118 
COMMON METHOD OF FORMING TABLES OF ANNUITIES. 
118. The following mode of computing tables of annuities was, until 
very recently, adopted by most authors on this subject;— 
Art. 112. 
l m -(-1 r + 4)7+2 T 2 + 4 
m-\- 2 
+ + &c. 
4« 4- l m +1 r 4- l m + 2 r 2 + l m+3 r 3 -f &c. l m r 
4, 
l m-1 
l + 
/«+1 r + Al+2?' 9 ~b l m +3 ? ' 3 + 4)1+4 ?‘ 4 + &C. \ l m r 
4)1-1 
(1 4- a m ) , p*, from which expression it appears that 
the value of an annuity at any age may be found, when the value is 
given at the age one year older. 
If we commence at the oldest age in the table, at which the value of 
the annuity is 0, and proceed through all the other ages to the time of 
birth, a table will be formed of the values of annuities; the rule expressed 
in words is to “ increase the value of an annuity at any age by unity, 
multiply the sum by the chance of a life one year younger completing 
that age, and by the present value of £l due at the end of one year; 
the result is the value of an annuity on a life one year younger than the 
given age.” 
119. As an example, let us find the values at 3 per cent by the rate 
of mortality among males at Chester, as given in Table 2 of Probability. 
4 00 S -m | 
a 99 •— —j— t (1 + a m ) 
<99 
4*9 , v 
tf B8 “ -T r (1 + a m ) 
¿98 
«97 = y- r (1 + «9») 
<97 
= — X .970874 x (l +0.) s= .7443 
30 
= — X .970874 X 1.7443 = 1.3731 
37 
= —- X.970874 X 2.3731 “ 1.9375 
44 
which results are found to agree with the values given in Table 3 (Pro 
bability), computed by the method described in Art. 113. 
It is scarcely necessary to state that the mode given in Art. 113 
is the more advantageous of the two, not only from the utility of the 
preparatory calculations, but also from its being a more expeditious plan 
of obtaining the values, as the trial of a few examples by each method 
will prove.