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1855.] 197 

On the Calculation of Annuities, and on some Questions in the 
Theory of Chances. By J. W. Lubbock, Esq., B.A.* 

[Extracted, by permission of the Author, from the Transactions of the 
Cambridge Philosophical Society.] 

1. ±HE object of the following investigation is to show how the 
probabilities of an individual living any given number of years are 
to be deduced from any table of mortality. All writers (with the 
exception of Laplace) have considered the probability of an indi- 
vidual dying at any age to be the number of deaths at that age 
recorded in the table, divided by the sum of the deaths recorded at 
all ages. This would be the case if the observations on which the 
table is founded were infinite; but the supposition differs the more 
widely from the truth the less extended are the observations, and 
cannot, I think, be admitted where the recorded deaths do not 
altogether exceed a few thousand, as is the case in the tables used 
in England. The number of deaths on which the Northampton 
Tables are founded is 4,689 (Price, vol. i. p. 357). The tables of 
Halley are founded upon the dea'ths which took place at Breslau, 
in Silesia, during five years, and which amounted to 5,869. 

If a bag contain an infinite number of balls of different colours 
in unknown proportions, a few trials or drawings will not indicate 
the proportion in which they exist in the bag, or the simple pro- 
bability of drawing a ball of any given colour; and not only the 
probability of drawing a ball of any given colour, calculated from a 
few observations, will be little to be depended on, but it will also 
differ the more from the ratio of the number of times a ball of the 
given colour has been drawn, divided by the number of the pre- 
ceding trials, the fewer the latter have been. 

Laplace (Theor. Anal, des Probabilites, p. 426) has investi- 
gated the method of determining the value of annuities. He there 
says — " Si l'on nomme y„ le nombre des individus de Page A dans 
la table de mortalite dont on fait usage, et y x le nombre des indi- 
vidus k l'age A + x, la probabilite de payer la rente a la fin de 

l'annee A + x sera — ." This hypothesis coincides with that I have 

y<> 

before alluded to, as adopted by all other writers. Laplace, how- 
ever, means this as an approximation, for he has investigated 
differently the probability of an individual of the age A living to 
the age A + a (p. 385 of the same work). He there considers two 
cases only possible ; but as an individual may die at any instant 

* Now Sir John William Lubbock, Bart., F.R.S., &c. 



198 On the Calculation of Annuities, [April 

during life, I think it may be doubted whether this hypothesis of 
possibility should be adopted. 

Captain John Graunt was the first, if I am not mistaken, who 
directed attention to questions connected with the duration of life. 
He published a book in 1661, entitled Observations on the Bills 
of Mortality, which contains many interesting details, although it 
is written in the quaint style which prevailed in those times. In 
this book, amongst other tables there is one showing in 229,250 
deaths how each arose ; and another showing of 100 births " how 
many die within six years, how many the next decad, and so for 
every decad till 76" — which is in fact a table of mortality, and is 
probably the first ever published. 

After Captain Graunt, Sir W. Petty published his Essays on 
Political Arithmetick. Halley, however, was the first who calcu- 
lated tables of annuities : he took the probabilities on which they 
depend, from a table of mortality founded on the deaths during 
five years at Breslau. Since his time a great number of writers 
have treated of these subjects, of whom a notice may be seen in 
the Encyclopaedia Britannica, or in the Report from the Committee 
on the Laws respecting Friendly Societies, 1827 (p. 94). It is to 
be regretted that those who have published tables of mortality 
should generally not only have altered the radix or number of 
deaths upon which the table is constructed, but also the number 
of deaths recorded at different ages, in order to render the decre- 
ments uniform; this is the case particularly with the Northamp- 
ton Tables, as published by Dr. Price (see Price on Reversionary 
Payments, vol. i. p. 358). For if observations were continued to a 
sufficient extent, they would probably show that some ages are 
more exposed to disease than others — that is, they would indicate 
the existence of climacterics, of which alterations such as these 
destroy all trace. 

I annex four tables,* which I have calculated, with the assist- 
ance of Mr. Deacon, from the Tables of Mortality for Males and 
Females at Chester, given by Dr. Price (vol. ii. p. 392). The first 
two tables show the probability of an individual at any age living 
any given number of years, as well as the expectation of life at any 
age. The last two show the value of £1 to be received by an 
individual of any age after any number of years, and the value of 
an annuity. The difference between these values for a male and 
female, is very great, and shows that tables which would be appli- 
cable for the one would not be for the other. 

* See page 207, and the note there. 



1855.] and on some Questions in the Theory of Chances. 199 

I have also subjoined a table comparing the values of annuities 
calculated from observations at Chester (according to the hypo- 
thesis of probability I have assumed), with some which have been 
calculated from observations at other places. Until lately, the 
Government of this country granted annuities, the price of which 
depended on the price of stock, which renders their tables com- 
plicated. I have given their values of a deferred annuity for five 
years, compared with those I have calculated from the observations 
at Chester : it will be seen that the former are much too high. 

2. Suppose a bag to contain a number of balls of p different 

colours, and that, having drawn m x -\-m 2 + m 3 + m p balls, m x 

have been of the first colour, m 2 of the second colour, m 3 of the 
third colour, m p of the p th colour. If oc X) x 2 , x 3 . . . x p are the 
simple probabilities of drawing in one trial a ball of any given 
colour, the probability of the observed event is x™y x x 2 m > • • • • x 

x p m p, multiplied by the coefficient of Xi m tx 2 m i x p m r in the 

development of {x x + x 2 . . . + x p ) m i +m z- •■■+ m P . The event being 
observed, the probability of this system of probabilities is 
x^i x ar 2 m * • • • • x x t m p, divided by the sum of all possible values 
of this quantity. 

The probability in n x -\-n 2 . . .. + n p subsequent trials of having 
w, balls of the first colour, n? of the second, n p of the p & , is a 
fraction of which the numerator is the sum of all the values of 
^.^,+w, x# 2 "» 4 +» s . . . .xx p m p +n p, and of which the denominator 
is the sum of all the values of j," 1 ! xx 2 m * .... xx p m p, multi- 
plied by the coefficient of ^"i x #2% .... xx p n p in the develop- 
ment of (Xi+X 2 + X 3 ....+X p ) n l +n *~~ +n P. 

Since x x + x 2 . . . .+x p =l, if x u x 2 , &c. be all supposed to 
vary from to 1, and all these values to be equally possible a 
priori, the numerator will be found by integrating the expression 

Xl m \+"\ x a^™ 2 " 1 "" 2 - • •(! — *i — x 2— «3 — x r -\T r + " p d*\ xdx 2 ...x dx p _ x 

first from x p _ 1 =0 to x p ^ 1 = l— x x — x 2 — x p _ 2 , then from 

x p - 2 =0 to :rp_2=l — x y — Xp-z, and so on. The deno- 
minator will be found in the same way. 

If the coefficient of x x m \ x x 2 m z . . . x x p m p in the development 
of (Xi + x 2 . . . . +x p ) n i +n f"+ n p be called C, these integrations 
give for the probability required 

(m i + l)(m 1 + 2)(m l + 3)...(m 1 +n i )(m 2 + l)(m i + 2)...(m 2 +n 2 )... 



Cx 



(mi + mi+nii + m p +p)(m 1 + m 2 +m 3 + m p +p+l) 

(m p +l)(m p +2) (m p +n p ) 

(mi+m 2 +p+n l + n 2 +n 3 — 1)' 



200 On the Calculation of Annuities, [April 

or if the product (m p +l) (m p + 2) .... (m p + n p ) be denoted by 
[m p +l]"p, which is the notation used by Lacroix (Traite du 
Calcul Differentiel, vol. iii. p. 121), the probability required is 

c . Qi + lM^ + l]" 2 - •••[>,+ 1]V 

[j», + W 2 m p +^]"l+"2+"3 • • •+", 

This probability is the same as if the simple probability of drawing 
a ball of the p & colour were m p + 1, with the difference of notation. 
When n 2 , n 3) « p _i, &c. = 0, and n p = 1, this expression 
gives for the chance of drawing a ball of the p 01 colour 

m p + l 



mi + m 2 .... + m p +p' 



and the probability that the index of the colour drawn is be- 
tween n— 1 and n + q + 1 is 

m n +m n+l .... m n+t +q 
mi + mz .... +m p +p 

If we suppose the law of the possibility of life to be such that p 
cases or ages are possible — a priori, m lt m n , &c. will be the number 
of recorded deaths in a table of mortality at those respective ages, 
and the chance of an individual living beyond the » th age will be 

»>»+>»„+i .... m p +p— n 



mi + nti .... +m p +p 



m n + m n _i + &c. + m p is the number given by the table as living at 
the n th year ; therefore, on the hypothesis of this law of possibility, 
the chance of an individual living beyond the w th year is a fraction 
of which the numerator is the number living at that age, +p—n, 
and the denominator is the whole population on which the table 
is founded, or the radix + p. The Tables I. and II. have been 
calculated from this formula, from observations at Chester given 
by Dr. Price (vol. ii. p. 107) : p was taken equal to 101 for a child 
at birth — that is, the chances of a child living beyond a hundred 
years, and of its dying in each intermediate year, were supposed 
to vary from to 1, all these values being equally probable, 
a priori. The value of any sum to be received after any number 
of years is equal to the sum itself, multiplied by the chance of 
the individual being alive to receive it : therefore these tables 
give the value of unity to be received after any number of years. 
Considering duration of life to be valuable in proportion to its 



Cx 



1855.] and on some Questions in the Theory of Chances. 201 

length, the value of the expectation of life to any individual is the 
sum of the chances of his living any number of years multiplied 
by the intervening time ; so that if P„. be the chance of an indivi- 
dual living exactly n years, the value of his expectation of life is 
2nP„, which is evidently equal to SP'„, if P'„ be the chance of an 
individual surviving n years : therefore the value of the expecta- 
tion of life of any individual is the sum of the numbers on the 
same line in Tables I. and II. The unity of expectation is here 
the expectation of an individual who is certain to live exactly one 
year. The Tables I. and II. give the values of contingencies 
depending on a single life, without discount ; the Tables III. and 
IV. are the same values, discounted at the rate of three per cent. 
compound interest. These tables give the values of annuities 
about six per cent, higher than those calculated from the North- 
ampton, and given by Dr. Price, vol. ii. p. 54. The only tables 
of annuities on female lives that I have met with are calculated 
from observations in Sweden, and are given by Dr. Price, vol. ii. 
p. 422 ; but they are calculated at four and five per cent, interest. 
It is not to be expected, however, that tables calculated from 
observations made in one country will serve in another, or even in 
different parts of the same country.* 

The probability of having ra, balls of the first colour in % + N 
trials, the colours of the other N balls being any whatever, is 

fxf^jl-x^fx^xfs (l— Xl — X2 x^-^pdx^x^ dx p _ x 

fxi'ix^" 2 . . . .(1 — Xy — x 2 . . . .x p _ l )'"pdx l dx 2 . . . .dx p _i ' 

multiplied by the coefficient of x n \ in the development of 
{ x i+y) n ^> the integrals being taken between the same limits 
as before. 

These integrations give for the probability required 

(»»i + l)(>»i + 2). ..(wi+7tiX»»2+'»3+'»4. ..+p— l)(m 2 +m 3 +m 4 . ••+/>). 
{mi+m 2 + m p +p)(m l + m 2 +m p +p+l) 

(m 2 +m 3 + m 4 +jp+N— 2) 

(j»,+m 2 + m 3 +p + m + N— 1)' 

C being equal to k+l> ^V///.^ ' Ado P tin S the 
same notation as before, this probability is equal to 

* Since writing the above, I find that Mr. Finlaison has given the values of 
annuities, distinguishing the 6exes, in the Report of the Committee on Friendly Societies, 
1825, p. 140. 

VOL. V. P 



202 On the Calculation of Annuities, [April 

[m l +iyi[m i +m 3 ..+m p +p— l] N 

[»ii + »n 2 + TOa «i p +jh]"i +n 

CQi + lJ"^-^ +m p +p— l] CT i +1 

~~ [m 2 +OT 3 ....+N+i»-l]"i + '" 1+1 

which probability, as before, is the same as if the simple probability 
of drawing a ball of the />*** colour were »ip + l. 

If m 2 + m 3 . . . . + m p +p—2=M., and if n, and N are in the 
same ratio as »i, and M, the chance that the number of balls of 
the first colour in «i + N trials is between the limits n x and n x ±z, 
by the reductions given in the Thiorie Anal, des Probabilites, 
p. 386, is 



1—21/ — — i—- +f — - / dze 2m 1 M(N+» 1 )(M + N+m 1 +n 1 ), 

e being the number of which the hyperbolic logarithm is unity, 
and the integral being taken from z=z, to ^=infinity. 

The question of determining the probability that the losses 
and gains of an Insurance Company on any class of life are 
contained within certain limits, is precisely similar to this. 

It will be seen from the formula "»*+ "Wi- •••"»»+? + g (p . 2 00, 

mt + nit + m p +p xr 

line 12), that if life were divided into an infinite number of ages or 
intervals (in which case p is infinite), the hypothesis of possibility 
remaining the same, the probability of an individual dying in any 
given interval would be the given interval divided by the whole 
duration of life, which coincides with that which is given by De 
Moivre's hypothesis. Thus if life were supposed to extend to a 
hundred years, the probability of an individual dying in any 
given year would be -j-^-g, and any finite number of observa- 
tions or recorded deaths would not influence the value of this 
probability. As diseases and other causes producing death are 
not equally distributed throughout life, the last hypothesis cannot 
be adopted. 

In order to investigate accurately the probability of death at 
any age, it would be necessary to know the law of possibility. 
Let <j>pX p be the probability of the possibility of x v : then the pro- 
bability in the former question of having n x balls of the first colour, 
w 2 of the second, &c, in n 1 + w 2 • • • • + w j> trials, is 

yy l+ "'(frgi>2 » + "Kf i ig!i)...(l— x l -x i ...x„_i)'"'+ n pdx i dx 2 — dx p _ x 
/*i"i(0i*i), «™s(^ 2 iF 2 ) (1 —xi —x 2 . . .x p ^) m p dxidxi —dx p ^ ' 



1855.] and on some Questions in the Theory of Chances. 203 

is a sign of function, and this function may be either continuous 
or discontinuous. 

This expression must be integrated between the same limits as 
before. 

The coefficients of the different powers of x p in <j>pX p , or the 
constants in <j>pX p , will generally be functions of the index p. If 
the probability of life were known at a great many places, and if 
x Pl were the value of x p at q v places, x P9 at q t places, &c, the law 
of possibility might be determined approximately by considering 
QpXp as a parabolic curve, of which x p is the abscissa, passing 
through the points, of which the ordinates are 

?1 ?2 



?1 + ?2 +> &c- ' 7i + ft +> &c. ' 

3. In the preceding investigations, the results of the preceding 
trials are supposed to be known ; it may be worth while to examine 
what the probability of any future event is when the results of the 
preceding trials are uncertain. 

Let a bag contain any number of balls of two colours, white 
and black: suppose m trials have taken place; and let e„ be the pro- 
bability that a white ball was drawn the /1 th trial, /„ the probability 
that a black ball was drawn. 

e„+f n =l. 

First let e lt e 2 . . . . e„ be all equal, and let x be the probability 
of drawing a white ball. If a white ball was drawn every time 
in the m trials which have taken place, the probability in «i + n 2 
future trials of having n x white balls and » 2 black balls is 

(«i+«2)(wi+n 2 -l)---("i + l)/^- | "''(l-^" t ^ 
1.2 n n faf'dx 

But the probability that a white ball was drawn every time is e m ; 
therefore the probability of drawing a white ball n t times and 
a black ball n 2 times, on this hypothesis, multiplied by the pro- 
bability of the hypothesis, is 

(nt+nzXnx + nz— l)...(ni + l) „ /^ m+ °» (1— x) n 2dx 
1.2 n 2 e ~JaFTx ; 

and the probability of drawing n x white balls and w 2 black balls 
will be the sum of the probabilities on every hypothesis, multi- 
plied respectively by the probability of the hypothesis, which is 

p2 



204 On the Calculation of Annuities, [April 

(wi + H^fa + n,-!).. .(«i + l) ( m f<r**jl-xpd* 
1.2 « 2 \ e /V<fo 



«/ /ar-\l—x)tb "*" 1.2 ^ faf-Xl-&yki ' 

This integral being taken from a;=0 to 03=1, is 

(» 1 +«g)(wi+n a — l)..(wi + l) r w 2 ,n 2 — 1,^— 2 l.m+1 

1.2 n 2 \m+n 1 + l.m+n 1 + 2...m+n 1 + ii2+l 

ni+\.iu.iu — 1....2 
+me m -]f. — 2.J! m+l.m+, &c. 

J» + »!.»» + Wi + l . . . .Wi+»]+W 2 

_ (wi + w 2 )(wi+w 2 — 1) (wi + 1) 1 

1.2. .. . n 2 m+2.w+3. . .w + » 1 + «2+i 

|n 2 .»! 2 — 1 .n 2 — 2. ..m+ni.m+ni — 1. ..m+1 .e m +n 2 -i-l ■ •• 
2.m+n x — 1. . .m+1 .m .me m ~]f+ , &c. 

_,,. . . lx rf n . + " 2 .« B 3 a^i(ex+/w)"' , , 

This series is equal to , - , . ±3d— when x and 

^ dx n \dy n i 

y are made equal to 1, and this is equal to 1 . 2 . 3 . . . . n x . 1 . 2 . 3 

. . . . ra 2 x coefficient of A"i k n *, in the development of 

(i+ hp(i + A)xi + eh +/ty 

(l +e h+/ky=l+m(eh+fk)+^^-(eh+/ky 

m.m — l.m — 2, , „,,„ 
+ j-^3 («*+/*)»+ , &c 

(1 + hfi(l+kp=h"iJ^+nih' , i- l k"2+ Wl '" 1 ~ 1 A"i-**-» 
+ n 2 &"i/E"2- ■ + » . w^".-'^- 1 + - '"' ' "' ~ * A-i- 2 ^-' + , &c. 

Coefficient of h" i A"* = 1 + Jn(n,e + n 2 /) 

m.m—l( w,.^— 1 n 2 .n 2 —l\ 



1855.] and on some Questions in the Theory of Chances. 205 

The probability required is 

1.2.3 «i+«2 



m+2.m+3. . . .m+ni + t^+l 



l+ m (n l e+T h f)+ — | - l - e i +2n l n 2 e/+ 2 - » g f A+, &c. 

If there are p different colour's, and if m trials have taken place, 
and e g , p is the chance that a ball of the p th colour was drawn 
the q th trial, the probability of drawing n x balls of the first colour, 
n 2 of the second, n P of the p^, in n t + n 2 . . . . +n p future trials, 
may be found in the same way. Let 

«1,1 +«1,* + «1,I+ &c «l,n = S 1 , «l, 

e l, 1 J *1, 2 "I" *1, 3 » e l, 4 < ^ C> = "2> *1> 

(the sum of the products of e 1 two and two together,) 

«1,1> «2,S + *1,8> «2,3+»&C. =S 1 « 1 , SAj* 

and so onj then it may be shown that this probability is equal to 

1.2.3.n 1+ n 2+ n 3 . ■ + n, (1+Sei) „ l(1 + S ^.. .. (1 + S ^, 
>n+^ w+w 1 +n 2 .. ..+n p +p— 1 

1 + (Sej), 1 + (Se 2 ), &c. being expanded by the binomial theorem, 
and the indices of S written at the foot. 

The method which was used for summing the series in the last 
page is of very general application, and depends, in fact, on this 
principle, that the generating function of the sum of any series 
is» the sum of the generating functions of each of the terms of the 
series. 

If in the last formula « 2 , n 3) &c.=0, and if there be only two 

events possible, and n x = 1, the probability required is — — ■— . In 

order to apply this, suppose an individual to have asserted m 
events to have taken place, of which the simple probabilities are 
equal, and equal to p ; and suppose it required to find the proba- 
bility of his telling the truth in another case, where the simple 
probability of the event he asserts to have taken place is not 
known. Let x be the veracity of the individual, the probability of 

his telling the truth on this hypothesis is — ,., — . ,., r : and 

° Jr px+(l—x)(l—p) 

the probability of his telling the truth is the sum of the pro- 
babilities of his telling the truth on each hypothesis, divided by 
the number of the hypotheses. 

* This is a method of notation which ohtains, but it is not meant to imply that 
S 1 e 1 S 1 e 2 =S,,e 1 x S„ e 2 . 



206 



On the Calculation of Annuities, 



[April 



Suppose x to vary from to 1, and all these values of x to be 
equally probable a priori, then the probability of his having told 

the truth and the event having taken place is / — i-r. — ttt r, 

r J px+(l'—p)(l—x) 

taken from x=0 to x=l, which integral is 



cM'-s^Ty- 



9 



If p = -^, this probability is '81601. Generally, Up > £, the 

assertion that the event has taken place (on this hypothesis of 
veracity) rather diminishes the probability that the event has taken 
place; if p= J, the assertion does not alter the probability; if p < \, 
the assertion rather increases it. 

9-1601 



9 1 + Se 

If P=^> «= -81601, let jh=10; then tlf^ 

r 10 m+x 



12 



which is the probability that the individual will tell the truth in 
another case. If the individual had told ten truths, the chance of 

his telling the truth in another case would have been — • 

All values of x between and 1 were supposed equally pos- 
sible : if they are not, let <j>x be the probability of the possibility 
of any value of x ; then the probability of an individual telling the 
ffxdx 



truth will be 



'px+(l — x) (1— p) 
grals being taken from x=0 to <e=1. 



divided by f$xdx, these inte- 



Table formed from the Burials in All Saints' Parish, Northampton, 
from 1735 to 1780. (Seepage 198, line 26.) 



Age. 


Actual 

Number of 

Burials. 


Reduced to radix 1 1050. 


By the 

Observations. 


As altered by 
Dr. Price. 


Under . . 2 

Between 2 and 5 

S „ 10 

„ 10 „ 20 

„ 20 „ 30 

30 „ 40 

„ 40 „ SO 

. ,, SO „ 60 

60 „ 70 

• „ 70 „ 80 

„ 80 „ 90 

„ 90 „ 100 

Total . . . 


1,529 
362 
201 
189 
373 
329 
365 
384 
378 
358 
199 
22 


37981 
899J 
499* 
469| 
926| 
817| 
906} 
954 
939J 
889| 
494| 
Mi- 


4367 
1034 
574 
543 
747 
750 
778 
819 
806 
763 
423 
46 


4,689 


ll 650 


11650 



1855.] and on some Questions in the Theory of Chances. 



207 



Table I. — Males. 



Table II. — Females. 





Expectation 

of Life, 

by the Author's 

method. 


Expectation 


Age. 


of Life, by 
usual method. 





29-75345 


2813 


1 


36-69541) 


35-76 


2 


40-21306 


39-42 


3 


42-54615 


41-97 


4 


43-83928 


43-33 


5 


44-09357 


43-20 


10 


42-75204 


41-92 


15 


38-95786 


38-05 


20 


35-82561 


34-86 


25 


32-99367 


32-00 


30 


30-27118 


29-25 


35 


27-04332 


25-97 


40 


2404172 


22-92 


45 


21-34876 


20-20 


50 


18-81444 


17-64 


55 


16-34857 


15-14 


60 


13-63392 


12-36 


65 


12-05917 


10-79 


70 


9-41263 


8-05 


75 


8-43636 


7-00 


80 


6-99009 


5-43 


85 


5-90384 


4-25 


90 


4-32000 


2-50 


95 


2-14285 


1-00 



Table III.— 


-Males. 


Age. 


Value of 
Annuity by 
the Author's 

method. 


Value of 

Annuity by 

usual method. 





13-96256 




1 


17-35468 




2 


19-17322 




3 


20-38907 




4 


21-15972 




5 


21-42118 


21-283 


10 


21-55443 


21-512 


15 


2038198 


20-283 


20 


19-42818 


19285 


25 


18-55566 


18-399 


30 


17-67138 


17-492 


35 


16-36473 


16134 


40 


15-05567 


14-667 


45 


13-81590 


13-493 


50 


12-59164 


12-316 


55 


11-28375 


10-866 


60 


9-63491 


9-140 


65 


8-77709 


8-220 


70 


6-94786 


6-260 


75 


6-17140 


5-291 


80 


5-11141 




85 


4-30505 







Expectation 


Expectation * 


Age. 


by the Author's 
method. 


of Life, by 
usual method. 





34-55535 


33-27 


1 


40-04475 


39-54 


2 


43-65276 


43-25 


3 


45-87700 


45-68 


4 


47-23533 


4711 


5 


47-99860 


47-44 


10 


45-69310 


45-17 


15 


41-96030 


41-36 


20 


38-76308 


38-10 


25 


35-49329 


34-78 


30 


32-79.565 


32-27 


35 


3000384 


29-26 


40 


27-13292 


26-37 


45 


24-29072 


23-50 


50 


21-43212 


20-62 


55 


18-35900 


17-52 


60 


15-09954 


14-20 


65 


12-83834 


11-94 


70 


9-78378 


8-81 


75 


8-12794 


7-14 


80 


6-30434 


5-20 


85 


5-97402 


4-81 


90 


4-55263 


3-46 


95 


2-07692 


1-71 



Table IV. — Females. 



Age. 


Value of 
Annuity by 
the Author's 

method. 


Value of 

Annuity by 

usual method. 





15-75290 




1 


18-42550 




2 


20-14850 




3 


21-32367 




4 


22-11858 




5 


22-64306 


22-624 


10 


22-41169 


22-439 


15 


21-25267 


21-235 


20 


20-36435 


20-323 


25 


19-33571 


19-265 


30 


18-65687 


18-583 


35 


17-62920 


17534 


40 


16-56779 


16-366 


45 


15-42043 


15-282 


50 


14-14872 


13-983 


55 


12-58232 


12-376 


60 


1067266 


10-410 


65 


9-37655 


9080 


70 


7-26287 


6-889 


75 


6-02689 


5-338 


80 


4-65536 




85 


4-34792 





Note. — We do not reprint all the tables originally given in this paper. They are of 
value now only as exhibiting the results of the methods laid down by the Author, and 
for that purpose those here quoted will suffice. — Ed. A. M.