Question $6$ A five$-$year $$50,000$ endowment insurance for $(60)$ has $$1,000$ underwriting expenses, $25\%$ of the first premium is commission for the agent of record and renewal expenses are $5\%$ of subsequent premiums. Write the gross future loss random variable: $\_()\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_($ P$)$resuming a portfolio of $10,000$ identical and independent policies, the expected loss and the variance of the loss of the portfolio are given below $($note that the premium basis is not given or needed$)$: Find the premium that results in a $97\mathrm{.}5\%$ probability of profit $($i$.$e$.).$ Premium: $\_()\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())\_(\_())$

Question $6$

A five$-$year $$50,000$ endowment insurance for $(60)$ has $$1,000$ underwriting expenses, $25\%$ of the first

premium is commission for the agent of record and renewal expenses are $5\%$ of subsequent premiums.

Write the gross future loss random variable:

Presuming a portfolio of $10,000$ identical and independent policies, the expected loss and the variance

of the loss of the portfolio are given below $($note that the premium basis is not given or needed$)$:

E$[$L$]=10,000(36,956\mathrm{.}49-3\mathrm{.}8786$P$)$

V$[$L$]=10,000(50,000+14\mathrm{.}52$P$)^(2)*0\mathrm{.}00095$

Find the premium that results in a $97\mathrm{.}5\%\{$:$\backslash$phi $^(-1)(0\mathrm{.}975)=1\mathrm{.}96).$

Premium:

Please show your work below