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%. It guarantees investors a minimum annual return of 6.5% and a maximum return (or rate cap) of 8.75%. This limits both the retirees' downside 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. How much money can Hometown expect to make on average on the contract? [10] What is the probability that Hometown would lose money on the contract? [10] Suppose that Hometown wants to identify the minimum guaranteed annual rate of return that provides a 2% chance of the company losing money on the contract. What should the minimum guaranteed annual rate of return he? [15] Each year, the Sch riber Corporation must determine how much to contribute to the company's pension plan. The company uses a ten-year planning horizon to determine the contribution which, if made annually in each ofthe next ten years, would allow for only a 10% chance of the fund running short of money. (Read that again.) The company then makes that contribution in the current year and repeats this process in each subsequent year to determine the specific amount to contribute each year. (Last year, the company contributed 523 million to the plan.) The pension plan covers two types of employees: hourly and salaried. In the current year, there will be 6,000 former hourly employees and 3,000 former salaried employees receiving benefits from the plan. The change in the number of retired hourly employees from one year to the next is expected to vary according to a normal distribution with a mean of 4% and standard deviation of 1%. The change in the number of retired salaried employees from one year to the next is expected to vary between 1% and 4% according to a truncated normal distribution with a mean of 2% and standard deviation of 1%. Currently, hourly retirees receive an average benefit of $15,000 per year, whereas salaried retirees receive an average annual benefit of $40,000. Both ofthese averages are expected to increase annually with the rate of inflation, which is assumed to vary between 2% and 7% according to a triangular distribution with a most likely value of 3.5%. The current balance in the company's pension fund is $1.5 billion. Investments in this fund earn an annual return that is assumed to be normally distributed with a mean of 12% and standard deviation of 2%. How much should the company contribute to the pension fund in the current year? [30]