916
*011
106
200
447
543
639
734
830
925
*020
115
210
456
552
648
744
839
935
*030
124
219
466
562
658
753
849
944
*039
134
229
475
571
667
703
858
954
*049
143
238
485
581
677
772
868
963
*058
153
247
495
591
686
782
877
973
*068
162
257
504
600
696
792
887
982
*077
172
266
T
10
1.0
460
276
285
295
304
314
323
332
342
351
361
2
3
2.0
3.0
61
62
63
64
65
66
67
68
69
370
464
558
652
745
839
932
67025
117
380
474
567
661
755
848
941
034
127
389
483
577
671
764
857
950
043
136
398
492
586
680
773
867
9GO
052
145
408
502
596
689
783
876
969
OG2
154
417
511
605
699
792
885
978
071
164
427
521
614
708
801
894
987
080
173
436
530
624
717
811
904
097
089
182
445
539
633
727
820
913
*006
099
191
455
549
642
736
829
922
*015
108
201
5
?
8
9
4.0
5.0
6.0
7.0
8.0
9.0
470
210
219
228
237
247
256
265
274
284
293
71
72
73
302
394
486
311
403
495
321
413
504
330
422
514
339
431
523
348
440
532
357
449
541
367
459
550
376
468
560
385
477
569
9
74
75
76
77
78
79
678
669
761
852
943
68034
587
679
770
861
952
043
596
688
779
870
961
052
605
697
788
879
970
061
614
706
797
888
979
070
624
715
800
897
988
079
633
724
815
906
997
088
642
733
825
916
*006
097
651
742
834
925
*015
106
660
752
843
934
*024
115
2
3
\
8
9
0.9
1.8
2.7
3.6
4.5
5.4
6.3
7.2
8.1
480
124
133
142
151
160
169
178
187
196
205
81
82
83
84
85
86
87
88
89
215
305
395
485
574
664
753
842
931
224
314
404
494
583
673
762
851
940
233
323
413
502
592
681
771
860
949
242
332
422
511
601
690
780
869
958
251
341
431
520
610
699
789
878
906
260
350
440
529
619
708
797
886
975
269
359
449
538
628
717
806
895
984
278
368
458
547
637
726
815
904
993
287
377
467
556
646
735
824
913
*002
296
386
476
565
, 655,
744
833
922
*011
"T~
3
8
0.8
1.6
2.4
490
69020
028
037
046
055
064
073
082
090
099
5
3.2
4.0
91
92
93
94
95
96
97
98
99
108
197
285
373
461
548
636
723
810
117
205
294
381
469
557
644
732
819
12G
214
302
390
478
566
653
740
827
135
223
311
399
487
574
662
749
836
144
232
320
408
496
583
671
758
845
152
241
329
417
504
592
679
767
854
161
249
338
425
513
601
688
775
862
170
258
346
434
522
609
697
784
871
179
267
355
443
531
618
705
793
880
188
276
364
452
539
627
714
801
888
6
8
9
4.8
5.6
C.4
7.2
500
897
906
914
923
932
940
049
958
966
975
N
1
2
3
4
5
6
7
8
I
P
' T I 8
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
PF
500
01
69897
906
914
923
932
940
949
958
966
975
984
992
*001
*010
018
027
036
044
*053
*062
02
70070
079
088
096
105
114
122
131
140
148
03
157
165
174
183
191
200
209
217
226
234
.04
243
252
260
269
278
286
295
303
312
321
05
329
338 |
346
355
364
372
381
389
398
406
06
415
424,
432
441
449
458
467
475
484
492
07
501
509
518
526
535
544
552
561
569
578
08
586
595
603
612
621
629
638
646
655
663
09
672
680
689
697
706
714
723
731
740
749
y
Og
510
757~
766
774
783
791
800
808
817
825
834
2
3
1.8
2.7
11
842
851
859
8G8
876
885
893
902
910
919
4
5
3.6
4 5
12
927
935
944
952
961
969
978
986
995
*003
6
5.4
13
71012
020
029
037
046
054
OG3
071
079
088
0.3
7.2
14
096
105
113
122
130
139
147
155
164
172
8.1
15
181
189
198
206
214
223
231
240
248
257
16
265
273
282
290
299
307
315
324
332
341
17
349
357
366
374
383
391
399
408
416
425
18
433
441
450
458
4GG
475
483
492
500
508
19
517
525
533
542
550
559
567
675
584
592
520
600~
609
617
625
634
642
650
659
667
675
21
684"
692
700
709
717
725
734
742
750
759
22
767
775
784
792
800
809
817
825
834
842
2?
850
858
8G7
875
883
892
900
908
917
925
8
1
24
933
941
950
958
966
975
983
991
999
*008
2
g
l'.6
2 4
25
72016
024
032
041
049
057
OG6
074
082
090
4
32
26
099
107
115
123
132
140
148
156
165
173
6
4.0
6
4.8
27
181
189
198
206
214
222
230
239
247
255
7
5.6
28
263
272
280
288
296
304
313
321
329
337
8
Q
6.4
7 2
29
346
354
362
370
378
387
395
403
411
419
530
428~
436
444
452
460
469
477
485
493
501
31
509"
518
526
534
542
550
558
5G7
575
583
32
591
599
607
616
624
632
640
648
656
665
33
673
681
689
697
705
713
722
730
738
746
34
754
762
770
779
787
795
803
811
819
827
35
835
843
852
800
868
876
884
892
900
908
30
916*
925
933
941
949
957
965
973
981
989
37
997
*006
*014
*022
*030
*038
*046
*054
*062
*070
7 n
.38
73078
086
094
102
111
119
127
135
143
151
1
O
0.7
1 4
39
159
167
175
183
191
199
207
215
223
231
3
21
540
239
247
255
263
272
280
288
296
304
312
5
2.8
3.5
41
320
328
336
344
352
360
368
376
384
392
6
11
42
400
408
416
424
432
440
448
456
464
472
g
5.Q
43
480
488
496
504
512
520
528
536
544
552
9
6.3
44
560
568
576
584
592
600
608
616
624
632
45
640
648
656
664
672
679
687
695
703
711
40
719
727
735
743
751
759
767
775
783
791
47
799
807
815-
823
830
838
846
854
862
870
48
878
886
894
O02
910
918
926
933
941
949
49
957
965
973
981
989
997
*005
*013
*020
*028
550
74036
044
052
060
068
076
084
092
099
10*
N
1
2
3
4
ft
6
7
S
9
PP
T 19
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
FF
65*
74036
044
052
060
068
076
084
092
099
107
51
115
123
131
139
147
155
162
170
178
186
62
194
202
210
218
225
233
241
249
257
265
63
273
28Q
288
296
304
312
320
327
335
343
54
351
359
367
374
382
390
398
406
414
421
55
429
437
445
453
461
468
476
484
492
500
56
607
515
523
531
539
547
554
562
670
578
57
586
593
601
609
617
624
632
640
648
656
58
663
671
679
687
695
702
710
718
720
733
59
741
749
757
7C4
772
780
788
796
803
811
660
819
827
834
842
850
858
865
873
881
889
61
896
904
912
920
927
935
943
950
958
9G6
62
974
981
989
997
*005
*012
*020
*028
*035
*043
63
75051
059
066
074
082
089
097
105
113
120
8
64
128
136
143
151
159
166
174
182
189
197
1 0.8
2 16
65
205
213
220
228
236
243
251
259
266
274
3 2A
66
282
289
297
305
312
320
328
335
343
351
4 3.2
67
358
366
374
381
389
397
404
412
420
427
4.0
4.8
68
435
442
450
458
405
473
481
488
496
504
7 5.6
8 64
69
511
519
520
534
542
549
557
505
572
580
9 1 7.2
570
587
595
603
610
618
626
633
641
648
656
71
664~
671
679
CS6
694
702
709
717
724
732
72
740
747
755
762
770
778
785
793
800
808
73
815
823
831
838
846
853
801
808
876
884
74
891
899
906
914
921
929
937
944
952
959
75
967
974
982
989
997
*005
*012
*020
*027
*035
76
76042
050
057
OC5
072
080
087
095
103
110
77
118
125
133
140
148
155
163
170/
178
185
78
193
200
208
215
223
230
238
245
253
200
79
268
275
283
290
298
305
313
320
328
335
580
343
350
358
305
373
380
388
395
403
410
81
418"
425
433
440
448
455
402
470
477
485
82
492
500
507
515
522
530
537
545
552
559
7
83
567
574
582
539
597
604
612
619
626
634
1 0.7
2 1.4
84
641
649
656
664
671
678
686
693
701
708
3 2.1
85
716
723
730
738
745
753
7CO
708
775
782
4 2.8
86
790
797
805
812
819
827
834
842
849
856
5 3.5
4.2
87
864
871
879
886
893
901
908
916
923
930
7 4.9
8 5.6
88
938
945
953
960
967
975
982
989
997
*004
9 6.3
89
77012
019
026
034
041
048
056
063
070
078
590
085~
093
100
107
115
122
129
137
144
151
91
159"
166
173
181
188
195
203
210
217
225
92
232
240
247
254
262
2(59
276
283
291
298
93
305
313
320
327
335
342
349
357
364
371
94
379
386
393
401
408
415
422
430
437
444
95
452
459
466
474
481
488
495
503
510
517
96
525
532
539
546
554
561
568
576
583
590
97
597
605
612
619
627
634
641
648
656
663
98
670
677
685
692
699
706
714
721
728
735
99
743
750
757
764
772
779
786
793
801
808
600
815~
822
830
837
844
851
859
866
873
880
N
1
%
3
4
5
6
7
8
9
FF
T I 10
TABLE I. COMMON LOGARITHMS OP NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
f
>P
600
77815
S22
830
837
844
851
859
866
873
880
01
02
03
04
05
06
07
08
09
887
960
78032
104
176
247
319
390
462
895
967
039
111
183
254
326
398
469
902
974
046
118
190
2G2
333
405
476
909
981
053
125
197
269
340
412
483
916
9S8
061
132
204
276
347
419
490
924
996
OGS
140
211
283
355
426
497
931
*003
075
147
219
290
362
433
504
938
*010
082
154
22G
297
3G9
440
512
945
*017
089
1C1
233
305
376
447
519
952
*025
097
168
240
312
3S3
455
52G
T~
3
08
610
533
540
547
554
561
5G9
57G
583
590
507
>
3
1 6
24
11
12
13
14
15
16
17
18
19
604
675
746
817
888
958
79029
099
169
611
682
753
824
895
965
036
106
176
61S
689
760
831
902
972
043
113
183
625
69b
767
838
909
979
050
120
190
633
704
774
845
916
986
057
127
197
640
711
781
852
923
993
064
134
204
647
718
789
859
930
*000
071
141
211
G54
725
796
8G6
937
*007
078
148
218
GG1
732
803
873
944
*014
055
155
225
GGS
739
810
880
951
*021
092
162
232
4
5
6
1
32
4 O
48
56
64
72
620
239
246
253
2GO
267
274
281
2S8
295
302
21
22
23
24
25
26
27
28
29
309
379
449
51S
588
657
727
796
865
316
386
456
525
595
664
734
803
872
323
393
4(53
532
602
671
741
S10
879
330
400
470
539
609
678
74S
817
8S6
337
4<V7
477
546
616
685
754
824
S93
344
414
484
553
623
692
761
831
900
351
421
491
560
630
699
768
837
90G
3.08
428
498
567
G37
70G
775
844
913
365
435
505
574
644
713
782
851
920
372
442
511
5S1
050
720
789
858
927
T"
2
3
4
5
8
9
7
07
2 1
2 8
3 5
4 2
4 y
56
63
630
934
941
94S
955
962
969
975
982
989
99G
31
32
33
34
35
36
37
38
39
80003
072
140
209
277
346
414
482
550
010
079
147
216
284
353
421
489
557
017
055
154
223
291
359
428
496
564
024
092
161
229
298
366
434
502
570
030
099
168
236
305
373
441
509
577
037
106
175
243
312
380
448
516
584
044
113
1S2
2r>o
318
3S7
455
523
591
051
120
188
257
325
393
462
530
598
058
127
195
2G4
332
400
4GS
536
G04
OG5
134
202
271
339
407
475
543
611
T~
3
6
06
1.2
1 8
640
618
625
632
G3S
645
652
659
GG5
672
679
4
5
2 4
30
41
42
43
44
45
46
47
48
49
686
754
821
889
956
81023
090
158
224
693
760
828
895
963
030
097
164
231
699
767
835
902
969
037
104
171
238
706
774
841
909
976
043
111
178
245
713
781
S48
916
983
050
117
184
251
720
787
855
922
990
057
124
191
258
726
794
862
929
99G
064
131
198
265
733
801
868
936
*003
070
137
204
271
740
808
875
943
*010
077
144
211
278
747
814
S82
949
*017
084
151
218
285
8
9
4 2
4.8
54
650
291
298
305
311
318
325
331
338
345
351
N
1
2
3
4
5
6
7
8
9
1
PP 1
T i 11
TABLE I. COMMON LOGARITHMS OP NUMBERS
To Five Decimal Places
N
O
1
9
3
4
5
6
7
8
9
PF
660
51
62
53
54
55
56
57
58
59
660
61
62
63
64
65
66
67
68
69
670
71
72
73
74
75
76
77
78
79
680
81
82
83
84
85
86
87
88
89
690
91
92
93
94
95
96
97
98
99
700
81291
358
425
491
558
624
690
757
823
889
298
305
311
318
325
331
338
345
351
7
1 0.7
2 1.4
t2.1
2.8
5 3.5
6 4.2
7 4,9
8 5.6
9 6.3
6
1 0.0
1.2
1.8
4 2.4
* S:8
I li
9 5.4
365
431
'498
5C4
631
697
763
829
895
371
438
505
571
637
704
770
836
902
378
445
511
578
644
710
776
842
908
385
451
518
584
651
717
783
849
915
391
458
525
591
657
723
790
856
921
398
465
531
598
604
730
796
862
928
405
471
538
604
671
737
803
809
935
411
478
544
611
677
743
809
875
941
418
485
551
617
684
750
816
882
948
954
82020
086
151
217
282
347
413
478
543
607
672
737
S02
866
930
995
83 059
* 123
187
251
315
378
442
506
569
632
696
759
822
885
961
908
974
981
987
994
*000
*007
*014
027
092
158
223
289
354
419
484
549
033
090
164
230
295
360
426
491
556
040
105
171
236
302
367
432
407
562
046
112
173
243
308
373
439
504
569
053
119
184
249
315
3SO
445
510
575
000
125
191.
256
321
387
452
517
682
066
132
197
263
328
303
458
523
583
073
138
204
269
334
400
465
530
595
079
145
210
276
341
406
471
536
601
614
620
627
633
640
G4G
653
659
666
679
743
808
872
937
*001
065
129
193
685
750
814
879
943
*008
072
136
200
692
756
821
885
950
*014
078
142
206
698
703
827
892
956
*020
085
149
213
705
769
834
898
963
*027
091
155
219
7)1
77G
840
905
969
*033
097
161
225
718
782
847
911
975
*Q40
104
168
232
724
789
853
918
982
*046
110
174
233
730
795
860
924
988
*052
117
181
245
257
264
270
276
283
289
296
302
308
321
385
448
512
575
639
702
765
828
327
301
455
518
582
645
708
771
835
334
398
461
525
588
651
715
778
841
340
404
467
531
594
658
721
784
847
347
410
474
537
601
664
727
790
853
353
417
480
544
607
670
734
797
860
359'
423
487
550
613
677
740
803
866
366
429
493
556
620
683
746
809
872
372
436
499
563
626
689
753
816
879
891
897
904
310
916
923
929
935
942
948
84011
073
136
198
261
323
386
448
954
017
080
142
205
267
330
392
454
960
023
086
148
211
273
336
398
460
967
029
092
155
217
280
342
404
466
973
036
098
161
223
286
348
410
473
979
042
105
167
230
292
354
417
479
985
048
111
173
236
298
361
423
485
902
055
117
180
242
305
367
429
491
998
061
123
186
248
311
373
435
497
*004
067
130
192
255
317
379
442
504
510
516
522
528
535
541
547
553
559
566
N
1
%
3
4
6
6
7
8
PP
T 112
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
3
3
4
5
6
7
8
9
F
p
700
S4510
516
522
528
535
541
547
553
559
566
01
02
03
04
05
06
07
08
09
672
634
696
757
819
880
942
85 003
005
578
640
702
763
825
887
948
009
071
584
646
708
770
831
893
954
016
077
590
652
714
776
837
899
960
022
083
597
658
720
782
844
905
967
028
089
603
665
726
788
850
911
973
034
095
609
671
733
794
856
917
979
040
101
615
677
739
800
862
924
985
046
107
621
683
745
807
868
930
901
052
114
628
689
751
813
874
936
997
058
120
T
7
(V7
710
126
132
138
144
150
156
163
169
175
181
2
3
1,4
2.1
11
12
13
14
15
16
17
18
19
187
248
309
370
431
491
552
612
673
193
254
315
376
437
497
558
618
679
199
260
21
382
443
503
564
625
685
205
266
327
388
449
509
570
631
691
211
272
333
394
455
516
576
637
697
217
278
339
400
461
622
582
643
703
224
285
345
406.
467
528
688
649
709
230
291
352
412
473
534
594
655
715
236
297
358
418
479
540
600
661
721
242
303
364
425
485
546
606
667
727
i
8
9
2.8
3.5
4.2
4.9
5.0
6.3
720
733
739
745
751
757
763
769
775
781
788
21
22
23
24
25
26
27
28
29
794
854
914
974
86034
094
153
213
273
800
860
920
980
040
100
159
219
279
806
866
926
986
046
106
165
225
285
812
872
932
992
052
112
171
231
291
818
878
938
998
058
118
177
237
297
824
884
944
*004
064
124
183
243
303
830
890
950
*010
070
130
189
249
308
836
896
956
*016
076
13G
195
255
314
842
902
962
*022
082
141
201
261
320
848
908
968
*028
088
147
207
267
326
2
3
5
6
6
o.<T
1.2
1.8
2.4
3.0
36
4.2
4.8
5.4
730
332
338
344
350
356
362
368
374
380
386
31
32
33
34
35
36
37
38
39
392
451
510
570
629
688
747
806
864
398
457
516
576
635
694
753
812
870
404
463
522
581
641
700
759
817
876
410
469
528
537
646
705
764
823
882
415
475
534
593
652
711
770
829
888
421
4S1
540
599
058
717
776
835
894
427
487
546
605
604
723
782
841
900
433
493
552
611
G70
729
788
847
906
439
499
558
G17
676
735
794
853
911
445
504
564
623
682
741
800
859
917
T
2
3
5
~oT
1.0
1.5
740
923
929
935
941
947
953
958
964
970
976
5
2.5
f\
41
42
43
44
45
46
47
48
49
982
87040
099
157
216
274
332
390
448
988
046
105
163
221
280
338
396
454
994
052
111
169
227
286
344
402
460
999
058
116
175
233
291
349
408
466
*005
064
122
181
239
297
355
413
471
*011
070
128
186
245
303
36U
419
477
*017
075
134
192
251
309
367
425
483
*023
081
140
198
256
315
373
431
489
*029
087
146
204
262
320
379
437
495
*035
093
151
210
268
326
384
442
500
i
3.5
U
750
506
512
518
623
529
535
541
547
552
558
N
1
3
3
4
5
6
7
8
9
]
fV
T 113
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
38
3
4
5
6
7
8
9
PP
750
87506
512
518
523
529
535
541
547
552
558
51
564
570
576
581
587
593
599
604
610
616
52
622
628
633
639
645
651
G56
602
668
574
53
679
685
691
697
703
70S
714
720
726
731
54
737
743
749
754
7GO
7GG
772
777
783
789
55
795
800
SOG
812
818
823
829
835
841
846
56
852
858
864
SG9
875
881
887
892
898
904
57
910
915
921
927
933
938
944
950
955
961
58
967
973
978
984
990
996
*001
*007
*013
*018
59
88024
030
036
041
047
053
058
004
070
076
760
6sT
087
093
098
104
110
116
121
127
133
61
138*
144
150
15G
1G1
167
173
178
184
190
62
195
201
207
213
218
224
230
235
241
247
63
252
258
264
270
275
281
287
292
298
304
6
1 06
64
309
315
321
326
332
338
343
349
355
3GO
2 1.2
65
366
372
377
383
389
395
400
400
412
417
3 1.8
66
423
429
434
440
446
451
457
463
468
474
4 2.4
5 30
67
480
485
491
497
502
508
513
519
525
530
6 3.6
7 4.2
68
536
542
547
553
559
504
570
570
581
587
8 4.8
69
593
598
604
610
615
G21
627
632
638
043
9 5.4
770
649
655
600
G66
672
G77
083
689
094
700
71
705
711
717
722
72S
734
739
745
750
756
72
762
707
773
779
784
790
795
801
807
812
73
818
824
829
835
840
84 G
852
857
803
80S
74
874
880
885
891
897
902
903
913
919
925
75
930
936
941
947
953
958
904
909
975
981
76
986
992
997
*003
*009
*014
*020
"=025
*031
*037
77
89042
048
053
059
004
070
070
OS1
087
092
78
098
104
109
115
120
120
131
137
143
148
79
154
159
165
170
170
182
187
193
198
204
780
209
215
221
22G
232
237
243
24S
254
260
81
265
271
276
282
287
293
298
304
310
315
82
321
326
332
337
343
348
354
3GO
3G5
371
5
83
376
382
387
393
398
404
409
415
421
426
1 O.5
2 1.0
84
432
437
443
448
454
459
4G5
470
476
481
3 1.5
85
487
492
498
504
509
515
520
520
531
537
4 2.0
86
542
548
553
559
5G4
570
575
581
580
592
5 2.5
6 3.0
87
597
603
609
614
C20
625
631
G30
642
647
7 3.5
8 4.0
88
653
658
G64
669
675
680
G86
691
697
702
9 4.5
89
708
713
719
724
730
735
741
740
752
757
790
763~
768
774
779
785
790
790
801
807
812
91
818
823
829
834
840
845
851
856
8G2
867
92
873
878
883
889
894
900
905
911
910
'922
93
927
933
938
944
949
955
9GO
9GG
971
977
94
982
988
993
998
*004
*009
*015
*020
*02G
*031
95
90037
042
048
053
059
064
069
075
080
086
96
091
097
102
10S
113
119
124
129
135
140
97
146
151
157
162
168
173
179
184
189
195
98
200
206
211
217
222
227
233
238
244
249
99
255
260
266
271
270
282
287
293
298
304
800
309
314
320
325
331
336
342
347
352
358
N
1
2
3
4
5
6
7
8
9
P P
T I 14
TABLE I. COMMON LOGARITHMS OP NUMBERS
To Five Decimal Places
N
1
*
3
4
&
6
7
8
9
PP
800
90309
314
320
325
331
336
342
347
352
358
01
363
369
374
380
385
390
396
401
407
412
02
417
423
428
434
439
445
450
455
461
466
03
472
477
482
488
493
499
504
509
515
520
04
26
531
36
542
547
553
558
563
569
574
05
80
585
590
596
601
607
612
617
623
628
06
634
639
644
650
655
660
666
671
677
682
07
687
693
698
703
709
714
720
725
730
736
08
741
747
752
757
763
7C8
773
779
784
789
09
795
800
806
811
816
822
827
832
838
843
810
849
854
859
865
870
875
881
886
891
897
11
902
907
913
918
924
929
934
940
945
950
12
956
901
966
972
977
982
988
993
998
*G04
13
91009
014
020
025
030
036
041
046
052
057
6
14
062
068
073
078
084
089
094
100
105
110
1 0.6
2 1.2
15
116
121
126
132
137
142
148
153
158
104
3 1.8
16
169
174
180
185
190
196
201
206
212
217
4 2.4
17
222
228
233
238
243
249
254
259
2f>5
270
5 3.0
6 3.6
7 4.2
18
275
281
286
291
297
302
307
312
318
323
8 4.8
19
328
334
339
344
350
355
300
305
371
376
9 5.4
820
381
387
392
307
403
403
413
418
124
429
21
434
440
445
450
455
461
466
471
477
482
22
487
492
498
503
508
514
519
524
529
535
23
540
545
551
556
561
566
572
577
582
587
24
593
598
603
609
614
619
624
630
635
640
25
645
651
656
661
666
672
677
682
687
693
26
698
703
709
714
719
724
730
735
740
745
27
751
756
761
766
772
777
782
787
793
798
28
803
808
814
819
824
829
834
840
845
850
29
855
881
8G6
871
876
882
887
892
897
903
830
908
913
918
924
929
934
939
944
950
955
31
960
965
971
976
981
98ft
991
907
*002
*007
32
92012
018
023
028
033
038
044
049
054
059
5
33
065
070
075
OSO
085
091
096
101
100
111
1 0.5
2 1.0
34
117
122
127
132
137
143
148
153
158
163
3 1.5
35
169
174
179
184
189
195
200
205
210
215
4 2.0
36
221
226
231
236
241
247
252
257
262
267
5 2.5
6 3.0
37
273
278
283
288
293
298
304
309
314
319
7 3.5
8 4.0
38
324
330
335
340
345
350
355
381
366
371
9 4.5
39
370
381
387
392
397
402
407
412
418
423
840
428~
433
438
443
449
454
459
464
469
474
41
480
485
490
495
500
505
511
516
521
526
42
531
536
542
547
552
557
562
507
572
578
43
583
588
593
598
603
609
614
619
624
629
44
634
639
645
650
655
660
665
670
675
6S1
45
686
691
696
701
700
711
716
722
727
732
46
737
742
747
752
758
763
70S
773
773
783
47
788
793
799
804
809
814
819
S?A
829
834
48
840
845
850
855
860
805
870
875
881
8SG
49
891
896
901
906
911
916
921
927
932
937
850
942
947
952
957
962
967
973
978
983
688
N
1
2
3
4
5
G
7
8
9
P*
T I 15
TABLE I. COMMON LOGABITHMS OP NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
P P
850
51
92942
993
947
952
957
962
967
973
978
983
988
998
*003
*008
*013
*018
*024
*029
*034
*039
52
93044
049
054
059
064
069
075
080
085
090
53
095
100
105
110
115
120
125
131
136
141
54
146
151
156
161
166
171
176
181
186
192
55
197
202
207
212
217
222
227
232
237
242
56
247
252
258
263
268
273
278
283
288
293
57
298
303
308
313
318
323
328
334
339
344
58
349
354
359
364
369
374
379
384
389
394
6
59
399
404
409
414
420
425
430
435
440
445
T~
0.6
860
450
455
4GO
4G5
470
475
480
485
490
495
\
1 .2
1.8
61
500
505
510
515
520
526
531
536
541
546
4
5
2.4
3.0
62
551
556
561
566
571
576
581
586
591
596
6
3.6
63
601
606
611
616
621
626
631
636
641
646
7
4.2
8
4.8
64
651
656
661
666
671
676
682
687
692
697
9
5.4
65
702
707
712
717
722
727
732
737
742
747
66
752
757
762
767
772
777
782
787
792
797
67
802
807
812
817
822
827
832
837
842
847
68
852
857
862
867
872
877
882
887
892
897
69
902
907
912
917
922
927
932
937
942
947
870
952"
957
962
9G7
972
977
982
987
992
997
71
94002
007
012
017
022
027
032
037
042
047
72
052
057
062
067
072
077
082
086
091
096
73
101
106
111
116
121
126
131
136
141
146
1
0.5
74
151
156
161
166
171
178
181
186
191
196
2
1.0
75
201
206
211
216
221
226
231
236
240
245
3
4
1.5
2 O
76
250
255
260
265
270
275
280
285
290
295
5
2.5
6
3-0
77
300
305
310
315
320
325
330
335
340
345
7
3.5
78
349
354
359
304
369
374
379
384
389
394
8
4.0
79
399
404
409
414
419
424
429
433
438
443
9
4-5
880
4"48~
453
458
463
4G8
473
478
483
488
498
81
498
503
507
512
517
522
527
532
537
542
82
547
552
557
562
567
571
576
581
586
591
83
596
601
606
611
616
621
626
630
635
640
84
645
650
055
660
665
670
675
680
685
689
85
694
699
704
709
714
719
724
729
734
738
86
743
748
753
758
763
768
773
778
783
787
87
792
797
802
807
812
817
822
827
832
836
4
88
841
846
851
856
861
866
871
876
880
885
T"
0.4
89
890
895
900
905
910
915
919
924
929
934
2
3
0.8
1.2
890
939
944
949
954
959
963
968
973
978
983
5
1.0
2.0
91
988""
993
998
*002
*007
*012~
*017
*022
*027
*032
6
2.4
2 8
92
95036
041
046
051
056
061
066
071
075
080
L
32
93
085
090
095
100
105
109
114
119
124
129
9
3.6
94
134
139
143
148
153
158
163
168
173
177
95
182
187
192
197
202
207
211
216
221
226
96
231
236
240
245
250
255
260
265
270
274
97
279
284
289
294
299
303
308
313
318
323
98
328
332
337
342
347
352
357
361
366
371
99
376
381
386
390
395
400
405
410
415
419
000
424
429
434
439
444
448
453
458
463
468
N
1
2
3
*
5
7
8
PF
T I 16
TABLE I. COMMON LOGARITHMS OP NUMBERS
To Five Decimal Places
N
1
9
3
4
&
6
7
8
9
PP
900
01
95424
472
429
434
439
487
444
448
453
458
463
468
477
482
492
497
501
506
511
516
02
521
525
530
535
540
545
550
554
559
564
03
509
574
578
583
588
593
598
602
607
612
04
617
622
626
631
636
641
646
650
656
660
05
665
670
674
679
684
689
694
698
703
708
06
713
718
722
727
732
737
742
746
751
756
07
761
766
770
775
780
785
789
794
799
804
08
809
813
818
823
828
832
837
842
847
852
09
856
861
866
871
875
880
885
890
895
899
910
904
909
914
918
923
928
933
938
942
947
11
952
957
961
966
971
976
980
985
990
995
12
999
*004
009
014
019
023
028
033
038
042
13
96047
052
057
061
006
071
076
080
085
090
5
1 O.5
14
095
099
104
109
114
118
123
123
133
137
2 1.0
15
142
147
152
150
161
166
171
175
180
185
3 1.5
16
190
194
199
204
209
213
218
223
227
232
4 2.0
5 2.5
17
237
242
246
251
256
201
265
270
275
280
6 3.0
7 3.5
18
284
289
204
208
303
308
313
317
322
327
8 4.0
19
332
336
341
346
350
355
300
305
3G9
374
9 4.5
920
379~
384
388
393
398
402
407
412
417
421
21
426
431
435
440
445
450
454
459
404
468
22
473
478
483
487
492
497
501
506
511
515
23
520
525
530
534
539
544
548
553
558
562
24
567
572
577
581
586
591
595
600
605
609
25
614
619
624
628
633
638
642
647
652
656
26
661
666
670
675
680
685
689
694
699
703
27
708
713
717
722
727
731
736
741
745
750
28
755
759
7G4
769
774
778
783
788
792
797
29
802
806
811
816
820
825
830
834
839
844
930
848~
453
858
802
807
872
876
881
886
890
31
895~
900
904
909
914
918
923
928
932
937
32
942
946
951
956
960
965
970
974
979
984
4
33
988
993
997
*002
*007
*011
*016
*021
*025
*030
T~ 04"
2 0.8
34
97035
039
044
049
053
058
003
007
072
077
3 1 2
35
081
086
090
095
100
104
109
114
118
123
4 1.6
36
128
132
137
142
146
151
155
100
165
109
5 2.O
6 24
37
174
179
183
188
192
197
202
206
211
210
7 2.8
8 3.2
38
220
225
230
234
239
243
248
253
257
202
9 3.6
39
267
271
276
280
285
290
294
299
304
308
940
313
317
322
327
331
336
310
345
350
351
41
359
364
368
373
377
382
387
301
300
400
42
405
410
414
419
424
428
433
437
412
447
43
451
456
400
465
470
474
479
483
488
493
44
497
502
50f>
fill
516
520
525
529
534
539
45
543
548
552
557
502
500
571
575
580
585
46
589
594
698
603
607
612
017
021
626
630
47
635
640
644
649
653
658
003
007
072
676
48
681
685
690
695
099
704
708
713
717
722
49
727
731
736
740
745
749
754
759
703
768
950
772
777
782
786
791
795
800
804
800
813
N
1
3
3
4
5
6
7
8
9
PP
T I 17
TABLE I. COMMON LOGARITHMS OF NUMBERS
To Five Decimal Places
N
1
2
3
4
5
6
7
8
9
P P
950
51
97772
818
777
7S2
786
791
795
800
804
809
813
823
827
832
836
841
845
8.30
855
859
52
864
868
873
877
882
880
891
896
900
905
53
909
914
918
923
928
1>32
937
941
940
950
54
955
959
964
90S
973
978
982
987
991
996
55
98000
005
009
014
019
023
028
032
037
041
56
046
050
055
059
064
068
073
078
082
087
57
091
096
100
105
109
114
118
123
127
132
58
137
141
146
150
155
159
1G4
168
373
177
59
182
186
191
195
200
204
209
214
218
223
960
227
232
236
241
245
250
254
259
203
268
61
272
277
281
286
290
295
299
304
308
313
62
318
322
327
331
336
340
345
349
354
358
63
303
367
372
376
381
385
390
394
399
403
5
64
408
412
417
421
426
430
435
439
444
448
1 O.5
2 1.0
65
453
457
462
466
471
475
480
484
489
493
3 1.5
66
498
502
507
511
516
520
525
529
534
53$
4 2.0
6 2.5
67
543
547
552
556
561
565
570
574
579
583
6 3.0
7 35
68
588
592
597
601
605
610
614
619
623
628
8 4^0
69
632
637
641
646
650
655
659
604
668
073
9 4.5
970
677"
682
G86
691
~G95~
700
704
709
713
717
71
722"
726
731
735
740
744
749
753
758
762
72
767
771
770
780
784
789
793
798
802
807
73
811
816
820
825
829
834
838
843
847
851
74
856
860
865
869
874
878
883
887
892
89G
75
900
905
909
914
918
923
927
932
936
941
76
945
949
954
958
903
907
972
976
981
985
77
989
994
998
*003
*007
*012
*016
*021
*025
*029
78
99034
038
013
047
052
05G
061
005
009
074
79
078
083
087
09^
096
100
105
109
114
118
980
iUsT
127
131
136
140
145
149
154
158
162
81
i&T
171
176
180
185
189
193
198
202
207~
82
211
216
220
224
229
233
238
242
247
251
4
83
255
260
264
2G9
273
277
282
286
291
295
"r-oT~
81
300
304
308
313
317
322
326
330
335
339
2 0.8
3 1.2
85
344
348
352
357
361
366
370
374
379
383
4 1.6
86
388
392
396
401
405
410
414
419
423
427
6 2.0
6 2.4
87
432
436
441
445
449
454
458
403
4G7
471
7 2.8
8 32
88
476
480
484
489
493
498
502
506
511
515
9 3.6
89
620
524
528
533
537
542
546
550
555
559
990
564~
568
572
577
581
585
590
594
599
603
91
607~
612
616
621
625
629
634
638
642
647
92
651
656
660
664
609
673
077
G82
686
691
93
695
699
704
708
712
717
721
726
730
734
94
739
743
747
752
756
7GO
765
769
774
778
95
782
787
791
795
800
804
803
813
817
822
96
826
830
835
839
843
848
852
856
861
865
97
870
874
878
883
887
891
890
900
904
909
98
913
917
922
926
930
935
939
944
948
952
99
957
961
965
970
974
978
983
987
991
996
1000
00660"
004
009
013
017
022
026
030
035
030
N
1
3
3
4
5
6
7
8
9
PP
T I 18
TABLE II. COMMON LOGARITHMS OF NUMBERS
From 1.00000 to 1.100000
To Seven Decimal Places
N
1
2
3
4
5
6
7
8
1000
1001
1002
1003
0000000
0434
OS09
1303
1737
2171
2005
3039
3473
3907
4341
8677
001 3009
4775
9111
3442
5208
9544
3875
56-12
9977
4308
6076
*04 1 1
4741
6510
*OXU
5174
6913
*1277
5607
7377
*1710
6039
7X10
*2143
G472
8244
*2576
6905
1004
1005
100G
7337
002 1G61
5980
7770
2093
6411
8202
2525
6843
8635
21)57
7275
9007
33X9
7706
9491)
3X21
8138
9932
42- r >3
8569
*0364
4685
9001
*0796
5116
9432
*1228
5548
9863
1007
1008
1009
1010
1011
1012
1013
003 0295
4005
8912
0726
50.J6
9342
1157
5407
9772
15RS
5X1 >H
*O203
2019
o:t2X
*oo:n
2451
075!)
*HW3
2X82
7190
*H93
3313
7020
*1924
3744
8051
*2354
4174
8481
*2784
004 3214
7512"
005 1805
6094
3044
4074
4504
4933
5363
5793
6223
6652
7082
7911
2234
C523
8371
2003
G952
8X00
3092
7380
1)229
3.VJ1
7809
9<i. r >9
3D5O
823S
*OOXX
4379
8666
*0. r >17
4XOS
9094
*0947
52ii7
9523
*137G
5666
9961
1014
1015
101G
006 0380
4600
8937
OSO8
5088
9365
1236
5516
9792
1 604
. r >944
*0219
2O92
G:t72
*0647
2521
6799
*1074
2949
7227
*1501
3377
7055
*1928
3805
80X2
*2355
4233
8510
*2782
1017
1018
1010
1020
1021
1022
1023
007 3210
7478
008 1742
C002
3637
7904
2168
4064
8.m
2594
4490
8757
3020
4917
91S4
JM4(i
5344
9O1O
3872
5771
*0037
429X
f>!98
*04G:t
4724
6624
*OX89
515O
7051
*131G
5570
6127
6853
7279
7704
8130
8556
8981
9407
9832
009 0257
4509
8750
0683
4934
9181
1108
5359
0605
1533
57X1
*0030
1959
G20S
*0454
2384
(il>:i3
*0878
2809
7058
*1303
3234
74X3
*1727
3659
7907
*2151
4084
8332
*2575
1024
1025
1020
010 3000
7239
Oil 1474
3424
7602
1897
3X4 S
8080
2320
4272
8.1H)
2743
4696
8933
3166
5120
9357
3590
5544
9780
4013
5967
*0204
4436
6391
*0627
4859
6815
*1050
5282
1027
1028
1029
1030
1031
1032
1033
5704
9931
0124154
6127
*0354
4576
6550
*0776
4998
6973
*11US
5420
7396
*1(21
5S42
7818
*2043
6264
8241
*24G5
6685
8664
*2SS7
7107
908G
*3310
7529
9509
*3732
7951
8372
8794
9215
9637
*()059
*0480
*0901
*1323
*1744
*2105
013 25S7
6797
014 1003
3008
7218
1424
3429
7639
1844
3850
8059
2264
4271
8480
2685
46D2
8901
3105
5113
9321
3525
5534
9742
3945
5955
*0162
4365
6376
*0583
4785
1034
1035
1036
5205
9403
015 3598
5625
9823
4017
6045
*0243
4436
6465
*0662
4855
6885
*10S2
5274
7305
*1501
5G93
7725
*1920
6112
8144
*2340
6531
8564
*2759
6950
8984
*3178
7369
1037
1038
1039
1040
1041
1042
1043
7788
016 1974
6155
8206
2392
6573
8625
2810
6991
6044
3229
7409
9462
3647
7S27
9831
4065
8245
*0300
4483
8063
*0718
4901
9080
*1137
5319
9498
*1555
5737
9916
017 0333
0751
1168
1586
2003
2421
2838
3256
3673
4090
4507
8677
018 2843
4924
9094
3259
5342
9511
3676
5759
9927
4092
6176
*0344
4508
6593
*0761
4925
7010
*1177
5341
7427
*1594
5757
7844
*2010
6173
8200
*2427
6589
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
4486
8037
*0747
4902
9052
1047
1048
1049
1050
9467
0203613
7755
02lT893~
9882
4027
' 8169
*0296
4442
8583
*0711
4856
8997
*1126
5270
9411
*1540
5684
9824
*1955
6099
*0238
*2369
6513
*0652
*2784
6927
*1066
*3198
7341
*1479
2307
2720
3134
354?
3961
4374
4787
5201
5614
N
O
1
2
3
4
5
6
7
8
TII 19
TABLE II. COMMON LOGARITHMS OF NUMBERS
From 1.00000 to 1.100000
To Seven Decimal Places
N
1
2
3
4
5
6
7
a
9
1050
1051
1052
1053
021 1893
2307
2720
3134
3547
3961
4374
4787
5201
5614
0027
0220157
4284
6440
0570
4696
6854
0983
5109
7267
1396
5521
7680
1808
5933
8093
2221
6345
8500
2034
6758
8919
3046
7170
9332
3459
7582
9745
3871
7994
1054
1055
1056
8406
023 2525
6639
8818
2936
7050
9230
3348
7462
9642
3759
7873
*0054
4171
8284
*04G6
4 r >82
8695
*0878
4994
9106
*1289
5405
9517
*1701
5817
9928
*2113
6228
*0339
1057
1058
1059
1060
1061
1062
10G3
024 0750
4857
8960
1161
5267
9370
1572
5678
9780
1982
0088
*0100
2303
6408
*0000
2804
C909
*1010
3214
7319
*1419
3625
7729
*1829
4036
8139
*2239
4440
8549
*2049
025 3059
7154
026 1245
5333
34 G8
3878
42S8
4697
5107
5516
5920
0335
6744
7503
1C54
5741
7972
2003
6150
8382
2472
6558
8701
2881
6907
9200
3289
7375
0009
3098
7783
*0018
4107
8192
*0-I27
45 if)
8000
*O83G
4924
0008
1064
1005
1066
941G
027 3496
7572
9824
3904
7979
*0233
4312
8387
*0611
4719
8794
*1Q49
5127
9201
*I457
5535
9009
*1865
5942
*0016
*2273
0350
*0423
*2680
6757
*0830
*3088
7105
*1237
1067
1068
1069
1070
1071
1072
1073
028 1644
5713
9777
2051
6119
*0183
2458
6526
*0590
2865
0032
*0996
3272
7339
*1402
3679
7745
*I808
4086
8152
*2214
4492
8558
*2620
4899
8964
*3026
5306
9371
*3432
029 3838
4244
4649
5055
5401
5807
6272
6678
7084
7489
7895
030 1948
5997
8300
2353
6402
8706
2758
6807
9111
3163
7211
9516
3508
7616
9922
3973
8020
*0327
4378
8425
*0732
4783
8830
*1138
5188
9234
*1543
5502
8038
1074
1075
1076
031 0043
4085
8123
0447
4489
8526
0851
4803
8930
1256
5206
9333
1660
5700
9737
2064
6104
*0140
2468
6508
*0544
2872
6912
*0947
3277
7315
*1350
3GS1
7719
*1754
1077
1078
1079
1080
1081
1082
1083
032 2157
6188
033 0214
2560
6590
0617
2963
6993
1019
3307
7396
1422
3770
7799
1821
4173
8201
2226
4576
8604
2629
4979
9007
3031
5382
9409
3433
5785
9812
3835
4238
8257"
034 2273
6285
4640
5042
5444
5846
6248
~*0265~
4279
8289
6650
7052
7453
7855
8659
2074
6686
9060
3075
7087
9402
3477
7487
9804
3878
7888
*0667
4680
8690
*1068
5081
9091
*1470
5482
9491
*187l
5884
9892
1084
1085
1086
035 0293
4297
8298
0693
4698
8698
1094
5098
9098
1495
5498
9498
1895
5898
9898
2296
6298
*0297
2696
6698
*0697
3096
7098
*1097
3497
7498
*1496
3897
7898
*1896
1087
1088
1089
1090
1091
1092
1093
036 2295
6289
037 0279
2695
6688
0678
3094
7087
1076
3494
7486
1475
3893
7885
1874
4293
8284
2272
4692
8083
2671
5091
9082
3070
5491
9481
3468
5890
9880
3867
4265
4663
5062
5460
5858
6257
*0237
4214
8188
6655
7053
7451
7849
8248
038 2226
6202
8646
2624
6599
9044
3022
6996
9442
3419
7393
9839
3817
7791
*0635
4612
8585
*1033
5009
8982
*1431
5407
9379
*1829
5804
9776
1094
1095
1096
039 0173
4141
8106
0570
4538
8502
0967
4934
8898
1364
5331
9294
1761
5727
9690
2158
6124
*0086
2554
6520
*04S2
2951
6917
*0878
3348
7313
*1274
3745
7709
*1670
1097
1098
1099
1100
040 2066
6023
9977
2462
6419
*0372
2858
6814
*0767
3254
7210
*1162
3650
7G05
*1557
4045
8001
*1952
4441
8396
*2347
4837
8791
*2742
5232
9187
*3137
5628
9582
*3532
041 3927
4322
4716
5111
5506
5900
6295
6690
7084
7479
N
1
2
3
4
5
6
7
8
9
T II 20
TABLE III. COMPOUND AMOUNT OF 1
n
H%
\%
%
!%
1%
i
3
3
&
1.0041 6667
1.0083 5069
1.0125 5216
1.0167 7112
1.0210 0767
1.0050 0000
1.0100 2500
1.0150 7513
1.0201 5050
1.0252 5125
1.0058 3333
1.0117 0069
1.0176 0228
1.0235 3830
1,0295 0894
1.0075 0000
1.0150 5625
1.0226 6917
1.0303 3919
1.0380 6673
1.0100 0000
1.0201 0000
1.0303 0100
1.0406 0401
1.0510 1005
6
8
9
10
1.0252 6187
1.0295 3379
.0338 2352
.0381 3111
.0424 5666
1.0303 7751
1.0355 2940
L0407 0704
1.0459 1058
1.0511 4013
1.0355 1440
1.0415 5490
1.0476 3064
1.0537 4182
1.0598 8865
1.0458 5224
1.0536 9613
1.0615 9885
1.0695 6084
1.0775 8255
1.0615 2015
1.0721 3535
1.0828 5671
1.0936 8527
1.1046 2213
11
12
13
14
15
.0468 0023
.0511 6190
.0555 4174
.0599 3983
.0643 5625
.0563 9583
.0616 77S1
.0609 8620
.0723 2113
.077t> 8274
1.0660 7133
1.0722 9008
1.0785 4511
1.0848 3G62
1.0911 6483
.0856 6441
.0938 0690
.1020 1045
.1102 7553
.1186 0259
1.1156 6835
1.1268 2503
1.1380 9328
1.1494 7421
1.1609 6896
16
17
IS
19
20
.0687 9106
.0732 4436
.0777 1621
.0822 0670
.0867 1589
.0830 7115
.0884 8651
.0939 2894
.0993 9858
.1048 9558
1.0975 2996
1.1039 3222
1.1103 7182
1.11C8 4899
1.1233 6395
.1269 9211
.1354 4455
.1439 6039
.1525 4009
.1611 8414
1.1725 7864.
1.1843 0443
1.1961 4748
1.2081 0895
1.2201 9004
21
22
23
24
25
.0912 4387
.0957 9072
.1003 5652
.1049 4134
.1095 4526
.1104 2006
.1159 7216
.1215 5202
.1271 5978
.1327 9558
1.1299 1690
1.1365 0808
1.1431 3771
1.1498 0602
1.1565 1322
.1698 9302
.1786 6722
.1875 0723
.1964 1353
.2053 8663
1.2323 9194
1.2447 1586
1.2571 6302
1.2697 3465
1.2824 3200
26
27
28
29
30
.1141 6836
.1188 1073
.1234 7244
.1281 5358
.1328 5422
.1384 5955
.1441 5185
.1498 7261
.1556 2197
.1614 0008
1.1632 5955
1.1700 4523
1.1768 7049
1.1837 3557
1.1906 4069
.2144 2703
.2235 3523
.2327 1175
.2419 5709
.2512 7176
1.2952 5631
1.3082 0888
1.3212 9097
1.3345 0388
1.3478 4892
31
32
33
34
35
.1375 7444
.1423 1434
.1470 7398
.1518 5346
.1566 5284
.1672 0708
.1730 4312
.1789 0833
.1848 0288
.1907 2689
1.1975 8610
1.2045 7202
1.2115 9869
1.2180 6634
1.2257 7523
.2606 5630
.2701 1122
.2796 3706
.2892 3434
.2989 0359
1.3613 2740
1.3749 4068
1.3886 9009
1.4025 7699
1.4166 0276
36
37
38
39
40
.1614 7223
.1663 1170
.1711 7133
.1760 5121
.1809 5142
.1966 8052
.2026 6393
.2086 7725
.2147 2063
.2207 9424
1.2329 2559
1.2401 17C5
1.2473 5167
1.2546 2789
1.2619 4655
.3086 4537
.3184 6021
.3283 4866
.3383 1128
,3483 4S61
1.4307 6878
1.4450 7647
1.4595 2724
1.4741 2251
1.4888 6373
41
42
43
44
45
.1858 7206
.1908 1319
.1957 7491
.2007 5731
1.2057 6046
.2268 9821
.2330 3270
.2391 9786
.2453 9385
.2516 2082
1.2693 0791
1.2767 1220
1.2841 51)69
1.2916 5062
1.2991 8525
.3584 6123
.3686 4969
.3789 1456
.3892 5642
.3996 7584
1.5037 5237
1.5187 8989
1.5339 7779
1.5493 1757
1.5648 1075
46
47
48
49
50
1.2107 8446
1.2158 2940
1.2208 9536
1.2259 8242
1.2310 9068
.2578 7892
.2641 6832
.2704 8916
.2768 4161
1.2832 2581
1.3067 6383
1.3143 8662
1.3220 5388
1.3297 6586
1.3375 2283
.4101 7341
.4207 4071
.4314 0533
.4421 4087
.4529 5693
1.5804 5885
1.59(52 6344
1.6122 2608
1,6283 4834
1.6446 3182
T III 21
TABLE III. COMPOUND AMOUNT OF 1
(1 + t)"
n
a%
1%
%
!%
1%
51
52
53
54
55
1.2362 2002
1.2413 7114
1.2405 4352
1.2517 3745
1.2569 5302
1.289G 4194
1.29GO 9015
1.3025 7060
1.3090 8346
1.3156 2887
1.3453 2504
1.3531 7277
1.3610 6628
1.3690 0583
1.3769 9170
1.4638 5411
1.4748 3301
1.4858 9426
1.4970 3847
1.5082 6626
1.6610 7814
1.G776 8892
1.6944 6581
1.7114 1047
1.7285 2457
56
57
58
69
60
1.2621 9033
1.2674 4 040
1.2727 3050
1.2780 3354
1.2833 5868
1.3222 0702
1.3288 1805
1.3354 6214
1.3421 3946
1.3488 5015
1.3850 2415
1.3931 0346
1.4012 2990
1.4094 0374
1.4176 2526
1.5195 7825
1.5309 7509
1.5424 5740
1.5540 2583
1.5656 8103
1.7458 0982
1.7632 6792
1.7809 0060
1.7987 0960
1.8166 9670
61
69
63
64
65
1.2887 0601
1.2940 7561
1.2994 6760
1.3048 8204
1.3103 1905
1.3555 9440
1.3G23-7238
1.3G91 8424
1.37GO 3016
1.3829 1031
1.4258 9474
1.4342 1246
1.4425 7870
1.4509 9374
1.4594 5787
1.5774 2363
1.5892 5431
1.6011 7372
1.6131 8252
1.6252 8139
1.8348 6367
1.8532 1230
1.8717 4413
1.8904 6187
1.9093 6649
66
67
68
69
70
1.3157 7872
1.3212 6113
1.3267 6G38
1.3322 9458
1.3378 4580
1.3898 2486
1.3967 7399
1.4037 5785
1.4107 7664
1.4178 3053
1.4679 7138
1.4765 3454
1.4851 4766
1.4938 1102
1.5025 2492
1.6374 7100
1.G497 5203
1.6621 2517
1.6745 9111
1.6871 5055
1.9284 6015
1.9477 4475
1.9672 2220
1.9868 9442
2.0067 6337
71
73
73
74
75
1.3434 2016
1.3490 1774
1.3546 3865
1.3602 8298
1.3659 5082
1.4249 1968
1.4320 4428
1.4392 0450
1.4464 0052
1.4536 3252
1.5112 8965
1.5201 0550
1.5289 7279
1.5378 9179
1.5468 6283
1.6998 0418
1.7125 5271
1.7253 9685
1.7383 3733
1.7513 7486
2.0268 3100
2.0470 9931
2.0675 7031
2.0882 4601
2.1091 2847
76
77
78
79
80
1.3716 4229
1.3773 5746
1.3830 9645
1.3888 5935
1.3946 4627
1.4609 0069
1.4682 0519
1.4755 4622
1.4829 2395
1.4903 3857
1.5558 8620
1.5649 6220
1.5740 9115
1.5832 7334
1.5925 0910
1.7645 1017
1.7777 4400
1.7910 7708
1.8045 1015
1.8180 4398
2.1302 1975
2.1515 2195
2.1730 3717
2.1947 6754
2.2167 1522
81
89
83
84
85
1.4004 5729
1.4062 9253
1.4121 5209
1.4180 3605
1.4239 4454
1.4977 9026
1.5052 7921
1.5128 0561
1.5203 6964
1.5279 7148
1.6017 9874
1.6111 4257
1.6205 4090
1.G299 9405
1.6395 0235
1.8316 7931
1.8454 1691
1.8592 5753
1.8732 0196
1.8872 5098
2.2388 8237
2.2612 7119
2.2838 8390
2.3067 2274
2.3297 8997
86
87-
88
89
90
1.4298 7764
1.4358 3546
1.4418 1811
1.4478 2568
1.4538 5829
1.5355 1134
1.5432 8940
1.5510 0585
1.5587 6087
1.5G65 5468
1.6490 6612
1.6586 8567
1.6683 6134
1.6780*9344
1.6878 8232
1.9014 0536
1,9156 6590
1.9300 3339
1.9445 0865
1.9590 9246
2.3530 8787
2.3766 1875
2.4003 8494
2.4243 8879
2.4486 3267
91
93
93
94
95
1.4599 1603
1.4659 9902
1.4721 0735
1.4782 4113
1.4844 0047
1.5743 8745
1.5822 5939
1.5901 7069
1.5981 2154
1.6061 1215
1.6977 2830
1.7076 3172
1.7175 9290
1.7276 1219
1.7376 8993
1.9737 8565
1.9885 8905
2.0035 0346
2.0185 2974
2.0336 6871
2.4731 1900
2.4978 5019
2.5228 2S69
2.5480 5698
2.5735 3755
96
97
98
99
100
1.4905 8547
1.4967 9624
1.5030 3289
1.5092 9553
1.5155 8426
1.6141 4271
1.6222 1342
1.6303 2449
1.6384 7611
1.6466 6849
1.7478 2646
1.7580 2211
1.7682 7724
1.7785 9219
1.7889 6731
2.0489 2123
2.0642 8814
2.0797 7030
2.0953 6858
2.1110 8384
2.5992 7293
2.6252 65G5
2.6515 1831
2.6780 3349
2.7048 1383
III 22
TABLE III. COMPOUND AMOUNT OF 1
n
%
1%
n%
!%
1%
101
102
103
104
105
.5218 9919
.5282 4044
.5346 0811
.5410 0231
.5474 2315
1.6549 0183
1.6631 7634
1.6714 9223
1.6798 4969
1.6882 4894
1.7994 0295
1.8098 9947
1.8204 5722
1.8310 7655
1.8417 6783
2.1269 1697
2.1428 G885
2.1589 4036
2.1751 3242
2.1914 4591
2.7318 6197
2.7591 8059
2.7867 7239
2.8146 4012
2.8427 8652
106
107
108
109
110
.5538 7075
.5603 4521
.5668 4665
.5733 7518
.5799 3091
1.6966 9018
1.7051 7363
1.7136 9950
1.7222 6800
1.7308 7934
1.8525 0142
1.8633 0768
1.8741 7697
1.8851 0967
1.8961 0614
2.2078 8175
2.2244 4087
2.2411 2417
2.2579 3260
2.2748 6710
2.8712 1438
2.8999 2653
2.9289 2579
2.9582 1505
2.9877 9720
111
112
113
114
115
.5865 1395
.5931 2443
.5997 6245
.6064 2812
.6131 2157
1.7395 3373
1.7482 3140
1.7569 7256
1.7657 5742
1.7745 8621
1.9071 6676
1.9182 9190
1.9294 8194
1.9407 3725
1.9520 5822
2.2919 2860
2.3091 1807
2.3264 3645
2.3438 8472
2.3614 6386
3.0176 7517
3.0478 5192
3.0783 3044
3.1091 1375
3.1402 0489
116
117
118
119
120
.6198 4291
.6265 9226
.6333 6973
.6401 7543
.6470 0950
1.7834 5914
1.7923 7644
1.8013 3832
1.8103 4501
1.8193 9673
1.9034 4522
1.9743 9865
1.9864 1890
1.9980 0634
2.0096 6138
2.3791 7484
2.3970 1865
2.4149 9629
2.4331 0876
2.4513 5708
3.1716 0693
3.2033 2300
3.2353 5623
3.2677 0980
3.3003 8689
121
122
123
124
125
.6538 7204
.6607 6317
.6676 8302
.6746 3170
.6816 0933
1.8284 9372
1.8376 3619
1.8468 5437
1.8560 5849
1.8653 3878
2.0213 8440
2.0331 7581
2.0450 3600
2.0569 6538
2.0689 6434
2.4697 4226
2.4882 6532
2.50G9 2731
2.5257 2927
2.5446 7224
3.3333 9076
3.3667 2467
3.4003 9192
3.4343 9584
3.4687 3980
126
127
128
129
130
.6886 1603
.6956 5193
.7027 1715
.7098 1181
.7169 3602
1.8746 6548
1.8840 3880
1.8934 5900
1.9029 2629
1.9124 4092
2.0810 3330
2.0931 7266
2.1053 8284
2.1176 6424
2.1300 1728
2.5637 5728
2.5829 8546
2.6023 5785
2.6218 7553
2.6415 3960
3.5034 2719
3.5384 6147
3.5738 4608
3.6095 84.54
3.6456 8039
131
132
133
134
135
.7240 8992
.7312 7363
.7384 8727
.7457 3097
.7530 0485
1.9220 0313
1.9316 1314
1.9412 7121
1.9509 7757
1.9607 3245
2.1424 4238
2.1549 3996
2.1675 1044
2.1801 5425
2.1928 7182
2.6613 5115
2.6813 1128
2.7014 2112
2.7216 8177
2.7420 9439
3.6821 3719
3.7189 5856
3.7561 4815
3.7937 0963
3.8316 4673
136
137
138
139
140
.7603 0903
.7676 4365
.7750 0884
.7824 0471
.7898 3139
1.9705 3612
1.9803 8880
1.9902 9074
2.0002 4219
2.0102 4340
2.2056 6357
2.2185 2994
2.2314 7137
2.2444 8828
2.2575 8113
2.7626 6009
2.7833 8005
2.8042 5540
2.8252 8731
2.8464 7697
3.8699 6319
3.9086 6282
3.9477 4945
3.9872 2695
4.0270 9922
141
142
143
144
145
.7972 8902
.8047 7773
.8122 9763
1.8198 4887
1.8274 3158
2.0202 9462
2.0303 9609
2.0405 4808
2.0507 5082
2.0610 0457
2.2707 5036
2.2839 9640
2.2973 1971
2.3107 2074
2.3241 9995
2.8678 2554
2.8893 3424
2.9110 0424
2.9328 3677
2.9548 3305
4.0673 7021
4.1080 4391
4.1491 2435
4.1906 1559
4.2325 2175
146
147
148
149
150
1.8350 4588
1.8426 9190
1.8503 6978
1.8580 7966
1.8658 2166
2.0713 0959
2.0816 6614
2.0920 7447
2.1025 3484
2,1130 4752
2.3377 5778
2.3513 9470
2.3651, 1117
2.3789 0765
2,3927 8461
2.9769 9430
2.9993 2175
3.0218 1667
3.0444 8029
3.0673 1389
4.2748 4697
4.3175 9544
4.3607 7139
4.4043 7910
4.4484 2290
T1II 23
TABLE III. COMPOUND AMOUNT OF 1
n
ll%
l%
1|%
lf%
2%
1
2
4
5
1.0112 5000
1.0226 2656
1.0341 3111
1.0457 6509
1.0575 2994
1.0125 0000
1.0251 5625
1.0379 7070
1.0509 4534
1.0640 8215
1.0150 0000
1.0302 2500
1.0456 7838
1.0613 6355
1.0772 8400
1.0175 0000
1.0353 0625
1.0534 2411
1.0718 5903
1.0906 1656
1.0200 0000
1.0404 0000
1.0612 0800
1.0824 3216
1.1040 8080
6
7
8
9
10
1.0694 2716
1.0814 5821
.0936 2462
.1059 2789
.1183 6958
1.0773 8318
1.0908 5047
1.1044 8610
1.1182 9218
1.1322 7083
1.0934 4326
1.1098 4491
1.1264 9259
1.1433 8998
1.1605 4083
1.1097 0235
1.1291 2215
1.1488 8178
1.1689 8721
1.1894 4449
1.1261 6242
1.1486 8567
1.1716 5938
1.1950 9257
1.2189 9442
11
12
13
14
15
.1309 5124
.1436 7444
.1565 4078
.1695 5186
.1827 0932
1.1464 2422
1.1607 5452
1.1752 6395
1.1899 5475
1.2048 2918
1.1779 4894
1.1956 1817
1.2135 5244
1.2317 5573
1.2502 3207
1.2102 5977
1.2314 3931
1.2529 8950
1.2749 16S2
1.2972 2786
1.2433 7431
1.2082 4179
1.2936 0663
1.3194 7876
1.3458 6834
16
17
18
19
20
.1960 1480
.2094 6997
.2230 7650
.2368 3611
.2507 5052
1.2198 8955
1.2351 3817
1.2505 7739
1.2662 0961
1.2820 3723
1.2689 8555
1.2880 2033
1.3073 4064
1.3269 5075
1,3468 5501
1.3199 2935
1.3430 2811
1.36G5 3111
1.3904 4540
1.4147 7820
1.3727 8571
1.4002 4142
1.4282 4625
1.4568 1117
1.4859 4740
21
22
23
24
25
.2648 2146
.2790 5071
.2934 4003
.3079 9123
.3227 0613
1.2980 6270
1.3142 8848
1.3307 1709
1.3473 5105
1.3641 9294
1.3670 5783
1.3875 6370
1.4083 7715
1.4295 0281
1.4509 4535
1.4395 3681
1.4647 2871
1.4903 6146
1.5164 4279
1.5429 8054
1.5156 6634
1.5459 7967
1.57G8 9926
1.6084 3725
1.6406 0599
26
27
28
29
30
.3375 8657
.3526 3442
1.3678 5156
1.3832 3989
1.3988 0134
1.3812 4535
1.3985 1092
1.4159 9230
1.4336 9221
1.4516 1338
1.4727 0953
1.4948 0018
1.5172 2218
1.5399 8051
1.5630 8022
1.5699 8269
1.5974 5739
1.6254 1290
1.6538 5762
1.6828 0013
1.6734 1811
1.70C8 8648
1.7410 2421
1.7758 4469
1.8113 6158
31
32
33
34
35
1.4145 3785
1.4304 5140
1.4465 4398
1.4628 1760
1.4792 7430
1.4697 5853
1.4881 3051
1.5067 3214
1.5255 6629
1.5446 3587
1.5865 2642
1.6103 2432
1.6344 7918
1.6589 9637
1.6838 8132
1.7122 4013
1.7422 1349
1.7727 0223
1.8037 2452
1.8352 8970
1.8475 8882
1.8845 4059
1.1)222 3140
1.9608 7603
1.9008 8955
36
37
38
39
40
1.4959 1613
1.5127 4519
1.5297 6357
1.5469 7341
1.5643 7687
1.5639 4382
1.5834 9312
1.6032 8678
1.6233 2787
1.6436 1946
1.7091 3954
1.7347 7663
1.7607 9828
1.7872 1025
1.8140 1841
1.8674 0727
1.9000 8689
1.9333 3841
1.9671 7184
2.0015 9734
2.0398 8734
2.0800 8509
2.1222 0879
2.1647 4477
2.2080 3966
41
42
43
44
45
1.5819 7611
1.5997 7334
1.6177 7079
1.6359 7071
1.6543 7538
1.6641 6471
1.6849 6677
1.7060 2885
1.7273 5421
1.7489 4614
1.8412 2868
1.8688 4712
1.896S 7982
1.9253 3302
1.9542 1301
2.0366 2530
2.0722 6024
2.1085 3000
2.1454 3019
2.1829 7522
2.2522 0046
2.2972 4447
2.3431 8936
2.3900 5314
2 4378 5421
46
47
48
49
50
1.6729 8710
1.6918 0821
1.7108 4105
1.7300 8801
1.7495 5150
1.7708 0797
1.7929 4306
1.8153 5485
1.8380 4679
1.8610 2237
1.9835 2621
2.0132 7010
2.0434 7829
2.0741 3046
2.1052 4242
2.2211 7728
2.2600 4789
2.2995 9872
2.3398 4170
2.3807 8893
2.4866 1129
2.5263 4351
2.5870 7039
2.6388 1179
2.6915 8803
T III 24
TABLE III. COMPOUND AMOUNT OF 1
n
1|%
ll%
ll%
1|%
2%
51
S3
53
64
65
1.7692 3305
1.7891 3784
1.8092 6564
1.8296 1988
1.8502 0310
1.8842 8515
1.9078 3872
1.9316 8670
1.9558 3279
1.9802 8070
2.1368 2106
2.1688 7337
2.2014 0647
2.2344 2757
2.2679 4398
2.4224 5274
2.4648 4566
2.5079 8046
2.5518 7012
2.5965 2785
2.7454 1979
2.8003 2819
2.8563 3475
2.9134 6144
2.9717 3067
56
57
58
59
60
1.8710 1788
1.8920 6G84
1.9133 5259
1.9348 7780
1.9566 4518
2.0050 3420
2.0300 9713
2.0554 7335
2.0811 6676
2.1071 8135
2.3019 6314
2.3364 9259
2.3715 3998
2.4071 1308
2.4432 1978
2.6419 6708
2.6882 0151
2.7352 4503
2.7831 1182
2.8318 1628
3.0311 6529
3.0917 8859
3.1536 2436
3.2166 9685
3.2810 3079
61
62
63
64
65
1.9786 5744
2.0009 1733
2.0234 2765
2.0461 9121
2.6092 1087
2.1335 2111
2.1601 9013
2.1871 9250
2.2145 3241
2.2422 1407
2.4798 6807
2.5170 6609
2.5548 2208
2.5931 4442
2.6320 4153
2.8813 7306
2.9317 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
2.0924 8949
2.1160 2999
2.1398 3533
2.1639 0848
2.1882 5245
2.2702 4174
2.2986 1976
2.3273 5251
2.3564 4442
2,3858 9997
2.6715 2221
2.7115 9504
2.7522 C896
2.7935 5300
2.8354 5629
3.1424 7319
3.1974 6647
3.2534 2213
3.3103 5702
3.3682 8827
3.6949 7357
3.7688 7304
3.B442 5050
3.9211 3551
3.9995 5822
71
72
73
74
75
2.2128 7029
2.2377 6508
2.2629 3994
*\2833 9801
2.3141 4249
2.4157 2372
2.4459 2027
2.4764 9427
2.5074 5045
2.5387 9358
2.8779 8814
2.9211 5796
2.9649 7533
3.0094 4996
3.0545 9171
3.4272 3331
3.4872 0990
3.5482 3607
3.6103 3020
3.6735 1098
4.0795 4939
4.1611 4038
4.2443 6318
4.3292 5045
4.4158 3546
76
77
78
79
80
2.3401 7659'
2.3665 0358
2.3931 2675
2.4200 4942
2.4472 7498
2.5705 2850
2.6026 6011
2.6351 9336
2.6681 3327
2.7014 8494
3.1004 1059
3.1469 1674
3.1941 2050
3.2420 3230
3.2906 6279
3.7377 9742
3.8032 0888
3.8697 6503
3.9374 8592
4.0063 9192
4.5041 5216
4.5942 3521
4.6861 1991
4.7798,4231
4.8754 3916
81
82
83
84
85
2.4748 0682
2.5026 4840
2.5308 0319
2.5592 7,4.73
2.5880 6657
2.7352 5350
2.7694 4417
2.8040 6222
2.8391 1300
2.8746 0191
3.3400 2273
3.3901 2307
3.4409 7492
3.4925 8954
3.5449 7838
4.0765 0378
4.1478 4260
4.2204 2984
4.2942 8737
4.3694 3740
4.9729 4794
5.0724 0690
5.1738 5504
6.2773 3214
5.3828 7878
86
87
88
89
90
2.6171 8232
2.6466 2562
2.6764 0016
2.7065 0966
2.7369 5789
2.9105 3444
2.9469 1612
2.9837 5257
3.0210 4948
3.0588 1260
3.5981 5306
3.6521 2535
3.7069 0723
.3.7625 1084
3.8189 4851
4.4459 0255
4.5237 0584
4.6028 7070
4.6834 2093
4.7653 8080
5.4905 3636
5.6003 4708
5.7123 5402
5.8266 0110
5.9431 3313
91
92
93
94
95
2.7677 4367
2.7988 8584
2.8303 7331
2.8622 1501
2.8944 1492
3.0970 4775
3.1357 6085
3.1749 5786
3.2146 4483
3.2548 2789
3.8782 3273
3.9343 7622
3,9933 9187
4.0532 9275
4.1140 9214
4.8487 7496
4.9336 2853
5.0199 6703
5.1078 1645
6.1972 0324
6.0619 9579
6.1832 3570
6.3069 0042
6.4330 3843
6.5616 9920
96
97
98
99
100
2.9269 7709
2.9599 0559
2,9932 0452
3.0268 7807
3.0609 3045
3.2955 1324
3.3367 0716
3.3784 1600
3.4206 4620
3.4634 0427
4.1758 0352
4.2384 4057
4.3020 1718
4.3665 4744
4.4320 4565
5.2881 5429
5.3806 9C99
5.4748 5919
5.5706 6923
5.6681 5594
6.6929 3318
6.8267 9184
6.9633 2768
7.1025 9423
7.2446 4612
T II I- -25
TABUS III. COMPOUND AMOUNT OF 1
(1 + 0"
n
2|%
2|%
2|%
3%
3|%
i
3
3
4
5
1.0225 0000
1.0455 0025
1.0690 3014
1.0930 8332
1.1176 7769
1.0250 0000
1.0506 2500
1.0768 9063
1.1038 1289
1.1314 0821
1.0275 0000
1.0557 5625
1.0847 8955
1.1146 2126
1.1452 7334
1.0300 0000
1.0609 0000
1.0927 2700
1.1255 0881
1.1592 7407
1.0350 0000
1,0712 2500
1.1087 1788
1.1475 2300
1.1876 8631
6
7
8
10
1.1428 2544
1.1685 3901
1.1948 3114
1.2217 1484
1.2492 0343
1.1596 9342
1.1886 8575
1.2184 0290
1.2488 6297
L2800 8454
1.1767 6836
1.2091 2949
1.2423 8055
1.2765 4602
1.3116 5103
1.1940 5230
1.2298 7387
1.2667 7008
1.3047 7318
1.3439 1638
1.2292 5533
1.2722 7926
1.3168 0904
1.3628 9735
1.4105 9876
11
13
13
14
15
1.2773 1050
1.3060 4999
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
1.4619 9413
1.5021 9896
1.3842 3387
1.4257 6089
1.4685 3371
1.5125 8972
1.5579 6742
1.4599 6972
1.5110 6866
1.5639 5600
1.6186 9452
1.6753 4883
16
17
18
19
30
1.4276 2146
1.4597 4294
1.4925 8716
1.5261 7037
1.5605 0920
1.4845 0562
1.5216 1826
1.5596 5872'
1.5986 5019
1.6386 1644
1.5435 0944
1.5859 5595
1.6295 6973
1.6743 8290
1.7204 2843
1.6047 0644
1.6528 4763
1.7024 3306
1.7535 0605
1.8061 1123
1.7339 8604
1.7946 7555
1.8574 8920
1.9225 0132
1.9897 8886
31
33
33
34
35
1.5956 2066
1.6315 2212
1.6682 3137
1.7057 6658
1.7441 4632
1.6795 81S5
1.7215 7140
1.7646 1068
1.8087 2595
1.8539 4410
1.7677 4021
1.8163 5307
1.8663 0278
1.9176 2610
1.9703 6082
1.8602 9457
1.9161 0341
1.9735 8651
2.0327 9411
2.0937 7793
2.0594 3147
2.1315 1158
2.2061 1448
2.2833 2849
2.3632 4498
36
37
38
39
30
1.7833 8962
1.8235 1588
1.8645 4499
1.9064 9725
1.9493 9344
1.9002 9270
1.9478 0002
1.9964 9502
2.04G4 0739
2.0975 6758
2.0245 4575
2.0802 2075
2.1374 2682
2.1962 0606
2.2566 0173
2.1565 9127
2.2212 8901
2.2879 2768
2.3565 6551
2.4272 6247
2.4459 5856
2.5315 6711
2.6201 7196
2.7118 7798
2.8067 9370
31
33
33
34
35
1.9932 5479
2.0381 0303
2.0839 6034
2.1308 4945
2.1787 9356
2.1500 0677
2.2037 5G94
2.2588 5086
2.3153 2213
2.3732 0519
2.3186 5828
2.3824 2138
2.4479 3797
2.5152 5626
2.5844 2581
2.5000 8035
2.5750 8276
2.6523 3524
2.7319 0530
2.8138 6245
2.9050 3148
3.0067 0759
3.1119 4235
3.2208 6033
3.3335 9045
36
37
38
39
40
2.2278 1642
2.2779 4229
2.3291 9599
2.3816 0290
2.4351 8897
2.4325 3532
2.4933 4870
2.5556 8242
2.6195 7448
2.6850 6384
2.6554 9752
2.7285 2370
2.8035 5810
2.8806 5595
2.9598 7399
2.8982 7833
2.9852 2668
3.0747 8348
3.1670 2698
3.2620 3779
3.4502 6611
3.5710 2543
3.6960 1132
3.8253 7171
3.9592 5972
41
43
43
44
46
2.4899 8072
2.5460 0528
2.6032 9040
2.6618 6444
2.7217 5639
2.7521 9043
2.8209 9520
2.8915 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 9589
3.5645 1677
3.6714 5227
3.7815 9584
4.0978 3381
4.2412 5799
4.3897 0202
4.5433 4160
4.7023 5855
46
47
48
49
50
2.7829 9590
2.8456 1331
2.9096 3961
2.9751 0650
3.0420 4640
3.1138 5086
3.1916 9713
3.2714 8956
3.3532 7680
3.4371 0872
3.4830 8606
3.5788 7093
3.6772 8988
3.7784 1535
3.8S23 2177
3.8950 4372
4.0118 9503
4.1322 5188
4.2562 1944
4.3839 0602
4.8669 4110
5.0372 8404
5.2135 8898
5.3960 6459
5 5849 2686
T III 26
TABLE III. COMPOUND AMOUNT OF 1
(1 -f t) w
n
*!%
2l%
2f%
3%
3|%
51
52
53
54
55
3.1104 9244
3.1804 7852
3.2520 3929
3.3252 1017
3.4000 2740
3.5230 3644
3.6111 1235
3.7013 9016
3.7939 2491
3.8887 7303
3,9890 8562
4.0987 8547
4,2115 0208
4.3273 1838
4.4463 1904
4.5154 2320
4.6508 8590
4.7904 1247
4.9341 2485
5.0821 4859
5.7803 9930
5.9827 1327
6.1921 0824
6.4088 3202
6.6331 4114
50
57
58
59
GO
3.4765 2802
3.5547 4990
3.6347 3177
3.7165 1324
3.8001 3479
3.9859 9236
4.0856 4217
4.1877 8322
4.2924 7780
4.3997 8975
4.5685 9343
4.6942 2975
4.8233 2107
4.9559 6239
5.0922 5136
5.2346 1305
5.3916 5144
5.5534 0098
5.7200 0301
5.8916 0310
6.8653 0108
7.1055 8662
7.3542 8215
7.6116 8203
7.8780 9090
61
63
63
64
65
3.8856 3782
3.9730 6467
4.0624 5862
4 1533 6394
4.2473 2588
4.5097 8449
4.6225 2910
4.7380 9233
4.8565 4464
4.9779 5826
5.2322 8827
5.3761 7620
5.5240 2105
5.6759 3162
5.8320 1974
6.0683 5120
6.2504 0173
6.4379 1379
6.6310 5120
6.8299 8273
8.1538 2408
8.4392 0793
8.7345 8020
9.0402 9051
9.3567 0068
60
67
68
69
70
. 4.3428 9071
4.4406 0576
4.5405 1939
4.6426 8107
4.7471 4140
5.1024 0721
5.2299 6739
5.3607 1658
5.4947 3149
5.6321 0286
5.9924 0029
6.1571 9130
6.3265 1406
6.5004 9319
6.6792 5676
7.0348 8222
7.2459 2868
7.4633 0654
7.6872 0574
7.9178 2191
9.6841 8520
10.0231 3168
10.3739 4129
10.7370 2924
11.1128 2526
71
72
73
74
75
4.8539 5208
4.9631 6600
V0748 3723
5.1890 2107
5.3057 7405
5.7729 0543
5,9172 2806
6.0651 5876
6.2167 8773
6.3722 0743
6.8629 3632
7.0516 6706
7.2455 8791
7.4448 4158
7.6495 7472
8.1553 5657
8,4000 1727
8.6520 1778
8.9115 7832
9.1789 2567
11.5017 7414
11.9043 3G24
12.3209 880 1
12.7522 2259
13.1985 5038
76
77
78
79
80
5.4251 5396
5.5472 1993
5.6720 3237
5.7996 5310
5.9301 4530
6.5315 1261
6.6948 0043
6.8621 7044
7.0337 2470
7.2095 6782
7.8599 3802
8.0760 8632
8.2981 7869
8.5263 7861
8.7608 5402
9.4542 9344
9.7379 2224
10.0300 5991
10.3309 6171
10.6408 9056
13.6604 9064
14.1386 1713
14.6334 6873
15.1456 4013
15.6757 3754
81
82
83
84
85
6.0635 7357
6.2000 0397
6.3395 0406
6.4821 4290
6.6279 9112
7.3898 0701
7.5745 5219
7.7639 1599
7.9580 1389
8.1569 6424
9.0017 7751
9.2493 2C39
9.5036 828G
9.7650 3414
10.0335 7258
10.9G01 1727
11.2889 2079
11.6275 8842
11.9764 1607
12.3357 0855
16.2243 8835
16.7922 4195
17.3799 7041
17.9882 6938
18.6178 5881
86
87
88
89
90
6.7771 2092
6.9296 0614
7.0855 2228
7.2449 4653
7.4079 5782
8.3608 8834
8.5C99 1055
8.7841 5832
9.0037 6228
9.2288 5633
10.3094 9583
10.5930 0690
10.8843 1465
11.1836 3331
11.4911 8322
12.7057 7981
13.0869 5320
13.4795 6180
13.8839 4865
14.3004 6711
19.2694 8387
19.9439 1580
20.6419 5285
21.3644 2120
22.1121 7595
91
92
93
94
95
7.5746 3688
7.7450 6621
7.9193 3020
8.0975 1512
8.2797 0921
9.4595 7774
9.6960 6718
9.9384 6886
10.1869 3058
10.4416 0385
11.8071 9076
12.1318 8851
12.4655 1544
12.8083 1711
13.1605 4584
14.7294 8112
15.1713 6556
15.6265 0652
16.0953 0172
16.5781 6077
22.8861 0210
23.6871 1568
24.5161 6473
25.3742 3049
26.2623 2850
96
97
98
99
too
8.4660 0267
8.6564 8773
8.8512 5871
9.0504 1203
9.2540 4630
10.7026 4395
10.9702 1004
11.2444 6530
11.5255 7693
11.8137 1635
13.5224 6085
13.8943 2852
14.2764 2255
14.6690 2417
15.0724 2234
17.0755 0559
17.5877 7076
18.1154 0388
18.6588 6600
19.2186 3198
27.1815 1006
28.1328 6291
29.1175 1311
30.1366 2607
31.1914 0798
T III 27
TABLE III. COMPOTTND AMOUNT OP 1
n
4%
i%
6%
5|%
6%
i
2
3
4
5
.0400 0000
.0816 0000
.1248 6400
.1698 5856
.2166 5290
1.0450 0000
1.0920 2500
1.1411 6613
1.1925 1860
1.2461 8194
1.0500 0000
1.1025 0000
1.1576 2500
1.2155 0625
1.2762 8156
1.0550 0000
1.1130 2500
1.1742 4138
1.2388 2465
1.3069 6001
1.0600 0000
1.1236 0000
1.1910 1600
1.2624 7696
1.3382 2558
6
7
8
9
10
.2653 1902
.3159 3178
.3685 6905
.42,13 1181
.4802 4428
1.3022 6012
1.3608 6183
1.4221 0061
1.4860 9514
1.5529 6942
1.3400 9564
1.4071 0042
1.4774 5544
1.5513 2822
1.6288 9463
1.3788 4281
1,4546 7916
1.5346 8651
1.6190 9427
1.7081 4446
1.4185 1911
1.5036 3026
1.5938 4807
1.6894 7896
1.7908 4770
11
1?
13
14
15
.5394 5406
.6010 3222
.6650 7351
.7316 7645
.8009 4351
1.6228 5305
1.6958 8143
1.7721 9610
1.8519 4492
1.9352 8244
1.7103 3936
1.7958 5633
1.8856 4914
1.9799 3160
2.0789 2818
1.8020 9240
1.9012 0749
2.0057 7390
2.1160 9146
2.2324 7649
1.8982 9856
2.0121 9647
2.1329 2826
2.2609 0396
2.3965 5819
16
17
18
19
20
.8729 8125
.9479 0050
2.0258 1652
2.1068 4918
2.1911 2314
2.0223 7015
2.1133 7681
2.2084 7877
2.3078 6031
2.4117 1402
2.1828 7459
2.2920 1832
2.4066 1923
2.5269 5020
2.6532 9771
2.3552 6270
2.4848 0215
2.6214 6627
2.7656 4691
2.9177 5749
2.5403 5168
2.6927 7279
2.8543 3915
3.0255 9950
3.2071 3547
21
22
23
24
25
2.2787 6807
2.3699 1879
2.4647 1554
2.5633 041(5
2.6658 3633
2.5202 4116
2.6336 5201
2.7521 6635
2.8760 1383
3.0054 3446
2.7859 6259
2.9252 6072
3.0715 2376
3.2250 9994
3.3863 5494
3.0782 3415
3.2475 3703
3.4261 5157
3.6145 8990
3.8133 9235
3.3995 6360
3.6035 3742
3.8197 4966
4.0489 3464
4.2918 7072
26
27
28
29
30
2.7724 6978
2.8833 6S58
2.9987 0333
3.1186 5145
3.2433 9751
3.1406 7901
3.2820 0956
3.4296 9999
3.5840 3G49
3.7453 1813
3.5556 7269
3.7334 5632
3.9201 2914
4.1161 3560
4.3219 4238
4.0231 2893
4.2444 0102
4.4778 4307
4.7241 2444
4.9839 5129
4.5493 8296
4.8223 4594
6.1116 8670
5.4183 8790
5.7434 9117
31
32
33
34
35
3.3731 3341
3.5080 5875
3.6483 8110
3.7943 1634
3.9400 8S99
3.9138 5745
4.0899 8104
4.2740 3018
4.4663 6154
4.6673 4781
4.5380 3949
4.7649 4147
5.0031 8854
5.2533 4797
5.5160 1537
5.2580 6861
5.5472 6238
5.8523 6181
6.1742 4171
6.5138 2501
6.0881 0064
6.4533 8668
6.8405 8988
7.2510 2528
7.68*60 8679
36
37
38
39
40
4.1039 3255
4.2680 8986
4.4388 1345
4.6163 6599
4.8010 2063
4.8773 7846
5.0968 6049
5.3262 1921
5.5658 9908
5.8163 6454
5.7918 1614
6.0814 0694
6.3854 7729
6.7047 5115
7.0399 8871
6.8720 8538
7.2500 5008
7.6488 0283
8.0694 8699
8.5133 0877
8.1472 5200
8.6360 8712
9.1542 6235
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
6.3516 1548
6.6374 3818
6.9361 2290
7.2482 4843
7.3919 8815
7.7615 8756
8.1496 6693
8.5571 5028
8.9850 0779
8.9815 4076
9.4755 2550
9.9966 7940
10.5464 9677
11.1265 5409
10.9028 6101
11.6570 3267
12.2504 5463
12.9854 8191
13.7646 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 5557
8.6436 7107
9.0326 3627
9.4342 5818
9.9059 7109
10.4012 6965
10.9213 3313
11.4673 9979
11.7385 1456
12.3841 3287
13.0652 6017
13.7838 4948
14.5419 6120
14.5904 8748
15.4659 1673
16.3938 7173
17.3775 0403
18.4201 5427
T III 28
TABLE III. COMPOUND AMOUNT OF 1
(1 + )
n
4%
*\%
6%
6|%
6%
1
53
53
54
55
7\3909 5068
7.6865 8871
7.9940 5226
8.3138 1435
8.6463 6692
9.4391 0490
9.8638 6463
10.3077 3853
10.7715 8077
11.2563 0817
12.0407 6978
12.6428 0826
13.2749 4868
13.9386 9611
14.6356 3092
15.3417 6907
16.1855 6637
17.0757 7252
18.0149 4001
19.0057 6171
19.5253 6353
20.6968 8534
21.9386 9846
23.2550 2037
24 6503 2159
56
57
58
59
60
8.9922 2160
9.3519 1046
9.7259 8688
10.1150 2635
10.5196 2741
11.7628 4204
12.2921 6993
12.8453 1758
13.4233 5687
14.0274 0793
15.3674 1246
16.1357 8309
16.9425 7224
17.7897 0085
18.6791 8589
20.0510 7860
21.1538 8793
22.3173 5176
23.5448 0611
24.8397 7045
26,1293 4089
27.6971 0134
29.3589 2742
31.1204 6307
32.9876 9085
61
62
63
64
65
10.9404 1250
11.3780 2900
11.8331 5016
12.3064 7617
12.7987 3522
14.6586 4129
15.3182 '8014
16.0076 0275
16.7279 4487
17.4807 0239
19.6131 4519
20.5938 0245
21.6234 9257
22.7046 6720
23.8399 0056
26.2059 5782
27.6472 8550
29.1678 8620
30.7721 1994
32.4645 8654
34.9669 5230
37.0649 6944
39.2888 6761
41.6461 9967
44.1449 7165
66
67
68
6
70
13.3106 8463
13.8431 1201
14.3968 3649
14.9727 0995
15.5716 1835
18.2673 3100
19.0893 6403
19.9183 8541
20.8460 6276
21.7841 3558
25.0318 9559
26.2834 9037
27.5976 6488
28.9775 4813
30.4264 2554
34.2501 3880
36,1338 9643
38.1212 6074
40.2179 3008
42.4299 1623
46.7936 6994
49.6012 9014
52.5773 6755
55.7320 0960
59.0759 3018
71
72
73
74
75
16.1944 8308
16.8422 6241
17.5159 5290
IS.2165 9102
18.9452 5406
22.7644 2168
23.7888 2066
24.8593 1759
25.9779 8688
27.1469 9629
31.9477 4681
33.5451 3415
35.2223 9086
36.9835 1040
38.8326 8592
44.7635 6163
47.2255 5751
49.8229 6318
52.5632 2615
55,4542 0359
62.6204 8599
60 3777 1515
70.3603 7806
74.5820 0074
79.0569 2079
70
77
78
79
80
19.7030 6485
20.4911 8744
21.3108 3494
22.1632 6834
23.0497 9907
28.3686 1112
29.6451 9862
30.9792 3256
32.3732 9802
33.8300 9643
40.7743 2022
42.8130 3623
44.9536 8804
47.2013 7244
49.5614 4107
58.5041 8479
61.7219 1495
65.1166 2027
68.6980 3439
72.4764 2628
83.8003 3603
88.8283 5620
94.1580 5757
99.8075 4102
105.7959 9348
81
83
83
84
85
23.9717 9103
24.9306 6267
25.9278 8918
20.9650 0175
28.0436 0494
35.3524 5077
36.9433 1106
38.6057 6006
40.3430 1926
42.1584 5513
52.0395 1312
54.6414 8878
57.3735 6322
GO. 2422 4133
63.2543 5344
76.4626 2973
80.6680 7436
85.1048 1845
89.7855 8347
94.7237 9056
112.1437 5309
118.8723 7828
126.0047 2097
133.5650 0423
141.5789 0449
86
87
88
89
90
29.1653 4914
30.3319 6310
31.5452 4163
32.8070 5129
34.1193 3334
44.0555 8561
46.0380 8696
48.1098 0087
50-2747 4191
52.5371 0530
66.4170 7112
69.7379 2467
73.2248 2091
76.8860 6195
80.7303 6505
99.9335 9904
105.4299 4698
111.2285 9407
117.3461 6674
123.8002 0591
150.0736 3875
159.0780 5708
168.6227 4050
178.7401 0493
189.4645 1123
91
92
93
94
95
35.4841 0668
36.9034 7094
38.3796 0978
39.9147 9417
41.5113 8594
54.9012 7503
57.3718 3241
59.9535 6487
62.6514 7529
65.4707 9168
84.7668 8330
89.0052 2747
93.4554 8884
98.1282 6328
103.0346 7645
130.6092 1724
137.7927 2419
145.3713 2402
153.3667 4684
161.8019 1791
200.8323 8190
212.8823 2482
225.6552 6431
239.1945 8017
253.5462 5498
96
97
98
99
100
43.1718 4138
44.8987 1503
46.6946 6363
48.5624 5018
50.5049 4818
68.4169 7730
71.4957 4128
74.7130 4964
78.0751 3687
81.5885 1803
108.1864 1027
113.5957 3078
119.2755 1732
125.2392 9319
131.5012 5785
170.7010 2340
180.0895 7969
189.9945 0657
200.4442 0443
211.4680 3567
268.7590 3028
284.8845 7209
301.9776 4642
320.0903 0520
339.3020 8351
T III 29
TABLE III. COMPOUND AMOTTNT OF 1
(1 + 0"
n
G\%
7%
7|%
8%
8|%
1
2
3
6
1.0650 0000
1.1342 2500
1.2079 4963
1.2864 6635
*-3700 8660
1.0700 0000
1.1449 0000
1.2250 4300
1.3107 9601
1.4025 5173
1.0750 0000
1 1556 2500
1.2422 9688
1.3354 6914
1.4356 2933
1.0800 0000
1.1604 0000
1.2597 1200
1.3604 8896
1.4693 2808
.0850 0000
.1772 2500
.2772 8913
.3858 5870
.5036 5669
e
7
8
10
1.4591 4230
1.5539 8655
1.6549 9567
1.7625 7039
1.8771 3747
1.5007 3035
1.6057 8148
1.7181 8618
1.8384 5921
1.9671 5136
1.5433 0153
1.6590 4914
1.7834 7783
1.9172 3866
2.0610 3156
1.5868 7432
1.7138 2427
1.8509 3021
1 9990 0463
2 1589 2500
.6314 6751
.7701 4225
.9206 0434
2.0838 5571
2.2609 8344
11
12
13
14
15
1.9991 5140
2.1290 9624
2.2674 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 7960
2.5604 1307
2.7524 4405
2.9588 7735
2.3316 3900
2.5181 7012
2.7196 2373
2.9371 9362
3.1721 6911
2.4531 6703
2.6616 8623
2.8879 2956
3.1334 0357
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 2754
3.8696 8446
3.1807 9315
3.4193 5264
3.6758 0409
3.9514 8940
4.2478 5110
3.4259 4264
3.7000 1805
3.9960 1950
4.3157 010G
4.6609 5714
3.6887 2102
4.0022 6231
4.3424 5461
4.7115 6325
5.1120 4612
21
22
23
24
25
3.7526 8199
3.9966 0632
4.2563 8573
4.5330 5081
4.8276 9911
4.1405 6237
4.4304 0174
4.7405 2986
5.0723 6695
5.4274 3264
4.5664 3993
4.9089 2293
5.2770 9215
5.6728 7406
6.0983 3961
5.0338 3372
5.4365 4041
5.8714 6365
6.3411 8074
6.8484 7520
5.5405 7005
6.0180 2850
6.5295 6092
7.0845 7360
7.6867 6236
26
27
28
29
30
6.1414 9955
5.4756 9702
5.8316 1733
C.2106 7245
6.6143 6616
5.8073 5292
6.2138 6763
6.6488 3836
7.1142 5705
7.6122 5504
6.5557 1508
7.0473 9371
7.5759 4824
8.1441 4436
8.7549 5519
7.3963 5321
7.9880 6147
8.G271 0639
9.3172 7490
10.0626 5689
8.3401 3716
9.0490 4881
9.8182 1796
10.6527 6649
11.5582 5164
31
32
33
34
35
7.0442 9990
7.5021 7946
7.9898 2113
8.5091 5950
9.0622 5487
8.1451 1290
8.7152 7080
9.3253 3975
9.9781 1354
10.6765 8148
9.4115 7683
10.1174 4509
10.8762 5347
11.6919 7248
12.5688 7042
10.8676 G944
11.7370 8300
12.G760 4964
13.6901 3361
14.7853 4429
12.5407 0303
13.6066 6279
14.7632 2913
16.0181 0300
17.3796 4241
36
37
38
39
40
9.0513 0143
10.2786 3003
10.9467 4737
11.6582 8595
12.4160 7453
11.4239 4219
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
1S.G252 7503
20.1152 9708
21.7245 2150
18.8509 1201
20.4597 4053
22.1988 2824
24.0857 2805
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 5678
18.3443 5475
19.6284 5959
21.0024 5176
19.3075 5689
20.8523 7366
22.4163 0168
24.0975 2431
25.9048 3863
23.4624 8322
25.3394 8187
27.3660 4042
29.5559 71GG
31.9204 4939
28.3543 2190
30.7644 3927
33.3794 16GO
36.21G6 6702
39.2950 8371
46
47
48
49
60
18.1168 1951
19.2944 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 0153
29.9362 7915
32.1815 0008
34.5951 1259
37.1897 4603
34.4740 8534
37.2320 1217
40.2105 7314
43.4274 1899
46.9016 1251
42.6351 6583
46.2591' 5492
50.1911 8309
54.4574 3365
59.0863 1551
T III 30
TABLE IV. PRESENT VALUE or 1
n
s%
1%
5%
1%
1%
i
3
3
5
0.9958 6062
0.9917 1846
0.9876 0345
0.9835 0551
0.9794 2457
0.9950 2488
0.9900 7450
0.9851 4876
0.9802 4752
0.9753 7067
0.9942 0050
0.9844 3463
0.9827 0220
0.9770 0302
0.9713 3688
0.9925 5583
0.9851 6708
0.9778 3333
0.9705 5417
0.9633 2920
0.9900 9901
0.9802 9605
0.9705 9015
0.9609 8034
0.9514 6569
6
7
8
10
0.9753 6057
0.9713 1343
0.9672 8308
0.9632 6946
0.9592 7249
0.9705 1808
0,9656 8963
0.9G08 8520
0.9561 0468
0.9513 4794
0.9657 0361
0.9G01 0301
0.9545 3489
0.9489 9907
0.9434 9534
0.9561 5802
0.9490 4022
0.9419 7540
0.9349 6318
0.9280 0315
0.9420 4524
0.9327 1805
0.9234 8322
0.9143 3982
0.9052 8695
11
12
13
14
15
0.9552 9211
0.9513 2824
0.9473 8082
0.9434 4978
0.9395 3505
0.9466 1489
0.9119 0534
0.9372 1924
0.9325 5646
0.9279 1688
0.9380 2354
0.9325 8347
0.9271 7495
0.9217 9780
0.9164 5183
0.9210 9494
0.9142 3815
0.9074 3241
0.9006 7733
0.8939 7254
0.8963 2372
0.8874 4923
0.8786 6260
0.8699 6297
0.8613 4947
16
17
18
19
20
0.9356 3656
0.9317 5425
0.9278 8805
0.9240 3789
0.9202 0371
0.9233 0037
0.9187 0684
0.9141 3616
0.9095 8822
0.9050 6290
0.9111 3686
0.9058 5272
0.9005 9923
0.8953 7620
0.8901 8346
0.8873 1766
0.8807 1231
0.8741 5614
0.8676 4878
0.8611 8985
0.8528 21556
0.8443 7749
0.8360 1731
0.8277 3992
0.8195 4447
21
22
23
24
25
0.9163 8544
0.9125 8301
0.9087 9636
0.9050 2542
0.9012 7012
0.9005 6010
0.8960 7971
0.8916 2160
0.8871 8567
0.8827 7181
0.8850 2084
0.8798 8816
0.8747 8525
0.8697 1193
0.8646 6803
0.8547 7901
0.8484 1589
0.8421 0014
0.8358 3140
0.8296 0933
0.8114 3017
0.8033 9621
0.7954 4179
0.7875 6613
0.7797 6844
26
27
28
29
30
0.8975 3041
0.8938 0622
0.8900 9748
0.8864 0413
0.8827 2610
0.8783 7991
0.8740 0986
0.8696 6155
0.8653 3488
0.8610 2973
0.8596 5339
0.8546 6782
0.8497 1118
0.8447 8327
0.8398 8395
0.8234 3358
0.8173 0380
0.8112 1966
0.8051 8080
0.7991 8690
0.7720 4798
0.7644 0392
0.7568 3557
0.7493 4215
0.7419 2292
31
32
33
34
35
0.8790 6334
0.8754 1577
0.8717 8334
0.8681 6599
0.8645 6364
0.8567 4600
0.8524 8358
0.8482 4237
0.8440 2226
0.8398 2314
0.8350 1304
0.8301 7038
0.8253 5581
0.8205 6915
0.8158 1026
0.7932 3762
0.7873 3262
0.7814 7158
0.7756 5418
0.7698 8008
0.7345 7715
0.7273 0411
0.7201 0307
0.7129 7334
0.7059 1420
36
37
38
39
40
0.8609 7624
0.8574 0372
0.8538 4603
0.8503 0310
0.8467 7487
0.8356 4492
0.8314 -8748
0.8273 5073
0.8232 3455
0.8191 3886
0.8110 7897
0.8063 7511
0.8016 9854
0.7970 4908
0.7924 2660
0.7641 4896
0.7584 6051
0.7528 1440
0.7472 1032
0.7416 4796
0.6989 2495
0.6920 0490
0.6851 5337
0.6783 6967
0.6716 5314
41
42
43
44
45
0.8432 6128
0.8397 6227
0.8362 7778
0.8328 0775
0.8293 5211
0.8150 6354
0.8110 0850
0.8069 7363
0.8029 5884
0.7989 6402
0.7878 3092
0.7832 6189
0.7787 1936
0.7742 0317
0.7697 1318
0.7361 2701
0.7306 4716
0.7252 0809
0.7198 0952
0,7144 5114
0.6650 0311
0.6584 1892
0.6518 9992
0.6454 4546
0.6390 5492
46
47
48
49
0.8259 1082
0.8224 8380
0.8190 7100
0.8156 7237
0.8122 8784
0.7949 8907
0.7910 3390
0.7870 9841
0.7831 8250
0.7792 8607
0.7652 4923
0.7608 1116
0.7563 9884
0.7520 1210
0.7476 5080
0.7091 3264
0.7038 5374
0.6986 1414
O.C934 1353
0.6882 5165
0.6327 2764
0.6264 6301
0.6202 6041
0.6141 1921
0.6080 3882
T IV 31
TABLE IV. PRESENT VALUE OP 1
v n
n
M*
1%
s%
1%
1%
51
52
53
54
55
0.8089 1735
0.8055 6084
0.8022 1827
0,7988 8956
0.7955 7467
0.7754 0902
0.7715 5127
0.7677 1270
0.7638 9324
0.7600 9277
0.7433 1480
0.7390 0394
0.7347 1809
0.7304 5709
0.7262 2080
0.6831 2819
0.6780 4286
0.6729 9540
0.6679 8551
0.6630 1291
0.6020 1864
0.5960 5806
0.5901 5649
0.5843 1336
0.5785 2808
56
57
58
59
60
0.7922 7353
0.7889 8608
0.7857 1228
0.7824 6207
0.7792 0538
0.7563 1122
0.7525 4847
0.7488 0445
0.7450 7906
0.7413 7220
0.7220 0908
0.7178 2179
0.7136 5878
0.7095 1991
0.7054 0505
0.6580 7733
06531 7849
0.6483 1612
0.6434 8995
0.6386 9970
0.5728 0008
0.5671 2879
0.5615 1365
0.5559 5411
0.5504 4962
61
62
63
64
65
0.7759 7216
0.7727 5236
0.7695 4591
0-7663 5278
0.7631 7289
0.7376 8378
0.7340 1371
0.7303 6190
0.7267 2826
0.7231 1269
0.7013 1405
0.6972 4678
0.6932 0310
0.6891 8286
0.6851 8594
0.6339 4511
0.6292 2592
0.6245 4185
0.6198 9266
0.6152 7807
0.5449 9962
0.5396 0358
0.5342 6097
0.5289 7126
0,5237 3392
66
67
G8
69
70
0.7600 0620
0.7568 5265
0.7537 1218
0.7505 8474
0.7474 7028
0.7195 1512
0.7159 3544
0.7123 7357
0.7088 2943
0.7053 0291
0.6812 1221
0.6772 6151
0.6733 3373
0.6694 2873
0.6655 4638
0.6106 9784
0.60C1 5170
0.6016 3940
0.5971 6070
0.5927 1533
0.5185 4844
0.5134 1429
0.5083 3099
0.5032 9801
0.4983 1486
71
72
73
74
75
0.7443 6874
0.7412 8008
0.7382 0423
0.7351 4114
0.7320 9076
0.7017 9394
0.6983 0243
0.6948 2829
0.6913 7143
0.6879 3177
0.6616 8654
0.6578 4909
0.6540 3389
0.6502 4082
0.6464 6975
0.5883 0306
0.5839 2363
0.5795 7681
0.5752 6234
0.5709 7999
0.4933 8105
0.4884 9609
0.4836 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
0.6845 0923
0.6811 0371
0.6777 1513
0.6743 4342
0.6709 8847
0.6427 2054
0.6389 9308
0.6352 8724
0.6316 0289
0.6279 3991
0.5667 2952
0.5625 1069
0.5583 2326
0.5541 6701
0.5500 4170
0.4604 3514
0.4647 8726
0.4601 8541
0.4556 2912
0.4511 1794
81
82
83
84
85
0.7140 5246
0.7110 8959
0.7081 3901
0.7052 0067
0.7022 7453
0.6676 5022
0.6643 2858
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
0.5338 4527
0.5298 7123
0.4466 5142
0.4422 2913
0.4378 5063
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 3522
0.6383 4350
0.6064 0384
0.6028 8700
0.5993 9056
0.5959 1439
0.5924 5838
0.5259 2678
0.5220 1169
0.5181 2575
0.5142 6873
0.5104 4043
0.4249 7350
0.4207 6585
0.4165 9985
0.4124 7510
0.4083 9119
91
92
93
94
95
0.6849 7088
0.6821 2868
0.6792 9827
0.6764 7960
0.6736 7263
0.6351 6766
0.6320 0763
0.6288 6331
0.6257 3464
0.6226 2153
0.5890 2242
0.5856 0638
0.5822 1015
0.5788 3363
0.5754 7668
0.5066 4063
0.5028 6911
0.4991 2567
0.4954 1009
0.4917 2217
0.4043 4771
0.4003 4427
0.3963 8046
0.3924 .5590
0.3885 7020
96
97
98
99
100
0.670B 7731
0.6680 9359
0.6653 2141
0.6625 6074
0.6508 1153
0.6195 2391
0.6164 4170
0.6133 7483
0.6103 2321
0.6072 8678
0.5721 3920
0.5688 2108
0.5655 2220
0.5622 4245
0.5589 8172
0.4880 6171
0.4844 2850
0.4808 2233
0.4772 4301
0.4736 9033
0.3847 2297
0.3809 1383
0.3771 4241
0.3734 0832
0.3697 1121
TV 32
TABLE IV. PBESENT VALTTE or 1
n
5%
*r,
2/0
W%
!%
1%
101
102
103
104
105
0.6570 7372
0.6543 4727
0.6516 3214
0.6489 2827
0.6462 3562
0.6042 6545
0.6012 5015
0.5982 6781
0.5952 9136
0.5923 2971
0.5557 3991
0.5525 1689
0.5493 1257
0.5461 2683
0.5429 5957
0.4701 6410
0.4666 6412
0.4631 9019
0.4597 4213
0.4563 1973
0.3600 5071
0.3624 2644
0.3588 3806
0.3552 8521
0.3517 6753
106
107
108
109
110
0.6435 5415
0.6408 8380
0.6382 2453
0.6355 7630
0.6329 3905
0.5893 8279
0.5864 5054
0.5835 3288
0.5806 2973
0.5777 4102
0.5398 1067
0.5366 8004
0.5335 6756
0.5304 7313
0.5273 9665
0.4529 2281
0.4495 5117
0.4462 0464
0.4428 8302
0.4395 8612
0.3482 8469
0.3448 3632
0.3414 2210
0.3380 4168
0.3346 9474
111
112
113
114
115
0.6303 1275
0.6276 9734
0.6250 9279
0.6224 9904
0.6199 1606
0.5748 6669.
0.5720 06G6
0.5691 6085
0.5663 2921
0.5635 1165
0.5243 3801
0.5212 9711
0.5182 7385
0.5152 6812
0.5122 7982
0.4363 1377
0.4330 6577
0.4298 4196
0.4266 4124
0.4234 6615
0.3213 8093
0.3280 9993
0.3248 5141
0.3216 3506
0.3184 5056
116
117
118
119
120
0.6173 4379
0.6147 8220
0.6122 3123
0.6096 9086
0.6071 6102
0.5607 0811
0.5579 1852
0.5551 4280
0.5523 8090
0.5496 3273
0.5093 0885
0.5063 5512
0.5034 1851
0.5004 9893
0.4975 9629
0.4203 1379
0.4171 8491
0.4140 7931
0.4109 9683
0.4079 3730
0.3152 9758
0.3121 7582
0.3090 8497
0.3060 2473
0.3029 9478
121
122
123
124
125
0.6046 4168
0.6021 3279
0.5996 3431
0.5971 4620
0.5946 6842
0.5468 9824
0.5441 7736
0.5414 7001
0.5387 7G12
0.53GO 9565
0.4947 1047
0.4918 4140
0.4889 8396
0.4861 5307
0.4833 3363
0.4049 0055
0.4018 8640
0.3988 9469
0.3959 2525
0.3929 7792
0.2999 9483
0.2970 2459
0.2940 8375
0.2911 7203
0.2882 8914
126
127
128
129
130
0.5922 0091
0.5897 4365
0.5872 9658
0.5848 5966
0.5824 3286
0.5334 2850
0.5307 7463
0.5281 3306
0.5255 0643
0.5228 9197
0.4805 3053
0.4777 4369
0.4749 7302
0.4722 1841
0.4694 7978
0.3900 5252
0.3871 4891
0.3842 6691
0.3814 0636
0.3785 6711
0.2854 3479
0.2826 0870
0.2798 1060
0.2770 4019
0.2742 9722
131
132
133
134
135
0.5800 1613
0.5776 0942
0.5752 1270
0.5728 2593
05704 4906
0.5202 9052
0.5177 0201
0.5151 2637
0.5125 6356
0.5100 1349
0.4667 5703
0.46-10 5007
0.4613 5881
0.4586 8316
0.4560 2303
0.3757 4899
0.3729 5185
0.3701 7553
0.3674 1988
0.3646 8475
0.2715 8141
0.2688 9248
0.2602 3018
0.2635 9424
0.2609 8439
136
137
138
139
140
0.5680 8205
0.5657 2486
0.5633 7745
0.5610 3979
0.5587 1182
0.5074 7611
0.5049 5135
0.5024 3916
0.4999 3946
0.4974 5220
0.4533 7832
0.4507 4895
0.4481 3483
0.4455 3587
0.4429 5198
0.3619 6997
0.3592 7541
0.3566 0090
0.3539 4630
0.3513 1147
0.2584 0039
0.2558 4197
0.2533 0888
0.2508 0087
0.2483 1770
141
142
143
144
145
0.5503 9351
0.5510 8483
0.5517 8572
0.5494 9615
0.5472 1609
0.4949 7731
0.4925 1474
0.4900 6442
0.4876 2628
0.4852 0028
0.4403 8308
0.4378 2908
0.4352 8989
0.4327 6542
0.4302 5560
0.3486 9625
0.3461 0049
0.3435 2406
0.3409 6681
0.3384 2860
0.2458 5911
0.2434 2486
0.2410 1471
0.2386 2843
0.2362 6577
146
147
148
149
150
0.5449 4548
0.5426 8429
0.5404 3249
0.5381 9003
0.5359 5688
0.4827 8835
0.4803 8443
0.4779 9446
0.4756 1637
0.4732 5012
0.4277 6033
0.4252 7953
0.4228 1312
0.4203 6102
0.4179 2313
0.3359 0928
0.3334 0871
0.3309 2676
0.3284 6329
0.3260 1815
0.2339 2650
0.2316 1040
0.2293 1723
0.2270 4676
0.2247 9877
T
TABLE IV. PBESENT VALTTE OF 1
= (!+ ,-)-
n
1|%
1|%
ll%
lf%
2%
1
2
3
4
5
0.9888 7515
0.9778 7407
0.96C9 9537
0.9562 3770
0.9455 9970
0.9876 5432
0.9754 6106
0.9634 1833
0.9515 2428
0.9397 7706
0.9852 2167
0.9706 6175
0.95G3 1699
0.9421 8423
0.9282 6033
0.9828 0098
0.9658 9777
0.9492 8528
0.9329 5851
0.91G9 1254
0.9803 9216
0.9611 6878
0.9423 2233
0.923S 4543
0.9057 3081
6
8
9
10
0.9350 8005
0.9246 7743
0.9143 9054
0.9042 -1808
0.8941 5881
0.9281 7488
0.9167 1593
0.9053 9845
0.8942 2069
0.8831 8093
0.9145 4219
0.9010 2679
0.8877 1112
0.8745 9224
0.8616 6723
0.9011 4254
0.8856 4378
0.8704 1157
0.8554 4135
0.8407 2860
0.8879 7138
0.8705 6018
0.8534 9037
0.8367 5527
0.8203 4830
11
13
13
14
15
0.8842 1142
0.8743 7470
0.8646 4742
0.8550 2835
0.8455 1629
0.8722 7746
0.8615 0860
0.8508 7269
0.8403 6809
0.8299 9318
0.84S9 3323
0.8363 8742
0.8240 2702
0.8118 4928
0.7998 5150
0.8262 6889
0.8120 5788
0.7980 9128
0.7843 6490
0.7708 7459
0.8042 6304
0.7884 9318
0.7730 3253
0,7578 7502
0.7430 1473
16
17
18
19
20
0.8361 1005
0.82G8 0846
0.8176 1034
0.8085 1455
0.7995 1995
0.8197 4635
0.809S 2602
0.7996 3064
0.7897 5866
0.7800 0855
0.7880 3104
0.7763 8526
0.7649 1159
0.7536 0747
0.7424 7042
0.7576 1631
0.7445 8605
0.7317 7990
0.7191 9401
0.7068 2458
0.7284 4581
0.7141 6256
0.7001 5937
0.6864 3076
0.6729 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 6796
0.7514 7453
0.7421 9707
0.7330 3414
0.7314 9795
0.7206 8763
0.7100 3708
0.6995 4392
0.6892 0583
0.6046 6789
0.6827 2028
0.6709 7817
0.6594 3800
O.G4SO 9632
0.6597 7582
0.64G8 3904
0.6341 5592
0.6217 2149
0.6095 3087
26
27
28
29
30
0.7476 1516
0.7392 9806
0.7310 7348
0.7229 4040
0.7148 9780
0.7239 8434
0.7150 4626
O.7062 1853
0.6974 9978
0.6888 8SG7
0.6790 2052
0.6G89 8574
0.6590 9925
O.G493 5887
O.G397 6213
0.6360 4970
0.6259 9479
0.6152 2829
0.6046 4697
0.5942 4764
0.5975 7928
0.5858 6204
0.5743 7455
0.5631 1231
0.5520 7089
31
32
33
34
35
0.7069 4467
0.6990 8002
0.6913 0287
0.6836 1223
0.6760 0715
O.G803 8387
0.6719 8407
0.6636 8797
0.6554 9429
0.6474 0177
O.G303 0781
0.6209 9292
0.6118 1568
O.G027 7407
0.5933 6G08
0.5840 2716
0.5739 8247
0.5G41 1053
0.5544 0839
0.5448 7311
0.5412 4597
0.5306 3330
0.5202 2873
0.5100 2817
0.5000 2761
36
37
3S
39
40
0.6684 8667
0.6610 4986
0.6536 9578
0.6464 2352
0.6392 3216
0.6394 0916
0.6315 1522
0.0237 1873
0.61GO 1850
0.6084 1334
0.5850 8974
0.5764 4309
0.5679 2423
0.5505 3126
0.5512 6232
0.5355 0183
0.5262 9172
0.5172 4002
0.5083 4400
0.4996 0098
0.4902 2315
0.4806 1093
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 5789
0.6044 5774
O.G009 0206
0.5934 8352
0.5SG1 5G56
0.5780 2006
0.5717 7290
0.5431 1559
0.53 r /0 8925
0.5271 8153
0.5193 0067
0.5117 1494
0.4910 0834
0.4825 6348
0.4742 6386
0.4661 0699
0.4580 9040
0.4440 1021
0.4353 0413
0.4267 6875
0.4184 0074
0.4101 9680
46
47
48
49
50
0.5977 3324
0.5910 8355
0.5845 0784
0.5780 0528
0.5715 7506
0.5647 1397
0.5577 4219
0.5508 5649
0.5440 5579
0.5373 3905
0.5041 5265
0.4967 0212
0.4893 6170
0.4821 2975
0.4750 0468
0.4502 1170
0.4424 6850
0.4348 5848
0.4273 7934
0.4200 2883
0.4021 5373
0.3942 6836
0.3865 3761
0.3789 5844
0.3715 2788
T IV 34
TABLE IV. PRESENT VALTTB OF 1
v = (1 + t)-"
n
*i%
l|%
1*cr
*2%
1|%
2%
51
59
53
54
55
0.5652 1637
0.5589 2843
0.5527 1044
0.5465 6162
0.5404 8120
0.5307 0524
0.5241 5332
0.5176 8229
0.5112 9115
0.5049 7892
0.4679 8491
0.4610 6887
0.4542 5505
0.4475 4192
0.4409 2800
0.4128 0475
0.4057 0492
0.3987 2719
0.3918 6947
0.3851 2970
0.3642 4302
0.3571 0100
0.3500 9902
0.3432 3433
0.3365 0425
50
57
58
50
60
0.5344 6843
0.5285 22f>6
0.5226 4282
0.5168 2850
0.5110 7887
0.4987 4461
0.4925 8727
0.4865 0594
0.4804 9970
0.4745 6760
0.4344 1182
0.4279 9194
0.4216 6694
0.4154 3541
0.4092 9597
0.3785 0585
0.3719 9592
0.3655 9796
0.3593 1003
0.3531 3025
0.3299 0613
0.3234 3738
0.3170 9547
0.3108 7791
0.3047 8227
61
69
63
64
65
0.5053 9319
0.4997 7077
0.4942 1090
0.4887 1288
0.4832 7602
0.4687 0874
0.4629 2222
0.4572 0713
0.4515 6259
0.4459 8775
0.4032 4726
0.3972 8794
0.3914 1669
0.3856 3221
0.3799 3321
0.3470 5676
0.3410 8772
0.3352 2135
0.3294 5587
0.3237 8956
0.2988 0614
0.2929 4720
0.2872 0314
0.2815 7170
0.2760 5069
66
67
68
69
70
0.4778 9965
0.4725 8309
0.4673 2568
0.4621 2675
0.4569 8566
0.4404 8173
0.4350 4368
0.4296 7277
0.4243 6817
0.4191 2905
0.3743 1843
0.3687 8663
0.3633 3658
0.3579 6708
0.3526 7692
0.3182 2069
0.3127 4761
0.3073 6866
0.3020 8222
0.2968 8670
0.2706 3793
0.2653 3130
0.2601 2873
0.2550 2817
0.2500 2761
71
73
73
74
75
0.4519 0177
0.4468 7443
0.4419 0302
0.4369 8692
0/321 2551
0.4139 5462
0.4088 4407
0.4037 9661
0.3988 1147
0.3938 8787
0.3474 6495
0.3423 3000
0.3372 7093
0.3322 8663
0.3273 7599
0.2917 8054
0.2867 6221
0.2818 3018
0.2769 8298
0.2722 1914
0.2451 2511
0.2403 1874
0.2356 0661
0.2309 8687
0.2264 5771
76
77
78
79
80
0.4273 1818
0.4225 6433
0.4178 6337
0.4132 1470
0.4086 1775
0.3890 2506
0.3842 2228
0.3794 7879
0.3747 9387
0.3701 6679
0.3225 3793
0.3177 7136
0.3130 7523
0.3084 4850
0.3038 9015
0.2675 3724
0.2629 3586
0.2584 1362
0.2539 6916
0.2496 0114
0.2220 1737
0.2176 6408
0.2133 9616
0.2092 1192
0.2051 0973
81
82
83
84
85
0.4040 7194
0.3995 7670
0.3951 3148
0.3907 3570
0.3863 8882
0.3655 9683
0.3610 8329
0.3566 2547
0.3522 2268
0.3478 7426
0.2993 9916
0.2949 7454
0.2906 1531
0.2863 2050
0.2820 8917
0.2453 0825
0.2410 8919
0.2369 4269
0.2328 6751
0.2288 6242
0.2010 8797
0,1971 4507
0.1932 7948
0.1894 8968
0.1857 7420
86
87
88
89
90
0.3820 9031
0.3778 3961
0.3736 3621
0.3694 7956
0.3653 6916
0.3435 7951
0.3393 3779
0.3351 4843
0.3310 1080
0.3269 2425
0.2779 2036
0.2738 1316
0.2697 6666
0.2657 7997
0.2618 5218
0.2249 2621
0.2210 5770
0.2172 5572
0.2135 1914
0.2098 4682
0.1821 3157
0.1785 6036
0.1750 5918
0.1716 2665
0.1682 6142
01
93
93
84
95
0.3613 0448
0.3572 8503
0.3533 1029
0.3493 7976
0.3454 9297
0.3228 8814
0.3189 0187
0.3149 6481
0.3110 7636
0.3072 3591
0.2579 8245
0.2541 6990
0.2504 1369
0.2467 1300
0.2430 6699
0.2062 3766
0.2026 9057
0.1992 0450
0.1957 7837
0.1924 1118
0.1649 6217
0.1617 2762
0.1585 5649
0.1554 4754
0.1523 9955
96
97
98
99
100
0.3416 4941
0.3378 4861
0.3340 9010
0.3303 7340
0.3266 9805
0.3034 4287
0.2996 9668
0.2959 9670
0.2923 4242
0.2887 3326
0.2394 7487
0.2359 3583
0.2324 4909
0.2290 1389
0.2256 2944
0.1891 0190
0.1858 4953
0.1826 5310
0.1795 1165
0.1764 2422
0.1494 1132
0.1464 8169
0.1436 0950
0.1407 9363
0.1380 3297
T IV 35
TABLE IV. PRESENT VALUE OF 1
n
2|%
!%
2|%
3%
3|%
i
2
3
5
0.9779 9511
0.9564 7444
0.9354 2732
0.9148 4335
0.8947 1232
0,9756 0976
0.9518 1440
0.9285 9941
0.9059 5064
0.8838 5429
0.9732 3601
0.9471 8833
0.9218 3779
0.8971 6573
0.8731 5400
0.9708 7379
0.9425 9591
0.9151 4166
0.8884 8705
0.8626 0878
0.9661 8357
0.9335 1070
0.9019 4271
0.8714 4223
0.8419 7317
6
8
10
0.8750 2427
0.8557 6946
0.8369 3835
0.8185 2161
0.8005 1013
0.8622 9687
0.8412 6524
0.8207 4657
0.8007 2836
0.7811 9840
0.8497 8491
0.8270 4128
0.8049 OC35
0.7833 6385
0.7623 9791
0.8374 8426
0.8130 9151
0.7894 0923
0.7664 1673
0.7440 9391
0.8135 0064
0.7859 9096
0.7594 1156
0.7337 3097
0.7089 1881
11
13
13
14
15
0.7828 9499
0.7656 6748
0.7488 1905
0.7323 4137
0.7162 2628
0.7621 4478
0.7435 5589
0.7254 2038
0.7077 2720
0.6904 6556
0.7419 9310
0.7221 3440
0.7028 0720
0.6339 9728
0.6656 9078
0.7224 2128
0.7013 7988
0.6809 5134
0.6611 1781
O.C>418 6195
0.6849 4571
0.6617 8330
0.6394 0415
0.6177 8179
0.5968 9062
16
17
18
19
20
0.7004 6580
0.6850 5212
0.6699 7763
0.6552 3484
0.6408 1647
0.6736 2493
0.6571 9506
0.6411 6591
0.6255 2772
0.6102 7094
0.6478 7424
0.6305 3454
0.6136 5892
0.5972 3496
0.5812 5057
0.6231 6694
0.6050 1645
0.5873 9461
0.5702 8603
0.5536 7575
0.5767 0591
0.5572 0378
0.5383 6114
0.5201 5569
0.5025 6588
21
23
23
24
26
0.6267 1538
0.6129 2457
0.5994 3724
0.5862 4668
0.5733 4639
0.5953 8629
0.5808 6467
0.5666 9724
0.5528 7535
0.5393 9059
0.5656 9308-
0.5505 5375
0.5358 1874
0.5214 7809
0.5075 2126
0.5375 4928
0.5218 9250
0.50G6 9175
0.4919 3374
0.4776 0557
0.4855 7090
0.4691 5063
0.4532 8563
0.4379 5713
0.4231 4699
36
27
28
2
30
0.5607 2997
0.5483 9117
0.5363 2388
0.5245 2213
0.5129 8008
0.5262 3472
0.5133 9973
0.5008 7778
0.4886 6125
0.4767 4269
0.4939 3796
0.4807 1821
0.4678 5227
0.4553 3068
0.4431 4421
0.4638 9473
0.4501 8906
0.4370 7675
0.4243 4636
0.4119 8676
0.408S 3767
0.3950 1224
0.3816 5434
0.3687 4815
0.3562 7841
31
32
33
34
35
0.5016 9201
0.4906 5233
0.4798 5558
0.4692 9641
0.4589 6960
0.4651 1481
0.4537 7055
0.4427 0298
0.4319 0534
0.4213 7107
0.4312 8301
0.4197 4103
0.4085 0708
0.3975 7380
0.3869 3314
0.3999 8715
0.3883 3703
0.3770 2625
0.3660 4490
0.3553 8340
0.3442 3035
0.3325 8971
0.3213 4271
0.3104 7605
0.2999 7686
36
37
38
39
49
0.4488 7002
0.4389 9268
0.4293 3270
0.4198 8528
0.4106 4575
0.4110 9372
0.4010 6705
0.3912 8492
0.3817 4139
0.3724 3062
0.3765 7727
0.3664 9856
0.3566 8959
0.3471 4316
0.3378 5222
0.3450 3243
0.3349 8294
0.3252 26 J 5
0.3157 5355
0.3065 5684
0.2898 3272
0.2800 3161
0.2705 6194
0.2614 1250
0.2525 7247
41
43
43
44
45
0.4016 0954
0.3927 7216
0.3B41 2925
0.3756 7653
0.3674 0981
0.3633 4695
0.3544 8483
0.3458 3886
0.3374 0376
0.3291 7440
0.3288 0995
0.3200 0968
0.3114 4495
0.3031 0944
0.2949 9702
0.2976 2800
0.2889 5922
0.2805 4294
0.2723 7178
0.2044 3862
0.2440 3137
0.2357 7910
0.2278 0590
0.2201 0231
0.2126 5924
43
47
48
49
50
0.3593 2500
0.3514 1809
0.3436 8518
0.3361 2242
0.3287 2608
0.3211 4576
0.3133 1294
0.3056 7116
0.2082 1576
0.2909 4221
0.2871 0172
0.2794 1773
0.2719 3940
0.2646 6122
0.2575 7783
0.2567 3653
0.2492 5876
0.2419 9880
0.2349 5029
0.2281 0708
0.2054 6787
0.1985 1968
0.1918 0645
0.1853 2024
0.1790 5337
T IV 36
TABLE IV. PKESENT VALUE OF 1
t," = (1 + ,T"
n
2|%
2|%
2f%
3%
3|%
51
53
II
55
0.3214 9250
0.3144 1810
0,3074 9936
0.3007 3287
0.2941 1528
0.2838 4606
0.2769 2298
0.2701 6876
0.2635 7928
0.2571 5052
0.2506 8402
0.2439 7471
0.2374 4497
0.2310 9000
0.2249 0511
0.2214 6318
0.2150 1280
0.2087 5029
0.2026 7019
0.1967 6717
0.1729 9843
0.1671 4824
0.1614 9589
0.1560 3467
0.1507 5814
56
57
58
59
60
0.2876 4330
0.2813 1374
0.2751 2347
0.2690 6940
0.2631 4856
0.2508 7855
0.2447 5956
0.2387 8982
0.2329 6568
0.2272 8359
0,2188 8575
0.2130 2749
0.2073 2603
0.2017 7716
0.1963 7679
0.1910 3609
0.1854 7193
0.1800 6984
0.1748 2508
0.1697 3309
0.1456 6004
0.1407 3433
0.1359 7520
0.1313 7701
0.1269 3431
61
63
63
64
65
0.2573 5801
0.2516 9487
0.2461 5635
0.2407 3971
0,2354 4226
0.2217 4009
0.2163 3179
0.2110 5541
0.2059 0771
0.2008 8557
0.1911 2097
0.1860 0581
0.1810 2755
0.1761 8253
0.1714 6718
0.1647 8941
0,1599 8972
0.1553 2982
0.1508 0565
0.1464 1325
0.1226 4184
0.1184 9453
0.1144 8747
0.1106 1591
0.1068 7528
66
67
68
69
70
0.2302 6138
0.2251 9450
0.2202 3912
0.2153 9278
0.2106 5309
0.1959 8593
0.1912 0578
0.1865 4223
0.1819 9241
0.1775 5358
0.1668 7804
0.1624 1172
0.1580 6493
0.1538 3448
0.1497 1726
0.1421 4879
0.1380 0853
0.1339 8887
0.1300 8628
0.1262 9736
0.1032 6114
0.0997 6922
0.0963 9538
0.0931 3563
0.0899 8612
71
73
73
74
75
0.2060 1769
0.2014 8429
0.1970 5065
0.1927 1458
0.:884 7391
0.1732 2300
0.1689 9805
0.1648 7615
0.1608 5478
0.1569 3149
0.1457 1023
0.1418 1044
0.1380 1503
0.1343 2119
0.1307 2622
0.1226 1880
0.1190 4737
0.1155 7998
0.1122 1357
0.1089 4521
0.0869 4311
0.0840 0300
0.0811 6232
0.0784 1770
0.0757 6590
76
77
73
79
80
0.1843 2657
0.1802 7048
0.1763 0365
0.1724 2411
0.1686 2993
0.1531 0389
0.1493 6965
0.1457 2649
0.1421 7218
0.1387 0457
0.1272 2747
0.1238 2235
0.1205 0837
0.1172 8309
0.1141 4412
0.1057 7205
0.1026 9131
0.0997 0030
0.0967 9641
0.0939 7710
0.0732 0376
0.0707 2827
0.0683 3650
0.0660 2560
0.0637 9285
81
83
83
84
85
0.1649 1925
0.1612 9022
0.1577 4105
0.1542 6997
0.1508 7528
0.1353 2153
0.1320 2101
0.1288 0098
0.1256 5949
0.1225 9463
0.1110 8917
0.1081 1598
0.1052 2237
0.1024 0620
0.0996 6540
0.0912 3990
0.0885 8243
0.0860 0236
0.0834 9743
0.0810 6547
0.0616 3561
0.0595 5131
0.0575 3750
0.0555 9178
0.0537 1187
86
87
88
89
90
0.1475 5528
0.1443 0835
0.1411 3286
0.1380 2724
0.1349 8997
0.1196 0452
0.1166 8733
0.1138 4130
0.1110 6468
0.1083 5579
0.0969 9795
0.0944 0190
0.0918 7533
0.0894 1638
0.0870 2324
0.0787 0434
0.0764 1198
0.0741 8639
0.0720 2562
0.0699 2779
0.0518 9553
0.0501 4060
0.0484 4503
0.0468 0679
0.0452 2395
91
93
93
94
95
0.1320 1953
0.1291 1445
0.1262 7331
0.1234 9468
0.1207 7719
0.1057 1296
0.1031 3460
0.1006 1912
0.0981 6500
0.0957 7073
0.0846 9415
0.0824 2740
0.0802 2131
0.0780 7427
0.0759 8469
0.0678 9105
0.0659 1364
0.0639 9383
0.0621 2993
0.0603 2032
0.0436 9464
0.0422 1704
0.0407 8941
0.0394 1006
0.0380 7735
96
97
98
99
100
0.1181 1950
0.1155 2029
0.1129 7828
0.1104 9221
0.1080 6084
0.0934 3486
0.0911 5596
0.0889 32C4
0.0867 6355
0.0846 4737
0.0739 5104
0.0719 7181
0.0700 4556
0.0681 7086
0.0663 4634
0.0585 6342
0.0568 5769
0.0552 0164
0.0535 9383
0.0520 3284
0.0367 8971
0.0355 4562
0.0343 4359
0.0331 8221
0.0320 6011
T TV 37
TABLE IV. PRESENT VALUE OF 1
n
4%
4|%
6%
5|%
6%
i
3
3
4
5
0.9615 3846
0.9245 5621
0.8889 9636
0.8548 0419
0.8219 2711
0.9569 3780
0.9157 2995
0.8762 9660
0.8385 6134
0.8024 5105
0.9523 8095
0.9070 2948
0.8638 3760
0.8227 0247
0.7835 2617
0.9478 6730
0.8984 5242
0.8516 1366
0.8072 1674
0.7651 3435
0.9433 9623
0.8899 9644
0.8396 1928
0.7920 9366
0.7472 5817
8
10
0.7903 1453
0.7599 1781
0.7306 9021
0.7025 8674
0.6755 6417
0.7678 9574
0.7348 2846
0.7031 8513
0.6729 0443
0.6439 2768
0.7462 1540
0.7106 8133
0.6768 3936
0.6446 0892
0.6139 1325
0.7252 4583
O.C874 3681
0.6515 9887
0.6176 2926
0.5854 3058
0.7049 6054
0.6650 5711
0.6274 1237
0.5918 9846
0.5583 9478
11
13
13
14
15
0.6495 8093
0.6245 9705
0.6005 7409
0.5774 7508
0.5552 6450
0.6161 9874
0.5896 6386
0.5642 7164
0.5399 7286
0.5167 2044
0.5846 7929
0.5568 3742
0.5303 2135
0.5050 6795
0.4810 1710
0.5549 1050
0.5259 8152
0.4985 6068
0.4725 6937
0.4479 3305
0.5267 8753
0.4969 6936
0.4688 3902
0.4423 0006
0.4172 6506
10
17
IS
19
30
0.5339 0818
0.5133 7325
0.4936 2812
0,4746 4242
0.4563 8695
0.4944 6932
0.4731 7639
0.4528 0037
0.4333 0179
0.4146 4286
0.4581 1152
0.4362 9669
0.4155 2065
0.3957 3396
0.3768 8948
0.4245 8109
0.4024 4653
0.3814 6590
0.3615 7906
0.3427 2896
0.3936 4628
0.3713 6442
0.3503 4379
0.3305 1301
0.3118 0473
21
23
23
24
25
0.4388 3360
0.4219 5539
0.4057 2633
0.3901 2147
0.3751 1680
0.3967 8743
0.3797 0089
0.3633 5013
0.3477 0347
0.3327 3060
0.3589 4236
0.3418 4987
0.3255 7131
0.3100 6791
0.2953 0277
0.3248 6158
0,3079 2567
0.2918 7267
0.27 06 5656
0.2622 3370
0.2941 5540
0.2775 0510
0.2617 9726
0.2469 7855
0.2329 9863
26
27
28
29
30
0.3606 8923
0.3468 1657
0.3334 7747
0.3206 6141
0.3083 1867
0.3184 0248
0.3046 9137
0.2915 7069
0.2790 1502
0.2670 0002
0.2812 4073
0,2678 4832
0.2550 9364
0.2429 4632
0.2313 7745
0.2485 6275
0.2356 0450
0.2233 2181
0.2116 7944
0.2006 4402
0.2198 1003
0.2073 6795
0.1956 3014
0.1845 5674
0.1741 1013
31
32
33
34
35
0.2964 6026
0.2850 5754
0.2740 9417
0.2635 5209
0.2534 1547
0.2555 0241
0.2444 9991
0.2339 7121
0.2238 9589
0.2142 5444
0.2203 5947
0.2098 6617
0.1998 7254
0.1903 5480
0.1812 9029
0.1901 8390
0.1802 6910
0.1708 7119
0.1619 6321
0.1535 1963
0.1642 5484
0.1549 5740
0.1461 8622
0.1379 1153
0.1301 0522
36
37
38
39
40
0.2436 6872
0.2342 9685
0.2252 8543
0.2166 2061
0.2082 8904
0.2050 2817
0.1961 9921
0.1877 5044
0.1796 6549
0.1719 2870
0.1726 5741
0.1644 3563
0.1566 0536
0.1491 4797
0.1420 4568
0.1455 1624
0.1379 3008
0.1307 3941
0.1239 2362
0.1174 6314
0.1227 4077
0.1157 9318
0.1092 3885
0.1030 6552
0.0972 2219
41
42
43
44
45
0.2002 7793
0.1925 7493
0.1851 6820
0.1780 4635
0.1711 9841
0.1645 2507
0.1574 4026
0.1506 6054
0.1441 7276
0.1379 6437
0.1352 8160
0.1288 3962
0.1227 0440
0.1168 6133
0.1112 9651
0.1113 3947
0.1055 3504
0.1000 3322
0.0948 1822
0.0898 7509
0.0917 1905
0.0865 2740
0.0816 2962
0.0770 0908
0.0726 5007
46
47
48
49
50
0.1646 1386
0.1582 8256
0.1521 9476
0.1463 4112
0.1407 1262
0.1320 2332
0.1263 3810
0.1208 9771
0.1156 9158
0.1107 0965
0.1059 9668
0.1009 4921
0.0961 4211
0.0915 6391
0.0872 0373
0.0851 8965
0.0807 4849
0.0765 3885
0.0725 4867
0.0687 6652
0.0685 3781
0.0646 5831
0.0609 9840
0.0575 4566
0.0542 8836
TABLE IV. PRESENT VALUE OF 1
v* (i +i)~ n
n
4%
4|%
5%
5l%
6%
61
52
53
54
55
0.1353 0059
0.1300 9672
0.1250 9300
0-1202 8173
0,1156 5551
0.1059 4225
0.1013 8014
0.0970 1449
0.0928 3683
0.0888 3907
0.0830 5117
0.0790 9635
0.0753 2986
0.0717 4272
0.0683 2640
0.0651 8153
0.0617 8344
0.0585 6250
0.0555 0948
0.0526 1562
0.0512 1544
0.0483 1645
0.0455 8156
0.0430 0147
0.0405 6742
56
57
68
59
60
0.1112 0722
0.1069 3002
0.1028 1733
0.0988 6282
0.0950 6040
0.0850 1347
0.0813 5260
0.0778 4938
0.0744 9701
0.0712 8901
0.0650 7276
0.0619 7406
0.0590 2291
0.0562 1230
0.0535 3552
0.0498 7263
0.0472 7263
0.0448 0818
0.0424 7221
0.0402 5802
0.0382 7115
0.0361 0486
0.0340 6119
0.0321 3320
0.0303 1434
61
62
63
64
65
0.0914 0423
0.0878 8868
0.0845 0835
0.0812 5803
0.0781 3272
0.0682 1915
0.0652 8148
0.0624 7032
0.0597 8021
0.0572 0594
0.0509 8621
0.0485 5830
0.0462 4600
0.0440 4381
0.0419 4648
0.0381 5926
0.0361 6992
0.0342 8428
0.0324 9695
0.0308 0279
0.0285 9843
0.0269 7965
0.0254 5250
0.0240 1179
0.0226 5264
66
67
68
69
70
0.0751 2762
0.0722 3809
0.0694 5970
0.0667 8818
0.0642 1940
0.0547 4253
0.0523 8519
0.0501 2937
0.0479 7069
0.0459 0497
0.0399 4903
0.0380 4670
0.0362 3495
0.0345 0948
0.0328 6617
0.0291 9696
0.0276 7485
0.0262 3208
0.0248 6453
0.0235 6828
0.0213 7041
0.0201 6077
0.0190 1959
0.0179 4301
0.0169 2737
71
72
73
74
75
0.0617 4942
0.0593 7445
0.0570 9081
0.0548 9501
(X0527 8367
0.0439 2820
0.0420 365.5
0.0402 2637
0.0384 9413
0.0368 3649
0.0313 0111
0.0298 1058
0.0283 9103
0.0270 3908
0.0257 5150
0.0223 3960
0.0211 7498
0.0200 7107
0.0190 2471
0.0180 3290
0.0159 6921
0.0150 6530
0.0142 1254
0.0134 0806
0.0126 4911
76
77
78
79
80
0.0507 5353
0.0488 0147
0.0469 2449
0.0451 1970
0.0433 8433
0.0352 5023
0.0337 3228
0.0322 7969
0.0308 8965
0.0295 5948
0.0245 2524
0.0233 5737
0.0222 4512
0.0211 8582
0.0201 7698
0.0170 0279
0.0162 0170
0.0153 5706
0.0145 5646
0.0137 9759
0.0119 3313
0.0112 5767
0.0106 2044
0.0100 1928
0.0094 5215
81
82
83
84
85
0.0417 1570
0.0401 1125
0.0385 6851
0.0370 8510
0.0356 5875
0.0282 8658
0.0270 6850
0.0259 0287
0.0247 8744
0.0237 2003
0.0192 1617
0.0183 0111
0.0174 2963
0.0165 9965
0.0158 0919
0.0130 7828
0.0123 9648
0.0117 5022
0.0111 3765
0.0105 5701
0.0089 1713
0.0084 1238
0.0079 3621
0.0074 8699
0.0070 6320
86
87
88
89
90
0.0342 8726
0.0329 6852
0.0317 0050
0.0304 8125
0.0293 0890
0.0226 9860
0.0217 2115
0.0207 8579
0.0198 9070
0.0190 3417
0.0150 5637
0.0143 3940
0.0136 5657
0.0130 0626
0.0123 8691
0.0100 0664
0.0094 8497
0.0089 9049
0.0085 2180
0.0080 7753
0.0066 6340
0.0062 8622
0.0059 3040
0.0055 9472
0.0052 7803
91
92
93
94
95
0.0281 8163
0.0270 9772
0.0260 5550
Q.0250 5337
0.0240 8978
0.0182 1451
0.0174 3016
0.0166 7958
0.0159 6132
0.0152 7399
0.0117 9706
0.0112 3530
0.0107 0028
0.0101 9074
0.0097 0547
0.0076 5643
0.0072 5728
0.0068 7894
0.0065 2032
0.0061 8040
0.0049 7928
0.0046 9743
0.0044 3154
0.0041 8070
0.0039 4405
96
97
98
99
100
0.0231 6325
0.0222 7235
0.0214 1572
0.0205 9204
0.0198 0004
0.0146 1626
0.0139 8685
0.0133 8454
0.0128 0817
0.0122 56C3
0.0092 4331
0.0088 0315
0.0083 8395
0.0079 8471
0.0076 0149
0.0058 5820
0.0055 5279
0.0052 6331
0.0049 8892
0.0047 2883
0.0037 2081
0.0035 1019
0.0033 1150
0.0031 2406
0.0029 4723
T IV 39
TABLE IV. PRESENT VALUE OF 1
v n - (1 + *)-"
n
6|%
7%
7l%
8%
8|%
i
2
0.9389 6714
0.8816 5928
0.8278 4909
0.7773 2309
0.7298 8084
0.9345 7944
0.8734 3873
0.8162 9788
0.7628 9521
0.7129 8618
0.9302 3256
0.8653 3261
0.8049 6057
0.7488 0053
0.6965 5863
0.9259 2593
0.8573 3882
0.7938 3224
0.7350 2985
0.6805 8320
0.9216 5899
0.8494 5529
0.7829 0810
0.7215 7428
0.6650 4542
8
9
10
0.6833 3412
0.6435 0621
0.6042 3119
0.5673 5323
0.5327 2604
0.6663 4222
0.6227 4974
0.5820 0910
0.5439 3374
0.5083 4929
0.6479 6152
0.6027 5490
0.5607 0223
0.5215 8347
0.4851 9393
0.6301 6963
0.5834 9040
0.5402 6888
0.5002 4897
0.4631 9349
0.6129 4509
0.5649 2635
0,5206 6945
0.4798 7968
0.4422 8542
11
12
13
14
15
0.5002 1224
0.4696 8285
0.4410 1676
0.4141 0025
0.3888 2652
0.4750 9280
0.4440 1196
0.4149 6445
0.3878 1724
0.3624 4602
0.4513 4319
0.4198 5413
0.3905 6198
0.3633 1347
0.3379 6602
0.4288 8286
0.3971 1376
0.3676 9792
0.3404 6104
0.3152 4170
0.4076 3633
0.3757 0168
0.3462 6883
0.3191 4178
0.2941 3989
16
17
18
19
20
0.3650 9533
0.3428 1251
0.3218 8969
0.3022 4384
0.2837 9703
0.3387 3460
0.3165 7439
0.2958 6392
0.2765 0832
0.2584 1900
0.3143 8699
0.2924 5302
0.2720 4932
0.2530 6913
0.2354 1315
0.2918 9047
0.2702 6895
0.2502 4903
0.2317 1206
0.2145 4821
0.2710 9667
0.2498 5869
0.2302 8450
0.2122 4378
0.1956 1639
21
22
23
24
25
0.2664 7608
0.2502 1228
0.2349 4111
0.2206 0198
0.2071 3801
0.2415 1309
0.2257 1317
0.2109 4688
0.1971 4662
0.1842 4918
0.2189 8897
0.2037 1067
0.1894 9830
0.1762 7749
0.1639 7906
0.1986 5575
0.1839 4051
0.1703 1528
0.1576 9934
0.1460 1790
0.1802 9160
0.1661 6738
0.1531 4965
0.1411 5176
0.1300 9378
20
27
28
29
30
0.1944 9579
0.1826 2515
0.1714 7902
0.1610 1316
0.1511 8607
0.1721 9549
0.1609 3037
0.1504 0221
0.1405 6282
OJ313 6712
0.1525 3866
0.1418 9643
0.1319 9668
0.1227 8761
0.1142 2103
0.1352 0176
0.1251 8682
0.1159 1372
0.1073 2752
0.0993 7733
0.1199 0210
0.1105 0885
0.1018 5148
0.0938 7233
0.0865 1828
31
32
33
34
35
0.1419 5875
0.1332 9460
0.1251 5925
0.1175 2042
0.1103 4781
0.1227 7301
0.1147 4113
0.1072 3470
0.1002 1934
0.0936 6294
0.10G2 5212
0.0088 3918
0.0919 4343
0.0855 2S77
0.0795 6164
0.0920 1G05
0.0852 0005
0.0788 8893
0.0730 4531
0.0676 3454
0.0797 4035
0.0734 9341
0.0677 3586
0.0624 2936
0.0575 3858
36
37
38
39
40
0.1036 1297
0.0972 8917
0.0913 5134
0.0857 7590
0.0805 4075
0.0875 3546
0.0818 0884
0.0764 5686
0.0714 5501
0.0667 8038
0.0740 1083
0.0688 4729
0.0640 4399
0.0595 7580
0.0554 1935
00626 2458
0.0579 8572
00536 9048
0.0497 1341
0.0460 3093
0.0530 3095
0.0488 7645
0.0450 4742
0.0415 1836
0.0382 6577
41
42
43
44
45
0.0756 2512
0.0710 0950
0.0666 7559
0.0626 0619
0.0587 8515
0.0624 1157
0.0583 2857
0.0545 1268
0.0509 4643
0.0476 1349
0.0515 5288
0.0479 5617
0.0446 1039
0.0414 9804
0.0386 0283
0.0426 2123
0.0394 6411
0.0365 4084
0.0338 3411
0.0313 2788
0.0352 6799
0.0325 0506
0.0299 5858
0.0276 1160
0.0254 4848
46
47
48
49
50
0.0551 9733
0.0518 2848
0.0486 6524
0.0456 9506
0.0429 0616
0.0444 9859
0.0415 8747
0.0388 6679
0.0363 2410
0.0339 4776
0.0359 0961
0.0334 0428
0,0310 7375
0.0289 0582
0.0268 8913
0.0290 0730
0.0268 5861
0.0248 6908
0.0230 2693
0.0213 2123
0.0234 5482
0.0216 1734
0.0199 2382
0.0183 6297
0.0169 2439
IV- 40
TABLE V. AMOUNT OP ANNUITY OF 1 PER PEKIOD
(1 4- t)* ~ 1
n
il%
1%
~%
1%
1%
i
2
3
4
5
1.0000 0000
2.0041 6667
3.0125 1736
4.0250 6952
5.0418 4064
1.0000 0000
2.0050 0000
3.0150 2500
4.0301 0013
5.0502 5063
1.0000 0000
2,0058 3333
3.0175 3403
4.0351 3031
5.0586 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 0501
6
7
8
9
10
6.0628 4831
7.0881 1018
8.1176 4397
9.1514 6749
10.1895 9860
6.0755 0188
7.1058 7939
8.1414 0379
9.1821 15S3
10.2280 2641
6.0S81 8354
7.1236 9794
S.I 652 5284
9.2128 8349
10.2666 2531
6.1136 3135
7.1594 8358
8.2131 7971
9.2747 7856
10.3443 3940
6.1520 1506
7.2135 3521
8.2856 7056
9.3085 2727
10.4022 1254
11
12
13
14
15
11.2320 5526
12.2788 5549
13.3300 1739
14.3855 5913
15.4454 9896
11.2791 6654
12.3355 6237
13.3972 4018
14.4642 2639
15.5365 4752
11.3265 1396
12.3925 8529
13.464$ 7537
14.5434 204S
15.6282 5710
11.4219 2194
12.5075 8G36
13.6013 9325
14.7034 0370
15.8136 7923
11.5668 3467
12.6S25 0301
13.8093 2804
14.9474 2132
16.0908 9354
16
17
18
19
20
16.5008 552O
17.57SO 4627
18.6518 9063
19.7296 0684
20.8118 1353
16.6142 3026
17.0973 0141
1S.7857 8791
19.8797 1685
20.9791 1544
16.7194 2193
17,8169 5189
18.9208 8411
20.0312 5593
21.14S1 0493
16.9322 8183
18.0592 7394
19.1947 1S49
20.3386 788S
21.4912 1897
17.2578 6449
18.4304 4314
19.6147 4757
20.8108 9504
22.0190 0399
21
22
23
24
25
21.8985 29 i2
22.9807 7330
210S55 6402
25.1S59 2054
26.2908 6187
22.0840 1101
23.1944 3107
24.3104 0322
25.4319 5524
26.5591 1502
22.2714 68S7
23.4013 S577
24.5378 9.iS6
25.0810 3157
20.8308 3759
22.6524 0312
23.8222 9011
25.0009 0336
26.1884 7059
27.3818 8412
23.2391 9403
24.4715 8598
25.7163 0183
20.9734 6485
28.2431 9950
26
27
28
29
30
27.4004 0713
28.5145 7549
29.6333 8G22
30.7568 5S67
31.8850 1224
27.C919 1059
28.8303 7015
29.9745 2200
31.1243 9461
32.2800 1658
27.9873 5081
29.1506 1035
30.3200 5558
31.4975 2007
32.6812 6104
28.5902 7075
29.8046 9778
31.0282 3301
32.2609 4476
33.5029 01S4
29.5256 3150
30.8208 8781
32.1290 9009
33.4503 8706
34.7848 9153
31
32
33
34
34
33.0178 6646
34.1554 4090
35.2977 5524
36.4448 2922
37.5966 8268
33.4414 1666
34.60S6 2375
35.7816 6686
30.9605 7520
38.1453 7S07
33.8719 0233
35.0694 8843
30.2740 6045
37.4856 5913
38.7043 2548
34.7541 7361
36.014S 2991
37.2849 4113
38.5645 7819
39.8538 1253
36.1327 4045
37.4940 6785
38.8690 0853
40.2576 9862
41.0602 7560
36
37
38
39
40
38.7533 3552
89.9148 0775
41.0811 1945
42.2522 9078
43,4283 4199
39.3361 0496
40.5327 8549
41.7354 4942
42.9441 2666
44.1588 4730
39.9301 0071
41.1630 2630
42.4031 4395
43.0504 9562
44.9051 2352
41.1527 1612
42.4613 6149
43.7798 2170
45.1081 7037
46.4464 8164
43.0768 7836
44.5076 4714
45.9527 2361
47.4122 5085
48.8863 7336
41
42
43
44
45
44.6092 9342
45.7951 6548
46.9859 7866
48.1817 5358
49.3825 1088
45.3796 4153
46.6065 3974
47.8395 7244
49.0787 7030
50.3241 6415
46.1670 7007
47.4363 7798
4S.7130 9018
49.9972 49S8
51.2889 0050
47.7948 3026
49.1532 9148
50.5219 4117
51.9008 5573
53.2901 1215
50.3752 3709
51.8789 8946
53.3977 7036
54.9317 5715
56.4810 7472
46
47
48
49
50
50.5882 7134
51.7990 5581
53.0148 8521
54.2357 8056
55.4617 6298
51.5757 8497
52.8336 6390
54.0978 3222
55.3683 2138
56.6451 6299
52.5S80 8575
53.8948 4950
55.2092 3621
56.5312 9009
57.8610 5595
54.6807 8799
56.0999 6140
57.5207 1111
58.9521 1644
60.3942 5732
58.0458 8547
59.6263 4432
01.2226 0777
02.8348 3385
64.4631 8218
T V 41
TABLE V. AMOUNT OF ANNUITY OF 1 PER PERIOD
(1 4- t) n - 1
n
5*
1%
%
1%
1%
51
52
53
54
55
56.6928 5366
67.9290 7388
59.1704 4503
60.4169 8855
61.6687 2600
57.9283 8880
59.2180 3075
60.5141 2090
61.8166 9150
63.1257 7496
59.1985 7877
60.5439 0381
61.8970 7659
63.2581 4287
64.6271 4870
61.8472 1424
63.3110 6835
64.7859 0136
66.2717 9562
67.7688 3409
66.1078 1401
67.7688 9215
69.4465 8107
71.1410 4688
72.8524 5735
56
57
58
59
60
62.9256 7902
64.1878 6935
65.4553 1881
66.7280 4930
68.0060 8284
64.4414 0384
65.7636 1086
67.0924 2891
68.4278 9105
69.7700 3051
66.0041 4040
67.3891 6455
68.7822 6801
70.1834 9791
71,5929 0165
69.2771 0035
70.7966 7860
72.3276 5369
73.8701 1109
75.4241 3693
74.5809 8192
76.3267 9174
78.0900 5966
79.8709 6025
81.6696 6986
61
62
63
64
65
69.2894 4152
70.5781 4753
71.8722 2314
73.1716 9074
74.4765 7278
71.1188 8066
72.4744 7507
73.8368 4744
75.2060 3 108
76.5820 6184
73.0105 2691
74.4364 2165
75.S706 3411
77.3132 1281
78.7642 0655
76.9898 1795
78.5672 4159
80.1564 9590
81.7576 6962
83.3708 5214
83.4863 6655
85.3212 3022
87.1744 4252
89.0461 8695
90.9366 4882
66
67
68
69
76
75.7868 9184
77.1026 7055
78.4239 3168
79.7506 9806
81.0829 9264
77.9649 7215
79.3547 9701
80.7515 7099
82.1553 28S5
83.5661 0549
80.2236 6442
81.6916 3579
83.1081 7034
84.6533 1800
86.1471 2902
84.9961 3353
86.6336 0453
88.2833 5657
89.9454 8174
91.6200 7285
92.8460 1531
94.7744 7546
96.7222 2021
98.6894 4242
100.6763 3684
71
72
73
74
75
82.4208 3844
83.7642 5S60
85.1132 7634
86,4679 1500
87.8281 9797
84.9839 3602
86.4083 5570
87.8408 9998
89.2S01 0448
90.7265 0500
87.6496 5394
89.1609 4359
90.6810 4909
92.2100 2188
93.7479 1367
93.3072 2340
95.0070 2758
96.7195 8028
98.4449 7714
100.1833 1446
102.6831 0021
104.7099 3121
106.7570 3052
108.8246 0083
110.9128 4684
76
77
78
79
80
89.1941 4880
90.5657 9109
91.9431 4855
93.3262 4500
94.7151 0436
92.1801 3752
93.C410 3S21
95.1092 4340
96.5847 8962
98.0677 1357
95.2947 7650
96.8506 6270
98.4156 2490
99.9897 1604
101.5729 8938
101.9346 8932
103.6991 9949
105.4769 4349
107.2680 2056
109.0725 3072
113.0219 7530
115.1521 9506
117.3037 1701
119.4767 5418
121.6715 2172
81
82
83
84
85
96.1097 5062
97.5102 0792
98.9165 0045
100.3286 5254
101.7466 8859
99.5580 5214
101.0558 4240
102.5611 2161
104.0739 2722
105.5942 9685
103.1654 9849
104.7672 9723
106.3784 3980
107.9989 8070
109.6289 7475
110.8905 7470
112.7222 5401
114.5676 7091
116.4269 2845
118.3001 3041
123.8882 3fe94
126.1271 1931
128.3883 9050
130.6722 7440
132.9789 9715
86
87
88
89
90
103.1706 3312
104.6005 1076
106.0363 4622
107.4781 6433
108.9259 9002
107.1222 6834
108.6578 7968
110.2011 6908
111.7521 7492
113.3109 3580
111.2684 7710
112.9175 4322
114.5762 2889
11G.2445 9022
117.9226 8307
120.1873 8139
122.0887 8675
124.0044 5265
125.9344 8604
127.8789 9469
135.3087 8712
137.6G18 7496
140.0384 9374
142.4388 7868
144.8632 6746
91
92
93
94
95
110.3798 4831
111.8397 6434
113.3057 6336
114.7778 7071
116.2561 1184
114.8774 9048
116.4518 7793
118.0341 3732
119.6243 0800
121.2224 2954
119.6105 0599
121.3082 9429
123.0159 2601
124.7335 1891
126.4611 3110
129.8380 8715
131.8118 7280
133.8004 6185
135.8039 6531
137.8224 9505
147.3119 0014
149.7850 1914
152.2828 6933
154.8056 9803
157.3537 5501
96
97
98
99
109
117.7405 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
135.2515 3903
139.8561 6377
141.9050 8499
143.9693 7313
146.0491 4343
148.1445 1201
159.9272 9256
162.5265 6548
165.1518 3114
167.8033 4945
170.4813 8294
T V 42
TABLE V. AMOUNT OF ANNUITY OF 1 PER PERIOD
(1 + - 1
n
M
1%
h%
S%
1%
101
10?
103
104
105
125.2558 0669
126.7777 0589
128.3059 4633
129.8405 5444
131.3815 5675
130.9803 6692
132,6352 6875
134.2984 4509
135.9699 3732
137.6497 8701
137.0405 0634
138.8399 0929
140.6498 0876
142.4702 6598
144.3013 4253
150.2555 9585
152.3825 1281
154.5253 8160
156.6843 2202
158.8594 5444
173.1861 0677
175.9180 5874
178.6772 3933
181.4640 1172
18^.2786 5184
106
107
103
109
110
132.9289 7990
134.4828 5065
130.0431 9586
137.6100 4251
139.1S34 1769
139.3380 3594
141.0347 2612
142.7398 9975
144.4535 9925
146.1758 6725
146.1431 0036
147.9956 0178
149.8589 0946
151.7330 8643
153.6181 9610
161.0509 0035
163.2587 8210
165.4832 2296
167.7243 4714
169.9822 7974
187.1214 3836
189.9926 5274
192.8925 7927
195.8215 0506
198.7797 2011
111
112
113
114
115
140.7633 4860
142.3498 6255
143.9429 8698
145.5427 4942
147.1491 7754
147.9067 4658
149.6462 8032
151.3945 1172
153.1514 8428
154.9172 4170
155.5143 0225
157.4214 6901
159.3397 6091
161.2692 4285
163.2099 8010
172.2571 4684
174.5490 7544
176.8581 9351
179.1846 2996
181.5285 1468
201.7675 1731
204.7851 9248
207.8330 4441
210.9113 7485
214.0204 8860
116
117
118
119
120
148.7622 9912
150.3821 4203
152.0087 3429
153.6421 0401
155.2822 7945
156.6018 2791
158.4752 8704
160.2676 6348
162.0690 0180
163.8793 4681
165.1620 3832
167.1254 8354
169.1003 8219
171.0868 0109
173.0848 0743
183.8899 7854
186.2691 5338
188.6661 7203
191.0811 6832
193.5142 7708
217.1606 9349
220.3323 0042
223.5356 2343
226.7709 7966
230.0386 8946
121
122
123
124
125
156.9292 8895
158.5831 6098
lt,0.2439 2415
lfil.9116 0717
163.5862 3887
165.6987 4354
167.5272 3726
169.3648 7344
171.2116 9781
173.0677 5630
175.0944 6881
177.1158 5321
179.1490 2902
181.1940 6502
183.2510 3040
195.9656 3416
198.4353 7642
200.9236 4174
203.4305 6905
205.9562 9832
233.3390 7635
236.6724 6712
240.0391 9179
243.4395 8370
246.8739 7954
126
127
128
129
130
165.2678 4819
166.9564 6423
16S.6521 1616
170.3548 3331
172.0646 4512
174.9330 9508
176.8077 6056
178.6917 9936
180.5852 5836
182.4881 8465
185.3199 9474
187.4010 2805
189.4942 0071
191.5995 8355
193.7172 4778
208.5009 7056
211.0647 2784
213.6477 1330
216.2500 7115
218.8719 4668
250.3427 1934
253.8461 4653
257.3846 0800
260.9584 5408
264.5680 3862
131
132
133
134
135
173.7815 8114
175.5056 7106
177.2369 4469
178.9754 3196
180.7211 6293
184.4006 2557
186.3226 2870
188.2542 4184
190.1955 1305
192.1464 9062
195.8472 6506
197.9897 0744
200.1446 4740
202.3121 5785
204.4923 1210
221.5134 8628
224.1748 3743
226.8561 4871
229.5575 6982
232.2792 5160
268.2137 1900
271.8958 5619
275.6148 1475
279.3709 6290
283.1616 7253
136
137
138
139
140
182.4741 6777
184.2344 7681
186.0021 2046
187.7771 2929
189.5595 3400
194.1072 2307
196.0777 5919
198.0581 4798
200.0484 3872
202.0486 8092
206.6851 8392
208.8908 4749
211.1093 7744
213.3408 4881
215.5853 3709
235.0213 4598
237.7840 0608
240.5673 8612
243.3716 4152
246.1969 2883
286.9063 1926
290.8C62 8245
294.7749 4527
298.7226 9473
302.7099 2167
141
142
143
144
145
191.3493 6539
193.1466 5441
194.9514 3214
196.7637 2977
198.5835 7865
204.0589 2432
206.0792 1894
208.1096 1504
210.1501 6311
212.2009 1393
217.8429 1822
220.1136 6858
222.3976 6498
224.6949 8469
227.0057 0544
249.0434 0580
251.9112 3134
254.8005 6558
257.7115 6982
260.6444 0659
306.7370 2089
310.8043 9110
314.9124 3501
319.0015 5936
323.2521 7495
146
147
148
149
150
200.4110 1023
202.2460 6610
204.0887 4800
205.9391 1779
207.7971 9744
214.2619 1850
216.3332 2809
218.4148 9423
220.5069 6870
222.6095 0354
229.3299 0538
231.6676 6317
234.0190 5787
236.3841 6904
238.7630 7669
263.5992 3964
266.5762 3394
269.5755 5569
272.5973 7236
275.6418 5265
327.4846 9670
331.7595 4367
336.0771 3911
340.4379 1050
344.8422 8960
T V 43
TABLE V. AMOUNT or ANNUITY or 1 PER PERIOD
n
l|%
l\%
l|%
l|%
2%
2
3
4
5
1.0000 0000
2.0112 5000
3.0338 7656
4.0680 0767
5.1137 7276
1.0000 0000
2.0125 0000
3.0376 5625
4.0756 2695
5.1265 7229
1.0000 0000
2.0150 0000
3.0452 2500
4.0909 0338
5.1522 6693
l.OOOO 0000
2.0175 0000
3.0528 0625
4.1062 3036
5.1780 8938
1.0000 0000
2.0200 0000
3.0604 0000
4.1216 0800
5.2040 4016
6
8
to
6.1713 0270
7.2407 2986
8.3221 8807
9.4158 1269
10.5217 4058
6.1906 5444
7.2680 3702
8.3588 8809
9.4633 7420
10.5S16 6637
6.2295 5093
7.3229 9419
8.4328 3911
9.5593 3169
10.7027 2167
6 2687 0596
73784 0831
8.5075 3045
9.6564 J224
10.8253 9945
6.3081 2096
7.4342 8338
8.5829 6905
9.7546 2843
10.9497 2100
11
12
13
14
15
11.6401 1016
12.7710 6140
13.9147 3584
15.0712 7662
16.2408 2848
11.7139 3720
12.8603 6142
14.0211 1594
15.1963 7988
16.3863 3463
11.8632 6249
13.0412 1143
14.2368 29GO
15.4503 8205
16.6821 3778
12.0148 4394
13.2251 0371
14.4565 4303
15.7095 3253
16.9844 4935
12.1687 1542
13.4120 8973
14.6803 3152
15.9739 3815
17.2934 1692
16
17
18
10
20
17.4235 3780
18.6195 5260
19.8290 2257
21.0520 9907
22.2889 3519
17.5911 6382
18.8110 5336
20.0461 9153
21.2967 6893
22.5629 7854
17.9323 6984
19.2013 5539
20.4893 7572
21.7967 1636
23.1236 6710
18.2816 7721
19.6016 0656
20.9446 3408
22.3111 6578
23.7016 1119
18.6392 8525
20.0120 7096
21.4123 1238
22.8405 5803
24.2973 6980
21
22
23
24
25
23.5396 8571
24.8045 0717
26.0835 5788
27.37G9 9790
28.6849 8913
23.8450 1577
25.1430 7847
26.4573 6695
27.78SO 8403
29.1354 3508
24.4705 2211
25.8375 7994
27.2251 4364
28.6335 2080
30.0630 2361
25.1163 8938
26.5559 2620
23.0208 5490
29.5110 1637
31.0274 5915
25.7833 1719
27.2989 8354
28.8449 6321
30.4218 6247
32.0302 9972
26
27
28
29
30
30.0C76 0526
31.3452 8183
32.6979 1025
34.0657 6731
35.4490 07C9
30.4996 2802
31.8808 7337
33.2793 8429
34.6053 7059
36.1290 68SO
31.5139 6896
32.0306 7850
34.4814 7807
35.9987 0035
37.53SG 8137
32.5704 3969
34.1404 2238
35.7378 7977
37.3632 9267
39.0171 5029
33.6709 0572
35.3443 2383
37.0512 1031
38.7922 3451
40.5680 7921
31
32
33
34
35
36.8478 0903
38.2623 46S8
39.6927 9829
41.1393 4227
42.6021 5987
37.5S06 8216
39.0504 40CO'
40.5385 7120
42.0453 0334
43.5708 6963
39.1017 6159
40.GS82 S801
42.2986 1233
43.9330 9152
45.5920 8789
40.6999 5042
42.4121 9955
44.1544 1305
45.9271 1527
47.7308 3979
42.3794 4079
44.2270 2961
46.1115 7020
48.0338 0160
49.9944 7763
36
37
38
39
40
44.0814 3417
45.5773 5030
47.0900 9549
48.6198 5906
50.1668 3248
45.1155 0550
46.6704 4932
48.2926 4243
49.8862 2921
51.4895 5708
47.2759 6921
48.9851 0874
50.7198 8538
52.4806 83C6
54.2078 9391
49.5661 2949
51.4335 3675
53.3336 2305
55.2069 6206
57.2341 3390
51.9943 6719
54.0342 5453
56.1149 3962
58.2372 3841
60.4019 8318
41
42
43
44
45
51.7312 0934
53.3131 8545
54.9129 5879
56.5307 2957
58.1667 0028
53.1331 7654
54.7973 4125
56.4823 0801
58.1883 3687
59.9156 9108
56.0819 1232
57.9231 4100
50,7919 8812
61.6888 6794
63.6142 0096
59.2357 3124
61.2723 5654
63.3446 2278
65.4531 5307
67.5985 8380
62.6100 2284
64.8022 2330
67.1594 6777
69.5026 5712
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
68.8817 8989
65.5684 1398
67.5519 4018
69.5652 1929
71.6086 9758
73.6828 2804
69.7815 5908
72.0027 3637
74.2627 8425
76.5623 8298
78.9022 2468
74.3305 6447
76.8171 7576
79.3535 1927
81.9405 8966
84.5794 0145
T V 44
TABLE V. AMOUNT OP ANNUITY OF 1 PER PERIOD
(1 + i) n - 1
-
n
l|%
l|%
ll%
lj%
2%
51
53
53
54
55
68.3763 5152
70.1455 8548
71.9347 2332
73.7439 8895
75.5736 0883
70.7428 12S>6
72.6270 9741
74.5349 3613
76.4666 2283
78.4224 5562
75.7880 7046
77.9248 9152
80.0937 6480
82.2951 7136
84.5295 9893
81.2830 1361
83.7054 6635
86.1703 1201
88.6782 9247
91.2301 6259
87.2709 8948
90.0164 0927
02.8167 3746
95.6730 7221
98.5865 3365
56
57
58
59
60
77.4238 1193
79.2948 2981
81.1868 906 5
83.1002 4923
85.0351 2704
80.4027 3631
82.4077 7052
84.4378 6765
86.4933 4099
88.5745 0776
86.7975 4292
89.0995 0606
91.4359 9S65
93.8075 3863
96.2146 5171
93.8266 9043
96.4686 5752
99.1568 5902
101.8921 0405
104.6752 1588
101.5582 6432
104.5894 2961
107.6812 1820
110.8348 4257
114.0515 3942
61
69
63
64
65
86.9917 7222
88.9704 2966
90.9713 4699
91.9947 7464
95.0409 6586
90.6816 8910
92.8152 1022
94.9754 0034
97.1625 9285
99.3771 2526
98.G578 7149
101.1377 3956
103.6548 0565
106.2096 2774
108.8027 7215
107.5070 3215
110.3884 0522
113.3202 0231
116.3033 0585
119.3386 1370
117.3325 7021
120.6792 2161
124.0928 0604
127.5746 6216
131.1261 5541
66
67
68
69
70
97.1101 7672
99.2026 6621
101.3186 9621
103.4585 3154
105.6224 4002
101.6193 3933
103.8895 8107
106.1882 0083
108.5155 5334
110.8719 9776
111.4348 1374
114.1063 3594
116.8179 3098
119.5701 9995
122.3637 5295
122.4270 3944
125.5695 1263
128.7669 7910
132.0204 0124
135.3307 5826
134.7486 7852
138.4436 5209
142.2125 2513
146.0567 7563
149.9779 1114
71
72
73
74
75
107.8106 9247
110.0235 6276
112.2613 2784
114.5242 6778
116.8126 6579
113.2578 9773
115.6736 2145
118.1195 4172
120.5960 3599
123.1034 8644
125.1992 0924
128.0771 973S
130.9983 5534
133.9633 3067
136.9727 8063
138.6990 4653
142.1262 7984
145.6134 8974
149.1617 2581
152.7720 5601
153.9774 6937
158.0570 1875
162.2181 5913
166.4625 2231
170.7917 7276
70
77
78
79
BO
119.1268 0828
121.4609 8487
123.8334 8845
126.2266 1520
128.6466 6462
125.6422 8002
128.2)28 0852
130.8154 6863
133.4506 6199
136.1187 9526
140.0273 7234
143.1277 8292
146.2746 9967
149.4688 2016
152.7108 5247
156.4455 6699
160.1833 6441
163.9865 7329
167.8563 3832
171.7938 2424
175.2076 0821
179.7117 6038
184.3059 9558
188.9921 1549
193.7719 5780
81
82
83
84
85
131.0939 3960
133.5687 4642
136.0713 9481
138.6021 9801
141.1614 7273
138.8202 8020
141.5555 3370
144.3249 7787
147.1290 4010
149.9081 5310
156.0015 1525
159.3415 3798
162.7316 6105
166.1726 3597
169.6652 2551
175.8002 1617
179.8767 1995
184.0245 6255
188.2449 9239
192.5392 7976
198.6473 9696
203.6203 4490
208.6927 5180
213.8666 0683
219.1439 3897
86
87
88
89
90
143.7495 3930
146.3667 2162
149.0133 4724
151.6897 4739
154.3962 5705
152.8427 5501
155.7532 8945
158.7002 0557
161.6839 5814
164.7050 0762
173.2102 0389
176.8083 5695
180.4604 8230
184.1673 8954
187.9299 0038
196.9087 1716
201.3546 1971
205.8783 2555
210.4811 9625
215.1646 1718
224.5268 1775
230.0173 5411
235.6177 0119
241.3300 5521
247.1566 5632
91
92
93
94
95
157,1332 1494
159.9009 6361
162.6998 4945
165.5302 2276
168.3924 3776
167.7638 2021
170.8008 6796
173.9966 2881
177.1715 8667
180.3862 3151
191.7488 4889
195.6250 8102
199.5594 5784
203.5528 4971
207.6061 4246
219.9299 9798
224.7787 7295
229.7124 0148
234.7323 6850
239.8401 8495
253.0997 8944
259.1617 8523
265.3450 2094
271.6519 2135
278.0849 5978
96
97
98
99
100
171.2868 5269
174.2138 2978
177.1737 3537
180.1669 3989
183.1938 1796
183.6410 5940
186.9365 7264
190.2732 7980
193.6516 9580
197.0723 4200
211.7202 3459
215.8960 3811
220.1344 7868
224.4364 9586
228.8030 4330
245.0373 8819
250.3255 4248
255.7062 3947
261.1810 9866
266.7517 6789
284.6466 5898
291.3395 9216
298.1663 8400
305.1297 1168
312.2323 0591
T V 45
TABLE V. AMOUNT OF ANNUITY OF 1 PER PERIOD
(1 + T - 1
s-r| =
n
2|%
2|%
2|%
3%
3|%
i
2
3
4
5
1.0000 0000
2.0225 0000
3.0680 0625
4.1370 3639
5.2301 1971
1.0000 0000
2.0250 0000
3.0756 2500
4.1525 1563
5.2563 2852
1.0000 0000
2.0275 0000
3.0832 5625
4.1680 4580
6.2826 6706
1.0000 0000
2.0300 0000
3.0909 0000
4.1836 2700
6.3091 3581
1.0000 0000
2.0350 0000
3.1062 2500
4.2149 4288
5.3624 6588
6
8
9
10
6.3477 9740
7.4906 2284
8.6591 6186
9.8539 9300
11.0757 0784
6.3877 3673
7.5474 3015
8.7361 1590
9.9545 1880
11.2033 8177
6.4279 4040
7.6047 0876
8.8138 3825
10.0562 1880
11.3327 6482
6.4684 0988
7.6624 6218
8.8923 3605
10.1591 0613
11.4638 7931
6.5501 5218
7.7794 0751
9.0516 8677
10.3684 9581
11.7313 9316
11
12
13
14
15
12.3249 1127
13.6022 2177
14.9082 7176
16.2437 0788
17.6091 9130
12.4834 6631
13.7955 5297
15.1404 4179
16.5189 5284
17.9319 2666
12.6444 1585
13.9921 3729
15.3769 2107
16,7997 8639
18.2617 8052
12.8077 9569
14.1920 2056
15.6177 9045
17.0863 2416
18.5989 1389
13.1419 9192
14.6019 6164
16.1130 3030
17.6769 8636
19.2956 8088
16
17
18
19
20
19.0053 9811
20.4330 1957
21.8^27 6251
23.3853 4966
24.9115 2003
19.3802 2483
20.8647 3045
22.3863 487 1
23.9460 0743
25.5446 5761
19.7639 7948
21.3074 8892
22.8934 4487
24.5230 1460
26.1973 9750
20.1568 8130
21.7615 8774
23.4144 3537
25.1168 6844
26.8703 7449
20.9710 2971
22.7050 1575
24.4996 9130
26,3571 8050
28.2796 8181
21
22
23
24
2$
26.4720 2923
28.0676 49S9
29.6991 7201
31.3674 0338
33.0731 6996
27.1832 7405
28.8028 5590
30.5844 2730
32.3490 3798
34.1577 6393
27.9178 2593
29.6855 6615
31.5019 1921
33.3082 2199
35.2858 4810
28.6764 8572
30.5367 8030
32.4528 8370
34.4264 7022
36.4592 6432
30.2604 7068
32.3289 0215
34.4604 1373
36.6665 2821
38.9498 5669
26
27
28
29
80
34.8173 1628
36.6007 0590
38.4242 2178
40.2887 6677
42.1952 6402
36.0117 0803
37.9120 0073
39.8598 0075
41.8562 9577
43.9027 03L16
37.2562 0892
39.2807 5467
41.3609 7542
43.4984 0224
45.6946 0830
38.5530 4225
40.7096 3352
42.9309 2252
45.2188 5020
47.5754 1571
41.3131 0168
43.7590 6024
46.2906 2734
48.9107 9930
51.6226 7728
31
32
33
34
35
44.1446 5746
46.1379 1226
48.1760 1528
50.2599 7563
52.3908 2508
46.0002 7074
48.1502 7751
50.3540 3445
52.6128 8531
54.9282 0744
47.9512 1003
50.2698 6831
52.6522 8969
55.1002 2765
57.6154 8391
50.0026 7818
52.5027 5852
55.0778 4128
57.7301 7652
60.4620 8181
54.4294 7098
57.3345 0247
00.3412 1005
63.4531 5240
66.6740 1274
36
37
38
39
40
54.5696 1864
56.7974 3506
59.0753 7735
61,4045 7334
63.7861 7624
67.3014 1263
59.7339 4794
62.2272 9664
64.7829 7906
67.4025 5354
60.1999 0972
62.8554 0724
65.5839 3094
68.3874 8904
71.2681 4499
63.2759 4427
66.1742 2259
69.1594 4927
72.2342 3275
75.4012 5973
70.0076 0318
73.4578 6930
77.0288 9472
80.7249 0604
84.5502 7775
41
42
43
44
45
66.2213 6521
68.7113 4592
71.2573 5121
73.8606 4161
76.5225 0605
70.0876 1737
72.8398 0781
75.6608 0300
78.5523 2308
81.5161 3116
74.2280 1898
77.2692 8950
80.3941 9496
83.6050 3532
86.9041 7379
78.6632 9753
82.0231 9645
85.4838 9234
89.0484 0911
92.7198 6139
88.6095 3747
92.6073 7128
96.8486 2928
101.2383 3130
105,7816 7290
46
47
48
49
50
79,2442 6243
82.0272 6834
84.8728 7165
87.7825 1126
90.7576 1776
84.5540 3443
87.6678 8530
90.8595 8243
94.1310 7199
97.4843 4879
90.2940 3857
93.7771 2463
97.3559 9556
101.0332 8544
104.8117 0079
96.5014 5723
100.3965 0095
104.4083 9598
108.5406 4785
112.7968 6729
110.4840 3145
115.3509 7255
120.3882 6659
125.6018 4557
130.9979 1016
T V-46
TABLE V. AMOUNT OF ANNUITY OF 1 PER PERIOD
(1 + i) n - 1
n
2|%
2|%
2f%
3%
3l%
51
52
63
54
55
93.7996 6416
96.9101 5661
100.0906 3513
103.3426 7442
106.6678 8460
100.9214 5751
104.4444 9395
108.0556 0629
111.7569 9645
115.5509 213Q
108.6940 2256
112.6831 0818
116.7818 9365
120.9933 9573
125.3207 1411
117.1807 7331
121.6961 9651
126.8470 8240
131.1374 9488
136.0716 1972
136.5828 3702
142.3632 3631
148.3459 4958
154.5380 5782
160.9468 8084
6
57
58
59
60
110.0679 1200
113.5444 4002
117.0991 8992
120.7339 2169
124.4504 3493
119.4396 9440
123.4256 8676
127.5113 2893
131.6991 1215
135.9915 8995
129.7G70 3375
134.3356 2718
139.0298 5692
143.8531 7799
148.8091 4038
141.1537 6831
146.3883 8136
151.7800 3280
157.3334 3379
163.0534 3680
167.5800 3099
174.4453 3207
181.5509 1869
188.9052 0085
196.5168 8288
61
62
63
64
65
128.2505 6972
132.1362 0754
136.1092 7221
140.1717 3083
144.3255 9477
140.3913 7970
144.9011 6419
149.5236 9330
154. 20 17 8563
159.1183 3027
153.9013 9174
159.1336 8002
104.5098 5622
170.0338 7726
175.7098 0889
168.9450 3991
175.0133 9110
181.2637 9284
187.7017 0662
194.3327 5782
204.3949 7378
212.5487 9786
220.9880 0579
229.7225 8599
238.7628 7650
66
67
68
69
70
148.5729 2066
152.9158 1137
157.3564 1713
161.8969 3651
166.5396 1758
164.0962 8853
169.1986 .9574
174.4286 6314
179.7803 7971
185.2841 1421
181.5418 2863
187.5342 2892
103.6914 2021
200.0179 3427
206.5184 2746
201.1627 4055
208.1976 2277
215.4435 5145
222.9068 5800
230.5940 6374
248.1195 7718
257.8037 6238
267.8268 9406
278.2008 3535
288.9378 6459
71
72
73
74
75
171.2867 5893
176.1407 1106
181,1038 7705
186.1787 1429
191.3677 3536
190.9162 1706
196.6891 2249
202.6063 5055
208.6715 0931
214.8882 9705
213.1976 8422
220.0606 2054
227.1122 8760
234.3578 7551
241.8027 1709
238.5118 8565
246.6672 4222
255.0672 6949
263.7192 7727
272.63C8 5559
300.0506 8985
311.5524 6400
323.4568 0024
335.7777 8824
348.5300 1083
76
77
78
79
80
196.6735 0941
202.0986 6337
207.6458 8329
213.3179 1567
219.1175 6877
221.2605 0447
227.7920 1709
234.4868 1751
241.3489 8795
248.3827 1265
249.4522 9181
257.3122 2983
265.3883 1615
273.6864 9485
282.2128 7345
281.8097 8126
201.2640 7469
301.0019 9693
311.0320 5684
321,3630 1855
361.7285 6121
375.3800 6085
389.5276 7798
404.1611 4671
419.3067 8685
81
82
83
84
85
225.0477 1407
231.1112 8763
237.3112 9160
243.6507 9567
250.1329 3857
255.5922 8047
262.9820 8748
270.5566 3966
278.3205 5566
286.2785 6955
290.9737 2747
209.9755 0498
309.2248 3137
318.7285 1423
328.4935 4837
332.0039 0910
342.9640 2638
354.2529 4717
365.8805 3558
377.8569 5165
434.9825 2439
451.2069 1274
467.9991 5469
485.3791 2510
503.3673 9448
86
87
88
89
90
256.7609 2969
263.5380 5060
270.4676 5674
277.5531 7902
284.7981 2555
294.4355 3379
302.7964 2213
311.3663 3268
320.1504 9100
329.1542 6328
338.5271 2095
348.8366 1678
359.4206 2374
370.3139 3839
381.4975 7170
300.1926 6020
402.8084 4001
415.9853 9321
429.4649 5500
443.3489 0365
521.9852 5329
541.2547 3715
561.1986 5295
581.8406 0581
603.2050 2701
91
92
93
94
95
292.2060 8337
299.7807 2025
307.5257 8645
315.4451 1665
323.5426 3177
338.3831 0961
347.8426 8735
357.5387 5453
367.4772 2339
877.6641 539S
392.9887 5492
404.7959 4568
416.9278 34 IS
429.3933 4962
442.20 1G 6074
457.6493 7076
472.3788 5189
487.5502 1744
03.1767 2397
519.2720 2569
625.3172 0295
648.2033 0506
671.8904 2073
696.4065 8546
721.7308 1595
96
97
98
99
100
331.8223 4099
340.2883 4366
348.9448 3139
357.7960 9010
366.8465 0213
388.1057 5783
398.8084 0177
409.7786 1182
421.0230 7711
432.5486 5404
455.3622 1257
468.8846 7342
482.7790 0194
497.0554 2449
511.7244 4867
535.8501 8645
552.9256 9205
570.5134 6281
588.6288 6669
607.2877 3270
748.0431 4451
775.2246 5457
803.3575 1748
832.4750 3059
862.6116 5666
V47
V. AMOUNT OF ANNUITY OF 1 PEB PBBIOD
(1 + Q* - 1
n
4%
*!%
5%
5|%
6%
i
3
6
1.0000 0000
2.0400 0000
3J216 0000
4.2464 6400
6.4163 2256
1.0000 0000
2.0450 0000
3.1370 2500
4.2781 9113
6.4707 0973
1.0000 0000
2.0500 0000
3.1525 0000
4.3101 2500
6.5256 3125
1.0060 oooo
2.0550 0000
3.1680 2500
4.3422 6638
6.5810 9103
1.0000 0000
2.0600 0000
3.1836 0000
4.3746 1600
6.6370 9296
e
7
8
10
6.6329 7546
7.8982 9448
9.2142 2626
10.5827 9531
12.0061 0712
6.7168 91C6
8.0191 5179
9.3800 1362
10.8021 1423
12.2882 0937
6.8019 1281
8.1420 0845
9.5491 0888
11.0265 6432
12.6778 9254
6.8880 5103
8.2668 9384
9.7215 7300
11.2562 5951
12.8753 5379
6.9753 1854
8.3938 3765
9.8974 6791
11,4913 1598
13.1807 9494
11
12
13
14
1$
13.4863 5141
15.0258 0546
16.6268 3768
18.2919 1119
20.0235 8764
13.8411 7879
15.4050 3184
17.1599 1327
18.9321 0937
20.7840 6429
14.2067 8716
15.9171 2652
17.7129 82S5
19.5986 3199
21.5785 6359
14.5834 9825
16.3855 9065
18.2867-9814
20.2925 7203
22.408Q 6350
14.9716 4264
16.8699 4120
18.8821 3767
21.0150 6593
23.2759 6988
16
17
18
19
20
21.8245 3114
23.6975 1239
25.6454 1288
27.6712 2940
29.7780 7858
22.7193 3673
24.7417 OGS9
2C.8550 8370
29.0635 6248
31.3714 2277
23.6574 9177
25.8403 6636
28.1323 8467
30.5390 0391
33.0059 6410
24.6411 3999
26.9964 0269
29.4812 0483
32.1026 7110
34.8683 1801
25.6725 2808
28.2128 7976
30.9056 5255
33.7599 9170
36.7855 9120
21
22
23
24
25
31.9692 0172
34,2479 6979
36.6178 8858
39.0826 0412
41.6459 0829
33.7831 3680
36.3033 7795
38.9370 2996
41.6891 9631
44.5652 1015
35.7192 5181
38.5052 1440
41.4304 7512
44.5019 9887
47.7270 9882
37.7860 7550
40.8643 0965
44.1118 4669
47.5379 9825
01.1525 8816
39.9927 2668
43.3922 9028
46.9958 2769
60.8155 7735
64.8645 1200
26
27
28
29
30
44.3117 4462
47.0842 1440
49.9675 8298
52.9662 8630
66.0849 3775
47.5706 4460
60.7113 2361
63.9933 3317
67.4230 3316
61.0070 6966
51.1134 6376
54.6691 2645
58.4025 8277
62.3227 1191
66.4388 4750
64.9659 8051
68.9891 0943
63.2335 1045
67.7113 6353
72.4351 7797
69.1563 8272
63.7057 6568
68.5281 1162
73.6397 9832
79.0581 8622
31
32
33
34
85
69.3283 3526
62.7014 6867
66.2095 2743
69.8579 0851
73.6522 2486
64.7523 8779
68.6662 4524
72.7562 2628
77.0302 6646
81.4966 1800
70.7607 8988
75.2988 2937
80.0637 7084
85.0669 5938
90.3203 0735
77.4194 2926
82.6774 9787
88.2247 6025
94.0771 2207
100.2513 6378
84.8016 7739
90.8897 7803
97.3431 6471
104.1837 5460
111.4347 7987
36
37
38
39
40
77.5983 1385
S1.7022 4640
85.&703 3626
90.4091 4971
95.0255 1570
86.1639 6581
91.0413.4427
96.1382 0476
101.4644 2398
107.0303 2306
95.8363 2272
101.6281 3886
107.7095 4580
114.0950 2309
120.7997 7424
106.7651 8879
113.6372 7417
120.8873 2425
128.5361 2708
136.6056 1407
119.1208 6666
127.2681 1866
135.9042 0578
145.0584 6813
164.7619 6562
41
42
43
44
45
99.8265 3633
104.8195 9778
110.0123 8169
115.4128 7696
121,0293 9204
112.8466 8760
118.9247 8854
125.2764 0402
131.9138 4220
138.8499 6510
127.8397 6295
135.2317 5110
142.9933 3866
151.1430 0559
159.7001 5587
145.1189 2285
154.1004 6360
163.5759 8910
173.5726 6850
184.119.1 6527
165.0476 8356
175.9505 4457
187.6075 7724
199.7580 3188
212.7435 1379
40
47
48
49
50
126.8705 6772
132.9453 9043
139.2632 0604
145.8337 3429
152,6670 8366
146.0982 1353
153.6726 3314
161.5879 0163
169.8593 5720
178.5030 2828
168.6851 6366
178.1194 2185
188.0253 9294
198.4266 6259
209.3479 9572
195.2457 1936
206.9842 3392
219.3683 6679
232.4336 2696
246.2174 7645
226.5081 2462
241.0986 1210
256.5645 2882
272.S584 0056
290.3369 0458
T V 48
TABLE V, AMOUNT OF ANNUITY OF 1 PER PERIOD
(1 + 0* - 1
n
4%
4%
6%
5|%
w
51
52
53
54
55
159.7737 6700
167.1647 1768
174.8513 0039
182.8453 6865
191.1591 7299
187.5356 6455
196.9747 6946
206.8386 3408
217.1463 7262
227.9179 6938
220.8153 9550
232.8561 6528
245.4989 7354
258.7739 2222
272.7126 1833
260.7594 3765
276.1012 0672
292.2867 7309
309.3625 4561
327.3774 8562
308.7560 5886
328.2814 2239
348.9783 0773
370.9170 0620
394.1720 2657
56
57
58
59
60
199.8055 3991
208.7977 6151
218.1496 7197
227.8756 5885
237.9906 8520
239.1742 6756
250.9371 0960
263.2292 7953
276.0745 9711
289.4979 5398
287.3482 4924
302.7156 6171
318.8514 4479
335.7940 1703
353.5837 1788
346.3832 4733
366.4343 2593
387.5882 1386
409.9055 6562
433.4503 7173
418.8223 4816
444.9516 8905
472.6487 9040
502.0077 1782
533.1281 8089
61
62
63
64
65
248.5103 1261
259.4507 2511
270.8287 5412
282.6619 0428
294.96S3 8045
303.5253 6190
318.1840 0319
333.5022 8333
349.5098 8608
366.2378 3096
372.2629 0378
391.8760 4897
412.4698 5141
434.0933 4398
456.7980 1118
458.2901 4217
484.4960 9999
512.1433 8549
541.3112 7170
572.0833 9164
566.1158 7174
601.0828 2405
638.1477 9349
677.4366 6110
719.0828 6076
66
67
68
69
70
307.7671 1507
321.0778 0030
334.9209 1231
349.3177 4880
364.2904 5S76
383.7185 3335
401.9858 6735
421,0752 3138
441.0236 1679
461.8696 7955
480.6379 1174
505.6698 0733
531.9532 9770
559.5509 6258
588.5285 1071
604.5479 7818
638.7981 1698
674.9320 1311
713.0532 7415
753.2712 0423
763.2278 3241
810.0215 0236
859.6227 9250
912.2001 6005
967.9321 6965
71
72
73
74
75
379.8620 7711
396.0565 6019
412.89S8 2260
430.4147 7550
448.6313 C652
483.6538 1513
506.4182 3681
530.2070 5747
555.0663 7505
581,044.3 6193
618.9549 3625
650.9026 8306
684.4478 1721
719.6702 0807
756.6537 1848
795.7011 2046
840.4646 8209
887.6902 3960
937.5132 0278
990.0764 2893
1027.0080 9983
1089.6285 8582
1156.00G3 0097
1226.3666 7903
1300.9486 7977
76
77
78
79
80
467.5766 2118
487.2796 8603
607.7708 7347
529.0817 0841
551.2449 7675
608.1913 5822
636.5599 6934
666.2051 6796
697.1844 0052
729.5576 9854
795.4864 0440
836.2607 2462
879.0737 6085
924.0274 4889
971.2288 2134
1045.5306 3252
1104.0348 1731
1165.7567 3226
1230.8733 5254
1299.5713 8693
1380.0056 0055
1463.8059 3659
1552.6342 9278
1646.7923 5035
1746.5998 9137
81
82
83
84
85
574.2947 75S2
598.2665 6685
623.1972 2952
649.1251 1870
676.0901 2345
763.3877 9497
798.7402 4575
835.6835 5680
874.2893 1686
914.6323 3612
1020.7902 6240
1072.8297 7552
1127.4712 6430
1184.8448 2752
1245.0870 6889
1372.0478 1321
1448.5104 4294
1529.1785 1730
1614.2833 3575
1704.0689 1921
1852.3958 8485
1964.5396 3794
2083.4120 1622
2209.4167 3719
2342.9817 4142
86
87
88
89
90
704.1337 2839
733.2990 7753
763.6310 4063
795.1762 8225
827.9833 3354
956.7907 9125
1000.8463 7685
1046.8844 6381
1094.9942 6468
1145.2690 0659
1308.3414 2234
1374.75S4 9345
1444.4964 1812
1517.7212 3903
1594.6073 0098
1798.7927 0977
1898.7263 0881
2004.1562 5579
2115.3848 4986
2232.7310 1660
2484.5606 4591
2634.6342 8466
2793.7123 4174
2962.3350 8225
3141.0751 8718
91
92
93
94
95
862.1026 6688
897.5867 7350
934.4902 4450
972.8698 5428
1012.7846 4845
1197.8061 1189
1252.7073 8692
1310.0792 1933
1370.0327 8420
1432.6842 5949
1675.3376 6603
1760.1045 4933
1849.1097 7680
1942.5652 6564
2040.6935 2892
2356.5312 2252
2487.1404 3976
2624.9331 6394
2770.3044 8796
2923.6712 3480
3330.5396 9841
3531.3720 8032
3744.2544 0514
3969.9096 6944
4209.1042 4961
96
97
99
99
100
1054.2960 3439
1097.4678 7577
1142.3665 9080
1189.0612 5443
1237.6237 0461
1498.1550 5117
1566.5720 2847
1 638.0677 6976
1712.7808 1939
1790.8559 5627
2143.7282 0537
2251.9146 1564
2365.5103 4642
2484.7858 6374
2610.0251 5593
3085.4731 5271
3256.1741 7611
3436.2637 5580
3626.2582 6237
3826.7024 6680
4462.6505 0459
4731.4095 3486
5016.2941 0696
5318.2717 5337
5638.3680 5857
T V-49
TABLE V. AMOUNT OF ANNUITY OF 1 PER PERIOD
(i + *)* - i
n
6|%
7%
?!%
8%
8l%
i
3
*
1.0000 0000
2.0650 0000
3.1992 2500
4.4071 7463
5.6936 4098
1.0000 0000
2.0700 0000
3.2149 0000
4.4399 4300
5.7507 3901
1.0000 0000
2.0750 0000
3.2306 2500
4.4729 2188
5.8083 9102
1.0000 0000
2.0800 0000
3.2464 0000
4.5061 1200
5.8666 0096
1.0000 0000
2.0850 0000
3.2622 2500
4,5395 1413
6.9253 7283
6
7
8
10
7.0637 2764
8.5228 6994
10.0768 5648
11.7318 5215
13.4944 2254
7.1532 9074
8.6540 2109
10.2598 0257
11.9779 8875
13.81G4 4796
7.2440 2034
8.7873 2187
10.4463 7101
12.2298 4883
14.1470 S750
7.3359 2904
8.9228 0336
10.6366 2763
12.4875 5784
14.4865 6247
7.4290 2952
9.0604 9702
10.8306 3927
12.7512 4361
14.8350 9932
11
12
13
14
15
15.3715 6001
17.3707 1141
19.4998 0765
21.7672 9515
24.1821 6933
15.7835 9932
17.S884 5127
20.1406 4286
22.5504 8786
25.1290 2201
16.2081 1906
18.4237 2799
20.8055 0759
23.3659 2068
26.1183 6470
16.6454 8746
38.9771 2646
21.4952 9658
24.2149 2030
27.1521 1393
17.0960 8276
19.5492 4979
22.2109 3603
25.0988 6559
28.2322 6916
16
17
18
19
20
26.7540 1034
29.4930 2101
32.4100 6738
35.5167 2176
38.8253 0867
27.8880 5355
30.8402 1730
33.9990 3251
37.3789 6479
40.9954 9232
29.0772 4206
32.2580 3521
35.6773 8785
39.3531 9194
43.3046 8134
30.3242 8304
33.7502 25C9
37.4502 4374
41.4462 6324
45.7619 6430
31.6320 1204
35.3207 3306
39.3229 9538
43.6654 4998
48.3770 1323
21
22
23
24
25
42.3489 5373
46.1016 3573
60.0982 4205
54.3546 2778
58.8876 7859
44.8651 7673
49.0057 3916
53.4361 4090
68.1766 7076
63.2490 3772
47.5525 3244
62.1189 7237
57.0278 9530
62.3049 8744
67.9778 6150
60.4229 2144
55.4567 5516
60.8932 9557
66.7647 5922
73.1059 3995
53.4890 5936
59.0356 2940
65.0536 6790
71.5832 1882
78.6677 9242
26
27
28
29
30
63.7153 7769
68.8568 7725
74.33?5 7427
80.1641 9159
86.3748 6405
68.6764 7036
,74.4838 2328
80.6976 9091
87.3465 2927
94.4607 8C32
74.0702 0112
80.6319 1620
87.6793 0991
95.2552 5816
103.3994 0252
79.9544 1515
87.3507 6836
95.3388 2983
103.9659 3622
113.2832 1111
86.3545 5478
94.6946 9193
103.7437 4075
113.5619 5871
124.2147 2520
31
32
33
34
35
92.9892 3021
100.0335 3017
107.5357 0963
115.5255 3076
124.0346 9026
102.0730 4137
110.2181 5426
118.9334 2506
128.2587 6481
138.2368 7835
112,1543 5771
121.5659 3454
131.6833 7963
142.5596 3310
154.2516 0558
123.3458 6800
134.2135 3744
145.9506 2044
158.6266 7007
172.3168 0368
135.7729 7684
148.3136 7987
161.9203 4266
176.6835 7179
192.7016 7539
36
37
38
39
40
133.0969 4513
142.7482 4656
153.0268 8259
163.9736 2995
175.6319 1590
148.9134 5984
160.3374 0202
172.5010 2017
185.6402 9158
199.6351 1199
166.8204 7600
180.3320 1170
194.8569 1258
210.4711 8102
227,2565 1960
187.1021 4797
203.0703 1981
220.3159 4540
238.9412 2303
269.0565 1871
210.0813 1780
2281.9382 2981
249.3979 7935
271.5968 0759
295.6825 3624
41
42
43
44
45
188.0479 9044
201.2711 0981
215.3537 3105
230.3517 2453
246.3245 8662
214.6095 6983
230.6322 3972
247.7764 9650
266.120S 6125
285.7,493 1084
245.3007 5857
264.6983 1546
285.5506 8912
307.9G69 9080
332.0645 1511
280.7810 4021
304.2435 2342
329.5830 0530
356-9496 4572
386.6056 1738
321.8155 5182
350.1698 7372
380.9343 1299
414.3137 2959
450.5303 9661
46
47
48
49
50
263.3a56 8475
281.4525 0426
300.7469 1704
321.2954 6665
343.1796 7198
306.7517 6260
329.2243 8598
353.2700 9300
378.9989 9951
406.5289 2947
357.9693 5375
385.8170 6528
415.7533 3442
447.9348 3451
482.5299 4709
418.4260 6677
452.9001 5211
490.1321 6428
530.3427 3742
673.7701 6642
489.8254 8032
532.4606 4615
578.7198 0107
628.9109 8416
683.3684 1782
V 60
TABLJB VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
n
5%
1%
H%
!%
1%
2
3
4
0.9958 6062
1.9875 6908
2.9751 7263
3.9586 7804
4.9381 0261
0.9950 2488
1.9850 9938
2.9702 4814
3.9504*9566
4.9258 6633
0.9942 0050
1.9S26 3513
2.9653 3733
3.9423 4034
4.9136 7723
0.9925 5583
1.9777 2291
2.9555 5624
3.9261 1041
4.8894 3961
0.9900 9901
1.9703 9506
2.9409 8521
3.9019 6555
4.8534 3124
6
8
9
10
5.9134 6318
6.8847 7661
7.8520 5969
8.8153 2915
9.7746 0164
5.8963 8441
6.8620 7404
7.8229 5924
8.7790 6392
9.7304 1186
' 5.8793 8084
6.8394 8385
7.7940 1875
8.7430 1781
9.6865 1315
5.8455 9763
6.7946 3785
7.7366 1325
8.6715 7642
9.5995 7958
5.7954 7647
6.7281 9453
7.6516 7775
8.5660 1758
9.4713 0453
11
12
13
14
15
10.7298 9374
11.6812 2198
12.6286 0280
13.5720 5257
14.5115 8762
10.6770 2673
11.6189 3207
12.5561 5131
13.4887 0777
14.4166 2465
10.6245 3669
11.5571 2016
12.4842 9511
13.4060 9291
14.3225 4473
10.5206 7452
11.4349 1267
12.3423 4508
13.2430 2242
14.1369 9495
10.3676 2825
11.2550 7747
12.1337 4007
13.0037 0304
13.8650 5252
16
17
18
10
20
15.4472 2418
16.3789 7843
17.3068 6648
18.2309 0438
19.1511 0809
15.3399 2502
10.2586 3186
17.1727 6802
18.0823 5624
18.9874 1915
15.2336 8160
16.1395 3432
17.0401 3354
17.9355 0974
18.8256 9320
15.0243 1261
15.9050 2492
16.7791 8107
17.6468 2984
18.5080 1969
14.7178 7378
15.5622 5127
16.3982 6858
17.2260 0850
18.0455 5297
21
2
23
24
25
20.0674 9352
20 9800 7653
21^888 7289
22.7938 9831
23.6951 6843
19.8879 7925
20.7840 5896
21.6756 8055
22.5628 6622
23.4456 3803
19.7107 1404
20.5906 0220
21.4653 8745
22.3350 9938
23.1997 6741
19.3627 9S70
20.2112 1459
21.0533 1473
21,8891 4614
22.7187 5547
18.8569 8313
19.6603 7934
20.4558 2113
21.2433 8728
22.0231 5570
26
27
28
29
30
24.5926 9884
25.4865 0506
26.3766 0254
27.2630 0668
28.1457 3278
24.3240 1794
25.1080 2780
26.0676 8936
26.9330 2423
27.7940 5397
24.0594 2079
24.9140 8862
25.7637 9979
26.6085 8307
27.4484 6702
23.5421 8905
24.3594 9286
25.1707 1251
25.9758 9331
26.7750 8021
22.7952 0366
23.5596 0759
24.3164 4310
25.0657 8530
25.8077 0822
31
32
33
34
35
59.0247 9612
29.9002 1189
30.7719 9524
31.6401 6122
32.5047 2486
28.6507 9997
29.5032 8355
30.3515 2592
31.1955 4818
32.0353 7132
28.2834 8006
29.1136 5044
29.9390 0625
30.7595 7540
31.5753 8566
27.5683 1783
28.3556 5045
29.1371 2203
29.9127 7621
30.6826 5629
26.5422 8537
27.2695 8947
27.9896 9255
28.7026 65S9
29.4085 8009
36
37
38
39
40
33.3657 0109
34.2231 0481
35.0769 5084
35.9272 5394
36.7740 2881
32.8710 1624
33.7025 0372
34.5298 5445
35.3530 8900
36.1722 2786
S2.3864 6463
33.1928 3974
33.9945 3828
34.7915 8736
35.5840 1396
31.446S 0525
32.2052 G576
32.9580 8016
33.7052 9048
34.4469 3844
30.1075 0504
30.7995 0994
31,4846 6330
32.1630 3298
32.8346 8611
41
42
43
44
45
37.6172 9009
38.4570 5236
39.2933 3013
40.1261 3788
40.9554 8999
36.0872 9141
37.7982 9991
38,6052 7354
39.4082 3238
40.2071 9640
36.3718 4487
37.1551 6676
37.9338 2612
38.7080 2929
39.4777 4248
35.1830 6545
35.9137 1260
36.6389 2070
37.3587 3022
38.0731 8136
33.4996 8922
34.1581 0814
34.8100 0806
35.4554 5352
36.0945 0844
46
47
48
49
50
41.7814 0081
42.6038 8461
43.4229 5562
44.2386 2799
45.0500 1582
41.0021 8547
41.7932 1937
42.5803 1778
43.3635 0028
44.1427 8635
40,2429 9170
41.0038 0287
41.7602 0170
42.5122 1380
43.2598 6460
38.7823 1401
39.4861 6774
40.1847 8189
40.8781 9542
41.5664 4707
36,7272 3608
37.3536 9909
37.9739 5949
38.5880 7871
39.1961 1753
T VI 51
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
n
15%
1%
%
!%
1%
51
45.8598 3317
44.9181 9537
44.0031 7940
42.2495 7525
39.7981 3617
59
46.6653 9401
45.6S97 4664
44.7421 8335
42.9276 1812
40.3941 9423
63
47.4676 1228
46.4.>74 5934
45.4769 0144
43.6006 1351
40.9843 5072
54
48.2G65 0184
47.2213 5258
46.2073 5853
44.2685 9902
41.5G86 6408
55
49.0G20 7651
47.9814 4535
46.9335 7933
44.9316 1193
42.1471 9216
56
49.8543 5003
48.7377 5657
47.G555 8841
45.5896 8926
42.7199 9224
57
50.6433 3612
49.4903 0505
48.3734 3020
46.2428 6776
43.2871 2102
58
51.4290 4840
50.2391 0950
49.0870 6898
4G.8911 8388
43.8486 3468
59
52.2115 0046
50.9841 8855
49.7065 8889
47.5346 7382
44.4045 8879
60
52.9907 0584
51.7255 6075
50.5019 9394
48.1733 7352
44.9550 3841
61
53.7666 7800
52.4632 4453
51.2033 0800
48.8073 1863
45.5000 3803
69
54.5394 3035
53.1972 5324
51.9005 5478
49.4365 4455
46.0396 4161
63
55.3089 7627
53.9276 2014
52.5937 5787
50.0610 8640
46.5739 0258
64
56.0753 2905
54.6543 4839
53.2829 4073
50.6809 7906
47.1028 7385
65
56.8385 0194
55.3774 6109
53.9681 2668
51.2962 5713
47.6266 0777
66
57.5985 0814
56.09C9 7621
54.6493 3888
51.9069 5497
48.1451 5621
67
58.3553 6078
56.8129 1165
55.3266 0040
52.5131 0667
48.6585 7050
68
59.1090 7296
57.5252 8522
55.9999 3413
53.1147 4607
49.1669 0149
69
59.8596 5770
58.2341 1465
56.6693 6287
53.7119 0677
49.6701 9949
70
60.6071 2798
58.9394 1756
57.3349 0925
54.3046 2210
50.1685 1435
71
61.3514 9672
59.6412 1151
57.9965 9579
54.8929 2516
50.6618 9539
72
62.0927 7680
60.3395 1394
58.6544 4488
65.4768 4880
51.1503 9148
73
62.8309 8103
61.0343 4222
59.3084 7877
56.0S64 2561
51.6340 5097
74
63.5661 2216
61.7257 1366
59.9687 1959
56.6316 8795
52.1129 2175
75
64.2982 1292
62.4136 4543
60.6051 8934
57.2026 6794
52.5870 5124
76
65.0272 6506
63.0081 54G6
61.2479 0988
57.7603 9746
53.0564 8637
77
65.7532 9388
63.7792 5836
61.8869 0297
58.3319 0815
53.5212 7364
78
66.4763 0924
64.45G9 7350
62.5221 9021
58.8902 3141
53.9814 5905
79
67.1963 2453
65.1313 1691
63.1537 9310
59.4443 9842
54.4370 8817
80
67.9133 5221
65.8023 0538
63.7817 3301
59.9944 4012
54.8882 0611
81
68.6274 0467
66.4699 5561
64.40GO 3118
60.5403 8722
55.3348 5753
89
69.3384 9426
67.1342 8419
G5.02G7 0874
61.0822 7019
55.7770 86GG
83
70.0466 3326
67.7953 07G5
65.G437 8G67
61.6201 1930
56.2149 3729
84
70.7518 3393
68.4530 4244
GG.2572 8585
62.1539 6456
56.6484 5276
85
71.4541 0846
69.1075 0491
66.8672 2705
62.6838 3579
57.0776 7600
80
72.1534 6898
69.7587 1135
67.4736 3089
63.2097 6257
57.5026 4951
87
72.8499 2759
70.40G6 7796
68.0765 1789
C3.7317 7427
57.9234 1535
88
73.5434 9633
71.0514 2086
68.6759 0845
64.2499 0002
58.3400 1520
89
74.2341 8720
71.6929 5608
69.2718 2283
64.7641 6875
58.7524 9030
90
74.9220 1212
72.3312 9958
69.8642 8121
65.2746 0918
59.1608 8148
91
75.6069 8300
72.9664 6725
70.4533 0363
65.7812 4981
59.5652 2919
99
76.2891 1168
73.5984 7487
71.0389 1001
66.2841 1892
59.9655 7346
93
76.9684 0995
74.2273 3818
71.6211 2017
66.7832 4458
60.3619 5392
94
77.6448 8955
74.8530 7282
72.1999 5379
67.2786 5467
60.7544 0982
95
78.3185 6218
75.4756 9434
72.7754 3047
67.7703 7685
61.1429 8002
96
78.9894 3950
76.0952 1825
73.3475 69fiV
68.2584 3856
61.5277 0299
97
79.6575 3308
76.7116 5995
73.9163 9075
68.7428 6705
61.9086 1682
98
80.3228 5450
77.3250 3478
74.4819 1294
69.2236 8938
62.2857 5923
99
80.9854 1524
77.9353 5799
75.0441 5539
69.7009 3239
62.6591 C755
190
81.6452 2677
78.5426 4477
75.6031 3712
70.1746 2272
63.0288 7877
I
T VI 52
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
1 - (1 + tT*
"
n
%
1%
55%
!%
1%
101
102
103
104
105
82.3023 0049
82.9566 4777
83.6082 7991
84.2572 0818
84.9034 4381
79.1469 1021
79.7481 6937
80.3464 3718
80.9417 2854
81.5340 5825
76.1588 7702
76.7113 9392
77.2607 0648
77.8068 3331 .
78.3497 9288
70.6447 8682
71.1114 5094
71.5746 4113
72.0343 8325
72.4907 0298
63.3949 2947
63.7573 5591
64.1161 9397
64.4714 7918
64.8232 4671
106
107
108
109
110
85.5469 9795
86.1878 8175
86.8261 0628
87.4616 8258
88.0946 2163
82.1234 4104
82.7098 9158
83.2934 2446
83.8740 5419
84.4517 9522
78.8896 0355
79.4262 8359
79.9598 5115
80.4903 2428
81.0177 2093
72.9436 2579
73.3931 7696
73.8393 8160
74.2822 6461
74.7218 5073
65.1715 3140
65.5163 6772
65.8577 8983
66.1958 3151
66.5305 2625
111
112
113
114
115
88.7249 3437
89.3526 3171
89.9777 2450
90.6002 2354
91.2201 3959
85.0266 6191
85.5986 6856
86.1678 2942
86.7341 5862
87.2976 7027
81.5420 5895
82.0033 5<>06
82.5816 2991
83.0968 9803
83.6091 7785
75.1581 6450
75.5912 3027
76.0210 7223
76.4477 1437
76.8711 8052
66.8619 0718
67.1900 0710
67.5148 5852
67.8364 9358
68.1549 4414
116
117
118
119
120
91.8374 8338
92.4522 6558
93.0644 9081
93.6741 8767
94.2813 4869
87.8583 7838
88.4162 9690
88.9714 3970
89.5238 2059
90.0734 5333
84.1184 3671
84.6248 4182
85.1282 6033
85.6287 5926
86.1263 5554
77.2914 9431
77.7086 7922
78.1227 5853
78,5337 5536
78.9416 9267
68.4702 4172
68.7824 1755
69.0915 0252
69.3975 2725
69.7005 2203
121
122
123
124
125
94.8859 9036
95.4881 2315
96.0877 5747
96.6849 0367
97.2795 7209
90.6203 5157
91.1615 2892
91.7059 9893
92.2447 7505
92.7808 7070
86.6210 6G02
87.1129 0742
87.6018 9638
88.0880 4946
88.5713 8308
79.3465 9322
79.7484 7962
80.1473 7432
80.5432 9957
80.9362 7749
70.0005 1686
70.2975 4145
70.5916 2520
70.8827 9722
71.1710 8636
126
127
128
129
130
97.8717 7301
98.4615 1606
99.0488 1324
99.6336 7290
100.2161 0576
93.3142 9920
93.8450 7384
94.3732 0780
94.8987 1422
95.4216 0619
89.0519 1361
89.5296 5731
90.0046 3032
90.4708 4873
90.9463 2851
81.3263 3001
81.7134 7892
82.0977 4583
82.4791 5219
82.8577 1929
71.4565 2115
71.7391 2985
72.0189 4045
72.2959 8064
72.5702 7786
131
132
133
134
135
100.7961 2189
101.3737 3131
101.9489 4401
102.5217 6994
103.0922 1899
95.9418 9671
96.4595 9872
96.9747 2509
97.4872 8S65
97.9973 0214
91.4130 8554
91.8771 3561
92.3384 9442
92.7971 7758
93.2532 0060
83.2334 6828
83.6064 2013
83.9765 9566
84.3440 1554
84.7087 0029
72.8418 5927
73.1107 5175
73.3769 8193
73.6405 7617
73.9015 6056
136
137
138
139
140
103.6603 0104
104.2260 2590
104.7894 0335
105.3504 4314
105.9091 5496
98.5047 7825
99.0097 2960
99.5121 (5875
100.0121 0821
100.5095 6041
93.7065 7892
94.1573 2787
94.6054 6270
95.0509 9857
95.4939 5056
85.0706 7026
85.4299 4567
85.7865 4657
86.1404 9288
86.4918 0434
74.1599 6095
74.4158 0293
74.6691 1181
74.9199 1268
75.1682 3038
141
142
143
144
145
106.4655 4847
107.0196 3330
107.5714 1902
108.1209 1517
108.6681 3126
101.0045 3772
101.4970 5246
101.9871 1688
102.4747 4316
102.9599 4344
95.9343 3364
96.3721 6272
96.8074 5261
97.2402 1804
97.6704 7364
86.8405 0059
87.1866 0108
87.5301 2514
87.8710 9195
88.2095 2055
75.4140 8948
75.6575 1434
75.8985 2905
76.1371 6747
76.3734 2324
146
147
148
149
150
109.2130 7674
109.7557 6103
110.2961 9353
110.8343 8356
111.3703 4044
103.4427 2979
103.9231 1422
104.4011 0868
104.8767 2505
105:3499 7518
98.0982 3397
98.5235 1350
98.9463 2663
99.3666 8765
99.7846 1078
88.5454 2982
88.8788 3854
89.2097 6530
89.5382 2858
89.8642 4673
76.6073 4974
76.8389 6014
77.0682 7737
77.2953 2413
77.5201 2290
TVI-~r>3
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
l - (1 + tT*
n
l|%
l|%
1|%
l|%
2%
i
3
4
5
0.9888 7515
1.9667 4923
2.9337 4460
3.8899 8230
4.8355 8200
0.9876 5432
1.9631 1538
2.9265 3371
3.8780 5798
4.8178 3504
0.9852 2167
1.9558 8342
2.9122 0042
3.8543 8465
4.7826 4497
0.9828 0098
1.9486 9875
2.8979. 8403
3.8309 4254
4.7478 5508
0.9803 9216
1.9415 6094
2.8838 8327
3.8077 2870
4.7134 5951
6
7
8
10
5.7706 6205
6.6953 3948
7.6097 3002
8.5139 4810
9,4081 0690
5.7460 0992
6.6627 2585
7.5681 2429
8.4623 4498
9.3455 2591
5.6971 8717
6.5982 1396
7.4859 2508
8.3605 1732
9.2221 8455
5.6489 9762
6.5346 4139
7.4050 5297
8.2604 9432
9.1012 2291
5.6014 3089
6.4719 9107
7.3254 8144
8.1622 3671
8.9825 8501
11
12
13
14
15
10.2923 1832
11.1666 9302
12.0313 4044
12.8863 6880
13.7318 8509
10.2178 0337
11.0793 1197
11.9301 8466
12.7705 5275
13.6005 4592
10.0711 1779
10.9075 0521
11.7315 3222
12.5433 8150
13.3432 3301
9.9274 9181
10.7395 4969
11.5376 4097
12.3220 0587
13.0928 8046
9.7868 4805
1O.5753 4122
11.3483 7375
12.1062 4877
12.8492 6350
16
17
IS
19
20
14.5679 9514
15.3948 0360
16.2124 1395
17.0209 2850
17.8204 4845
14.4202 9227
15.2299 1829
16.0295 4893
16.8193 0759
17.5993 1613
14.1312 6405
14.9076 4931
15.6725 6089
16.4261 6837
17.1686 3879
13.8504 9677
14.5950 8282
15.3268 6272
16.0460 5673
16.7528 8130
13.5777 0931
14.2918 7188
14.9920 3125
15.6784 6201
16.3514 3334
21
22
23
24
25
18.6110 7387
19.3929 0371
20.1660 3580
20.9305 6693
21.6865 9276
18.3696 9495
19.1305 6291
19.8820 3744
20.6242 3451
21.3572 6865
17.9001 3673
18.6208 2437
19.3308 6145
20.0304 0537
20.7196 1120
17.4475 4919
18.1302 6948
18.8012 4764
19.4606 8565
20.1087 8196
17.0112 0916
17.6580 4820
18.2922 0412
18.9139 2560
19.5234 5647
26
27
28
29
30
22.4342 0792
23.1735 0598
23.9045 7946
24.6275 1986
25.3424 1766
22.0812 5299
22.79.62 9925
23.5025 1778
24.2000 1756
24.8889 0623
21.3986 3172
22.0676 1746
22.7267 1671
23.3760 7558
24.0158 3801
20.7457 3166
21.3717 2644
21.9869 5474
22.5916 0171
23.1858 4934
20.1210 3576
20.7068 9780
24.2812 7236
21.8443 8466
22.3964 5555
31
32
33
34
35
26.003 6233
26.7484 4236
27.4397 4522
28.1233 5745
28.7993 6460
25.5692 9010
26.2412 7418
26.9049 6215
27.5604 5644
28.2078 5822
24.6461 4582
25.2671 3874
25.8789 6442
26.4817 2849
27.0755 9458
23.7698 7650
24.3438 5897
24.9079 6951
25.4623 7789
26.0072 5100
22.9377 0152
23.4683 3482
23.9885 6355
24.4985- 9172
24.9986 1933
36
37
38
39
40
20.4678 5127
30.1289 0114
30.7825 9692
31.4290 2044
32.0682 5200
28.8472 6737
29.4787 8259
30.1025 0133
30.7185 1983
31.3269 3316
27.6606 8431
28.2371 2740
28.8050 5163
29.3G45 8288
29.9158 4520
26.5427 5283
27.0690 4455
27.5862 8457
28.0946 2857
28.5942 2955
2*5.4888 4248
25.9694 5341
26.4406 4060
26.9025 8883
27.3554 7924
41
42
43
44
45
32.7903 7340
33.3254 6195
33.9435 9649
34.5548 5438
35.1593 1212
31.9278 3522
32.5213 1874
33.1074 7530
33.6863 9536
34.2581 6825
30.4589 6079
30.9940 5004
31.5212 3157
32.0406 2223
32.5523 3718
29.0852 3789
29.5678 0135
30.0420 6522
30.5081 7221
30.9662 6261
27.7994 8945
28.2347 9358
28.6615 6233
29.0799 6307
29.4901 5937
46
47
48
49
50
35.7570 4536
36.3481 2891
36.9326 3674
37.5106 4202
38.0822 1708
34.8228 8222
35.3806 2442
35.9314 8091
36.4755 3670
37.0128 7574
33.0564 8983
33.5531 9195
34.0425 5365
34.5246 8339
34.9996 8807
31.4164 7431
31.8589 4281
32.2938 0129
32.7211 8063
33.1412 0946
29.8923 1360
30.2865 8196
30.6731 1957
31.0520 7801
31.4236 0589
T VI 64
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 TBB PERIOD
1 - (I + tT*
051 = :
n
if*
1|%
l|%
l|%
2%
51
53
53
54
55
38.6474 3345
39.2063 6188
39.7590 7232
40.3056 3394
40.8461 1514
37.5435 8099
38.0677 3431
38.5854 1660
39.0967 0776
39.6016 8667
35.4676 7298
35.9287 4185
36.3829 9690
36.8305 3SS2
37.2714 6681
33.5540 1421
33.9597 1913
34.3584 4633
34.7503 1579
35.1354 4550
31.7878 4892
32.1449 4992
32.4950 4894
32.8382 8327
33.1747 8752
56
57
58
59
60
41.3805 8358
41.9091 0613
42.4317 4896
42.9485 7746
43.4596 5633
40.1004 3128
40.5930 1855
41.0795 2449
41.5600 2419
42.0345 9179
37.7058 7863
38.1338 7058
38.5555 3751
38.9709 7292
39.3802 6889
35.5139 5135
35.8359 4727
36.2515 4523
36.6108 5526
36.9639 8552
33.5046 9365
33.8281 3103
34.1452 2650
34.4561 0441
34.7608 8668
61
63
63
64
65
43.9650 4952
44.4648 2029
44.9590 3119
45.4477 4407
45,9310 2009
42.5033 0054
42.9662 2275
43.4234 2988
43.8749 9247
44.3209 8022
39.7835 1614
40.1808 0408
40.5722 2077
40.9578 5298
41.3377 8618"
37.3110 4228
37.6521 3000
37.9873 5135
38.3168 0723
38.6405 9678
35.0596 9282
35.3526 4002
35.6398 4316
35.9214 1486
36.1974 6555
66
67
68
60
70
46.4089 1975
6.8815 0284
47.3488 2852
47.8109 5527
48.2679 4094
44.7614 6195
45.1965 0563
45.6261 7840
46.0505 4656
46.4696 7562
41.7121 0461
42.0808 9125
42.4442 2783.
42.8021 9490
43.1548 7183
38.9588 1748
39.2715 6509
39.5789 3375
39.8810 1597
40.1779 0267
36.4681 0348
36.7334 3478
36.9935 6351
37.2485 9168
37.4986 1929
71
73
73
74
75
48.7198 4270
49.1667 1714
49.b086 2016
60.0456 0708
50,4777 3259
46.8836 3024
47.2924 7431
47.6962 7093
48.0950 8240
48.4889 7027
43.5023 3678
43.8446 6677
44.1819 3771
44.5142 2434
44,8416 0034
40.4696 8321
40.7564 4542
41.0382 7560
41.3152 5857
41.6874 7771
37.7437 4441
37.9840 6314
38.2196 6975
38.4506 5662
38.6771 1433
76
77
78
79
80
50.9050 5077
51.3276 1510
51.7454 7847
52,1586 9317
62.5673 1092
48.8779 9533
49.2622 1761
49.6416 9640
50.0164 9027
50.3866 5706
45.1641 3826
45.4819 0962
45.7949 8485
46.1034 3335
46.4073 2349
41.8550 1495
42.1179 5081
42.3763 6443
.42.6303 3359
42.8799 3474
38.8991 3170
39.1167 9578
39.3301 9194
39.5394 0386
39.7445 1359
81
82
83
84
85
52.9713 8286
63.3709 5957
63.7660 9104
64.1568 2674
64.5432 1557
50.7522 5389
51.1133 3717
61.4699 6264
51.8221 8532
52.1700 5958
46.7067 2265
47.0016 9720
47.2923 1251
47.5786 3301
47.8607 2218
43.1252 4298
43.3663 3217
43.6032 7486
43.8361 4237
44.0650 0479
39.9456 0156
40.1427 4663
40.3360 2611
40.6255 1579
40,7112 8999
86
87
88
89
90
64.9253 0588
55.3031 4549
55.6767 8169
56.0462 6126
56.4116 3041
52.5136 3909
52.8529 7688
53.1881 2531
53.5191 3611
53.8460 6035
48.1386 4254
48.4124 5571
48.6822 2237
48.9480 0234
49.2098 5452
44.2899 3099
44.5109 8869
44.7282 4441
44.9417 6355
45.1516 1037
40.8934 2156
41.0719 8192
41.2470 4110
41,4186 6774
41.5869 2916
91
93
93
94
95
66.7729 3490
57.1302 1992
57.4835 3021
57.8329 0997
68.1784 0294
54.1689 4850
64.4878 5037
64.8028 1518
65.1138 9154
65.4211 2744
49.4678 3696
49.7220 0686
49.9724 2055
50.2191 3355
60.4622 0054
45.3578 4803
45.5605 3860
45.7597 4310
45.9555 2147
46.1479 3265
41.7518 9133
41.9136 1895
42.0721 7545
42.2276 2299
42.3800 2264
96
11
18
68.5200 5235
68.8579 0096
59.1919 9106
59.6223 6446
69.8490 6251
55.7245 7031
56.0242 6698
66.3202 6368
56.6126 0610
56.9013 3936
50.7016 7541
50.9376 1124
51.1700 6034
51.3990 7422
51.6247 0367
46.3370 3455
46.5228 8408
46.7055 3718
46.8850 4882
47.0614 7304
42.5294 3386
42.6759 1555
42.8195 2505
42.9603 1867
43.0983 5164
T VI 55
TABU; VI. PRESENT VAIAJE OF ANNUITY OF 1 PER PERIOD
i - (i + ir*
71
2\%
2|%
2?or
4%
3%
3|%
1
3
3
4
5
0.9779 9511
1.9344 6955
2.8698 9687
3.7847 4021
4.6794 5253
0.9756 0976
1.9274 2415
2. 8560 2356
3.7619 7421
4.6458 2850
0.9732 3601
1.9204 2434
2.8422 6213
3.7394 2787
4.6125 8183
0.9708 7379
1.9134 6970
2.8286 1135
3.7170 9840
4.5797 0719
0.9661 8357
1.8996 9428
2.8016 3698
3.6730 7921
4.5150 5238
6
8
9
10
5.5544 7680
6.4102 4626
7.2471 8461
8.0657 0622
8.8662 1635
5.5081 2536
6.3493 UOGO
7.1701 3717
7.9708 6553
8.7520 6393
5.4623 6678
6.2894 0806
7.0943 1441
7.8776 7826
8.6400 7616
5.4171 9144
6.2302 8296
7.0196 9219
7.7861 0892
8.5302 0284
5.3285 5302
6.1145 4398
6.8739 5554
7.6076 8651
8.3166 0532
11
12
13
14
15
9.6491 1134
10.4147 7S82
11.1635 9787
11.8959 3924
12.6121 6551
9.5142 0871
10.2577 6460
10.9831 8497
11.6909 1217
12.3813 7773
9.3820 6926
10.1042 0366
10.S070 1086
11.4910 0814
12.1566 9892
9.2526 2411
9.9510 0399
10.0349 5533
11.2960 7314
11.9379 3509
9.0015 5104
9.6G33 3433
10.3027 3849
10.9205 2028
11.5174 1090
16
17
18
19
29
13.3126 3131
13.9976 8343
14.6676 6106
15.3228 9590
15.9637 1237
13.0550 0266
13.7121 9772
14.3533 6363
14.9788 9134
. 15.5891 6229
12.8045 7315
13.4351 0769
14.0487 6G61
14.64GO 0157
15.2272 5213
12.5611 0203
13.1G61 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
16.5904 2775
17.2033 5232
17.8027 8955
18.3800 3624
18.9623 8263
16.1845 4857
16.7C54 1324
17.3321 1048
17.8849 8583
18.4243 7G42
15.7929 4612
16.3434 9987
16.8793 1861
17.4007 9670
17.9083 1795
15.4150 2414
15.9369 1664
16.4436 0839
16.9355 4212
17.4131 4769
14.6979 7420
15.1671 2484
15.6204 1047
16.0583 6760
16.4815 1459
26
27
28
29
39
19.5231 1260
20.0715 0376
20.6078 2764
21.1323 4977
21.6453 2985
18.9506 1114
19.4640 1087
19.9648 8866
20.4535 4991
20.9302 9259
18.4022 5592
18.8829 7413
19.3508 2640
19.8061 5708
20.2493 0130
17.8768 4242
18.3270 3147
18.7641 0823
19.1884 5459
19.6004 4135
16.8903 5226
17.2853 6451
17.6670 1885
18.0357 6700
18.3920 4541
31
32
33
34
35
22.1470 2188
22.6376 7419
23.1175 2977
23.5868 2618
24.0457 9577
21.3954 0741
21.8491 7796
22.2918 8094
22.7237 8628
23.1451 5734
20.0805 8520
21.1003 2623
21.5088 3332
21.9064 0712
22.2933 4026
20.0004 2849
20.3887 6553
20.7657 9178
21.1318 36G8
21.4872 2007
18.7362 7576
19.0688 6547
19.3902 0818
19.7006 8423
20.0006 6110
36
37
38
39
49
24.4946 6579
24.9336 5848
25.3629 9118
25.7828 7646
26.1935 2221
23,5562 5107
23.9573 1812
24.3486 0304
24.7303 4443
25.1027 7505
22.6699 1753
23.0364 1609
23.3931 0568
23.7402 4884
24.0781 0106
21.8322 5250
22.1672 3544
22.4924 6159
22.8082 1513
23.1147 7197
20.2904 9381
20.5705 2542
20.8410 8736
21.1024 9987
21.3550 7234
41
42
43
44
45
26.5951 3174
26.9879 0390
27.3720 3316
27.7477 0969
28.1151 1950
25.4061 2200
25.8206*0683
26.1664 4569
26.5038 4945
26.8330 2386
24.4069,1101
24.7269 2069
25.0383 6563
25.3414 7507
25.6364 7209
23.4123 9997
23.7013 5920
23.9819 0213
24.2542 7392
24.5187 1254
21.5991 0371
21.8348 8281
22.0626 8870
22.2827 0102
22.4954 5026
46
47
48
49
59
28.4744 4450
28.8258 6259
29.1695 4777
29.5056 7019
29.8343 9627
27.1541 6962
27.4674 8255
27.7731 5371
28.0713 6947
28.3623 1168
25.9235 7381
26.2029 9154
26.4749 3094
26.7395 9215
26.9971 6998
24.7754 4907
25.0247 0783
25.2667 0664
25.5016 5693
25.7297 6401
22.7009 1813
22.8994 3780
23.0912 4425
23.2765 6450
23.4556 1757
T VI 56
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
1 -(1 +i)~ n
n
2|%
2|%
2|%
3%
3l%
51
52
53
54
55
30.1558 8877
30.4703 0687
30.7778 0623
31.0785 3910
31.3726 5438
28.64G1 5774
28.9230 8072
29.1932 4948
29.45G8 2876
29.7139 7928
27.2478 5400
27.4918 2871
27.7292 7368
27.9003 6368
28.1852 6879
25.9512 2719
26.1062 3999
20.3749 9028
26.5776 6047
20.7744 2764
23.6286 1630
23.7957 6454
23.9572 6043
24.1132 9510
24.2640 5323
50
57
58
59
60
31.6602 9768
31.9416 1142
32.2167 3489
32.4858 0429
32.7489 5285
29.9648 5784
30.2096 1740
30,448-1 0722
30.6813 7290
30,9080 5649
28.4041 5454
28.0171 8203
28.8245 0800
20.0262 8522
29.2226 6201
26.9654 6373
27.1509 3566
27.3310 0549
27.5058 3058
27.6755 6367
24.4097 1327
24.5504 4760
24.6864 2281
24.8177 9981
24.9447 3412
61
62
63
64
65
33.0063 1086
33.2580 0573
33.5041 6208
33.7449 0179
33.9803 4405
31.1303 9657
31.34G7 2836
31.5577 8377
31.7036 9148
31.9645 7705
29.4137 8298
29.5997 8879
29.7808 1634
29.9569 9887
30.1284 6605
27.8403 5307
28.0003 4279
28,1556 7261
28.3064 7826
28.4528 9152
25.0673 7596
25.1858 7049
25.3003 5796
25.4109 7388
25.5178 491G
66
67
68
69
70
34.2106 0543
34.4357 9903
34.6560 3905
34.8714 3183
35.0820 8492
32.1005 6298
32.3517 6876
32.5383 1099-
32.7203 0310
32.8978 5098
30.2953 4409
30.4577 5581
30.0158 2074
30.7096 5522
30.9193 7247
28.5950 4031
28.7330 4884
28.8670 3771
28.9971 2399
29.1234 2135
25.6211 1030
25.7208 7951
25.8172 "7489
25.9104 1052
26.0003 9664
71
72
73
74
75
35.2S81 0261
35.4895 8691
35.fS66 3756
35.S793 5214
36.0678 2605
33.0710 7998
33.2400 7803
33.4049 5417
33.5058 0895
33.7227 4044
31.0650 8270
31.2008 9314
31.3449 0816
31.4702 2936
31.0099 5558
29.2460 4015
29.3050 8752
29.4806 0750
29.5928 8106
29.7018 2028
26.0873 3975
26.1713 4275
26.2525 050H
26.3309 2278
26.4066 8868
76
77
78
79
80
36.2521 5262
36.4324 2310
36.6087 2675
36.7811 5085
36.9497 8079
33.8758 4433
34.0252 1308
34.1709 4047
34.3131 12G5
34.4518 1722
31.7371 8304
3 1.8(510 0540
31.9815 1377
32.0087 9085
32.2129 4098
29.8075 9833
29.9102 89G4
30.0099 8994
30.1067 8635
30.2007 6345
26.4798 9244
26.5506 2072
20.6189 5721
26.0849 8281
26.7487 7567
81
82
83
84
85
37.1147 0004
37.2759 9026
37.4337 3130
37.588Q 0127
37.7388 7655
34.5871 3875
34.7191 5970
34.8479 6074
34.9736 2023
35.0962 I486
32.3240 3015
32.4321 4013
32.5373 68.50
32.6397 7409
32.7394 4009
30.2920 0335
30.3805 8577
30.4665 8813
30.5500 8556
30.6311 5103
26.8104 1127
26.8099 6258
20.9275 0008
26.9830 9186
27.0368 0373
86
87
88
89
00
37.8864 3183
38.0307 4018
38.1718 7304
38.3099 0028
38.4448 9025
35.2158 1938
35.3325 0071
35.4463 4801
35.5574 12G9
35.6657 6848
32.8304 3804
32.9308 3994
33.0227 1527
33.1121 3165
33.1991 5489
30.7098 5537
30.7862 6735
30.8004 5374
30.9324 7936
31.0024 0714
27.0886 9926
27.1388 3986
27.1872 8489
27.2340 9168
27.2793 15C4
91
92
03
94
95
38.5769 0978
38.7000 2423
38.8322 9754
38.9557 9221
39.0765 6940
35.7714 8144
35.8746 1004
35.9752 3516
36.0734 0016
36.1691 7089
33.2838 4905
33.3002 7044
33.4404 9776
33.5245 7202
33.6005 5071
31.0702 9820
31.1362 1184
31.2002 05G7
31.2623 3560
31.3226 5592
27.3230 1028
27.3052 2732
27.4060 1073
27.4454 2080
27.4835 0415
96
97
98
99
100
39.1946 8890
39.3102 0920
39.4231 8748
39.5336 7968
39.6417 4052
36.2626 0574
36.3537 6170
3G.442G 9434
36.5294 5790
30.6141 0526
33.0745 0775
33.7404 7956
33.8165 2512
33.8846 9598
33.9510 4232
31.3812 1934
31.4380 7703
31.4032 7867
31.5408 7250
31.5989 0534
27.5202 9387
27.5558 3948
27.5901 8308
27.6233 6529
27.6554 2540
T VI 67
TABLE VI. PRESENT VALUE OP ANNUITY OF 1 PER PERIOD
It
4%
4f%
I 6%
6 1%
6%
1
2
3
5
0.9615 3846
1.8860 9467
2.7750 9103
3.6298 9522
4.4518 2233
0.9569 3780
1.8726 6775
2.7489 6435
3.5875 2570
4.3899 7674
0.9523 8095
1.8594 1043
2.7232 4803
3.5459 5050
4.3294 7667
0.9478 6730
1.8463 1971
2.6979 3338
3.5051 5012
4.2702 8448
0.9433 9623
1.8333 9267
2.6730 1195
3.4651 0561
4.2123 6379
6
8
9
10
5.2421 3686
6.0020 5467
6.7327 4487
7.4353 3161
8.1108 9578
5.1678 7248
5.8927 0094
6.5958 8607
7.2687 9050
7.9127 181
5.0756 9206
5.7863 7340
6.4632 1276
7.1078 2168
7.7217 3493
4.9955 3031
5.6829 6712
6.3345 6599
6.9521 9525
7.5376 2583
4.9173 2433
5.5823 8144
6.2097 9381
6.8016 9227
7.3600 8705
11
12
13
14
15
8.7604 7671
9.3850 7376
9.9856 4785
10.5631 2293
11.1183 8743
8,5289 1692
9.1185 8078
9.6828 5242
10.2228 2528
10.7395 4573
8.3064 1422
8.8632 5164
9.3935 7299
9.8986 4094
10.3796 5804
8.0925 3633
8.6185 1785
9.1170 7853
9.5896 4790
10.0375 8094
7.8868 7458
8.3838 4394
8.8526 8296
9.2949 8393
9.7122 4899
16
17
18
19
20
11.6522 9561
12.1656 6885
12.6592 9697
13.1339 3940
13.5903 2634
11.2340 1505
11.7071 9143
12.1599 9180
12.5932 9359
13.0079 3645
10.8377 6956
11.2740 6625
11.6895 8690
12.0853 2086
12.4622 1034
10.4621 6203
10.8646 0856
11.2460 7447
11.6076 5352
11.9503 8249
10.1058 9527
10.4772 5969
10.8276 0348
11.1581 1649
11.4699 2122
21
22
23
24
25
14,0291 5995
14.4511 1533
14.8568 4167
15.2469 6314
15.6220 7994
13.4047 2388
13.7844 2476
14.1477 7489
14.4954 7837
14.8282 0896
12.8211 5271
13.1630 0258
13.4885 7388
13.7986 4179
14.0939 4457
12.2752 4406
12.5831 6973
12.8750 4240
13.1516 9895
13.4139 3266
11.7640 7662
12.0415 8172
12.3033 7898
12.5503 5753
12.7833 5616
26
27
28
29
30
15.9827 6918
16.3295 8575
16.6630 6322
16.9837 1463
17.2920 3330
15.1466 1145
15.4513 0282
15.7428 7351
16.0218 8853
16.2888 8854
14.3751 8530
14.6430 3362
14.8981 2726
15.1410 7358
15.3724 5103
13.6624 9541
13.8980 9991
14.1214 2172
14.3331 0116
14.5337 4517
13.0031 6619
13.2105 3414
13.4061 G428
13.5907 2102
13.7648 3115
31
32
33
34
35
17.5884 9356
17.8735 5150
18.1476 4567
18.4111 9776
18.6646 1323
16.5443 9095
16.7888 9086
17.0228 0207
17.2467 5796
17.4610 1240
15.5928 1050
15.8026 7667
16,0025 4921
16.1929 0401
16.3741 9429
14.7239 2907
14.9041 9817
IS 0750 6936
15.2370 3257
15.3905 5220
13.9290 8599
14.0840 4339
14.2302 2961
14.3681 4114
14.4982 4636
36
37
38
39
40
18.9082 8195
19.1425 7880
19.3678 6423
19.5844 8484
19.7927 7338
17.6660 4058
17.8622 3979
18.0499 9023
18.2296 5572
18.4015 8442
16.5468 5171
16.7112 8734
16.8678 9271
17.0170 4067
17.1590 8635
15.5360 6843
15.6739 9851
15.8047 3793
15.9286 6154
16.0461 2469
14.6209 8713
14.7367 8031
14.8460 1916
14.9490 7468
15.0462 9687
41
42
43
44
45
19.9930 5181
20.1856 2674
20.3707 9494
20.5488 4129
20.7200 3970
18.5661 0949
18.7235 4975
18.8742 1029
19.0183 8305
19.1563 4742
17.2943 6796
17.4232 0758
17.5459 1198
17.6627 7331
17.7740 6982
16.1574 6416
16.2629 9920
16.3630 3242
16.4578 5063
16.5477 2572
15.1380 1592
15.2245 4332
15.3061 7294
15.3831 8202
15.4558 3209
46
47
48
49
50
20.8846 5356
21.0429 3612
21.1951 3088
21.3414 7200
21.4821 8462
19.2883 7074
19.4147 0884
19.5356 0654
19.6512 9813
19.7620 0778
17.8800 6650
17.9810 1571
18.0771 6782
18.1687 2173
18.2559 2546
16.6329 1537
16.7136 6386
16.7982 0271
16.8627 5139
16.9315 1790
15.5243 6990
15.5890 2821
15.6500 2661
15.7075 7227
15.7618 6064
T VI 58
TABLE VI. PRESENT VALUE OF ANNUITY OF 1 PER PERIOD
It
4%
4|%
6%
|%
6%
51
52
63
54
55
21.6174 8521
21.7475 8193
21.8720 7493
21.9929 5667
22.1086 1218
19.8679 5003
19.9693 3017
20.0663 4466
20.1591 8149
20.2480 2057
18.3389 7663
18.4180 7298
18.4934 0284
18.5651 4556
18.6334 7196
16.9966 9943
17.0584 8287
17 1170 4538
17 1725 5486
172251 7048
15.8130 7607
15.8613 9252
15.9069 7408
15.9499 7554
15.9905 4297
56
57
58
59
GO
22.2189 1940
22.3267 4943
22.4295 6676
22.5284 2957
22.6234 8997
20.3330 3404
20.4143 8664
20.4922 3602
20.5067 3303
20.6380 2204
18.6985 4473
18.7605 1879
18.8195 4170
18.8757 5400
18.9292 8952
17.2750 4311
17.3223 1575
17.3671 2393
174095 9614
174498 5416
16.0288 1412
16.0649 1898
16.0989 8017
16.1311 1337
16.1614 2771
61
62
63
64
65
22.7148 9421
22.8027 8289
22.8872 9124
22.9685 4927
23.0466 8199
20.7062 4118
20.7715 2266
20.8339 9298
20.8937 7319
20.9509 7913
18.9802 7574
19.0288 3404
19.0750 8003
19.1191 2384
19 1610 7033
174880 1343
17.5241 8334
17 5584 6762
17.5909 6457
17 6217 6737
16.1900 2614
16.2170 0579
16.2424 5829
16.2664 7009
16.2891 2272
66
67
68
69
70
23.1218 0961
23.1940 4770
23.2635 0740
23.3302 9558
23.3945 1498
21.0057 2165
21.0581 0684
21 1082 3621
21.1562 0690
21.2021 1187
19.2010 1936
19.2390 6606
19.2753 0101
19.3098 1048
19.3426 7665
17.6509 6433
17.6786 3917
177048 7125
177297 3579
17 7533 0406
16.3104 9314
16.3306 5390
16.3496 7349
16.3676 1650
16.3845 4387
71
72
73
74
75
23.4562 6440
23.5156 3885
23.572? 2966
23.6276 2468
23.6804 0834
21.2460 4007
21.2880 7662
21.3283 0298
21.3667 9711
21.4036 3360
19.3739 7776
19.4037 8834
19.4321 7937
19.4592 1845
19.4849 6995
17 7756 4366
177968 1864
17.8168 8970
17.8359 1441
17 8539 4731
16.4005 1308
16.4155 7838
16.4297 9093
16.4431 9899
16.4558 4810
76
77
78
79
80
23.7311 6187
23.7799 6333
23.8268 8782
23.8720 0752
23.9153 9185
21.4388 8383
21.4726 1611
21. -6048 9579
21.5357 8545
21.5653 4493
19.5094 9519
19.5328 5257
19.5550 9768
19.5762 8351
19.5964 6048
17 8710 4010
17.8872 4180
17.9025 9887
17.9171 5532
179309 5291
16.4677 8123
16.4790 3889
16.4896 5933
16.4996 7862
16.5091 3077
81
82
83
84
85
23.9571 0754
23.9972 1879
24.0357 8730
24.0728 7240
24.1085 3116
21.5936 3151
21.6207 0001
21.6466 0288
21.6713 9032
21.6951 1035
19.6156 7665
19.6339 7776
19.6514 0739
19.6680 0704
19.6838 1623
17.9440 3120
17 9564 2768
179681 7789
17.9793 1554
179898 7255
16.5180 4790
16.5264 6028
16.5343 9649
16.5418 8348
16.5489 4668
86
87
88
89
90
24.1428 1842
24.1757 8694
24.2074 8745
24.2379 6870
24.2672 7759
21.7178 0895
21.7395 3009
21.7603 1588
21.7802 0658
21.7992 4075
19.6988 7260
19.71 $2 1200
19.7268 6857
19.7398 7483
19.7522 6174
17.9998 7919
18.0093 6416
18.0183 5466
18.0268 7645
18.0349 5398
16.5556 1008
16.5618 9630
16.5678 2670
16.5734 2141
16.5786 9944
91
92
93
94
95
24.2954 5923
24.3225 5695
24.3486 1245
24.3736 6582
24.3977 5559
21.8174 5526
21.8348 8542
21.8515 6499
21.8675 2631
21.8828 0030
19.7640 5880
19.7752 9410
19.7859 9438
19.7961 8512
19.8058 9059
18.0426 1041
18.0498 6769
18.0567 4662
18.0632 6694
18.0694 4734
16.5836 7872
16.5883 7615
16.5928 0769
16.5969 8839
16.6009 3244
96
97
98
99
100
24.4209 1884
24.4431 9119
24.4646 0692
24.4851 9896
24.5049 9900
21.8974 1655
21.9114 0340
21.9247 8794
21.9375 9612
21.9498 5274
19.8151 3390
19.8239 3705
19.8323 2100
19.8403 0571
19.8479 1020
18.0753 0553
18.0808 5833
18.0861 2164
18.0911 1055
18.0958 3939
16.6046 5325
16.6081 6344
16.6114 7494
16.6145 9900
16.6175 4623
T VI 59
TABLE VI. PRESENT VALTTE OF ANNUITY OF 1 PEK PERIOD
n
6f%
7%
7|%
8%
i%-
l
2
3
4
6
0.9389 6714
1.8206 2642
2.6484 7551
3.4257 9860
4.1556 7944
0.9345 7944
1.8080 1817
2.6243 1604
3.3872 1126
4.1001 9744
0.9302 3256
1.7955 6517
2.6005 2574
3.3493 2627
4.0458 8490
0.9259 2593
1.7832 6475
2.5770 9699
3.3121 2684
3.9927 1004
0.9216 5899
1.7711 1427
2.5540 2237
3.2755 9666
3.9406 4208
6
7
8
9
10
4.8410 1356
5.4845 1977
6.0887 5096
6.6561 0419
7 1888 3022
4.7665 3966
5.3892 8940
5.9712 9851
6.5152 3225
7.0235 8154
4.6938 4642
5.29G6 0132
5.8573 0355
6.3788 8703
6.8640 8096
4.6228 7966
5.2063 7006
5.7466 3894
6.2468 8791
6.7100 8140
4.5535 8717
5.1185 1352
5.6391 8297
6.1190 6264
6.5613 4806
11
12
13
14
15
7.6390 4246
8.1587 2532
8.5997 4208
9.0138 4233
9.4026 6885
7.4986 7434
7.9426 8630
8.3576 5074
8.7454 6799
9.1079 1401
7.3154 2415
7.7352 7827
8.1258 4026
8.4891 5373
8.8271 1974
7.1389 6426
7.5360 7802
7.9037 7594
8.2442 3698
8.6594 7869
6.9689 8439
7.3446 8607
7.6909 5490
8.0100 9668
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
9.7632 2299
10.0590 8691
10.3355 9524
10.5940 1425
9.1415 0674
9.4339 5976
9.70GO 0908
9.9590 7821
10.1944 9136
8.8513 6916
9.1216 3811
9.3718 8714
9.6035 9920
9.8181 4741
8.5753 3325
8.8251 9194
9.0554 7644
9.2677 2022
9.4633 3661
21
2*
23
24
25
11.2849 8333
11.5351 9562
11.7701 3673
11.9907 3871
12.1978 7672
10.8355 2733
11.0612 4050
11.2721 8738
11.4693 3400
11.0535 8318
10.4134 8033
10.6171 9101
10.8066 8931
10.9829 6680
11.1469 4586
10.0168 0316
10.2007 4366
10.3710 5895
10.5287 5828
10.6747 7619
9.6436 2821
9.8097 9559
9.9629 4524
10.1040 9700
10.2341 9078
26
27
28
29
30
12.3923 7251
12.5749 9766
12.7464 7668
12.9074 8984
13.0586 7591
11.8257 7867
11.9867 0904
12.1371 1125
12.2776 7407
12.4090 4118
11.2994 8452
11.4413 8095
11.5733 7763
11.6961 6524
11.8103 8627
10.8099 7795
10.9351 6477
11.0510 7849
11.1584 0601
11.2577 8334
10.3540 9288
10.4846 0174
10.5664 5321
10.6603 2554
10.7468 4382
31
32
33
34
35
13.2006 3465
13.3339 2925
13.4590 8850
13.5766 0892
13.6869 5673
12.5318 1419
12.6465 5532
12.7537 9002
12.8540 0936
12.9476 7230
11.9166 3839
12.0154 7757
12.1074 2099
12,1929 4976
12.2725 1141
11.3497 9939
11.4349 9944
11.5138 8837
11.5869 3367
11.6545 6822
10.8265 8416
10.9000 7757
10.9678 1343
11.0302 4279
11.0877 8137
36
37
38
39
40
13.7905 6970
13.8878 5887
13.9792 1021
14.0649 8611
14.1455 2687
13.0352 0776
13.1170 1660
13.1934 7345
13.2649 2846
13.3317 0884
12.3465 2224
12.4153 6953
12.4794 1351
12.5389 8931
12.5944 0866
11.7171 9279
11.7751 7851
11.8288 6899
11.8785 8240
11.9246 1333
11.1408 1233
11.1896 8878
11.2347 3620
11.2762 5457
11.3145 2034
41
42
43
44
45
14.2211 5199
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 2615
12.8186 2898
11.9672 3457
12.0066 9867
12.0432 3951
12.0770 7362
12.1084 0150
11.3497 8833
11.3822 9339
11.4122 5197
11.4398 6357
11.4653 1205
46
47
48
49
50
14.5354 2575
14.5872 5422
14.6359 1946
14.6816 1451
14.7245 2067
13.650O 2018
13.6916 0764
13.7304 7443
13.7667 9853
13.8007 '4629
12.8545 3858
12.8879 4287
12.9190 1662
12.9479 2244
12.9748 1157
12.1374 0880
12.1642 6741
12.1891 3649
12.2121 6341
12.2334 8464
11.4887 6686
11.5103 8420
11.5303 0802
11.5486 7099
11.5655 9538
T VI 60
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
1 1
n
1-2%
1%
%
!%
1%
i
3
5
1.0041 V 6667
0.5031 2717
0.3361 1496
0.2526 0958
0.2025 0093
1.0050 0000
0.5037 6312
0.3366 7221
0.2531 3279
0.2030 0997
1.0058 3333
0.5043 7924
0.3372 2976
0.2536 6644
0.2035 1357
1.0075 0000
0.5056 3200
0.3383 4579
0.2547 0501
0.2045 2242
1.0100 0000
0.5075 1244
0.3400 2211
0.2562 8109
0.2060 39SO
7
8
9
10
0.1691 0564
0.1452 4800
0.1273 5512
0.1134 3876
0.1023 0596
0.1695 9546
0.1457 2854
0.1278 2886
0.1139 0736
0.1027 7057
0.1700 8594
0.1462 0986
0.1283 0351
0.1143 7698
0.1032 3632
0.1710 6891
0.1471 7488
0.1292 5552
0.1153 1929
0.1041 7123
0.1725 4837
0.1486 2828
0.1306 9029
0.1167 4037
0.1055 820$
11
12
13
14
15
0.0931 9757
0.0856 0748
0.0791 8532
0.0736 8082
0.0689 1045
0.0936 5903
0.0860 6043
0.0796 422-4
0.0741 3609
0.0693 6436
0.0941 2175
0.0865 2675
0.0801 0064
0.0745 9295
0.0698 1999
0.0950 5094
0.0874 5148
0.0810 2188
0.0755 1146
0.0707 3639
0.0964 6408
0.0888 4879
0.0824 1482
0.0769 0117
0.0721 2378
16
17
13
19
20
0.0647 3655
0.0610 5387
0.0577 8053
0.0548 5191
0.0522 1630
0.0651 8937
0.0615 0579
0.05S2 3173
0.0553 0253
0.0526 6645
0.0650 4401
0.0619 5966
0.0586 8499
0.0557 5532
0.0531 1889
0.0665 5879
0.0628 7321
0.0595 9766
0.0566 6740
0.0540 3063
0.0679 4460
0.0642 5806
0.0609 8205
0.0580 5175
0,0554 1532
21
22
23
24
25
0.0498 3183
0.0476 6427
0.0456 8531
0.0438 7139
0.0422 0270
0.0502 8163
0.0481 1380
0.0461 3465
0.0443 2061
0.0426 5186
0.0507 3383
0.0485 6585
0.0465 8663
0.0447 7258
0.0431 0388
0.0516 4543
0.0494 7748
0.0474 9846
0.0456 8474
0.0440 1650
0.0530 3075
0.0508 6371
0.0488 8584
0.0470 7347
0.0454 0675
26
27
28
29
30
0.0406 6247
0.0392 3645
0.0379 1239
0.0366 7974
0.0355 2936
0.0411 1163
0.0396 8565
0.0383 6167
0.0371 2914
0.0359 7892
0.0415 C376
0.0401 3793
0.0388 1415
0.0375 8186
0.0364 3191
0.0424 7693
0.0410 5176
0.0397 2871
0.0384 9723
0.0373 4816
0.0438 6888
0.0424 4553
0.0411 2444'
0.0398 9502
0.0387 4811
31
33
33
31
35
0.0344 5330
0.0334 4458
0.0324 9708
0.0316 0540
0.0307 6476
0.0349 0304
0.0338 9453
0.0329 4727
0.0320 5586
0.0312 1550
0.0353 6633
0.0343 4815
0.0334 0124
0.0325 1020
0.0316 7024
0.0362 7352
0.0352 6634
0.0343 2048
0.0334 3053
0.0325 9170
0.0376 7573
0.0366 7089
0.0357 2744
0.0348 3997
0.0340 0368
36
37
38
39
40
0.0299 7090
0.0292 2003
0.0285 0875
0.0278 3402
0.0271 9310
0.0304 2194
0.0296 7139
0.0289 6045
0.0282 8607
0.0276 4552
0.0308 7710
0.0301 2698
0.0294 1G49
0.0287 4258
0.0281 0251
0.0317 9973
0.0310 5082
0.0303 4157
0.0296 6893
0.0290 3016
0.0332 1431
0.0324 6805
0.0317 6150
0.0310 9160
0.0304 6560
41
42
43
44
45
0.0263.8352
O.OSf-O 0303
0.0254 4961
0.0249 2141
0.0244 1675
0.0270 3631
0.0264 5622
0.0259 0320
0.0253 7541
0.0248 7117
0.0274 9379
0.02G9 1420
0.0263 6170
0.0258 3443
0.0253 3073
0.0284 2276
0.0278 4452
0.0272 9338
0.0267 6751
0.0262 6521
0.0298 5102
0.0292 7663
0.0287 2737
0.0282 0441
0.0277 0605
46
47
48
49
50
0.0239 3409
0.0234 7204
0.0230 2929
0.0226 0468
0.0221 9711
0.0243 8894
0.0239 2733
0.0234 8503
0.0230 6087
0.0226 5376
0.0248 4905
0.0243 8798
0.0239 4624
0.0235 2265
0.0231 1611
0.0257 8495
0.0253 2532
0.0248 8504
0.0244 6292
0.0240 6787
0.0272 2775
0.0267 7111
0.0283 3384
0.0259 1474
0,0255 1273
T VII 61
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
1 1 j.,-
__ -j- i
n
a*
1%
5%
!%
1%
51
53
53
54
55
0.0218 0557
0.0214 2916
0.0210 6700
0.0207 1830
0.0203 8234
0.0222 6269
0.0218 8675
0.0215 2507
0.0211 7686
0.0208 4139
0.0227 2563
0.0223 5027
0.0219 8919
0.0216 4157
0.0213 0671
0.0236 6888
0.0232 9503
0.0229 3546
0.0225 8938
0.0222 5605
0.0251 2680
0.0247 5603
0.0243 9956
0.0240 5658
0.0237 2637
56
57
58
59
60
0.0200 5843
0.0197 4593
0.0194 4426
0.0191 5287
0.0188 7123
0.0205 1797
0.0202 0598
0.0199 0481
0.0196 1392
0.0193 3280
0.0209 8390
0.0206 7251
0.0203 7196
0.0200 8170
0.0198 0120
0.0219 3478
0.0216 2496
0.0213 2597
0.0210 3727
0.0207 5836
0.0234 0823
0.0231 0156
0.0228 0573
0.0225 2020
0.0222 4445
61
63
64
65
0.0185 98S8
0.0183 3536
0.0180 8025
0.0178 3315
0.0175 9371
0.0190 6096
0.0187 9796
0.0185 4337
0.0182 9681
0.0180 5789
0.0195 2999
0.0192 6762
0.0190 1366
0.0187 6773
0.0185 2946
0.0204 8873
0.0202 2795
0.0199 7560
0.0197 3127
0.0194 9460
0.0219 7800
0.0217 2041
0.0214 7125
0.0212 3013
0.0209 9667
66
67
68
69
70
0.0173 6156
0.0171 3639
0.0169 1788
0.0167 0574
0.0164 9971
0.0178 2627
0.0176 0163
0.0173 8366
0.0171 7206
0.0169 6657
0.0182 9848
0.0180 7449
0.0178 5716
0.0176 4622
0.0174 4138
0.0192 6524
0.0190 4286
0.0188 2716
0.0186 1785
0.0184 1464
0.0207 7052
0.0205 5136
0.0203 3888
0.0201 3280
0.0199 3282
71
73
73
74
75
0.0162 9952
O.0161 0493
0.0159 1572
0.0157 3165
0.0155 5253
0.0167 6693
0.0165 7289
0.0 163 8422
0.01G2 0070
0.01GO 2214
0.0172 4239
0.0170 4001
0.0168 6100
0.01G6 7814
0.0165 0024
0.0182 1728
0.0180 2554
0.0178 3917
0.0176 5796
0.0174 8170
0.0197 3870
0.0195 5019
0.0193 6706
0.0191 8910
0.0190 1609
76
77
78
79
80
0.0153 7816
0.0152 0836
0*0150 4295
0.0148 8177
0.0147 2464
0.0158 4832
0.0156 7908
0.0155 1423
0.0153 53GO
0.0151 9704
0.0163 2709
0.0161 5851
0.0159 9432
0.0158 3436
0.0156 7847
0.0173 1020
0.0171 4328
0.0169 8074
0.01G8 2244
0.01G6 6821
0.0188 4784
0.0186 8416
0.0185 2488
0.0183 6984
0.0182 1885
81
83
83
84
85
0.0145 7144
0.0144 2200
0.0142 7620
0.0141 3391
0.0139 9500
0.0150 4439
0.0148 9552
0.0147 5028
0.0146 0855
0.0144 7021
0.0155 2650
0.0153 7830
0.0152 3373
0.0150 9268
0.0149 5501
0.0165 1790
0.0163 7136
0.0162 2847
0.0160 8908
0.0159 5308
0.0180 7180
0.0179 2851
0.0177 8886
0.0176 5273
0.0175 1998
86
87
89
89
90
0.0138 5935
0.0137 2685
0.0135 9740
0.0134 7088
0.0133 4721
0.0143 3513
0.0142 0320
0.0140 7431
0.0139 4837
0.0138 2527
0.0148 2060
0.0146 8935
0.0145 6115
0.0144 3588
0.0143 1347
0.0158 2034
0.0156 9076
0.0155 6423
0.0154 4064
0.0153 1989
0.0173 9050
0.0172 6417
0.0171 4089
0.0170 2056
0.0169 0306
91
93
93
94
95
0.0132 2629
0.0131 0803
0.0129 9234
0.0128 7915
0.0127 6837
0.0137 0493
0.0135 8724
0.0134 7213
0.0133 5950
0.0132 4930
0.0141 9380
0.0140 7679
0.0139 6236
0.0138 5042
0.0137 4090
0.0152 0190
0.0150 8657
0.0149 7382
0.0148 6356
0.0147 5571
0.0167 8832
0.0166 7624
0.0165 6673
0.0164 5971
0.0163 5511
96
97
98
99
100
0.0126 6992
0.0125 5374
0.0124 4976
0.0123 4790
0.0122 4811
0.0131 4143
0.0130 3583
0.0129 3242
0.0128 3115
0.0127 3194
0.0136 3372
0.0135 2880
0.0134 2608
0.0133 2549
0.0132 2696
0.0146 5020
0.0145 4696
0.0144 4592
0.0143 4701
0.0142 5017
0.0162 5284
0.0161 5284
0.0160 5503
0.0159 5936
0.0158 6574
T VII 62
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
n
&%
i<r
% %
%
!%
1%
101
102
103
104
105
0.0121 5033
0.0120 5449
0.0119 6054
0.0118 6842
0.0117 7809
0.0126 3473
0.0125 3947
0.0124 4611
0.0123 5457
0.0122 6481
0.0131 3045
0.0130 3587
0.0129 4319
0.0128 6234
0.0127 6238
0.0141 5533
0.0140 6243
0.0139 7143
0.0138 8226
0.0137 9487
0.0157 7413
0.0156 8446
0.0155 9668
0.0155 1073
0.0154 2658
106
107
108
109
110
0.0116 8948
0.0116 0256
0.0115 1727
0.0114 3358
0.0113 5143
6.0121 7679
0.0120 9045
0.0120 0575
0.0119 2264
0.0118 4107
0.0126 7594
0.0125 9029
0.0125 0028
0.0124 2385
0.0123 4298
0.0137 0922
0.0136 2524
0.0135 4291
0.0134 6217
0.0133 8296
0.0153 4412
0.0152 6336
0.0151 8423
0.0151 0669
0.0150 3069
111
112
113
114
115
0.0112 7079
aOlll 9161
0.0111 1386
0.0110 3750
0.0109 6249
0.0117 6102
0.0116 8242
0.0116 0526
0.0115 2948
0.0114 5506
0.0122 6361
0.0121 8571
0.0121 0923
0.0120 3414
0.0119 6041
0.0133 0527
0.0132 2905
0.0131 5425
0.0130 8084
0.0130 0878
0.0149 5620
0.0148 8317
0-0148 1156
0.0147 4133
0.0146 7243
116
117
118
119
120
0.0108 8880
0.0108 1639
0.0107 4524
0.0106 7530
0.0106 0655
0.0113 8195
0.0113 1013
0.0112 3956
0.0111 7021
0.0111 0205
0.0118 8799
0.0118 1686
0.0117 4698
0.0116 7832
0.0116 1085
0.0129 3803
0.0128 6857
0.0128 0037
0.0127 3338
0.0126 6758
0.0146 0488
0.0145 3860
0.0144 735Q
0.0144 0973
0.0143 4709
121
122
123
124
125
O.0105 3896
0.0104 7251
0.0104 0715
O.0103 4288
0.0102 7965
0.0110 3505
0.0109 6918
0.0109 0441
0.0108 4072
0.0107 7808
0.0115 4454
0.0114 7936
0.0114 1528
0.0113 5228
0.0112 9033
0.0126 0294
0.0125 3942
0.0124 7702
0.0124 1568
0.0123 5540
0.0142 8561
0.0142 2525
0.0141 6599
0.0141 0780
0.0140 5065
126
127
128
129
130
O.0102 1745
0.0101 6625
O.0100 9603
O.0100 3677
O.0099 7844
0.0107 1647
0.0106 5586
0.0105 9623
0.0105 3755
0.0104 7981
0.0112 2940
0.0111 6948
0.0111 1054
0.0110 5255
0.0109 9550
0.0122 9614
0.0122 3788
0.0121 8060
0.0121 2428
0.0120 6888
0.0139 9452
" 0.0139 3939
0.0138 8524
0.0138 3203
0.0137 7975
131
132
133
134
135
O.0099 2102
O.0098 6149
0.0098 0883
O.0097 5403
O.0097 0005
0.0104 2298
0.0103 6704
0.0103 1197
0.0102 5775
0.0102 0436
0.0109 3935
0.010S 8410
0.0108 2972
0.0107 7619
0.0107 2349
0.0120 1440
0.0119 6080
0.0119 0808
0.0118 5621
0.0118 0516
0.0137 2837
0.0136 7788
0.0136 2825
0.0135 7947
0.0135 3151
136
137
138
139
140
0.0096 4689
0.0095 9453
0.0095 4295
0.0094 9213
0.0094 4205
0.0101 5179
0.0101 0002
0.0100 4902
0.0099 9879
0.0099 4930
0.0106 7161
0.0106 2052
0.0105 7021
0.0105 2007
0.0104 7187
0.0117 5493
0.0117 0550
0.0116 5684
0-0116 0894
0.0115 6179
0.0134 8437
0.0134 3801
0.0133 9242
0.0133 4759
0.0133 0349
141
142
143
144
145
0.0093 9271
0.0093 4408
0.0092 9615
0.0092 4890
0.0092 0233
0*0099 0055
0.0098 5250
0.0098 0616
0.0097 5850
0.0097 1252
0.0104 2380
0.0103 7644
0.0103 2978
0.0102 8381
0.0102 3851
0.0115 1536
0.0114 6965
0.0114 2464
0.0113 8031
0.0113 3664
0.0132 6012
0.0132 1746
0.0131 7549
0.0131 3419
0.0130 9356
146
147
148
149
150
0.0091 5641
0.0091 1114
0.0090 6650
0.0090 2247
0.0089 7905
0.0096 6719
0.0096 2250
0.0095 7844
0.0095 3500
0.0094 9217
0.0101 9386
0.0101 4986
0.0101 0649
0.0100 6373
0.0100 2159
0.0112 9364
0.0112 5127
0.0112 0953
0.0111 6841
0.0111 2790
0.0130 5358
0.0130 1423
0.0129 7551
0.0129 3739
0.0128 9988
T VII 63
TABLE VII. PERIODICAL, PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
Offl
n
il%
il%
il%
i|%
2%
3
5
1.0112 5000
0.5084 5323
0.3408 6130
0.2570 7058
0.2068 0034
1.0125 0000
0.5093 9441
0.3417 0117
0.2578 6102
0.2075 6211
1.0150 0000
0.5112 7792
0.3433 8296
0.2594 4478
0.2090 8932
1.0175 0000
0.5131 6295
0.3450 6746
0.2610 3237
0.2106 2142
1.0200 0000
0.5150 4950
0.3467 5467
0.2626 2375
0.2121 5839
6
7
8
10
0.1732 9034
0.1493 5762
0.1314 1071
0.1174 5432
0.1062 9131
0.1740 3381
0.1500 8872
0.1321 3314
0.1181 7055
0.1070 0307
0.1755 2521
0.1515 5616
0.1335 8402
0.1196 0982
0.10S4 3418
0.1770 2256
0.1530 3059
0.1350 4292
0.1210 5813
0.1098 7534
0.1785 2581
0.1545 1196
0.1365 0980
0.1225 1544
0.1113 2653
11
13
14
15
0.0971 5984
0.0895 5203
0.0831 1626
0.0776 0138
0.0728 2321
0.0978 6839
0.0902 5831
0.0838 2100
0-0783 0515
0.0735 2646
0.0992 9384
0.0916 7999
0.0852 4036
0.0797 2332
0.0749 4436
0.1007 3038
0.0931 1377
0.0866 7283
0.0811 5562
0.0763 7739
0.1021 7794
0.0945 5960
0.0881 1835
0.0826 0197
0.0778 2547
16
17
18
19
20
0.0686 4363
0.0649 5698
0.0616 8113
0.0587 5120
0.0561 1531
0.0693 4672
0.0656 6023
0.0623 8479
0.0594 5548
0.0568 2039
0.0707 6508
0.0670 7966
0.0638 0578
0.0608 7847
0.0582 4574
0.0721 9958
0.0685 1623
0.0652 4492
0.0623 2061
0.0596 9122
0.0736 5013
0.0699 6984
0.0667 0210
0.0637 8177
0.0611 5672
21
22
23
24
25
0.0537 3145
0.0515 6525
0.0495 8833
0.0477 7701
0.0461 1144
0.0544 3748
0.0522 7238
0.0502 9666
0.0484 8665
0.0468 2247
0.0558 6550
0.0537 0331
0.0517 3075
0.0499 2410
0.0482 6345
0.0573 1464
0.0551 5638
0.0581 8796
0.0513 8565
0.0497 2952
0.0587 8477
0.0566 3140
0.0546 6810
0.0528 7110
0.0512 2044
26
27
28
29
30
0.0445 7479
0.0431 5273
0.0418 3299
0.0406 0498
0.0394 5953
0.0452 729
0.0438 "6677
0.0425 4863
0.0413 2228
0.0401 7854
0.0467 3196
0.0453 1527
0.0440 0108
0.0427 7878
0.0416 3919
0.0482 0269
0.0467 9079
0.0454 8151
0.0442 6424
0.0431 'i975
0.0496 9923^
0.0482 9309
0.0469 8967
0.0457 7836
0.0446 4992
31
32
33
34
35
0.0383 8806
0.0373 8535
0.0364 4349
0.0355 5763
0.0347 2299
0.0391 0942
0.0381 0791
0.0371 6786
0.0362 8387
0.0354 5111
0.0405 7430
0,0395 7710
0.0386 4144
0.0377 6189
0.0369 3363
0.0420 7005
0.0410 7812
0.0401 4779
0.0392 7363
0.0384 5082
0.0435 9635
0.0426 1061
0.0416 8653
0.0408 1867
0.0400 0221
36
37
i
0.0339 3529
0.0331 9072
0.0324 8589
0.0318 1773
0.0311 8349
0.0346 6533
0.0339 2270
0.0332 1983
0.0325 5365
0.0319 2141
0.0361 5240
0.0354 1437
0.0347 1613
0.0340 5463
0.0334 2710
0.0376 7507
0.0369 4257
0.0362 4990
0.0355 9399
0.0349 7209
0.0392 3285
0.0385 0678
0.0378 2057
0.0371 7114
0.0365 5575
41
43
44
45
0.0305 8069
0.0300 0709
0.0294 6064
0.0289 3949
0.0284 4197
0.0313 2063
0.0307 4906
0.0302 0466
0.0296 8557
0.0291 9012
0.0328 3106
0.0322 6426
0.0317 2465
0.0312 1038
0.0307 1976
0.0343 8170
0.0338 2057
0.0332 8666
0.0327 7810
0.0322 9321
0.0359 7188
0.0354 1729
0.0348 8993
0.0343 8794
0.0339 0962
46
47
48
49
50
0.0279 6652
0.0275 1173
0.0270 7632
0.0266 5910
0.0262 5898
0.0287 1675
0.0282 6406
0.0278 3075
0.0274 1563
0.0270 1763
0.0302 5125
0.0298 0342
0.0293 7500
0.0289 6478
0.0285 7168
0.0318 3043
0.0313 8836
0.0309 6569
0.0305 6124
0.0301 7391
0.0334 5342
0.0330 1792
0.0326 0184
0.0322 0396
0.0318 2321
VII 64
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
J--JL.M
~
n
1|%
lj%
ll%
l|%
2%
51
52
S3
54:
55
0.0258 7494
0.0255 0606
0.0251 5149
0.0248 1043
0.0244 8213
0.0266 3571
0.0262 6897
0.0259 1653
0.0255 7760
0.0252 5145
0.0281 9469
0.0278 3287
0.0274 8537
0.0271 5138
0.0268 3018
0.0298 6269
0.0294 4665
0.0291 0492
0.0287 7672
0.0284 6129
0.0314 5856
0.0311 0909
0.0307 7302
0.0304 5226
0.0301 4337
56
67
58
59
60
0.0241 G592
0.0238 6116
0.0235 6726
0.0233 8366
0.0230 0985
0.0249 3739
0.0246 3478
0.0243 4303
0.0210 6158
0.0237 8993
0.0265 2106
0.0262 2341
0.0259 3661
0.0256 6012
0.0253 9343
0.0281 5795
0.0278 6606
0.0275 8503
0.0273 1430
0.0270 5336
0.0298 4656
0.0295 6120
0.0292 8667
0.0290 2243
0.0287 6797
61
62
63
64
65
0.0227 4534
0.0224 89G9
0.0222 4247
0.0220 0329
0.0217 7178
0.0235 3758
0.0232 7410
0.0230 2904
0.0227 9203
0.0225 G2C8
0.0251 3604
0.0248 8751
0.0216 4741
0.0244 1534
0.0241 9094
0.0268 0172
0.02G5 5892
0.0263 2455
0.02(50 9821
0.0258 7952
0.0285 2278
0.0282 8643
0.0280 5848
0.0278 3855
0.0276 2624
66
67
63
69
70
0.0215 4758
0.0213 3037
0.0211 1985
0.0209 1571
0.0207 1769
0.0223 40G5
O.021U 2560
0.0219 1724
0.0217 1.^7
0.0215 1941
0.0239 7386
0.0237 6376
0.0235 6033
0.0233 6329
0.0231 7235
0.0256 6813
0.0254 6372
0.0252 6596
0.0250 7459
00248 8930
0.0274 2122
0.0272 2316
0.0270 3173
0.0268 4665
0.0266 6765
71
72
73
74
75
0.0205 2552
0.0203 3896
0.0201 5779
0.0199 8177
0.0198 1072
0.0213 2941
0.0211 4501
0.0209 6600
0.0207 9215
0.0206 2325
0.0229 8727
0.0228 0779
0.0226 33G8
0.0224 6473
0.0223 0072
0.0247 0985
0.0245 3600
0.0243 6750
0.0212 0413
0.0240 4570
0.0264 9446
0.0263 2f>83
0.0261 6454
0.0260 0736
0.0258 5508
76
77
78
79
80
0.0196 4442
0.0194 8269
0.0193 2536
0.0191 7226
O.0190 2323
0.0204 5910
0.0202 9953
0.0201 4435
0.0199 9341
0.0198 4652
0.0221 4146
0.0219 8676
0.0218 3645
0.0216 9036
0.0215 4832
0.0238 9200
0.0237 4284
0.0235 9806
0.0234 5748
0.0233 2093
0.0257 0751
0.0255 6447
0.0254 2576
0.0252 9123
0.0251 6071
81
82
83
84
85
O.0188 7812
0.0187 3678
0.0185 9908
0.0184 6489
0.0183 3409
0.0197 0356
0.0195 6437
0.0194 2881
0.0192 9675
0.0191 6808
0.0214 1019
0.0212 7583
0.0211 4509
0.0210 1784
0.0208 9396
0.0231 8828
0.0230 5936
0.0229 3406
0.0228 1223
0.0226 9375
0.0250 3405
0.0249 1110
0.0247 9173
0.0246 7581
0.0245 6321
86
87
88
89
90
0.0182 0654
0.0180 8215
0.0179 6081
0.0178 4240
0.0177 2684
0.0190 4267
0.0189 2041
0.0188 0119
0.0186 8490
0.0185 7146
0.0207 7333
0.0206 5584
0.0205 4138
0.0204 2984
0.0203 2113
0.0225 7850
0.0224 6636
0.0223 5724
0.0222 5102
0.0221 4760
0.0244 5381
0.0243 4750
0.0242 4416
0.0241 4370
0.0240 4602
91
93
93
94
95
0.01 7ff 1403
0.0175 0387
0.0173 9629
0.0172 9119
0.0171 8851
0.0184 6076
0.0183 5271
0.0182 4724
0.0181 4425
0.0180 4366
0.0202 1516
0.0201 1182
0.0200 1104
0.0199 1273
0.0198 1681
0.0220 4690
0.0219 4882
0.0218 5327
0.0217 6017
0.0216 G944
0.0239 5101
0.0238 5859
0.0237 6868
0.0236 8118
0.0235 9602
96
97
98
99
100
0.0170 8816
0.0169 9007
0.0168 9418
0.0168 0041
0.0167 0870
0.0179 4540
0.0178 4941
0.0177 5560
0.0176 6391
0.0175 7428
0.0197 2321
0.0196 3186
0.0195 4268
0.0194 5560
0.0193 7057
0.0215 8101
0.0214 9480
0.0214 1074
0.0213 2876
0.0212 4880
0.0235 1313
0.0234 3242
0.0233 5383
0.0232 7729
0,0232 0274
VII 65
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
OB] $a"|
n
2\%
2l%
2f%
3%
3|%
i
3
4
5
1.0225 0000
0.5169 3758
0.3484 4458
0.2642 1893
0.2137 0021
1.0250 0000
0.5188 2716
0.3501 3717
0.2658 1788
0.2152 4686
1.0275 0000
0.5207 1825
0.3518 3243
0.2674 2059
0.2167 9832
1.0300 0000
0.5226 1084
0.3535 3036
0.2690 2705
0.2183 5457
1.0350 0000
0.5264 0049
0.3569 3418
0.2722 5114
0.2214 8137
6
7
8
10
0.1800 3496
a 1560 0025
0.1379 8462
0.1239 8170
0.1127 8768
0.1815 4997
0.1574 9543
0.1304 6735
0.1254 5689
0.1142 5876
0.1830 7083
0.1589 9747
0.1409 5795
0.1269 4095
0.1157 3972
0.1845 9750
0.1605 0635
0.1424 5639
0.1284 3386
0.1172 3051
0.1876 6821
0.1635 4449
0.1454 7665
0.1314 4601
0.1202 4137
It
12
13
14
15
0.1036 3649
0.0960 1740
0.0895 7686
0.0840 6230
0.0792 8852
0.1051 0596
0.0974 8713
0.0910 4827
0.0855 3653
0.0807 6646
0.1065 8629
0.0989 6871
0.0925 3252
0.0870 2457
0.0822 5917
0.1080 7745
0.1004 6209
0.0940 2954
O.0885 2634
0.0837 6658
0.1110 9197
0.1034 8395
0.0970 6157
0.0915 7073
0.0868 2507
16
17
18
19
20
0.0751 1663
0.0714 4039
0.0681 7720
0.0652 6182
0.0626 4207
0.0765 9899
0.0729 2777
0.0696 7008
0.0667 6062
0.0641 4713
0.0780 9710
0.0744 3186
0.0711 8063
6.0682 7802
0.0656 7173
0.0796 1085
0.0759 5253
0.0727 0870
0.0698 1388
0.0672 1571
0.0826 8483
0.0790 4313
0.0758 1684
0.0729 4033
0.0703 6108
21
22
23
24
25
0.0002 7572
0.0581 2821
0.0561 7097
0.0543 8023
0.0527 3599
0.0817 8733
0.0596 4661
0.0576 9638
0.0559 1282
0.0542 7592
0.0633 1941
0.0611 8640
0.0592 4410
0.0574 6863
0.0558 3997
0.0648 7178
0.0627 4739
0.0608 1390
0.0590 4742
0.0574 2787
0.0680 3659
0.0659 3207
0.0640 1880
0.0622 7283
0.0606 7404
26
27
28
29
30
0.0512 2134
0.0498 2188
0.0485 2525
0.0473 2081
0.0461 9934
0.0527 6875
0.0513 7687
0.0500 8793
0.0488 9127
0.0477 7764
0.0543 4116
0.0529 5776
0.0516 7738
0.0504 8935
0.0493 8442
0.0559 3829
0.0545 6421
0.0532 9323
0.0521 1467
0.0510 1926
0.0592 0540
0.0578 5241
0.0566 0265
0.0554 4538
0.0543 7133
SI
32
33
34
35
0.0451 5280
0.0441 7415
0.0432 5722
0.0423 9655
0.0415 8731
0.0467 -3900
0.0457 6831
0.0448 5938
0.0440 0675
0.0432 0558
0.0483 5453
0.0473 9263
0.0464 9253
0.0456 4875
0.0448 5645
0.0499 0893
0.0490 4662
O.0481 5612
0.0473 2196
0.0465 3929
0.0533 7240
0.0524 4150
0.0515 7242
0.0507 59G6
0.0499 9835
36
37
38
30
40
0.0408 2522
0.0401 0643
0.0394 2753
0.0387 8543
0.0381 7738
0.0424 5158
0.0417 4090
0.0410 7012
0.0404 3615
0.0398 3623
0.0441 1132
0.0434 0953
O.0427 4764
0.0421 2256
0.0415 3151
0.0458 0379
0.0451 1162
0.0444 5934,
0.0438 4385
0.0432 6238
0.0492 8416
0.0486 1325
0.0479 8214
0.0473 8775
0.0468 2728
41
42
43
44
45
0.0376 0087
0.0370 5364
0.0365 3364
0.0360 3901
0.0355 6805
0.0392 6786
0.0387 2876
0.0382 1688
0.0377 3037
0.0372 6752
0.0409 7200
0.0404 4175
0.0399 3871
0.0394 6100
0.0390 0693
0.0427 1241
0.0421 9167
0.0416 9811
0.0412 2985
0.0407 8518
0.0462 9822 -
0.0457 9828
0.0453 2539
0.0448 7768
0.0444 5343
46
47
48
4t
50
0.0351 1921
0.0346 9107
0.0342 8233
0.0338 9179
0,0335 1836
0.0368 2676
0.0364 0669
0.0360 0599
0.0356 2348
0.0352 5806
0.0385 7493
0.0381 6358
0.0377 7158
0.0373 9773
0.0370 4092
0.0403 6254
0.0399 6051
0.0395 7777
0.0392 1314
0.0388 6550
0.0440 5108
0.0436 6919
0.0433 0646
0.0429 6167
0.0426 3371
T VII 66
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
n
2j%
2l%
2|%
3%
3|%
51
52
53
4
55
0.0331 6102
0.0328 1884
0.0324 9094
0.0321 7654
0.0318 7489
0.0349 0870
0.0345 7446
0.0342 6449
0.0339 4799
0.0336 5419
0.0367 0014
0.0363 7444
0.0360 6297
0.0357 6491
0.0354 7953
0.0385 3382
0.0382 1718
0.0379 1471
0.0376 2558
0.0373 4907
0.0423 2156
0,0420 2429
0.0417 4100
0.0414 7090
0.0412 1323
56
57
58
59
60
0.0315 8530
0.0313 0712
0.0310 3977
0.0307 82G8
0.0305 3533
0.0333 7243
0.0331 0204
0.0328 4244
0.0325 9307
0.0323 5340
0.0352 0612
0.0349 4404
0.0346 9270
0.0344 5153
0.0342 2002
0.0370 8447
0.0368 3114
0.0365 8848
0.0363 5593
0.0361 3296
0.0409 6730
0.0407 3245
0.0405 0810
0.0402 9366
0.0400 8802
61
63
63
64
65
0.0302 9724
0.0300 6795
0.0298 4704
0.0296 3411
0.0294 2878
0.0321 2294
0.0319 012G
0.0316 8790
0.0314 8249
0.0312 8463
0.0339 9767
0.0337 8402
0.0335 7866
0.0333 8118
0.0331 9120
0.0359 1908
0.0357 1385
0.0355 1682
0.0353 2760
0.0351 4581
0.0398 9249
0.0397 0480
0.0395 2513
0.0393 5308
0.0391 8826
66
67
68
69
70
0.0292 3070
0.0290 3955
0.0288 5500
0.0286 7677
0.0285 0458
0.0310 9398
0.0309 1021
0.0307 3300
0.0305 6200
0.0303 97.12
0.0330 0837
0.0328 3236
0.0326 6285
.0.0324 9955
0.0323 4218
0.0349 7110
0.0348 0313
0.0346 4159
0.0344 8618
0.0343 3663
0.0390 3031
0.0383 7892
0.0387 3375
0.0385 9453
0.0384 6095
71
73
73
.74
75
0.0283 3816
0.0281 77?8
0.0280 2169
0.0278 7118
0.0277 2554
0.0302 3790
0.0300 8417
0.0299 3568
0.0297 9222
0.0296 5358
0.0321 9048
0.0320 4420
0.0319 0311
. 0.0317 6698
0.0316 3560
0.0341 9266
0.0340 5404
0.0339 2053
0.0337 9191
0.0330 6796
0.0383 3277
0.0382 0973
0.0380 9160
0.0379 7816
0.0378 6919
76
77
78
79
80
0.0275 8457
0.0274 4808
0.0273 1589
0.0271 8784
0.0270 6376
0.0295 1956
0.0293 8997
0.0292 6463
0.0291 4338
0.0290 2605
0.0315 0878
0.0313 8633
0.0312 6806
0.0311 5382
0.0310 4342
0.0335 4849
0.0334 3331
0.0333 2224
0.0332 1510
0.0331 1175
0.0377 6450
0.0376 6390
0.0375 6721
0.0374 7426
0.0373 8489
81
855
83
84
85
0.0269 4350
0.0268 2692
0.0267 1387
.0.0266 0423
0.0264 9787
0.0289 1248
0.0288 0254
0.0286 9608
0.0285 9298
0.0284 93vLO
0.0309 3674
0.0308 3361
0.0307 3389
0.0306 3747
0.0305 4420
0.0330 1201
0.0329 1576
0.0328 2284
0.0327 3313
0.0326 4650
0.0372 9894
0.0372 1628
0.0371 3676
0.0370 6025
0.0369 8662
86
87
88
89
90
0.0263 9467
0.0262 9452
0.0261 9730
0.0261 0291
0.0260 1126
0.0283 96^3
0.0283 0255
0.0282 1165
0.0281 2353
0.0280 3809
0.0304 5397
0.0303 6667
0.0302 8219
0-0302 0041
0.0301 2125
0.0325 0284
0.0324 8202
0.0324 0393
0.0323 2848
0.0322 5550
0.0369 1576
0.0368 4756
0.0367 8190
0.0367 1868
0.0366 6781
91
92
93
94
95
0.0259 2224
0.0258 3577
0.0257 517,0
0.0256 7012
. 0.0255 9078
0.0279 5523
0.0278 7486
0.0277 9690
0.0277 2126
0.0276 4786
0.0300 4460
0.0299 7038
0.0298 9850
0.0298 2887
0.0297 6141
0.0321 8508
0.0321 1694
0.0320 5107
0.0319 8737
0.0319 2577
0.0365 9919
0.0365 4273
0.0364 8834
0.0364 3594
0.0363 8546
96
97
98
99
100
0.0255 1366
0.0254 3868
0.0253 6578
0.0252 9489
0.0252 2594
0.0275 7662
0.0275 0747
0.0274 4034
0.0273 7517
0.0273 1188
0.0296 9605
0.0296 3272
0.0295 7134
0.0295 1185
0.0294 5418
0.0318 6619
0.0318 0856
0.0317 5281
i 0.0316 9886
0.0316 4667
0.0363 3682
0.0362 8995
0.0362 4478
0.0362 0124
0.0361 5927
VII 67
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
1
1
= -ft
n
4%
4%
5%
4%
6%
1
3
4
6
1.0400 0000
0.5301 9608
0.3603 4854
0.2754 9005
0.2246 2711
1.0450 0000
0.5339 9756
0.3637 7336
0.2787 4365
0.2277 9164
1.0500 0000
0.5378 0488
0.3672 0856
0.2820 1183
0.2309 7480
1.0550 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 9640
6
8
9
10
0.1907 G100
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
0.1547 2181
0,1406 9008
0.1295 0458
0.2001 7895
0.1759 6442
0.1578 6401
0.1438 3946
0.1326 6777
0.2033 6263
0.1791 3502
0.1610 3594
0.1470 2224
0.1358 6796
11
12
13
14
15
0.1141 4904
0.1065 5217
0.1001 4373
0.0946 6897
0.0899 4110
0.1172 4818
0.1096 6619
0.1032 7535
0.0978 2032
0.0931 1381
0.1203 8889
0.1128 2541
0.1064 5577
0.1010 2397
0.0963 4229
0.1235 7065
0.1160 2923
0.1096 8426
0.1042 7912
0.0996 2560
0.1267 9294
0.1192 7703
0.1129 6011
0.1075 8491
0.1029 6276
16
17
18
19
20
0.0858 2000
0.0821 9852
0.0789 9333
0.0761 3862
0.0735 8175
0.0890 1537
0.0854 1758
0.0822 3690
0.0794 0734
0.0768 7614
0.0922 6991
0.0886 9914
0.0855 4622
0.0827 4501
0.0802 4259
0.0955 8254
0.0920 4197
0.0889 1992
0.0861 5006
0.0836 7933
0.0989 5214
0.0954 4480
0.0923 5654
0.0896 2086
0.0871 8456
21
22
23
24
25
0.0712 8011
0.0691 9881
0.0673 0906
0.0655 8683
0.0640 1196
0.0746 0057
0.0725 4565
0.0706 8249
0.0689 8703
0.0674 3903
0.0779 9611
0.0759 7051
0.0741 3682
0.0724 7090
0.0709 5246
0.0814 6478
0.0794 7123
0.0776 6965
0.0760 3580
0.0745 4935
0.0850 0455
0.0830 4557
0.0812 7848
0.0796 7900
0.0782 2672
26
27
28
29
30
0.0625 6738
0.0612 3854
0.0600 1298
0.0588 7993
0.0578 3010
0.0660 2137
0.0647 1946
0.0635 2081
0.0624 1461
0.0613 9154
0.0695 6432
0.0682 9186
0.0671 2253
0.0660 4551
0.0650 5144
0.0731 9307
0.0719 5228
0.0708 1440
0.0697 6857
0.0688 0539
0.0769 0435
0.0756 9717
0.0745 9255
0.0735 7961
0.0726 4891
31
32
33
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
0.0579 8191
0.0572 7045
0.0641 3212
0.0632 8042
0.0624 9004
0.0617 5545
0.0610 7171
0.0679 1665
0.0670 9519
0.0663 3469
0.0656 2958
0.0649 7493
0.0717 9222
0.0710 0234
0.0702 7293
0.0695 9843
0.0689 7386
36
37
38
39
40
0.0528 8688
0.0522 3957
0.0516 3192
0.0510 6083
0.0505 2349
0.0566 0578
0.0559 8402
0.0554 0169
0.0548 5567
0.0543 4315
0.0604 3446
0.0598 3979
0.0592 8423
0.0587 6462
0.0582 7816
0.0643 6635
0.0637 9993
0.0632 7217
0.0627 7991
0.0623 2034
0.0683 9483
0.0678 5743
0.0673 5812
0.0668 9377
0.0664 6154
41
42
43
44
45
0.0500 1738
0.0495 4020
0.0490 8989
0.0486 6454
0.0482 6246
0.0538 6158
0.0534 0868
0.0529 8235
0.0525 8071
0.0522 0202
0.0578 2229
0.0573 9471
0.0569 9333
0.0566 1625
0.0562 6173
0.0618 9090
0.0614 8927
0.0611 1337
0.0607 6128
0.0604 3127
0.0660 5886
0.0656 8342
0.0653 3312
0.0650 0606
0.0647 0050
46
47
48
49
50
0.0478 8205
0.0476 2189
0.0471 8065
0.0468 5712
0.0465 5020
0.0518 4471
0.0515 0734
0.0511 8858
0.0508 8722
0.0506 0215
0.0559 2820
0.0556 1421
0.0553 1843
0.0550 3965
0.0547 7674
0.0601 2175
0.0598 3129
0.0595 5854
0.0593 0230
0.0590 6145
0.0644 1485
0.0641 4768
0.0638 9766
0.0636 6356
0.0634 '4429
T VII 68
TABLE VII. PERIODICAL PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
n
4%
*!%
5%
6|%
6%
si
53
53
54
55
0.0462 5885
0.0459 8212
0.0457 1915
0.0454 6910
0.0452 3124
0.0503 3232
0.0500 7679
0.0498 3469
0.0496 0519
0.0493 8754
0.0545 2867
0.0542 9450
0.0540 7334
0.0538 6438
0.0536 6686
0.0588 3495
0.0586 2186
0,0584 2130
0.0582 3245
0.0580 5458
0.0632 3880
0.0630 4617
0,0628 6551
0.0626 9602
0.0625 3696
56
57
58
59
60
0.0450 0487
0.0447 8932
0.0445 8401
0.0443 8836
0.0442 0185
0.0491 8105
0.048Gb 8506
0.0487 9897
0.0486 2221
0.0484 5426
0.0534 80 JO
0.0533 0343
0.0531 3626
0.0529 7802
0.0528 2818
0.0578 8698
0.0577 2900
0.0575 8006
0.0574 3959
0.0573 0707
0.0623 8765
0.0622 4744
0,0621 1574
0.0619 9200
0,0618 7572
61
63
63
64
65
0.0440 2398
0.0438 5430
0.0436 9237
0.0435 3780
0.0433 9019
0.0482 9462
0.0481 4284
0.0479 9848
0.0478 6115
0.0477 3047
0.0526 8627
0.0525 5183
0.0524 2442
0.0523 0365
0.0521 8915
0.0571 8202
0.0570 6400
0.0569 5258
0.0568 4737
0.0507 4800
0.0617 6642
0,0616 6366
0.0615 6704
0.0614 7615
0.0613 9066
66
67
68
6
70
0.0432 4921
0.0431 1451
0.0429 8578
0.0428 6272
0.0427 4506
0.0476 0608
0.0474 8765
0.0473 7487
0.0472 6745
0.0471 G511
0.0520 8057
0.0519 7757
0.0518 7986
0.0517 8715
0.0516 9915
0.0560 5413
0.0565 6544
0.0564 8163
0.0564 0242
0.0563 2754
0.0613 1022
0.0612 3454
0.0611 6330
0.0610 9625
0.0610 3313
71
72
73
74
75
0.0426 3253
0.0425 2489
0.0424 2190
0.0423 2334
0.0422 2900
0.0470 6759
0.0469 7463
0.04G8 8606
0.0468 0159
0.0467 2104
0.05J6 1563
0.0515 3633
0.0514 6103
0.0513 8953
0.0513 2161
0.0562 5675
0.0561 8982
0.0561 2652
0.0560 6665
0.0560 1002
0.0609 7370
0.0609 1774
0.0608 6505
0.0608 1542
0.0607 6867
76
77
78
70
80
0.0421 3869
0.0420 5221
0.0419 6939
0.0418 9007
0.0418 1408
0.0466 4422
0.0465 7094
0.0465 0104
0.0464 3434
0.0463 7069
0.0512 5709
0.0511 9o80
0.0511 3756
0.0510 8222
0.0510 2962
0.0559 6645
0.0659 0577
0.0558 5781
0.0558 1243
0.0557 6948
0.0607 2463
0.0606 8315
0.0606 4407
0.0609 0724
0.0605 7254
81
83
83
84
85
0.0417 4X27
0.0416 7150
0.0416 0463
0.0415 4054
0.0414 7909
0.0463 0995
0.0462 5197
0.0461 9663
0.0461 4379
0.0460 9334
0.0509 7963
0.0509 3211
0.0508 8694
0.0508 4399
0.0508 0316
0.0557 2884
0.0558 9036
0.0556 5395
0.0556 1947
0,0555 8683
0.0605 3984
0.0605 0903
0.0604 7998
0.0604 5261
0.0604 2G81
86
87
88
89
90
0.0414 2018
0.0413 6370
0.0413 0953
0.0412 5758
0.0412 0775
0.0460 4516
0.0459 9915
0.0459 5522
0.0459 1325
0.0458 7316
0.0507 6433
0.0507 2740
0.0506 9228
0.0506 5888
0.0506 2711
0.0555 5593
0.0555 2667
0.0554 9896
0.0554 7273
0.0554 4788
0.0604 0249
0.0603 7956
0.0603 5795
0.0603 3767
0.0603 1836
91
93
93
94
95
0.0411 5995
0.0411 1410
0.0410 7010
0.0410 2789
0.0409 8738
0.0458 3486
0.0457 9827
0.0457 6331
0.0457 2991
0.0456 9799
0.0505 9689
0.0505 6815
0.0505 4080
0.0505 147S
0.0504 9003
0.0554 2435
0.0554 0207
0.0553 8096
0.0553 6097
0.0553 4204
0.0603 0025
0.0602 8318
0.0602 6708
0.0802 5190
0.0602 3758
96
97
98
l8
0.0409 4850
0.0409 1119
0.0408 7538
0.0408 4100
0.0408 0800
0.0456 6749
0.0456 3834
0.0456 1048
0.0455 8385
0.0455 5839
0.0504 6648
0.0504 4407
0.0504 2274
0.0504 0245
0.0503 8314
0.0553 2410
0.0553 0711
0.0552 9101
0.0552 7577
0.0552 6132
0.0602 2408
0.0602 1135
0.0601 9935
0.0601 8803
0.0601 7736
X VII 9
TABLE VII. PERIODICAL. PAYMENT OF ANNUITY WHOSE
PRESENT VALUE is 1
1 _ 1
ami Sn\
n
<%
7%
71%
8%
8|%
2
3
5
1.0650 0000
0.5492 6150
0.3775 7570
0.2919 0274
0.2406 3454
1.0700 0000
0.5530 9179
0.3810 5166
0.2952 2812
0.2438 9069
1.0750 0000
0.5569 2771
0.3845 3763
0.2985 6751
0.2471 6472
1.0800 0000
0.5607 6923
0.3880 3351
0.3019 2080
0.2504 5645
1.0850 0000
0.5646 1631
0.3915 3925
0.3052 8789
0.2537 6575
6
7
8
10
0.2065 6831
0.1823 3137
0.1642 3730
0.1502 3803
0.1391 0469
0.2097 9580
0.1855 5322
0.1674 6776
0.1534 8647
0.1423 7750
0.2130 4489
0.1888 0032
0.1707 2702
0.1537 6716
0.1456 8593
0.2163 1539
0.1920 7240
0.1740 1476
0.1600 7971
0.1490 2949
0.2196 0708
0.1953 6922
0.1773 3065
0.1634 2372
0.1524 0771
11
12
13
14
15
0.1300 5521
0.1225 6817
0.1162 8256
0.1109 4048
0.1063 5278
0.1333 5690
0.1259 0199
0.1196 5085
0.1143 4494
0.1097 9462
0.1366 9747
0.1292 7783
0.1230 6420
0.1177 9737
0.1132 8724
0.1400 7634
0.1326 9502
0.1265 2181
0.1212 9685
0.1168 2954
0.1434 9293
0.1361 5286
0.1300 2287
0.1248 4244
0.1204 2046
16
17
18
19
20
0,1023 7757
0.0989 0633
0.0958 5461
0.0931 5575
0.0907 6640
0.1058 5765
0.1024 2519
0.0994 1260
0.09G7 5301
0.0943 9293
0.1093 9116
0.1060 0003
0.1030 2896
0.1004 1090
0.0980 9219
0.1129 7687
0.1096 2943
0.1067 0210
0.1041 2763
0.1018 5221
0.1166 1354
0.1133 1198
0.1104 3041
0.1079 0140
0.1056 7097
21
22
23
24
25
0.0886 1333
0.0866 9120
0.0849 6078
0.0833 9770
0.0819 8148
0.0922 8900
0.0904 0577
0.0887 1393
0.0871 8902
0.0858 1052
0.0960 2937
0.0941 8687
0.0925 3528
0.0910 5008
0.0897 1067
0.0998 3225
0.0980 3207
0.0964 2217
0.0949 7796
0.0936 7878
0.1036 9541
0.1019 3892
0.1003 7193
0.0989 6975
0.0977 1168
26
27
28
29
30
0.0806 9480
0.0795 2288
0.0784 5305
0.0774 7440
0.0765 7744
0.0845 6103
0.0834 2573
0.0823 9193
0.0814 4865
0.0805 8640
0.0884 9961
0.0874 0204
0.0864 0520
0.0854 9811
0.0846 7124
0.0925 0713
0.0914 4809
0.0904 8891
0.0896 1854
0.0888 2743
0.0965 8016
0.0955 6025
0.0946 3914
0.0938 0577
0.0930 5058
31
32
33
34
35
0.0757 5393
0.0749 9665
0.0742 9924
0.0736 5610
0.0730 6226
0.0797 9691
0.0790 7292
0.0784 0807
0.0777 9674
0.0772 3396
0.0839 1628
0.0832 2599
0.0825 9397
0.0820 1461
0.0814 8291
0.0881 0728
0.0874 5081
0.0868 5163
0.0863 0411
0.0858 0326
0.0923 6524
0.0917 4247
0.0911 7588
0.0906 5984
0.0901 8937
36
37
38
39
40
0.0725 1332
0.0720 0534
0.0715 3480
0.0710 9854
0.0706 9373
0.0767 1531
0.0762 3685
0.0757 9505
0.0753 8676
0.0750 0914
0.0809 9447
0.0805 4533
0.0801 3197
0.0797 5124
0.0794 0031
0.0853 4467
0.0849 2440
0.0845 3894
0.0841 8513
0.0838 6016
0.0897 6006
0.0893 6799
0.0890 0966
0.0886 8193
0.0883 8201
41
42
43
44
45
0.0703 1779
0.0699 6842
0.0696 4352
0.0693 4119
0.0690 5968
0.0746 5962
0.0743 3591
0.0740 3590
0.0737 5769
0.0734 9957
0.0790 7663
0.0787 7789
0.0785 0201
0.0782 4710
0.0780 1146
0.0835 6149
0.0832 8684
0.0830 3414
0.0828 0152
0.0825 8728
0.0881 0737
0.0878 5576
0.0876 2512
0.0874 1363
0.0872 1961
46
47
48
49
50
0.0687 0743
0.0685 5300
0.0683 2506
0.0681 1240
0.0679 1393
0.0732 5996
0.0730 3744
0.0728 3070
0.0726 3853
0.0724 5985
0.0777 9353
0.0775 9190
0.0774 0527
0.0772 3247
0.0770 7241
0.0823 8991
0.0822 0799
0.0820 4027
0.0818 8557
0.0817 4286
0.0870 4154
0.0868 7807
0.0867 2795
0.0865 9005
0.0864 6334
T VII 70
TABIJB VIII. COMPOUND AMOUNT OF 1 FOB FRACTIONAL PEBIODS
p
5%
1*
5*
:*
1%
2
3
4
6
12
13
26
1.0020 8117
1.0013 8000
.0010 4004
.0006 9324
.0003 4656
.0003 1990
.0001 5994
1.0024 9688
1.0016 6390
1.0012 4766
1.0008 3160
1.0004 1571
1.0003 8373
1.0001 9185
.0029 1243
.0019 4068
.0014 5515
.0009 6987
.0004 8482
.0004 4751
1.0002 2373
1.0037 4290
1.0024 9378
1.0018 6975
1.0012 4611
1.0006 2286
1.0005 7494
1.0002 8743
.0049 8756
.0033 2228
.0024 9068
.0010 5977
.0008 2954
.0007 6570
.0003 8276
P
1|%
i|%
l|%
lj%
2%
2
3
6
12
13
26
.0056 0927
.0037 3602
.0028 0081
.0018 6627
.0009 3270
.0008 6092
1.0004 3037
.0062 3059
.0041 4943
.0031 1046
.0020 7257
.0010 3575
.0009 5604
,0004 7790
1.0074 7208
.0049 7521
.0037 2909
.0024 8452
.0012 4149
.0011 4594
.0005 7280
1.0087 1205
1.0057 9963
1.0043 4658
1.0028 9562
1.0014 4677
1.0013 3540
1.0006 6748
1.0099 5050
1.0066 2271
1.0049 6293
1.0033 0589
1.0016 5158
1.0015 2444
1.0007 6193
P
2|%
2|%
2?<7T
4%
3%
3|%
2
3
4
6
12
26
52
1.0111 8742
.0074 4444
.0055 7815
.0037 1532
.0018 5594
.0008 5616
,0004 2799
.0124 2284
.0082 648t
.0061 9225
.0041 2392
.0020 5984
.0009 5017
.0004 797
1.0136 5675
1.0090 8390
1.0068 0522
1.0045 3168
1.0022 6328
1.0010 4396
1.0005 2184
1.0148 8916
.0099 0163
.0074 1707
.0049 3862
.0024 6627
.0011 3752
.0005 6800
1.0173 4950
1.0115 3314
1.0086 3745
1.0057 5004
1.0028 7090
1.0013 2401
1.0006 6179
P
4%
*!%
6%
5|%
6%
2
3
4
6
12
26
52
.0198 0390
.0131 5941
.0098 5341
.0065 5820
.0032 7374
.0015 0963
.0007 5453
.0222 5242
.0147 8046
.0110 6499
.0073 6312
.0036 7481
.0016 9439
1.0008 4684
.0246 9508
.0163 9636
.0122 7224
.0081 6485
.0040 7412
1.0018 7831
1.0009 3871
1.0271 3193
1.0180 0713
1.0134 7518
1.0089 6340
1.0044 7170
1.0020 6138
1.0010 3016
1.0295 6302
1.0196 1282
1.0146 7385
1.0097 5880
1.0048 6755
1.0022 4363
1.0011 2118
P
6^%
7%
7|%
8%
8|%
2
3
4
6
12
26
52
1.0319 8837
1.0212 1347
1.0158 6828
1.0105 5107
1.0052 6169
1.0024 2504
1.0012 1179
.0344 0804
.0228 0912
.0170 5853
.0113 4026
.0056 5415
.0026 0564
1.0013 0197
1.0368 2207
1.0243 9981
1.0182 4460
1.0121 2638
l.OOGO 4492
1.0027 8544
1.0013 9175
1.0392 3048
1.0259 8557
1.0194 2655
1.0129 0946
1.0064 3403
1.0029 6443
1.0014 8112
1.0416 3333
1.0275 6644
1.0206 0440
1.0136 8952
1.0068 2149
1.0031 4202
1.0015 7008
T VIII 71
TABLE IX. NOMINAL RATH j WHICH IF CONVERTED p TIMES
PER YEAR GIVES EFFECTIVE RATE i
P
%
S*
H%
!%
1%
2
3
4
6
12
13
26
.0041 6234
.0041 6089
.0041 6017
.0041 5945
.0041 5873
.0041 5868
.0041 5834
.0049 9377
.0049 9169
.0049 9065
.0049 8962
.0049 8858
,0049 8850
.0049 8802
.0058 2485
.0058 2203
.0058 2062
.0058 1921
.0058 1780
.0058 1769
.0058 1704
.00.74 8599
.0074 8133
.0074 7900
.0074 7667
.0074 7434
.0074 7416
.0074 7309
.0099 7512
.0099 6685
.0099 6272
.0099 5859
.0099 5446
.0099 5414
.0099 5224
P
1|%
1|%
l\%
l|%
2%
2
3
4
6
12
13
26
.0112 1854
.0112 0807
.0112 0285
.0111 9763
.0111 9241
.0111 9200
.0111 8960
.0124 6118
.0124 4828
.0124 4183
.0124 3539
.0124 2895
.0124 2846
.0124 2549
.0149 4417
.0149 2562
.0149 1636
.0149 0710
.0148 9785
.0148 9714
.0148 9288
.0174 2410
.0173 9890
.0173 8631
.0173 7374
.0173 6119
.0173 6022
.0173 5443
.0199 0099
.0198 6813
.0198 5173
.0198 3534
.0198 1898
.0198 1772
.0198 1017
P
2|%
I%
2f%
3%
3|%
2
3
4
12
26
52
.0223 7484
.0223 3333
.0223 1261
.0222 9192
.0222 7125
.0222 6013
.0222 5537
.0248 4507
.0247 9451
.0247 6809
.0247 4349
.0247 1S04
.0247 0434
.0246 9848
.0273 1349
.0272 5170
.0272 2087
.0271 9009
.0271 5936
.0271 4283
.0271 3575
.0297 7831
.0297 0490
.0296 6829
.0296 3173
.0295 9524
.0295 7561
.0295 6721
.0346 9S99
.0345 9943
.0345 4978
.0345 0024
.0344 5078
.0344 2420
.0344 1281
/>
4%
4|%
5%
5|%
6%
2
3
4
6
12
26
52
.0306 0781
.0394 7821
.0394 1363
.0393 4918
.0392 8488
.0392 5031
.0392 3551
.0445 0483
.0443 4138
.0442 5990
.0441 7874
.0440 9771
.0440 5417
.0440 3552
.0493 9015
.0491 8907
.0490 8894
.0489 8908
.0488 8949
.0488 3597
.0488 1306
.0542 3386
.0540 2139
.0539 0070
.0537 8036
.0536 6039
.0535 9593
.0535 6834
.0591 2603
.0588 3847
.0586 9538
.0585 5277
.0584 1061
.0583 3425
.0583 0157
P
6|%
7%
1\%
8%
8|%
2
3
4
12
26
52
.0639 7674
.0636 4042
.0034 7314
.0633 0644
.0631 4033
.0630 6113
.0630 1295
.0688 1609
.0684 2737
.0682 3410
.0680 4156
.0678 4974
.0677 4676
.0677 0268
.0736 4414
.0731 9942
.0729 7840
.0727 5827
.0725 3903
.0724 2134
.0723 7098
.0784 6097
.0779 5670
.0777 0619
.0774 5674
.0772 0836
.0770 7506
.0770 1802
.0832 6607
.0826 9933
.0824 1758
.0821 3712
.0818 5792
.0817 0811
.0816 4401
T IX 72
E X. THE VALUE OF THE CONVERSION FACTOR
I
3p
p
M%
1%
5%
!%
1%
2
3
4
6
12
13
26
.0010 4058
.0013 8761
.0015 6115
.0017 3471
.0019 0829
.0019 2164
.0020 0176
1.0012 4844
1.0016 6482
1.0018 7305
1.0020 8131
1.0022 89GO
1.0023 0563
1.0024 2182
1.0014 5621
.0019 4193
.0021 8485
.0024 2781
.0026 7080
.0026 8950
1.0028 0166
1.0018 7150
1.0024 9585
1.0028 0812
1.0031 2046
1.0034 3286
1.0034 5690
1.0036 0111
1.0024 9378
.0033 2596
.0037 4223
.0041 5861
.0045 7510
.0046 0714
1.0047 9941
P
1|%
1|%
1|%
l|%
2%
2
3
4
6
12
13
26
.0028 0463
.0037 4068
.0042 0892
.0046 7730
.0051 4583
.0051 8188
.0053 9818
1.0031 1529
1.0041 5516
1.0046 7537
1.0051 9575
1.0057 1632
1.0057 5637
1.0059 9669
1.0037 3604
1.0049 8346
1.0056 0755
1.0062 3191
1.0068 5652
1.0069 0458
1.0071 9296
1.0043 6176
1.0058 1084
1.0065 3878
1.0072 6707
1.0079 9571
1.0080 5177
1.0083 8820
1.0049 7525
1.0066 3733
1.0074 6856
1.0083 0125
1.0091 3389
1.0091 9790
1.0095 8243
P
2|%
2|%
2f%
3%
s|%
2
3
6
12
26
52
1.0055 9371
1.0074 6292
1.0083 9839
1.0093 3444
1.0102 7107
1.0107 7565
1.0109 9195
1.0062 1142
1.0082 8761
1.0093 2677
1.0103 6665
1.0114 0725
1.0119 6786
1.0122 0819
.0068 2837
.0091 1141
.0102 5422
.0113 9789
.0125 4243
.0131 5908
.0134 2343
.0074 4458
.0099 3431
.0111 8072
.0124 2816
.0136 7662
.0143 4929
.0146 3757
1.0086 7475
1.0115 7748
1.0130 3094
1.0144 8578
1.0159 4203
1.0167 2674
1.0170 6316
j>
4%
4|%
5%
6|%
6%
2
3
6
12
26
52
1.0099 0195
1.0132 1713
1.0148 7744
1.0165 3957
1.0182 0351
1.0191 0023
1.0194 8470
1.0111 2621
1.0148 5328
1.0167 2026
1.0185 8953
1.0204 6109
1.0214 6980
1.0219 0231
.0123 4754
.0164 8597
.0185 5942
.0206 3570
.0227 1479
.0238 3548
.0243 1602
.0135 6596
.0181 1522
.0203 9495
.0226 7810
.0249 6465
.0261 9729
1.0267 2586
1.0147 8151
1.0197 4104
1.0222 2688
1.0247 1676
1.0272 1070
1.0285 5526
1.0291 3186
P
6l%
7%
7|%
8%
B\%
2
3
4
6
12
26
52
.0159 9419
.0213 6348
.0240 5523
.0267 5172
.0294 5294
1.0309 0941
1.0315 3404
1.0172 0402
1.0229 8254
1.0258 8002
1.0287 8298
1.0316 9143
1.0332 5978
1.0339 3242
1.0184 1103
1.0245 9826
1.0277 0129
1.0308 1059
1.0339 2617
1.0356 0640
1.0363 2705
1.0196 1524
1.0262 1065
1,0295 1904
1.0328 3456
1.0361 5721
1.0379 4927
1.0387 1794
.0208 1667
.0278 1974
.0313 3332
.0348 5492
.0383 8455
1.0402 8845
1.0411 0511
T X 73
TABLE XI. AMERICAN EXPERIENCE TABLE OF MORTALITY
Age
X
Num-
ber
living
'*
Num-
ber
of
deaths
**
Yearly
proba-
bility of
dying
*x
Yearly
proba-
bility of
living
P*
Age
X
Num-
ber
living
'*
Num-
ber
of
deaths
i,
Yearly
proba-
bility of
dying
*x
Yearly
proba-
bility of
living
P X
10
100,000
749
0.007 490
0.992 510
53
66,797
1,091
0.016 333
0.983 067
11
99,251
746
0-007 516
0.992 484
54
65,706
1,143
0,017 396
0.982 604
12
98,505
743
0.007 543
0.992 457
55
64,563
1,199
0.018 571
0.981 429
13
97,762
740
0.007 569
0.992 431
56
63,364
1,260
0.019 885
0.980 115
14
97,022
737
0.007 596
0.992 404
57
62,104
1,325
0.021 335
0.978 665
15
96,285
735
0.007 634
0.992 366
58
60,779
1,394
0.022 936
0.977 064
16
95.550
732
0.007 661
0.992 339
59
59,385
1,468
0.024 720
0.975 280
17
94,818
729
0.007 688
0.992 312
60
57,917
1,516
0.026 693
0.973 307
18
94,089
727
0.007 727
0.992 273
61
50,371
1,628
0.028 880
0.971 120
19
93,362
725
0.007 765
0.992 235
62
54,743
1,713
0.031 292
0.968 708
20
92,637
723
0.007 805
0,992 195
63
53,030
1,800
0.033 943
0.966 057
21
91,914
722
0.007 855
0.992 145
64
51,230
1,889
0.036 873
0.963 127
22
91,192
721
0.007 906
0.992 094
65
49,341
1,980
0.040 129
0.959 871
23
90,471
720
0.007 958
0.992 042
66
47,301
2,070
0.043 707
0.956 293
24
89,751
719
0.008 Oil
0.991 989
67
45,291
2,158
0.047 647
0.952 353
25
89,032
718
0.008 065
0.991 935
68
43,133
2,243
0.052 002
0.947 998
26
88,314
718
0.008 130
0.991 870
69
40,890
2,321
0.056 762
0.943 238
27
87,596
718
0.008 197
0.991803
70
38,509
2,391
0.061 903
0.938 007
28
86,878
718
0.008 264
0.991 736
71
36,178
2,448
0.067 665
0.932 335
20
86,160
719
0.008 345
0.991 655
72
33,730
2,487
0.073 733
0.926 267
30
85,441
720
0.008 427
0.991 573
73
31,243
2,505
0.080 178
0.919 822
31
84,721
721
0.008 610
0.991 490
74
28,738
2,501
O.OS7 028
0.912 972
32
84,000
723
0.008 607
0.991 393
75
26,237
2,476
0.094 371
0.905 629
33
83,277
726
0.008 718
0.991 282
76
23,761
2,431
0.102311
0.897 689
34
82,551
729
0.008 831
0.991 169
77
21,330
2,369
0.111 OG4
0.888 936
35
81,822
732
0.008 946
0.991 054
78
18,961
2,291
0.120 827
0.879 173
36
81,090
737
0.009 089
0.900911
79
16,670
2,196
0.131 734
0.868 266
37
80,353
742
0.009 234
0.990 766
80
14,474
2,091
0.144 466
0.855 534
38
79,611
749
0.009 408
0.990 592
81
12,383
1,964
0.1 58 005
0.841 395
38
78,862
756
0.009 586
0.990 414
82
10,419
1,816
0.174297
0.825 703
40
78,106
765
0.009 794
0.990 206
83
8,603
1,648
0.191 561
0.808 439
41
77,341
774
0.010 008
0.989 992
84
6,955
1,470
0.211 359
0.788 641
42
76,567
785
0.010 252
0.989 748
85
5,485
1,292
0.235 552
0.764 448
43
75,782
797
0.010 517
0.989 483
86
4,193
1,114
0.265 681
0.734 319
44
74,985
812
0.010 829
0.989 171
87
3,079
933
0.303 020
0.696 980
45
74,173
828
0.011 163
0.988 837
88
2,146
744
0.346 692
0.653 308
46
73,345
848
0.011 562
0.988 438
89
1,402
555
0.395 863
0.604 137
47
72,497
870
0.012 000
0.988 000
90
847
385
0.454 545
0.545 455
48
71,627
896
0.012 509
0.987 491
91
462
246
0.532 468
0.467 534
49
70,731
027
0.013 106
0.986 894
92
216
137
0.634 259
0.365741
50
69,804
962
0.013 781
0.986 219
93
79
58
0.734 177
0.265 823
51
68,842
1,001
0.014 541
0,985 459
94
21
18
0.857 143
0.142 857
52
67,841
1,044
0.015383
0.084 611
95
a
3
1.000 000
0.000 000
T XI 74
TABLE XII. COMMUTATION COLUMNS, SINGLE PREMIUMS, AND ANNUITIES
DUE. AMERICAN EXPERIENCE TABLE, 3>6 PER CENT
Age
X
D x
NX
c*
M x
a x
I+a x
A x
10
11
12
13
14
70891.9
67981.5
65189.0
62509.4
59938.4
1575 535
1504 643
1436 662
1371 473
1308 963
613.02
493.69
475.08
457.16
439.91
17612.9
17099.9
16606.2
16131.1
15674.0
22.2245
22.1331
22.0384
21.9403
21.8385
0.24845
0.25154
0.25474
0.25806
0.26151
15
16
17
18
19
57471.6
55104.2
52832.9
50653.9
48562.8
1249 025
1191 553
1136 449
1083 616
1032 962
423.88
407.87
392.47
378.15
364.36
15234.1
14810.2
14402.3
14009.8
13631.7
21.7329
21.6236
21.5102
21.3926
21.2707
0.26508
0.26877
0.27261
0.27659
0.28071
20
21
22
23
24
46556.2
44630.8
42782.8
41009.2
39307.1
984 400
937 843
893 213
850430
809 421
351.07
338.73
326,82
315.33
304.24
13267.3
12916.3
12577.5
12250.7
11935.4
21.1443
21.0134
20.8779
20.7375
20.5922
0.28497
0.28940
0.29399
0.29873
0.30365
25
26
27
28
29
37673.6
36106.1
34601.5
33157.4
31771.3
770 113
732 440
696 334
661 732
628 575
293.55
283.62
274.03
264.76
256.16
11631.1
11337.6
11054.0
10779.9
10515.2
20.4417
20.2858
20.1244
19.9573
19.7843
0.30873
0.31401
0.31947
0.32512
0.33097
30
31
32
33
34
30440.8
29163.5
27937.5
26760.5
25630.1
596 804
566 363
537 199
509 262
482 501
247.85
239.797
232.331
225.406
218.683
10259.0
10011.2
9771.38
9539.04
9313.64
19.6054
19.4202
19.2286
19.0304
18.8256
0.33702
0.34328
0.34976
0.35646
0.36339
35
36
37
38
39
24544.7
23502.5
22501.4
21539.7
20615.5
456 871
432 326
408 824
386 323
364783
212.157
206.383
200.757
195.798
190.945
9094.96
8882.80
8676.42
8475.66
8279.86
18.6138
18.3949
18.1688
17.9354
17.6946
0.37055
0.37795
0.38560
0,39349
0.40163
40
41
42
43
44
19727.4
18873.6
18052.9
17263.6
16504.4
344 167
324 440
305 566
287 513
270 250
186.684
182.493
178.828
175.421
172.680
8088.92
7902.23
7719.74
7540.91
7365.49
17.4461
17.1901
16.9262
16.6543
16.3744
0.41003
0.41860
0.42762
0.43681
0.44628
45
46
47
48
49
15773.6
15070.0
14392.1
13738.5
13107.9
253 745
237 972
222 902
208 510
194 771
170.127
168.345
166.872
166.047
165.983
7192.81
7022.68
6854.34
6687.47
6521.42
16.0867
15.7911
15.4878
15.1770
14.8591
0.45600
0.46600
0.47626
0.48677
0.49752
50
51
52
12498.6
11909.6
11339.5
181 663
169 165
157 252
166.424
167.316
168.601
6355.44
6189.01
6021.70
14.5346
14.2041
13.8679
0.50849
0.51967
0.53104
T XII 75
TABLE XII. COMMUTATION COLUMNS, SINGLE PREMIUMS, AND ANNUITIES
DUE. AMERICAN EXPERIENCE TABLE, 3H PER CENT
Age
X
D x
N x
c x
M x
ax*
l+a x
A x
53
64
10787.4
10252.4
145916.
135128.
170.234
172.317
5853.10
5682.86
13.5264
13.1801
0.54258
0.55430
55
56
57
58
59
9733.40
9229.60
8740.17
8264.44
7801.82
124876.
115142.
105912.8
97172.6
88908.2
174.646
177.325
180.168
183.139
186.340
5510.54
5335.90
5158.57
4978.40
4795.27
12.8296
12.4753
12.1179
11.7579
11.3958
0.56615
0.57813
0.59022
0.60239
0.61463
60
61
62
63
64
7351.65
6913.44
6486.75
6071.27
5666.85
81106.4
73754.7
66841.3
60354.5
54283.3
189.604
192.909
196.117
199.109
201.887
460S.93
4419.32
4J2C.41
4030.30
3831.19
11.0324
10.6683
10.3043
9.9410
9.5791
0.62692
0.63924
0.65155
0.66383
0.67607
65
66
67
68
69
5273.33
4890.55
4518.65
4157.82
3808.32
48616.4
43343.1
38452.5
33933.9
29776.1
204.457
208.522
208.022
20S.903
208.858
3629.30
3424.84
3218.32
3010.30
2801.40
9.2193
8.8626
8.5097
8.1615
7.8187
0.68824
0.70030
0.71223
0.72401
0.73560
70
71
72
73
74
3470.67
3145.43
2833.42
2535.75
2253.57
25967.7
22497.1
19351.6
16518.2
13982.5
207.881
205.639
201.851
196.436
189.491
2592.54
2384.66
2179.02
1977.17
1780.73
7.4820
7.1523
6.8298
6.5141
6.2046
0.74698
0.75813
0.76904
0.77972
0.79018
75
76
77
78
79
1987.87
1739.39
1508.63
1295.73
1100.647
11728.9
9741.02
8001.63
6493.00
5197.27
181.253
171.940
161.889
151.2646
140.0891
1591.24
1409.99
1238.05
1076.158
924.894
5.9002
5.6002
5.3039
5.0111
4.7220
0.80048
0.81062
0.82064
0.83054
0.84032
80
81
82
83
84
923.338
763.234
620.465
494.995
386.641
4096.62
3173.29
2410.05
1789.59
1294.59
128.8801
116.9588
104.4881
91.6152
78.9565
784.806
655.924
538.966
434.478
342.862
4.4368
4.1577
3.8843
3.6154
3.3483
0.84997
0.85940
0.86865
0.87774
0.88677
85
86
87
88
89
294.610
217.598
154.383
103.963
65.6231
907.95
613.34
395.74
241.36
137.398
67.0490
55.8566
45.1992
34.82426
25.09929
263.906
196.857
141.000
95.8011
60.9768
3.0819
2.8187
2.5634
2.3216
2.0937
0.89578
0.90468
0.91332
0.92149
0.92920
90
91
92
93
94
38.3047
20.18692
9.11888
3.22236
0.827611
71.775
33.4700
13.2831
4.16420
0.94184
16.82244
10.385393
5.588150
2.285484
0.685393
35.8775
19.05509
8.66970
3.08155
0.79576
1.8738
1.6580
1.4567
1.2923
1.1380
0.93664
0.94393
0.95074
0.95630
0.96152
95
0.114232
0.114232
0.110369
0.110369
1.0000
0.96618
T XII 76
CENT
Age
JC
u x
kx
Age
X
Ux
k x
10
11
12
13
14
1.042 811
1.042 838
1.042 866
1.042 894
1.042 922
0.007 546
0.007 573
0.007 600
0.007 627
0.007 654
53
54
55
56
57
1.052 185
1.053 323
1.054 585
1.055 999
1.057 563
0.016 604
0.017 704
0.018 922
0.020 289
0.021 800
15
16
17
18
19
1.042 962
1.042 990
1.043 019
1.043 059
1.043 100
0.007 692
0.007 720
0.007 748
0.007 787
0.007 826
58
59
60
61
62
1.059 296
1.061 234
1.063 385
1.065 780
1.068 433
0.023 474
0.025 347
0.027 425
0.029 739
0.032 303
20
21
22
23
24
1.013 141
1.043 195
1.043 248
1.043 303
1.043 358
0.007 866
0.007 917
0.007 969
0.008 022
0.008 076
63
64
65
66
67
1.071 365
1.074 625
1.078 270
1.082 304
1.086 782
0.035 136
0.038 285
0.041 807
0.045 704
0.050 031
25
26
27
28
29
1.043 415
1.043 484
1.043 554
1.043 625
1.043 710
0.008 130
0.008 197
0.008 264
0.008 333
0.008 415
63
69
70
71
72
1.091 774
1.097 284
1.103 403
1.110 117
1.117 388
0.054 855
0.060 178
0.066 090
0.072 576
0.079 602
30
31
32
33
34
1.043 796
1.043 884
1,043 986
1.044 102
1.044 221
0.008 498
0.008 583
0.008 682
0.008 795
0.008 910
73
74
75
76
77
1.125 218
1.133 660
1.142 852
1.162 960
1.164 314
0.087 167
0.095 323
0.104 204
0.113 971
0.124 941
35
36
37
38
39
1,044 343
1.044 493
1.044 647
1.044 830
1.045 018
0.009 027
0.009 172
0.009 320
0.009 498
0.009 679
78
79
80
81
82
1.177 243
1.192 031
1.209 771
1.230 099
1.253 477
0.137 433
0.151 720
0.168 861
0.188 502
0.211 089
40
41
42
43
44
1.045 238
1.045 463
1.045 721
1.046 001
1.046 331
0.009 891
0.010 109
0.010 359
0.010 629
0.010 947
83
84
85
86
87
1.280 245
1.312 384
1.353 917
1.409 469
1.484 979
0.236 952
0.268 004
0.308 133
0.361 806
0.434 762
45
46
47
48
49
1.046 684
1.047 106
1.047 571
1.048 111
1.048 745
0.011 289
0.011 697
0.012 146
0.012 668
0.013 280
88
89
90
91
92
1.584 244
1.713 188
1.897 500
2.213 750
2.829 873
0.630 671
0.655 254
0.833 333
1.138 889
1.734 177
50
51
52
1.049 463
1.050 272
1.051 177
0.013 974
0.014 755
0.015 629
93
94
95
3.893 571
7.245 000
2.761 905
6.000000
T XIII 77
ANSWERS
TO
EXERCISES AND PROBLEMS
Chapter I
Paged
2. I - $625.00; S - $5,625.00.
6. $799.14. 8. 7%.
6. 5K years. 9. $4,500.00.
7. 7%. 10. 3Ji years.
1. (a) la - $3.25; I e - $3.21.
(b) /<> - $3.24; I e - $3.19.
(c) / a -$1.31; 7 fl = $1.29.
(d) /<> - $4.52; J e = $4.46.
2. 7 - $44.80; 7 e - $44.19.
3. $13.27.
11. 5%.
12. $256.00.
13. 9%.
Pages 6-7
3. $9.93.
4. $29.89.
6. 55 days.
6. 75 days.
7. 9%.
1. (a) $7.50.
(b) $6.04.
(c) $8.75.
(d) $18.27.
6. $155.33.
1. $2,200.00.
2. $312.00.
8. $986.84.
9. 5%.
1. $1,479.75.
2. $381.61.
3. $2,024.17.
4. $569.09.
2. (a) $9.38.
(b) $7.54.
(c) $10.94.
(d) $22.84.
6. $155.20.
Pages 8-9
3.
(a) $14.53.
(b) $11.05.
(c) $38.47.
7. $153.20.
Pages 11-12
3. P - $5,769.23; Disc. = $230.77.
4. $1,818.18.
10. (a) $1,035.00.
(b) $1,024.75.
(c) 7.4%.
4. 1M years.
14. $452.40.
15. $256.00.
16. $26,250.00.
8. $14.60.
9. $21.60.
10. $1.06.
11. $28.80.
4. (a) $19.64.
(b) $14.93.
(c) $52.00.
8. $0.69.
5. $288.46.
7. $990.10.
11. (b) $1,021.38.
(c) 6.41%.
Pages 16-17
6. $1,352.13.
6. $1,267.69.
7. $2,480.83.
8. $2,556.46.
245
9. $255.10,
11. 6%.
12. $1,015.23.
13. 67 days.
14. $5,019.73.
16. $2,000.00.
16. 8%.
246
Answers
Pages 16-17 Continued
17. $2,072.54.
18. S - $800.00; Face
19. $1,216.93.
$788.18.
20. Ji year.
21. $1,000.00.
22. 0.
1. i - 6.383%.
2. i - 6.185%.
3. i = .0869; .0833; .0816; .0808.
Pages 19-20
4. d
5. d
6. $803.74.
7. $800.95.
1. (a) .0712.
(b) .0759.
(c) .0619.
(d) .0822.
2. 8.74%.
3. (a) .0688.
(b) .0779.
(c) .0583.
4. 9.89%.
6. 7.41%.
1. (a) $492.61; $497.61.
(b) 507,50; 512.32.
(c) 522.50; 527.80.
8. (a) $501.58.
(b) $501.65.
Pages 21-22
.0741; .0769; .0784; .0792.
15%; i = 15.4%.
9. (a) $1,004.50.
(b) $6.83.
6. i 12.4% or 3.1% per 90 days.
7. 16%%; 13.92%.
8. 4% cash discount is best.
9. 18.56%; $78.47 at end of 60 days.
10. 6.88%; 5.88%; 4.82%.
11. %o is best.
12. %o is best.
13. 6.12%,
14. 6% cash discount is best.
Pages 31-32
2. (a) $489.00; $487.12.
(b) 511.25; 508.56.
(c) 533.75; 531,70.
3. $619.65; $619.77.
4. $620.67; $618.75.
6. $912.66, F.D. at 8 mo.; $912.55, F.D. at 12 mo.
6. $437.93. 7. $938.08, F.D. at 12 mo.
8. $1,873.22, F.D. at 9 mo.; $1,873.31, F.D. at 8 mo.
9. May 28.
10. April 22.
11. 4 mo. 7 days.
12. 6J^ months.
13. Dec. 9.
14. March 2.
15. Sept. 12.
16. May 11.
17. Oct. 3.
18. Jan. 15.
19. July 16.
Pages 33-34
1. $2,000.00; $2,500.00.
2. $2,500.00; $4,000.00.
3. $3,000.00; $7,000.00; $5,000.00.
4. $1,000.00; $1,500.00; $2,500.00.
6. $12,000.00.
6. 4 days.
7.
hours.
8. 17.867 Ibs.
9. 115 Ibs.
Answers 247
Pages 33-34 Continued
10. $1,182.27 for 3 mos.; $1,182.07 for exact days.
11. B.D. - $25.00; T.D. - $24.39. 13. $506.11. 16. 6.89%.
12. (a) $2,520.96. 14. $730.00. 17. $1.79.
(b) $2,520.46. 16. $1,459.06. 18. $400.00.
19. $87.80. 22. $1,470.59. 25. 8.5302.
20. $1,666.67. 23. $12,200.00. 26. 37.8 yrs.
21. $2,317.60. 24. 13.18. 27. 6.45.
28. 26^% if all amis, are focalized at 10 mos.
29. 48% if all amts. are focalized at 5 mos.
30. 26%J% if all amts. are focalized at 10 mos.
Chapter II
Page 38
1. $1,800.94. 3. $1,198.28. 6. $2.63.
2. $2,012.20. 4. $442.94. 6. $2,278.77.
Pages 41-42
1. $1,181.96. 3. $1,670.40. 6. (a) 6.09%.
2. (a) $1,187.60. 4. $1,638.62. (b) 6.136%.
(b) $1,190.50. 6. $2,695.97. (c) 6.168%.
8. 5.18%. 16. (a) 7.23%.
9. u = 5.58%; i 2 = 5.12%. (b) 7.19%.
10. $1,155.48. (c) 7.12%.
13. $3,639.70. 16. (a) 3.94136%.
14. Better to pay cash. (b) 4.90889%.
(c) 5.86954%.
Pages 44-46
1. $140.99. 2. $2,343.60. 3. $1,137.75. 4. $4,226.67.
6. $1,106.12. 10. $1,337.72.
6. (a) $334.99 and $334.84. 12. $1,688.91.
(b) $377.04 and $376.87. 13. $193.07.
7. Yes. 14. $387.35.
8. $2,883.67. 16. PI - $6,417.63; P 2 - $6,455.35.
9. $243.76. 16. $61.55.
Page 48
1. 22.35 years. 4. 16. 6. (a) 14.2. 8. J 2 - 5.5%.
2. 6.3%. .30103 (b) 11.9. 9. 12.9 years.
3.5.14%. 6 * log(l +t)' 7. 20.2 years. 10.6.054%.
248
Answers
Pages 56-66
1. (a) $1,175.29.
(b) $1,360.54.
(c) $1,575.00.
2. (a) $1,579.49.
(b) $1,828.46.
(c) $2,116.67.
3. For the $500 debt:
(a) $519.32.
(b) $631.24.
(c) $695.94.
12. $1,024.51.
13. 5.81 years.
5.74%.
10. $1,159.94.
11. 0.66 years.
18. J 2 = 5.91%; /,
19. 44.13%.
20. j* 12.24%; i - 12.89%.
21. / lt - 23.53%.
22. j 4 * 5.955664%.
23. 6.045%.
For the $750 debt:
(a) $533.01.
(b) $647.88.
(c) $714.29.
6. Pi - $5,250.09; P 2
6. $2,723.25.
7. (a) $3,152.50.
(b) $2,723.25.
8. $332.96.
9. $721.80.
$5,238.41.
14. 5.86 years.
16. 39*7 years.
24. (a) 8.48%.
(b) 8.48%.
26. (a) 11.89 years.
(b) 11.72 years.
(c) 17.5 years.
26. 7.25%.
16. $709.26.
Chapter in
Page 60
1. $3,601.83.
2. $16,532.98.
3. $1,293.68.
4. $1,977.12.
6. $2,564.54. ?
6. $79,840.69.
10. Si - $5,920.98; S 2
$6,003.05.
8. $14,045.45.
9. $416.45.
1. $2,978.85.
2. $12,088.47.
Page 63
3. $2,710.33.
4. $36,919.78.
7. Si = $12,006.11; S z - $11,748.01. 9. $2,983.81.
8. 3.2878%. 10. $2,987.18.
6. $15,303.59.
6. $3,037.04.
1. $10,379.66.
2. $8,832.09.
Pages 63-69
3. $27,084.63.
4. $577.18.
6. $1,228.03.
6. $4,680.04.
1. $3,637.50.
2. $16,839.82.
3. $7,334.80.
Page 72
4. $10,507.65.
6. $5,825.65.
6. $23,742.48.
7. $3,655.42.
6. $16,737.12.
9. $1,692.16.
Answers
249
Pages 76-78
1. $7,325.48.
2. (a) $7,310.84.
(b) $7,332.96.
3. $30,705.23.
4. $30,774.62.
6. Annuity
Payable
Annually
Interest Convertible
Semi~ann.
Quarterly
Annually
Semi-ann.
Quarterly
$4,507.74
4,552.38
4,574.80
$4,518.10
4,563.28
4,585.98
$4,523.39
4,568.85
4,591.70
6. Annuity
Payable
Annually
Interest Convertible
Semi-ann.
Quarterly
Annually
Semi-ann.
Quarterly
$4,775.14
4,834.10
4,863.76
$4,792.45
4,852.36
4,882.50
$4,801.35
4,861.74
4,892.13
7. Annuity
Payable
Annually
Interest Convertible
Semi-ann.
Quarterly
Annually
Semi-arm.
Quarterly
$4,639.51
4,691.13
4,717.08
$4,652.77
4,705.11
4,731.64
$4,659.72
4,712.43
4,738.94
8. $3,474.59.
9. $3,461.61.
10. $3,566.07.
11. $15,157.30.
12. $1,463.14.
13. $13,498.73.
14. $158.26.
15. $18,779.88 if payment at age 60 is included.
16. $18,822.76 if payment at age 60 is included.
17. $624.49.
18. $1,595.30.
19. $1,598.46.
Pages 82-83
1. $7,265.76.
2. $7,235.16.
3. $4,768.81.
4. $3,561.46.
5. $3,596.72.
6. $10,659.30.
_ ($9,177.71 by interpolation.
7 * 1 $9,176.77 by logarithms.
11. Annuity
Payable
Annually
Semi-ann.
Quarterly
12. Annuity
Payable
Annually
Semi-ann.
Quarterly
14. $19,010.68.
15, $5,167.18.
Annually
$811.09
819.12
823.16
Annually
$772.17
781.71
786.50
9. $63,417.98.
10. $63,028.88.
Interest Convertible
Semi-ann. Quarterly
$809.48 $808.66
817.57 816.78
821.64 820.87
Interest Convertible
Semi-ann.
$769.84
779.46
784.30
16. $88,632.52.
18. $9,048.57.
Quarterly
$768.64
778.31
783.17
19. $5,712.91.
20. $2,561.26.
250 Answers
Page 86
1. $447.11. 3. See 15, p. 77. 6. $624.49. 7. $1,626.89.
2. $4,129.86. 4. See 16, p. 77. 6. $1,678.57. 8. $1,630.59.
Page 88
1. $367.84. 3. $4,369.52. 5. $10,329.22. 7. $21,412.19.
2. $3,985.39. 4. $3,887.56. 6. $21,280.01. 8. $4,198.60.
Pages 91-92
1. $6,134.82. 3. $6,171.81. 6. (a) $5,974.89.
2. $6,149.34. 4. $6,018.89. (b) $5,952.48.
6. A' = $7,811.63; Tax - $390.58. 7. $320,957.26
8. $13,949.28. 9. $638.28. 10. $1,863.49.
Page 95
1. 5.33%. 3. 19.7% with F.D. at 12 mo.
2. 6.88%. 4. 4.76%.
Page 97
1. 9 full payments with a partial payment at end of 10 years.
4. 9 full payments; $255.53 at end of 10 years.
6. 14 full payments; $402.39 at end of 24 years.
Pages 99-100
1. $250.44. 3. (a) $533.05. 4. $1,567.74; $4,067.74.
2. $532.09. (b) $531.59. 6. $1,563.39; $4,063.39.
6. $609.11. 7. $2,195.89. 8. $2,221.75.
Pages 104-106
1. 0.67. 3. $2,355,465.79. 5. $174,951.78.
2. $2,400,000. 4. $5,128.45. 6. $1,010.21.
Pages 107-109
2.
$1,093.38.
6.
$116.
10.
$55,454.05.
15. $55,325.34.
3.
$1,288.00.
7.
$6,944.59.
11.
$19,753.09.
16. (a) $55,256.31.
4.
4.905%.
8.
$1,536.81.
12.
$1,456.93.
(b) $55,360.76.
6.
130.
9.
$8,480.01.
13.
$2,276.27.
17. $2,638.80.
18.
$871.85; $684.58.
22. 5.45%.
26.
Yes.
19.
$535.39.
23. $3,056.70.
27.
14 years.
20.
$914.67.
24. 4.66%.
31.
$25,435.38.
21.
14; $5,267.97.
25. 19.75%.
32.
$23,968.84.
Answers 251
Page 110
1. (a) $1,011.59. 6. $29.13; 34.95%.
(b) $1.69. 7. $4,542.09.
2. Yes, by 2 cents. 8. $299.68.
3. j fl = 12.24%; i 12.88%. 9. $238.63.
4. $1,732.02. 10. 44^% using simple interest.
5. $2,382.98.
Chapter IV
Page 113
1. $372.57. 3. $1,358.68. 5. $1,232.50; $2732.50.
2. $523.61. 4. $260.21. 6. $228.49.
Page 115
1. $1,219.14. 4. $69.67; $6,037.46; $8,255.66.
2. $3,351.75. 6. $2,821.36.
3. $1,883.18.
Page 118
1. $81 a year in favor of (b). 4. (a) $796.72 and $831.12.
2. $748.21. (b) $796.72 and $796.72.
3. $732.57. (c) $796.72 and $765.25.
Page 120
1. $1,142.59. 6, (a) $456.85. 9. $13,329.09.
2. $872.31. (b) $442.86. 10. $20,855.57.
3. $2,067.01. 7. (a) $3,670.08. 11. $4,693.60.
4. $321.43; $3,834.72. (b) $3,777.69. 12. $4,503.09.
5. $1,610.70. 8. 7; $147.15; $1,406.93.
Page 121
Problems
1. 53.8% by simple interest theory. 6. $5,680.18.
2. 28.2% by simple interest theory. 7. 138; $97.58.
3. 53% by simple interest theory. 8. $640.12.
9. $2,619,923.28.
(a) m - og og*i*-og 1 . $956.50.
W m log
log (1 + i)
252 Answers
Chapter V
Page 183
1. $27.50. 6. R = $318.02.
2. $124.81. (a) $2,410.68 and $1,963.19.
3. 44.5%, rate of depreciation. (b) $447.49.
4. (a) $1,620.66 and $1,379.73. 6. -$196.25.
(b) $240.93. 7. 42- units.
9. $453.04. 11. $391.58. 13. $356.25.
10. 213 -. 12. $103.76.
Pages 136-136
1. $185,898.00. 2. $901,286.91. 3. $460.98. 4. $78,008.97.
Page 138
1. 20.2 years. 2. 20.4 years. 3. 38.6 years. 4. 39.11 years.
Pages 139-140
1. $278.63. 3. 9.32%.
2. 20.63%; $952.44; $755.95; $600.00. 4. $800.69.
6. $79,563.85. 6. $5,615.60; 30 years. 7. $316,956.82.
8. $46,298.95; R = $1,846.27; Amt. in S.F. $19,216.09.
9. $28,505.24. 10. $62,955.62.
Page 140
1. $75,578.04. 2. $8.69.
4. Amortization plan better by $565.07 per year.
6. $40,250.97. 6. $1,666.40. 7. 20.57%; $3,154.56.
Chapter VI
Page 144
1. $538.97. 4. $5,541.38. 9. $1,781.97.
2. $939.92. 6. $480.92. 10. Yes; P $92.56.
3. $9,110.50. 8. $1,766.01. 11. $5,719.47.
Page 147
1. $940.25. 3. $9,062.53. 6. $12,587.75.
2. $5,335.16. 4. $470.44. 6. $982.24.
Answers
253
1. P - $943.52.
2. P $1,039.56.
Page 160
3. P $538.97.
4. P - $982.24.
6. P - $504.75.
6. P - $5,609.40.
1. $986.83.
2. Po - $961.96;
P = $975.24.
3. Po - $512.63;
P - $520.46;
Q.P. - $512.13.
Page 152
4. Yes; P - $90.75.
5. Po = $92.29;
P - $93.98;
Q.P. - $92.45.
6. Po =- $1,027.02;
P - $1,043.96;
Q.P. - $1,025.96.
7. Po = $1,013.65;
P - $1,031.00;
Q.P. - $1,012.33.
1. $6,063.69.
2. $26,084.46.
Pages 163-164
3. $19,006.41.
4. $17,237.05.
6. $1,932.61.
1. $467.26.
2. (a) $574.79.
(b) $535.75.
(c) $437.25.
(d) $384.43.
Page 165
3. (a) $510.47.
(b) $451.44.
4. (a) $531.93.
(b) $470.04.
1. 0.0473.
2. 0.04195.
Page 158
3. 0.0739.
4. 0.0579.
6. 0.0474.
6. 0.0471.
7. 0.0326.
1. 0.0469; 0.0517.
2. 0.0474; 0.0718; 0.0469.
Page 160
Exercises
3. 0.0420; 0.0577; 0.0468.
4. 0.0521.
6. 0.0367.
1. $968.85.
2. $1,035.85.
3. $305,753.73.
( By interpolation 0.0571.
1 By formula 0.0568.
Page 160
Problems
6. 0.0517.
6. Po - $1,043.76;
P $1,050.43.
7. $93.18.
8. $95.69.
254
Answers
i. (a) K 2 ; (b)
2.
Chapter VII
Pages 163-164
4.
5. 0.4.
6. H.
7. (a) H; (b)
8. KG; Ks;
10.
11.
9. (a)
(b)
(d) %.
12. Former.
13.
1. 0.0085.
Page 166 (Top)
2. 0.514.
3. 0.18.
1. 4.
2. 8.
Page 166 (Bottom)
3. 36. 6. 504.
4. 288. 6. 2,730.
Pages 166-167
7. 3,024.
1. 24.
4. 2,730.
9. 34,650.
13. (a) 362,880.
2. (a) 360.
6. 325.
10. 2,520.
(b) 725,760.
(b) 720.
6. 10.
11. 48.
(c) 725,760.
3. 840.
8. 30,240.
12. 720.
(d) 2,903,040.
2. 126
3. 560.
4. 31.
6. (a) % 76
(b) % 8 .
(C)
21. 6.
Pages 168-169
9. 45.
10. 63.
11. (a) 126.
(b) 84.
13. 302,400.
14. 878,948,939.
22. 10. 23. 7.
16. 31.
17. 3,600.
18. (a) 700.
(b) 1,408.
20. n = 11, r - 2.
26. 711,244,800.
4. He-
6. (a)
(b)
(c)
6. (a)
(b)
(c)
Pages 171-172
7. (a) 0.7624.
(b) 0.8378.
(c) 0.0205.
8. 0.0570.
9.
Answers 255
Pages 171-172 Continued
10. (a) 0.06. 11. %. 13. (a)
(b) 0.56. 12. (a) ^62- (b)
(c) 0.38. (b) MT- H.
(d) 0.44. (c) io% 31 . 15.
Pages 175-176
1. 15. 9. (a) 5,040.
2. 500. (b) 840.
3. 85,680. (c) 13,699.
4. (a) 4 ^o2- 10. (a) 0.015.
(b) %2. (b) 0.42.
(c) ^02. (c) 0.425.
(d) Jio2. (d) 0.845.
5. 675,675. 11.
6. 216. 12.
7. (a) 180. 13. 0.0081; 0.0756; 0.2646; 0.3483.
(b) 120. 14. 0.743.
(c) 6. 16. $10.
8. 720. 16. 10 <Ao(.91914)%08086) 80 .
Page 178
1. 0.5775; 0.4225; 1. 4. (a) $8.43. 6. (a) $13.78.
2. 0.3753. (b) $4.46. (b) $11.58.
3. $7.49. (c) $6.51. (c) $14.37.
Page 180
Exercises
6. 0.0104. 6. 0.5775. 7. 0.08098; 0.00822.
Page 180
Problems
1. 0.4938. 6. 0.01979. 10. (a) 0.77124.
2. 0.7138; 0.001201. 7. $4,900; $4.90. (b) 0.01477.
3. $19,092.07. 9. $8,249.20. (c) 0.11479.
6. 0.8264; 0.9920. (d) 0.09920.
11. $2,802.61. 15. (a) 0.5542.
12. 0.55253. (b) 0.9856.
13. (a) npx-nPv 16. (a) ioo<Ao P* M< V.
(b) (1 ~ nP*)(l - nPy). 10
(C) n p x + nPv " 2 nPx - nPy . (b) y^ ioQoCr mQ -r r,
(d) Same as (c). f^fi
256
Answers
2. $2,261.72.
3. $21,597.29.
4. $1,285.30.
Chapter VIH
Page 184
6. $14,956.01.
6. $16,469.28.
7. $6,555.76.
8. $24,355.37.
9. (a) $7,754.46.
(b) $7,297.62.
10. $7,144.18.
2. $12,038.88.
3. $738.84.
Page 186
6. $6,019.44.
7. $1,847.10.
1. $712.83.
2. $6,167.04.
Page 188
3. $1,541.01.
4. $9,559.39.
12. $2,348.54.
1. $70,147.19.
2. $28,116.41; $2,548.53.
3. $471.83.
4. $176.57.
6. $5,141.72.
Page 192
6. $3,889.75.
7. $1,960.54.
8. $1,363.77.
9. $117.11.
10. $2,568.60.
11. $662.39.
13. $48,752.88.
16. $129.53.
2. $7,592.16.
Page 195
3. $7,991.04. 4. $7,917.36.
6. $17,071.10.
2. $11,376.75.
3. A - $117,632.40; Tax
4. $114,882.40.
5. $14,644.05.
6. $2,323.50.
7. $1,470.32.
8. $1,558.90.
9. $2,552.70.
Page 196
10. $218.59, first payment immediately
$4,705.30. 12. $25,805.64.
13. $74,822.32.
16. Yes.
17. $1,872.19.
18. $21,834.77.
19. $1,199.00.
20. $9,266.29.
Page 201
1. $1,887.86.
2. $3,370.15.
3. $171.90.
4. $123.56.
7. $134.78; $137.72; $349.85; $365.86.
8. $477.69.
9. $1,806.51.
11. $225.25; $229.29; $408.20; $421.97.
12. $242.04.
13. $222.78.
Answers
257
2. $2,196.79.
3. $7.64; $7.79; $8.14; $8.64; $9.46.
4. $107.97.
Page 203
6. $13.52.
7. $221.81.
8. $40.69.
2. $237.37.
Page 206
3. $1,614.80. 4. $188.65.
6. $7,279.34.
1. (a) $35.60; $36.38.
(b) $35.91; $37.08.
Page 207
3. (a) $76.93.
(b) $77.25.
1. $118.27.
2. $129.66; $531.56.
3. $286.48.
4. $218.97.
Page 210
11. $8.14; $26.02.
12. $8.14; $17.68.
13. $410.73.
14. $102.15.
15. $948.94.
16. $541.32.
17. $324.32.
Page 214
1. $183.40; $374.72; $574.33; $782.62; $1,000.00.
2. $33.89; $69.19; $105.94; $144.22; $184.10; $225.64; $268.93; $314.04; $361.05;
$410.06.
Page 216
3. $153.07; $312.47; $478.50; $651.47; $831.64.
4. $726.72; $790.57; $857.29; $927.04; $1,000.00.
6. $306.67; $341.77.
6. $351.53.
2. $2,642.46.
3. $4,088.92.
Page 218
4. $7,933.18.
6. $356.85.
6. $741.45.
1. $409.15.
Pages 219-220
4. 1.
8. $364.33.
10. $17.30; $35.27; $53.94; $73.32; $93.46; $114.39; $136.11; $158.69; $182.12; $206.47.
11. $10.76; $21.89; $33.39; $45.27; $57.54; $70.19; $83.25; $96.70; $110.57; $124.85.
12. $81.97; $167.47; $256.64; $349.66; $446.72.
18. $22.26; $46.30; $69.17; $93.88; $119.46; $145.93; $173.31; $201.62; $230.88; $261.10.
258
Answers
Pages 219-220 Continued
14. $13.42; $27.28; $41.55; $56.27; $71.42; $87.03; $103.07; $119.56; $136.46; $153.71
15. $1.55. 16. $13.29.
1.
Page 227
Automatic Extension
At end of
Reserve
Paid-up
Insurance
Years
Months
1st year
$ 14.67
1
6
$ 35.00
2nd "
29.81
3
1
70.00
3rd "
45.39
4
7
104.00
4th "
61.43
6
138.00
5th "
77.92
7
4
171.00
6th "
94.86
8
6
204.00
7th "
112.25
9
7
236.00
8th "
130.07
10
5
267.00
9th "
148.29
11
2
298.00
10th "
166.88
11
9
328.00
2.
Automatic Extension
At end of
Reserve
Paid-up
Insurance
Years
Months
1st year
$ 22.25
2
4
$ 54.00
2nd "
45.30
4
9
106.00
3rd "
69.17
7
1
158.00
4th "
93.88
9
3
210.00
5th "
119.46
11
2
262.00
6th "
145.93
12
10
313.00
7th "
173.31
14
4
364.00
8th "
201.62
15
7
414.00
9th "
230.88
16
8
464.00
10th "
261.10
17
7
513.00
Answers
Page %ytContinued
259
4 , j f
Automatic
Extension
Pure
Paid-up
At end of
Reserve
Years
Months
Endowment
Insurance
1st year
2nd "
$ 33.15
67.59
3
7
6
2
$ 58.78
116.63
3rd "
103.38
10
7
173.55
4th "
140.58
13
8
229.54
5th "
179.23
16
3
$33.08
284.54
6. $316.58.
6. $13,493.46.
7. $30.29.
8. $3,456.10.
Page 236
1. $306.79, Net Level Reserve; $301.48, F.P.T. Reserve.
2. $18.75; $53.71; $90.03.
3. $10.90; $46.14; $82.75.
5. $32.09; $34.88; $38.26; $42.37.
6. $18.47; $20.64; $23.42; $27.04.
8. $66.26; $66.75; $67.44; $68.42.
1. $13,534.60.
2. $10,169.80.
Page 236
3. (a) $7,986.16.
(b) $14,376.56.
4. $23,074.00.
5. $1,000.00.
6. 14 yrs. 7 mos. 7. $7.79; $15.48. 8. $7.79; $23.68. 9. $7.79; $41.61.
1. 80%.
2. 60%; $3,900.
3. $12; $14.40; $18.
11. $1,192.31.
12. $396.04.
13. Jones' offer by $24.18.
14. 24.49%.
Review Problems
Pages 237-243
21. $3,101.89.
22. $279.76.
23. 8.347
4. Single discount; $4.
6. 66%; $66; 34%.
6. $72.60; $90.75.
7. $79.40.
9. 0.80.
10. $675.
15. $515.46.
16. $548.90.
17. $651.81.
18. 7% months.
19. 8H months.
20. (a) $1,031.45.
(b) $5.35.
47%. 25. $2,316.61.
68%. 26. $1,029.12.
27. $4,004.13.
28. 8.16%; 8.41%.
260 Answers
Pages 237-243 Continued
29. d - 5.66%, J4 5.87%, /4 - 5.78%. 45. $270.33.
30. t - 6.38%, j 4 6.24%, / 4 - 6.14%. 46. n - 14; $479.20.
31. i - 6.23%, d - 5.86%, J4 - 6.09%. 47. $12,177.03.
32. * - 6.14%, d - 5.78%, / 4 - 5.91%. 48. 41 full payments; $125.90.
33. 1.4778%. 49. $3,997.64.
36. Yes, and save $133.21. 60. Yes.
37. (a) $7,721.73. 61. $16.04.
(b) $7,052.25. 62. R = $243.89.
38. $350.36. 63. $24,649.90; $54,649.90.
39. 8; $244.57. 64. $13,329.09; 5.6%.
40. $3,648.80. 66. 1st method better by $344.66 a year.
42. j s - 5.40%; i - 5.47%. 66. 4 full payments; $1,073.71 at end of
43. i = 5.51. 5 years.
44. $10,106.20. 67. $8,348.40.
68. 27.522%; $1,000. 60. $972.40. 62. $6,319.55.
69. R = $399.80. 61. (a) $17,626.51. 63. 1,620 units.
B.V. = $2,834.56. (b) $2,227.60. 64. 1,862 units.
66. $925.61. 69. $9,444.17.
66. $927.66. 70. $10,518.61 by Bankers' Rule.
67. $914.51. 71. j, = 5.3914%.
68. P - $9,376.97.
Miscellaneous
72. 5 years. 84. Yes. About 41 yrs. to exhaust prin-
74. B's offer. cipal.
log 0.5 86. $3,391.75; $437.09.
log (1 - d) ' 6- 18 + years.
87. $188,687.20. 92. .05827. 99. .094.
88. Yes. 94. .0285. 100. .053.
89. 11.26%. 97. $1,491.83.
91. .06184. 98. .0805.
INDEX
[Numbers Refer to Pages]
Account, equated date of, 28
Accrued dividend on a bond, 150
Accumulated value:
of an annuity, 58
of a principal, 9
Accumulation of discount, 149
Accumulation schedule, 111
American Experience Table, 164, 176,
Amortization :
compared with sinking fund, 116
of a premium on a bond, 147
of a principal (debt), 112
Amount:
at compound interest, 35, 39
at simple interest, 1
in a sinking fund, 113
of an annuity, 57
Annual premium, 200
Annual rent, 56
Annuity bond, 152
Annuity certain:
amount of, 57
annual rent of, 56
deferred, 89
denned, 56
due, 83
interest on, 92
periodic payment of, 97
periodic rent of, 56
present value of, 63
term of, 56, 95
Annuity due:
certain, 83
life, 186
Bank discount, 12
Beneficiary, 198
Benefits of an insurance, 198
.Bond:
accrued dividend on, 150
accumulation of discount on, 149
amortization of premium on, 147
defined, 141
dividend of, 141
face value of, 141
purchase price of, 141
quoted price of, 150
redemption price of, 141
tables, 154
183
Book value:
of a bond, 147
of a debt, 113
of a depreciating asset, 123
Capitalized cost, 100, 102
Combination, 167
Commutation symbols, 183
Composite life, 136
Compound amount, 35, 36, 39
Compound discount, 42, 53
Compound events, 169
Compound interest, 35
Contingent annuity, 57
Continuous conversion, 243
Conversion period, 35
Date:
equated 25, 28
focal, 25
Decreasing annuity, 105
Deferred annuity:
certain, 89
life, 186
Dependent events, 171
Depreciation :
defined, 122
fixed percentage method, 125
of mining property, 134
reserve, 122
sinking fund method, 128
straight line method, 123
unit cost method, 130
Discount:
accumulation of, 149
bank, 12
compound, 42, 53
rate, 12
simple, 15
true, 9
Dividend rate, 141
Dividends, 141
Effective rate, 38, 53
Endowment:
261
insurance, 204
period, 204
pure, 182
262
Index
Equated date, 25
of an account, 29
Equated time, 17, 50
Equation of value, 24, 48
Equivalent debts, 23
Events:
dependent, 170
independent, 170
mutually exclusive, 169
Exact simple interest, 3
Expectation, mathematical, 172
Extended insurance, 224
Face of note, 12
Face value of a bond, 141
Fackler's accumulation formula, 215
Factorial, 166
Forborne temporary life annuity due, 189
Force of discount, 243
Force of interest, 243
Full preliminary term, 227
Gross premiums, 199, 221
Illinois Standard, 231
Increasing annuity, 105
Independent events, 170
Insurance:
definitions, 198
endowment, 204
limited payment life, 201
ordinary life, 200
term, 202
whole life, 199
Interest:
compound, 35
effective rate of, 38
force of, 243
nominal rate of, 38
simple, 1
Investment rate, 141
Life annuity:
deferred, 186
due, 186
present value of, 185
temporary, 187
Life insurance, see Insurance
Loading, 121
Makeham's formula, 144
Mathematical expectation, 172
Maturity value, 9
Mining property, depreciation of, 134
Modified preliminary term, 231
Mortality table, 176
Natural premium^ 203
Net annual premiums:
for an endowment policy, 205
Net annual premiums (Continued) :
for a limited payment policy, 201
for an ordinary life policy, 200
for a term policy, 203
Net premiums, 198
Net single premiums:
for endowment insurance, 204
for whole life insurance, 199
for term insurance, 202
Nominal rate, 38
Non-forfeiture table, 223
Ordinary life policy, 200
Ordinary interest, 3
Paid-up insurance, 225
Par value of a bond, 141
Period, conversion, 38
Periodic rent, 57
Perpetuity, 100
Policy:
endowment, 204
holder of, 198
limited payment life, 201
options, 223
ordinary life, 200
surrender or loan value of, 223
term, 202
whole life, 199
Premium :
amortization of, 147
annual, 200, 203, 204
gross, 221
natural, 203
net, 198
net single, 199
on a bond, 144
Present value:
of an annuity certain, 63
of a debt, 9, 42
of a life annuity, 185
Price, redemption, 141
Principal, 1
Probability:
a priori, 162
defined, 163
empirical, 164
history of, 161
Proceeds, 12
Prospective method of valuation, 216
Purchase price of bonds, 141
Pure endowment, 182
Rate:
dividend, 141
investment, 141
of depreciation, 125
of discount, 12, 53
of interest, 1, 38
Index
263
Redemption price, 141
Rent, periodic, 57
Reserve:
meaning of, 212
terminal, 213
Retrospective method, 213
Scrap value, 123
Serial bonds, 153
Simple discount, 15
Simple interest, 1
Sinking fund:
accumulation schedule, 111
compared with amortization, 116
defined, 111
method of depreciation, 128
Straight line method of depreciation, 123
Surrender or loan value, 223
Temporary life annuity, 187
Terminal reserve, 213
Term insurance, 202
Time:
equated, 27
methods of counting, 4
True discount, 9
Unit Cost Method, 130
Valuation:
full level premium method, 227
full preliminary term plan, 227
Illinois Standard plan, 231
modified preliminary term plan, 231
of bonds, 141
prospective method, 216
retrospective method, 213
Value:
book, 113, 123, 147
equation of, 24, 48
face value, 12, 141
maturity value, 9
present value, 9, 42, 63
scrap, 123
wearing, 123
Wearing value, 123
Whole life insurance, 199
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