You hold a $12 million stock portfolio with a beta of 1.

You hold a $12 million stock portfolio with a beta of 1.0. You believe that the risk-adjusted abnormal return on the portfolio (the alpha) over the next three months is 2%. The S&P 500 index currently is at 1,200 and the risk-free rate is 1% per quarter.

a. What will be the futures price on the three-month maturity S&P 500 futures contract?

b. How many S&P 500 futures contracts are needed to hedge the stock portfolio?

c. What will be the profit on that futures position in three months as a function of the value of the S&P 500 index on the maturity date?

d. If the alpha of the portfolio is 2%, show that the expected rate of return (in decimal form) on the portfolio as a function of the market return is r_p=.03-1.0X (r_M-.01).

e. Call ST the value of the index in three months. Then S_T /S₀= ST/1,200 =1+ r_M. (We are ignoring dividends here to keep things simple.)Substitute this expression in the equation for the portfolio return, r_p, and calculate the expected value of the hedged stock-plus-futures portfolio in three months as a function of the value of the index.

f. Show that the hedged portfolio provides an expected rate of return of 3% over the next three months.

g. What is the beta of the hedged portfolio? What is the alpha of the hedged portfolio?