LIFE ANNUITIES. 
113 
when there are two lives, aged m and m,, years, the expression becomes 
V 
1 (I Pm,n) (1 P«i l , n) “—Pm, « "4" P«ij , n P (tn, nij ) } n 
(«I, «11), n 
when there are lives, aged in, m 1} it becomes 
PRESENT VALUES OF LIFE ANNUITIES. 
112. To find the present value of an annuity payable at the find of 
every year during the existence of a single life. 
Let the annuity be ¿£l*, and m the age of the individual during whose 
life it is to continue : then the present value of the first year’s payment 
of the annuity is found by multiplying the present value of £ I due at 
the end of one year by the chance of the life living one year (Art. 103), 
the present value of the second payment by multiplying the present value 
of ¿£l due at the end of two years by the chance of the life living two 
years, and finding in the same manner the present value of each year’s 
payment to the extremity of life; the sum is the present value of the 
annuity. 
denote the present value of an annuity of ¿£*1 
during a life aged in years. 
{ 
Let a m 
the present value of an annuity of 1 during 
the joint existence of the lives aged m, in y , m 2 
&c., years. 
present value of an annuity of £l until the 
failure of the joint existence of the last v sur 
vivors of lives aged respectively in, in 2 , &c., 
years. 
a 
V 
l aged m years. 
{ present value of an annuity of £l for n years, 
depending on the joint existence of the lives 
aged m, w 1} m 2 , &c., years. 
a (in, m l , m. 2 , &c.), „1 
* The formulae in all cases are given on the supposition that the annuity is £1 ; 
from which the present value of an annuity of any other amount may be found hy 
multiplying the present value of £1 per annum by the yearly' income of which the 
value is required. 
i