110
LIFE ANNUITIES.
At the age of (14, the number who survive is 6335, of whom 6047
attain the age of 21 : the sum which must be paid at the age of 14, to
provide £l to each of these individuals on attaining the age of 21, is
6047 X 1.03 -7 , and the sum to be contributed on behalf of each is
6047 X 1.03“ 7
”6335 ’
This sura is less than 1.03~ 1 , which any individual would have
paid to secure an absolute right to ¿£l at the end of 7 years: the
difference arises from there being some chance of the individual not
surviving the term which would entitle him to the sum; and it is but
equitable that he should pay that fraction only of the present value which
expresses the chance of his receiving it. In the present case of 6335
persons living at the age of 14, only 6047 reach the age of 21, and, as we
may suppose every individual has the same chance of being one of these
survivors, and 6047 is the number of chances divided amongst 6335
individuals,
the chance of each individual is
6047
6335'
(Probability,
Art. 4.)
104. The difference between 6335 and 6047 is 288, the number who
die between the ages of 14 and 21 years, out of 6335 persons; and, as
each has the same chance of being one of the 288, the chance at the
age of 14 of an individual dying before he attains the age of 21 is
288
6335’
If we make r n = present value of £l due at the end of n years,
Pm,« — probability of a life aged m living n years,
{ do. of the joint existence of any num
ber of lives aged respectively m, m l5
m 2 , &c., years, continuing n years,
V v __ ido. of the joint existence of the last
(m, mj , m 2 , &c.),n — survivors,
l m number living at the age m according to the Tables,
the probability of a life aged in living n years is
Pm, n
n+n
Rule, The probability of an individual surviving any number of
years is found by dividing the number living in the Tables at the ad
vanced age, by the number living at the present age.
Example. What is the probability of a male aged 36 completing the
age of 53, according to the rate of mortality at Chester ? (Probability,
Table 2.)
n = 53 — 36 = 17
_ — l » _ 3396 _
Pmn -P*"-4187 _
.7094.
