Q$9$: A financial institution has issued a large number of $15-$year policies. Under these policies:

Premiums are payable annually in advance

A maturity benefit of Rs$.30,000$ is payable at the end of $15$ years

Premiums are calculated such that the present value of benefits payable and expenses incurred equal that of the premiums under the policies. In doing this calculation, the financial institution makes the following assumptions:

Interest

$4\%$ per annum effective

Expenses at the start of the policy

Ongoing expenses

$3\%$ of the maturity benefit

$2\%$ of each premium paid

$($a$)$ Calculate the annual premium for each of these policies assuming that premiums under all policies continue to be paid for each of the $15$ years.

$($b$)$ The financial institution observes that on each premium due date, $3\mathrm{.}7037\%$ of the policies that had paid premium in the preceding year, defaulted on their premiums. It has been decided that policies that defaulted in premiums would be immediately refunded the premiums already paid under them, without any interest. Also, they would not be eligible for the maturity benefit of Rs$.30,000$

The financial institution wishes to recalculate the premiums for these policies to allow for these premium defaults Calculate the revised annual premium.