payable at the end of a year — ; then will the 
value of an annuity on the given life, on the 
supposition of a certainty, be -f y X N = 
x T+”n7 But the fact is, that it is uncer- 
r ■ 
lain whether the given life will exist to the end 
of the vear or not . this last value therefore, 
must be diminished in the proportion of this 
uncertainty; that is, it must be multiplied by 
the probability that the given life will survive 
one vear, or supposing — - to- express this pro- 
•' a 
b 
bability, it will be — X 1 -j- N. In the same 
manner the values of annuities on the joint con- 
tinuance of lives may be found : Call the value of 
any two joint lives M, the probability that two 
' J ‘ . Id 
lives one year younger will exist a year , 
and - as above, the value of 1/., payable at 
r 
the end of the year. Then, by reasoning as be- 
fore, the value of the joint continuance of two 
lives one year younger will be expressed by 
id 
X 1 -f- M. 
acr 
By these theorems, tables may be calculated 
of the values of single or joint lives, according 
to any table of the probabilities ot life, and by 
the use of logarithms, and computing upwards, 
/ from the oldest to the youngest life, the labour 
of forming such tables is not very great; few 
persons, however have occasion to undertake it, 
as the tables published by Dr. Price, Mr. Mor- 
gan, and Mr Maseres, shew the values of life- 
annuities as accurately as the present knowledge 
of the decrements and duration of human life 
will admit, and are sufficient for almost every 
useful purpose. 
TABLE I. 
Shewing the Value of an Annuity of £.\ on a 
Single Life at every Age according to the 
probabilities of the duration of Human Life 
at Northampton, reckoning Interest at 5 per 
Cent. 
Ages. 
Value. 
Age 
Value. , 
Age. 
Value. 
Birth. 
8.863 
33 
12.740 
66 
7.034 
1 year 
2 
1 1 .563 
34 
12.623 
67 
6.787 
13 420 
35 
1 2.502 
68 
6.536 
3 
14.135 
36 
12 377 
69 
6.281 
4 
14.613 
37 
12._49 
70 
6 023 
5 
14.827 
38 
12.116 
71 
5.764 
6 
15.041 
39 
11.979 
72 
5.504 
7 
15.166 
40 
11.837 
73 
5.245 
8 
15 226 
41 
1 1.695 
74 
4.990 
9 
15.210 
42 
11.551 
75 
4.744 
10 
15.139 
43 
11.407 
76 
4.511 
11 
15.043 
44 
11.258 
77 
4 277 
12 
14.937 
45 
11.105 
78 
4.035 
13 
14.826 
46 
10.947 
79 
3.776 
14 
14.710 
47 
10.784 
80 
3.515 
15 
14.588 
48 
10.616 
81 
3.263 
16 
14.460 
49 
10.443 
82 
3.020 
17 
14.334 
50 
10.269 
83 
2.797 
18 
14.217 
51 
10.097 
84 
2.627 
19 
14 108 
52 
9.925 
85 
2.471 
20 
14.007 
53 
9.748 
86 
2.328 
21 
13 917 
54 
9.567 
87 
2.193 
22 
13.833 
55 
9.382 
83 
2.080 
23 
13.746 
56 
9.193 
89 
1.924 
24 
13.658 
57 
8.999 
90 
1.723 
25 
13.567 
58 
8.801 
91 
1.447 
26 
13.473 
59 
8.599 
92 
1.153 
27 
13.377 
60 
8.3.92 
93 
0.816 
28 
13.278 
61 
8.181 
94 
0 524 
29 
13.177 
62 
7.966 
95 
0.238 
30 
13.072 
63 
7.742 
96 
0.000 
31 
12.965 
64 
7.514 
32 
12.854 
65 
7.276 
LIFE ANNUITIES. 
These values suppose the payments to he 
made yearly, and to begin at the end of the first 
year ; if the payments are to be made half-yearly, 
the value in the table will be increased about 
one-fifth of a year’s purchase. 
In order to find the present value of an an- 
nuity during any given life, it is only necessary 
to multiply the value in the table corresponding 
with the age, by the given annuity. 
Exam hie.. What should a person, aged 45, give, 
to purchase an annuity of 50/. during his life ? 
The value in the table against 45 years is 
1 1.105, which multiplied by 50, gives the answer 
5551. 5s. 
TABLE II. 
Shewing the Value of an Annuity during the 
joint continuance of Two Lives, according to 
the probabilities of Life at Northampton ; i 
reckoning Interest at 5 per Cent. 
Ages. 
Value, j 
Ages. 1 
Value. 
Ages . 
Value. 
5-5 
11.984 
20-25 
10.989 
40-45 
8.643 
5-10 
12.315 
20-30 
10.707 
40-50 
8.177 
5-15 
1 3 .954 
10-35 
10.363 
40-55 
7.651 
5-20 
11.561 
20-40 
9.937 
40-60 
7.015 
5-25 
11.281 
20-45 
9.448 
40-65 
6.240 
5-30 
10.959 
20-50 
8.861 
40-70 
5.298 
5-35 
10.572 
20-55 
S.216 
40-75 
4.272 
5-40 
10.102 
20-60 
7.463 
40-80 
3.236 
5-45 
9.571 
20-65 
6.576 
45-45 
8.312 
5-50 
8.941 
20-70 
5.532 
O 
l Q 
1 
-T 
7.891 
5-55 
8.256 
20—75 
4.424 
45-55 
7.41 1 
5-50 
7.466 
20-80 
3.325 
45-60 
6.822 
5-65 
6.546 
25-25 
10.764 
45-65 
6.094 
5-70 
5.472 
25-30 
10.499 
45-70 
5.195 
5-75 
4.362 
25-35 
10.175 
45-75 
4.206 
5-80 
3.238 
25-40 
9.771 
45-80 
3.197 
10-10 
12.665 
25-45 
9.304 
50-50 
7.522 
10-15 
12.302 
25-50 
8.739 
50-55 
7.098 
10-20 
11 906 
25-55 
8.116 
50-60 
6.568 
1 0-25 
11.627 
25—60 
7.383 
50-65 
5.897 
10-30 
1 1 .304 
25-65 
6.515 
50-70 
5 054 
10-35 
10.916 
25-70 
5.489 
50-75 
4.112 
10-40 
10.442 
25-75 
4.396 
50-80 
3.140 
10-45 
9.900 
25-80 
3.308 
55-55 
6.735 
10-50 
9.260 
30-30 
10.255 
55-60 
6.272 
10-55 
8 560 
30-35 
9 954 
55-65 
5.671 
10-60 
7.750 
30-40 
9.576 
55-70 
4.893 
10-65 
6.803 
30-45 
9' 135 
55-75 
4.006 
10-70 
5.700 
30-50 
8 596 
55-80 
3.076 
10-75 
4.522 
30-55 
7.999 
60-60 
5.888 
10-80 
3.395 
30-60 
7.292 
60-65 
5.372 
15-15 
11.960 
30-65 
6.447 
60-71 
4.680 
15-20 
11.585 
30-70 
5.442 
60-75 
3.866 
15-25 
11,324 
30-75 
4 3 65 
60-80 
2.992 
15-30 
11.021 
30-80 
3.290 
65-65 
4.960 
15-35 
10.655 
35-35 
9 680 
65-70 
4.378 
15-40 
10.205 
35-40 
9.331 
65-75 
3.665 
15-45 
9.690 
35-45 
8 921 
65-80 
2.873 
15-50 
9.076 
35-50 
8.415 
70-70 
3.930 
15-55 
8 403 
’ 35-55 
7.849 
70-75 
3.347 
15-60 
7.622 
35-60 
7 174 
70-80 
2.675 
15-65 
6.705 
35-65 
6.360 
75-75 
2.917 
15-70 
5.631 
; 35-70 
5.382 
75-80 
2 381 
15-75 
4.495 
35-75 
4.327 
80-80 
2.018 
15-80 
3.372 
I 35-80 
3 268 
85-85 
1.256 
20-20 
1 1 .232 
I 40-40 
9.016 
90-90 
i 0.909 
It is unnecessary to insert a Table of the 
values of the longest of two lives, as it may be 
easily found from the values given in the 
above tables by the following general rules : 
“ From the sum of the values of the single 
lives subtract the value of an annuity on the 
joint lives, and the remainder will give the 
value of an annuity on the continuance of the 
longest of two such lives.” 
Example. What is the value of an annuity 
on the longest of two lives whose ages are 
thirty and forty ? 
By Table I. the value of a single life of 30 
6(J 
is 13.071', and by the same Table the value 
of a single life of 40 is 11.837. 1 heir sum 
■therefore is 24.909, from which 9 57(1 (the 
value of the joint lives of 30 and 40 by 1 able 
11.) being subtracted, gives 15.333 for the 
number of years purchase required. 
' The value of an annuity on three joint 
lives may be found from the preceding tables, 
by the following rule: 
“ Let A be the youngest, and C the 
oldest of the three proposed lives. r l ake the 
value of the two joint lives B and C, and find 
the age of a single life D of the same value. 
Then find the value of the joint lives A and 
D, which will be the answer.” 
Example. Let the threg, given lives be 
20,30, and 40. The value of the two oldest 
joint lives B and C will (by Table II.) |>e 
9.576, answering in Table 1. to a single lite 
I) of 54 years; and the value of the joint 
lives A an*d D, or the ages in the Table which 
come nearest to them, gives 8.216 for the 
value sought. 
The value of three joint lives being 
known, the value of the longest of any three 
lives may be computed by the following 
rule: 
“ From the- sum of the values of all the 
single lives, subtract the sum ol the values of 
all the jo -.lit lives combined two and two. 
Then to the remainder add the value o; the 
three joint lives; and this last sum will be 
the value of the longest of the three lives. ’ 
Example. '1 he sum of the values of three 
single lives whose ages are 20, 30, and 40, is 
(by Table I.) 38.916. The value of two 
joint lives, whose ages are 20 and 30, is (by 
Fable II.) 10.707 \ of two joint lives whose 
aces are 20 and 40, is 9 937, and twojoint lives 
whose ages are 30 and 40 is 9.576 ; the sum of 
these three values is 30.220. This sum sub- 
tracted from 38.916, leaves. 8.696, which re- 
mainder added to 8.216 (the value ot the 
three joint lives in the last example), gives 
16.912, the value of the longest of the three 
lives. The answers in this and the preceding 
example are not quite exact, inconsequence 
i of the table of joint lives being confined t® 
the combinations of every fifth year of age; 
those who have occasion to make such com- 
• putatioris, will find more extensive tables of 
; the values ofjoint lives in Dr. Price’s excel- 
| lent Treatise on Reversionary Payments; but 
| a general table of the values of two joint lives*- 
! for every possible difference of age, at dif- 
j ferent rates of interest, has long been very 
j desirable. 
| The solutions of the following Problems, in 
j addition to the rules already given, will com- 
prehend all the cases which most commonly 
occur relating to the values of annuities on 
lives or survivorship. 
Prcb. I. To detemine the value of an an— 
! nuitv on a given life for any number of years. 
| Solution . Find the value of a life as many 
years older than the" given life as are equal to 
the term for which the annuity is proposed. 
Multiply this value by 1/., payable at the 
end ot this term, and also by the probability 
that the life will continue so long. Subtract. 
I the product from the present value of the 
! given life, and the remainder multiplied by 
the annuity will be the answer. 
Example. Let the annuity be 20/. the 
age of tne given life 35 years,, and the term 
proposed 14 years. The value of a life aged. 
49 years (or 14 years older than the given 
