LIFE ANNUIT1E-S. 
115 
On comparing the expressions for these two values, we observe that in 
finding the value at the age of 95 every term is introduced which was 
employed in finding the value at the age of 96 ; so that it costs very 
little more trouble to find the value at both the ages than to find the 
value at one of them only; but, had the first expression for a m been used, 
the operation employed in finding the value at the age of 96 -would not 
have afforded direct assistance in finding the value at the age of 95 ; the 
method which has been adopted has also other important advantages, 
the preparatory operations being of great use in abridging the labour of 
finding the values of Temporary and Deferred Annuities and Assurances. 
The following example, in numbers, of the values of annuities at 
4 per cent, by the Carlisle Rate of Mortality (Table 1), will show the 
process of forming a table of the values of annuities on single lives. 
l m r m - IX.01692512 = 
.01692512 
.01692512 
/ I03 ?- I03 = 3x. 01760212= 
.05280636 
<-6 03’— 
* 
.05280636“ 
Nl02 “ 
.06973148 
.06973148 
/ ]02 r ,03 = 5 X. 01830625 = 
.09153125 
^102 
.09153125“ 
N 101 = 
.16126273 
.16126273 
/ 101 r 10l = 7 X. 01903850= 
.13326950 
u m — 
.133269507 
.29453223 
.29453223 
l m ?’ IOO = 9 x. 01980004= 
.17820036 
¿boo — 
.17820036“ 
n 99 = 
.47273259 
.47273259 
/ 99 r" =11X. 02059204= 
.2265124 
IJ oo — ‘ 
.2265124 
n 98 = 
.6992450 
.6992450 
/«, r 58 =14X .02141572 = 
.2998201 
— 
.2998201 “ 
n 97 = 
.9990651 
.9990651 
k 7 r 7 = 18X .02227235 = 
.4009023 
a 97 
.4009023 “ 
.32051 
.76183 
1.21005 
1.65282 
2.08700 
2.33222 
2.49204 
N M =1.3999674 
4 a r 96 =23 X .02316325 = .5327548 
N 95 = 1.9327222 
l BS r 95 -30X .02408978= .7226934 
1.3999674 „ „ 
a aK =—^—:- = 2.62779 
O95 
.5327548 
1.9327222 
‘“7226934 
— 2.67433 
114. In forming a table of annuities great care must be taken that 
the products of the present value of £l and the number of living at each 
age are accurately obtained, since an error at any one age will evidently 
affect the results at all the younger ages. A good method of guarding 
against inaccuracy is to have the products computed, either by two 
different methods or by two different individuals, and the results care 
fully compared: this being done, we find the sum of all the products 
above each age, and check them by finding the sums for every five or 
ten years, or any other convenient interval; if they agree we may 
assume the intermediate sums to be correct*, and then proceed to the 
divisions. 
i 2 
* A balance of errors may possibly exist.