132 
LIFE ANNUITIES. 
By Example 5, page 130, 10(l+a (36) ) =80.2909 
Ditto «(86)— 8.82668 
v 
8.82668 : 1 :: 80.2909 
80.2909 
8.82668 
9.096 =£9 1 11 
ENDOWMENTS. 
142. From the above expressions we may find the annual premium re 
quired to secure a sum upon an individual attaining any particular year. 
By Art, 105, the present value of ¿£l to be received at the end of n 
years is 
L 
L 
yjn-^n 
D„ 
l m l m r m ~~ D m ' 
If we suppose n payments, the first paid immediately, the annual 
. .-.-i -i Urn 
premium will be — 
D a 
D X N 
l-'m m—I 
2719.999 
X 100= 
271999.9 
12.055 = 
N 13 -N 20 75523.846-52960.516 22563. 33 
annual premium to secure £100 at the end of 7 years to a child aged 
14. (Northampton 3 per cent.) 
143. ,To find the value of an annuity granted on the longest of any 
number of lives. 
Let there he any number of lives aged m, ni x m 2 , &c., respectively, 
then, by Art. Ill, the probability of some one or more of these lives 
being in existence at the end of any year from the present time, as the 
nth, on which the receipt of the payment of the annuity at the end of 
that year depends, is 1 — (1 — (1 — p mi ,n) (1 — Pm i>n ), &c.: 
if n be made equal to unity the expression will give the probability of one 
or more of the lives being in existence at the end of the first year, which, 
multiplied by the present value of £l due at the end of one year, will 
show the present value of the payment to be received at the end of the 
first year; if n be 2, and the value of the expression in this case he 
multiplied by the present value of £l. due at the end of two years, the 
result will be the present value of the payment to be received at the end 
of the second year ; and the sum of the present values of each payment 
for every age to the end of the Table will evidently be the present value 
of the annuity. # 
{ P'n, 1 "t" Pm, 
,l+Pn 
1 
6 
Po* 
)> 1 PCn. 
-Pen, 
.»n 2 )n- &C - + 
Pin, 
Wj , r, 
»2 )>1 ) &C.} 
l {Pm,i+P mi 
,2 + P« 
&c ‘- 
'Pim 
,m, )) 2 Pirn, 
Wg )}2~ 
-*<V 
m s ))2 &C.~h 
P(m,m j , 
m ) 5 2> &C. } 
{ Pm,3~hPm, 
.a^Pn 
V 3. &C.- 
-PC 
n.m, )J3 — Pim, 
m 3 )>8‘ 
—Pi m l 
, nig )i 8> &C. + 
Po* 
Wlj , 5 
m a )J 3> &C.} 
&c. 
&c. 
&C.