annuities. 
Example 1. — What is the present value 
of an annuity of 631. to continue for 21 
years ? 
The value in the table against 21 years is 
12,821153, which multiplied by 63 gives the 
answer 807 l. 14s. 7d. 
Example 2. — What present sum is equi- 
valent to a nett rent of 201. per annum for 
69 years ? 
The value in the table against 69 years is 
19,309810, which multiplied by 20 gives the 
answer 3861. 3s. lid. 
If any of the annuities in the above table, 
instead of being for an absolute term of 
years, had been subject to cease if a given 
life should fail during the term, it is evident 
that the value would have been lessened 
in proportion to the probability of the life 
failing ; and, that if instead of being for a 
certain number of years, the annuity de- 
pended wholly on the uncertain continuance 
of a given life or lives, its value must be as- 
certained by the probable duration of such 
life or lives. In order to compute the value 
of Life Annuities, therefore, it is neces- 
sary to have recourse to tables that exhibit 
the number of persons, which, out of a cer- 
tain number born, are found to be living at 
the end of every subsequent year of human 
life, which thus shew what are termed the 
probabilities of life. 
Various tables of this kind have been 
formed by the different writers on this sub- 
ject, as Dr. Halley, Mr. Thomas Simpson, 
M. Kersseboom, M. De Parcieux, Dr. 
Price, Dr. Haygarth, Mr. Wargentin, M. 
Susmilch, and others ; and the true method 
of computing the value of life annuities ac- 
cording to the probabilities of any table of 
mortality is laid down by Mr. William 
Morgan as follows : 
“ Was it certain that a person of a given 
age would live to the end of a year, the 
value of an annuity of 1 1. on such a life 
would be the present sum that would in- 
crease in a year to the value of a life one 
year older, together with the value of the 
single payment of 1 1. to be made at the end 
of a year; that is, it would be 1 1. together 
with the value of a life aged one year older 
than the given life, multiplied by the value 
of 11. payable at the end of a year. Call 
the value of a life one year older than the 
given life N, and the value of 1 1. payable at 
the end of a year then will the value of 
an annuity on the given life, on the supposi- 
11 1 
tion of a certainty, be --(--xN = )i X 
i -j- N. But the fact is, that it is uncertain 
whether the given life will exist to the' end 
of the year 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 
. b t 
survive one year, or supposing - to express 
this probability, it will be —X 1 — IN. ” 
/ ar 
The values of annuities on the joint conti- 
nuance of two lives are found by reasoning 
in a similar manner ; and such values, both 
for single and joint lives, are given in the 
following tables. 
TABLE III. 
Shewing the value of an annuity of 1Z. on 
a single life, at every age, according to 
the probabilities of the duration of life 
at Northampton, reckoning interest at 
5 per cent, per annum. 
Age. 
Value. 
Age. 
Value. 
Age. 
Value. 
Birth. 
8,863 
33 
12,740 
66 
7,034 
1 year 
11,563 
34 
12,623 
67 
6,787 
2 
13,420 
35 
12,502 
68 
6,536 
3 
14,135 
36 
12,377 
69 
6,281 
4 
14,613 
37 
12,249 
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 
11,695 
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 
1 3 
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 
88 
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 
1,133 
27 
13,377 
60 
8,392 
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 
0,060 
31 
12,965 
64 
7,514 
32 
12,854 
65 
7,276 
The values in this and the following tables, 
suppose the payments to be made yearly, 
and to begin at the end of a year ; but if 
all the payments are to be half-yearly pay- 
R 2 
