Question 3
[20 marks]
i. If the force of mortality and force of interest are constant, show that the benefit reserve at the end
of t years for t > 0 for a fully continuous whole life insurance on (x) is 0. (Hint: When force of
mortality and force of interest are constant, $A_x = \frac{\mu}{\delta + \mu}$).
ii. For a fully discrete 5-year term insurance of 1 on (50) the annual premium at the beginning of each
year is 0.15. Annual interest rate is 5.00% and $q_x = 0.2$ for all x ? 50. Calculate the reserve at the
end of the 4th year by prospective method.
iii. For a triple decrement model you are given $\mu_x^{(1)}(t) = 0.02$, $\mu_x^{(2)}(t) = 0.03$ and $\mu_x^{(3)}(t) = 0.05$,
t > 0. Calculate probability that (x) will die at age x + 5 due to cause 2.
iv. Consider the following double decrement table.
Age
$q_x^{(1)}$
$q_x^{(2)}$
30
0.02
0.01
31
0.04
0.03
32
0.05
0.04
Calculate the probability that (30) die between 1 and 3 years due to cause 1.
[5 marks per each question]
