Your friend would like your assistance in determining how much to save for retirement. He is planning to invest $ 10,000 in a tax-sheltered retirement fund at the end of each year (i.e., appreciation of the fund is not taxed).

The rate of return each year can be modeled as a normally distributed random variable with a mean of 12 percent and a standard deviation of 2 percent and can vary from year to year. He is 45 years old now.

a) Please start with the base case, assuming the rate of return is deterministic at 12 percent each year. How much money should he expect to have in his retirement fund at the end of age 60 (including the investment made at the end of that year)? Please calculate the cash flow year by year and do not use any existing financial function in Excel, as you will need to incorporate uncertain return rates in each year in the following parts.

b) Considering the uncertainty in the rate of return each year, how much money should he expect to have in his retirement fund at the end of age 60? What is the standard deviation of that amount? What is the probability that he will have more than $ 450,000 in his retirement fund when he reaches age 60? Please show the data or chart to support your argument.

c) How much should he invest each year if he wants the mean value of his portfolio to be at least $ 500,000 at age 60? Please show the data or chart to support your argument.

d) How much should he invest each year if he wants there to be a 90 percent chance of having at least $ 500,000 in his retirement fund at age 60? Please show the data or chart to support your argument.