A $1-$year term insurance on $(58)$ has a death benefit of $100,000$ payable at the

moment of death. A quarterly premium of $200$ is payable in advance throughout the

term. The policyholder is subject to the following mortality pattern:

In conducting a profit test, the following reserve values are used:

Other assumptions different from those used in pricing and reserving are:

Initial expenses: $300$ plus $50\%$ of the first quarterly premium.

Renewal expenses: $10\%$ of each subsequent premium.

Interest rate: $8\%$ per annum effective.

Assuming that all deaths occur at the middle of each quarter, calculate:

$($a$)$ the profit signature $({}_0,{}_{0\mathrm{.}25},{}_{0\mathrm{.}5},{}_{0\mathrm{.}75},{}_1)$;

$[$Hint: Adapt the profit formula for quarterly intervals.$]$

$($b$)$ the net present value per policy sold, assuming a hurdle rate of $15\%.$