
146 Life insurance and net premiums
and its expected value is
p
Z
m
0
e
−ρs
1 −F(x + s)
1 −F(x)
ds. (12.9)
Therefore as an equation for the net premium we obtain
p
Z
m
0
e
−ρs
1 −F(x + s)
1 −F(x)
ds = c
Z
∞
0
e
−ρs
f (x + s)
1 −F(x)
ds,
and hence
p
x,ρ
= c
R
∞
0
e
−ρs
f (x + s) ds
R
m
0
e
−ρs
[1 −F(x + s)] ds
. (12.10)
Since the denominator now is smaller than in (12.5), the premium is
now higher. This is of course as expected: the same benefit is being
paid for over what is usually going to be a shorter period.
Term life insurance. In fixed term life insurance contracts the
amount c is payable at the moment of death, if a person A of age x
dies before the age of x + m. Otherwise, the insured ...