Question $3$

$120$ marks $1$

i$.$ If the force of mortality and force of interest are constant, show that the benefit reserve at the end of $c$ 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, $\frac{?}{\text{b}\text{a}\text{r}}(A{)}_x=\frac{}{+}.$

ii$.$ For a fully discrete $5-$year term insurance of $1$ on $(50)$ the annual premium at the beginning of each year is $0\mathrm{.}15.$ Annual interest rate is $5\mathrm{.}00\%$ and $q_x=0\mathrm{.}2$ for all $x50.$ Calculate the reserve at the end of the $4^{th}$ year by prospective method.

iii. For a triple decrement model you are given ${}_x^{(1)}(t)=0\mathrm{.}02,{}_x^{(2)}(t)=0\mathrm{.}03$ and ${}_x^{(3)}(t)=0\mathrm{.}05,t>0.$ Calculate probability that $(x)$ will die at age $x+5$ due to cause $2.$

iv$.$ Consider the following double decrement table.

$\backslash$table$[[$Age$,q_x^{(1)},q_x^{(2)}$