ANNUITIES ON JOINT LIVES, 
123 
the tables; in which case the chance of receiving the payment at the 
age of 70 is discounted for 40 years, and for a greater period at every 
age above 70. 
The present value of an annuity certain for the term of years that 
an individual has the expectation of enjoying is greater therefore than 
the value of the same annuity to cease on the failure of that individual’s 
existence. At the age of 45, Chester rate of mortality amongst males. 
Table 2, the expectation is 22 years, for which term at 3 per cent the 
value of an annuity certain is 15.937, and the value of the life annuity 
at that age is 14.382. (Prob. Table 3.) 
ANNUITIES ON JOINT LIVES. 
128. When an annuity depends on the joint existence of any number 
of lives aged respectively m, m 2 , &c., years, the present value of the 
annuity is represented by the symbol a m> mi , m2> &0 . 
By Art. 106, the present value of the expectation of receiving the 
annuity at the end of the first year when there are two lives aged m and 
w, is ——-— 
• 'mi 
the second year’s payment of the annuity is r®; and if the 
value of the expectation of receiving each year’s payment be found to 
the greatest age in the table, and the several values be summed to 
gether, the total will be the present value of the annuity. 
i'mA-l • ^ ”b • ^mi+2 ^ T +3 ^ "P &C. 
(tm > ™l“ J~1 
• l mi 
multiplying the numerator and denominator by r m we have 
_ L +l .C 1+1 -r m+1 +L + *. ¿ mi+2 .r-+® + l m+3 . L i+3 r m+3 + &c. 
Km, m l 11 m 
• ' 
hence the following rule ‘. 
Multiply the number living opposite each age in the table by the 
present value of £l due the same number of years as the oldest age, 
then again each of these products by the number living at the corre 
sponding age of the other life; thus, in finding the values of annuities 
on two joint lives when the difference of age is 5 years, the correspond 
ing ages of the lives at one period of existence will be 36 and 41, in 
which case we find the product of the number of living given in the 
tables at the age of 41, and the present value of £1 due at the end of 
41 years, and multiply this result by the number living at 36. 
Having found the products at all the ages of a given difference from 
birth to the extremity of life, we begin at the oldest ages and find suc 
cessively the sums of all the products above each combination. 
Then the value of an annuity during the joint existence of two lives