andmaturing at age $65$ to a group of individuals aged $35$ exact.The policy benefits were:

a lump sum of $$20,000$ on survival to age $55$

a further lump sum of $$40,000$ on survival to maturity

a lump sum of $$25,000$ at the end of the year of death, provided the death occurs after age $40.$

Level gross premiums of $$1,456\mathrm{.}82$ are payable annually in advance throughout of e policy term, or until earlier death.

$($i$)($a$)$ Calculate, showing all working, the net premium for this policy.

$($b$)$ Explain why the net premium calculated in part $($i$)($a$)$ is smaller than the actual gross premium payable.

You are given that the gross premium reserve per policy in force at the end of the $17$ th policy year is $$27,113.$

$($ii$)$ Calculate, showing all working and using the recursive method, the gross premium reserve per policy in force at the end of the $18$ th year of the policy.

$[4]$

Reserving basis:

Mortality

Interest

Initial expenses

Renewal expenses

AM$92$ Ultimate

$4\%$ p$.$a$.$ effective

$$150$ plus $60\%$ of the first premium

$5\%$ of the second and subsequent premiums

The company had $500$ policies still in force at the start of the $18$ th policy year. Information about actual experience over the $18$ th policy year of the policy is as follows:

return on the company's investments was $4\mathrm{.}2\%$ p$.$a$.$

total renewal expense was $$41,000$ incurred at the start of the year

there were $3$ deaths

there were $5$ surrenders, each receiving a payout of $$22,000$ at the end of the year.

$($iii$)$ Calculate, showing all working, the total profit over the $18$th policy year arising from this group of policies.

$($iv$)$ Comment on the sources of profit or loss contributing to your answer to part $($iii$).$

$[4]$

$[$Total $21]$