You hold an $ 8 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 percent. The S& P/ TSX 60 index currently is at 800 and the risk- free rate is 1 percent per quarter.
a. What will be the futures price on the three- month maturity S& P/ TSX 60 futures contract?
b. How many S& P/ TSX 60 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/ TSX 60 index on the maturity date?
d. If the alpha of the portfolio is 2 percent, show that the expected rate of return (in decimal form) on the portfolio as a function of the market return is rp = .03 + 1.0 × (rM -.01).
e. Let ST be the value of the index in three months. Then ST/ S0 = ST/ 800 = 1 + rM. (We are ignoring dividends here to keep things simple.) Substitute this expression in the equation for the portfolio return, rp, 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 percent over the next three months.
g. What is the beta of the hedged portfolio? What is the alpha of the hedged portfolio?