Hometown Insurance sells $10-$year annuities to retirees who are looking for stable

sources of investment income. Hometown invests the annuity funds it receives in an

equity index fund with annual returns that are normally distributed with a mean of

$9\%$ and standard deviation of $3\mathrm{.}5\%.$ It guarantees investors a minimum annual

return of $6\%$ and a maximum return $($or rate cap$)$ of $8\mathrm{.}5\%.$ This limits both the

retirees' down$-$side risk and up$-$side return potential. Of course, Hometown makes its

money on these contracts when the actual return exceeds the rate cap. Suppose a

retiree invests $$50,000$ in such an annuity contract. Assume investment earnings are

credited at the end of the year and are reinvested.

a$.$ Build a spreadsheet model for this problem that computes the profit Hometown

would make on the contract.

b$.$ How much money can Hometown expect to make on average on the contract?

c$.$ What is the probability that Hometown would lose money on the contract?