/
134
LIFE ANNUITIES,
What is the present value of an annuity of £50 on the longest of
two lives; one a male aged 35, the other a female aged 40 ? (Chester
5 per cent.)
(Tables. Prob.)a 35 = 13.1892
do. a i0 = 13.3287
26. 5179
Table 23. «35.«= 10.6690
15.8489
50
792.445 = £792 8 11.
147. To find the present value of a deferred annuity on the longest
of any number of lives.
If the annuity be deferred n years, the first payment will have to be
received at the end of n +1 years ; the present value of which is found
by multiplying the present value of £l due at the end of ?i+l years
by the probability of the existence of the survivor at the end of that
term; and the present value of any other payment is found in like
manner by multiplying the probability of the event on which the pay
ment depends taking place, by the present value of £l due the num
ber of years that must lapse before the payment will be due; the several
terms of the series in Art. 143 represent these values; and the sum of
them all after the first n terms will be the value of the deferred annuity;
the sum of these terms in the first perpendicular column is
Pm, n+1 7' n+1 +p m< n+2 r n+2 + p m , n+3 r n + 3 + &c,,
which, by Art. 133, is the present value of an annuity deferred n years
on a life aged m, and the suras in the other columns also evidently re
present values of deferred annuities ; if therefore, in the formula obtained
for the value of an annuity to be continued for the whole term of life,
we substitute the present values of deferred annuities for the present
values of immediate annuities for the term of each life, the expression
for the required value of the deferred annuity will be obtained.
148. When there are three lives the formula is
^(m) d" ^(m.) 4“ &(m 9 ) ^(m, m.) m 2 ) U(m, t m \ 4“ ¿?(m, m,, m 9 ) •
1« In "Jn 1 In -]n * 1 n 1 in
V,
149. And for the longest of two lives the expression is
^(m) 4" m.) ■>
150. If the annuity depend on the joint existence of the lives during
the n years that the annuity is deferred, the formula will be
