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.