210 
LIFE ANNUITIES. 
V (m. m\, m%, &c.), n 
P 
(m, mi, m2, &c.), n 
Qm, m, 
■0) 
2 
il 
2 
= W» • ^i-f» • &c j „ the probability of any number 
l-m • l m x ' lm 2 
of lives aged m, m x , m. 2 , &c, jointly surviving n 
[¡years. 
=: the probability of v or more of the lives aged m, m,, 
fm 2 &c., surviving n years. 
~ probability of a life aged m dying before another 
aged m x . 
— probability of a life aged m dying before another 
aged m l within the next t years, 
= the present value of £l due n years hence. 
S prefixed to an expression denotes the sum of the 
values of the variable quantity from the present 
ages to the extreme tabular period of existence. 
= the sum of the first t values. 
= sum of all after the first t values. 
D m , N,„, M m , R m , S m , represent the number opposite age m in the 
columns so marked. 
FORMULAE. 
Two joint lives and the suRvivoR (aged m, mj). 
a m +a mi — a m , TOl “ value of an annuity for the above period. 
Three lives : 
The value of an annuity payable so long as there shall be at least 
two out of three lives in existence aged m, m x , m 2 , &c., is 
®m, m x H” ^m, >«2 4" ®mi, m 2 “ • 
The value of an annuity payable until the death of the survivor is 
a m 4" a m, 4" a rn 2 a m, m\ ~ a m, m. 2 a m v rn 2 4" Cl m> r „ v 
TEMPORARY ANNUITIES. 
The present value of an annuity for n years on a life aged m is 
L 
a o») 
«1 
N —N 
«' 1,1 m+n l m+n n 
or a m ^ r n . a m+n 
D„ 
L 
The present value of an annuity for n years on two joint lives aged 
m and m u is 
/ / «' 
n — n — m+n m \+ n » 
■— u 'm, m x ] ' 1 ,u m+n, m x +n • 
L 
The present value of an annuity for n years on the survivor of two 
lives aged m and m x is 
^(m) 4" <*(«, mp 1 
«1 nl «1