Problem set lecture 5 (CAPM)(1)

Problem set for presentation:

Problem 1

Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively. The beta of stock A is .8, while that of stock B is 1.5. The T- bill rate is currently 6%, while the expected rate of return on the S& P 500 index is 12%. The standard deviation of stock A is 10% annually, while that of stock B is 11%. If you currently hold a passive index portfolio, would you choose to add either of these stocks to your holdings?

Problem 2

A portfolio manager summarizes the input from the macro and micro forecasters in the following table:

a.Calculate expected excess returns, alpha values, and residual variances for these stocks.

b.Construct the optimal risky portfolio.

c.What is Sharpe’s measure for the optimal portfolio and how much of it is contributed by the active portfolio?

d.What should be the exact makeup of the complete portfolio for an investor with a coefficient of risk aversion of 3?

e.Recalculate for a portfolio manager who is not allowed to short sell securities. What is the cost of the restriction in terms of Sharpe’s measure? What is the utility loss to the investor ( A= 2.8) given his new complete portfolio?

f.Suppose that on the basis of the analyst’s past record, you estimate that the relationship between forecast and actual alpha is:

Actual abnormal return =.3 X Forecast of alpha

How much is expected performance affected by recognizing the imprecision of alpha forecasts?

Problem 3

a.A mutual fund with beta of .8 has an expected rate of return of 14%. If r f 5 5%, and you expect the rate of return on the market portfolio to be 15%, should you invest in this fund? What is the fund’s alpha?

b.What passive portfolio comprised of a marketindex portfolio and a money market account would have the same beta as the fund? Show that the difference between the expected rate of return on this passive portfolio and that of the fund equals the alpha from part (a).

Problem 4

Consider the following table, which gives a security analyst's expected return on two stocks for two particular market returns:

a.What are the betas of the two stocks?

b.What is the expected rate of return on each stock if the market return is equally likely to be 6% or 21%?

c.If the T- bill rate is 3% and the market return is equally likely to be 6% or 21%, draw the SML for this economy.

d.Plot the two securities on the SML graph. What are the alphas of each?