$5.(2$ points$)$ For a special fully discrete $30-$year term insurance on $(40)$:

$($i$)$ The force of mortality is

$$

$$

$=$

$0\mathrm{.}0001$

$($

$1\mathrm{.}0$

$7$

$$

$)$

$\backslash$mu

x

$$

$=0\mathrm{.}0001(1\mathrm{.}07$

x

$).$

$($ii$)$

$$

$=$

$0\mathrm{.}04$

i$=0\mathrm{.}04.$

$($iii$)$ The death benefit is $10,000$ in the first $15$ years and $5,000$ afterwards.

$($iv$)$ Death benefits are paid at the end of the year of death.

$($v$)$ Premiums are paid quarterly for at most $10$ years.

$($vi$)$ Annual expenses are:

Year Per Premium Per Policy

First year $30\%8\mathrm{.}00$

Renewal $3\%0\mathrm{.}70$

$($vii$)$ Per premium and per policy expenses are payable at the beginning of each year. Per premium expenses are calculated based on the sum of premiums scheduled to be paid during the year.

Using a spreadsheet, calculate the annual gross premium using the equivalence principle.