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 3 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 3-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 3 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 rp _ .03 _ 1.0 _ (rM _ .01).
e. Call ST the value of the index in 3 months. Then ST /S0 _ ST /1,200 _ 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 3 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?
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