1. The following is part of a computer output from regression of monthly returns on Waterworks stock against the S&P500 Index. A hedge fund manager believes that Waterworks is underpriced, with an alpha of 2% over the coming month.

Beta= 7.5, R-Square = .65, Standard deviation = .o6 (i.e., 6% monthly)

• If he holds a $3 million portfolio of waterworks stock and wishes to hedge market exposure for the next month using one month maturity S&P 500 futures contract, how many contract should he enter? Should he buy or sell contract? The S&P 500 is currently at 1000, and the contract multiplier is $250.
• What is the standard deviation of the monthly return of the hedge portfolio?
• Assuming that monthly return are approximately normal distributed, what is the probability that this market neutral strategy will lose money over the next month? Assume the risk free rate is .5% per month.

2. Return to previous problem. Suppose you hold a weighted portfolio on 100 stocks with the same alpha alpha, beta, and residual standard deviation as Waterworks. Assume the residual return on each of the stocks are independent of each other. What is the residual standard deviation of the portfolio?

b) Recalculate the probability of a loss on market neutral strategy involving equally weighted, market-hedge position in 100 stocks over the next month.

3. Return to previous problem. Now suppose that the manager misestimates the beta of Waterworks stocks, believing it is .50 instead of .75. The standard deviation of the monthly market rate is 5 %.

(a) What is the Standard deviation of the (now improperly) hedged portfolio.

(b) What is the probability of incurring a loss over the next month if the monthly market return has an expected value of 1% and a standard deviation of 5%? Compare your answer to the probability you found in problem 1.

4. Historical data suggest that the standard deviation of an all equity strategy is about 5.5% per month. Suppose the risk free rate is now 1% per month and market volatility is at its historical level. What would be a fair monthly fee to a perfect market timer, according to the black Scholes formula.