Problem set lecture 3 (contd)

Problem set for presentation:

Problem 1

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:

The correlation between the fund returns is 0.10.

a-1.What are the investment proportions in the minimum-variance portfolio of the two risky funds.

a-2.What is the expected value and standard deviation of its rate of return?

b.Draw the investment opportunity set of the two risky funds

c.Draw a tangent from the riskfree rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio?

d.Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.

e.What is the Sharpe ratio of the best feasible CAL?

f.You require that your portfolio yield an expected return of 14%, and that it be efficient, on the best feasible CAL. What is the standard deviation of your portfolio? What is the proportion invested in the T-bill fund and each of the two risky funds?

g.If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem f. What do you conclude?

Problem 2

Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a 3-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of 3-year strategies. (All rates are annual, continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a SD of 20%. The hedge fund risk premium is estimated at 10% with a SD of 35%. The return on each of these portfolios in any year is uncorrelated with its return or the return of any other portfolio in any other year. The hedge fund management claims the correlation coefficient between the annual returns on the S&P 500 and the hedge fund in the same year is zero, but Greta believes this is far from certain.

a.Compute the estimated 3-year risk premiums, SDs, and Sharpe ratios for the two portfolios.

b.Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation?

c.What is the expected return on the portfolio?

d.What should be Greta’s capital allocation?

e.If the correlation coefficient between annual portfolio returns is 0.3, what is the annual covariance?

f.With correlation of 0.3, what is the covariance between the 3-year returns?

g.Using an annual correlation of 0.3, calculated the standard deviation of the rate of return on this investment if you have a project that has a 0.7 chance of doubling your investment in a year and a 0.3 chance of halving your investment in a year.