Question

Return to the previous problem, as below.

a. Suppose you hold an equally weighted portfolio of 100 stocks with the same alpha, beta, and residual standard deviation as Waterworks. Assume the residual returns (the e terms in Equations 20.1 and 20.2) on each of these stocks are independent of each other. What is the residual standard deviation of the portfolio?

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

Question: The following is part of the computer output from a regression of monthly returns on Waterworks stock against the S&P 500 Index. A hedge fund manager believes that Waterworks is underpriced, with an alpha of 2% over the coming month.

a. If he holds a \(3 million portfolio of Waterworks stock and wishes to hedge market exposure for the next month using one-month maturity S&P 500 futures contracts, how many contracts should he enter? Should he buy or sell contracts? The S&P 500 currently is at 1,000 and the contract multiplier is \)250.

b. What is the standard deviation of the monthly return of the hedged portfolio?

c. Assuming that monthly returns are approximately normally 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.

Short answer

a. 6%

b. Unlikely

Step-by-step solution

Calculation of residual standard deviation of the portfolio ‘a’

Since the residual standard deviation is smaller than each stock’s SD by a factor =10. Hence instead of a standard residual SD of 6%, residual standard deviation is now .6%

Calculation of the probability of loss ‘c’

Since ERR of market neutral position = the risk free rate + Alpha

= .005 + .02 =.025

= 2.5%

The z value for a rate of return of zero = X －$\mu /\sigma$

= 0 – 0.25 / .06

= -.4167

Therefore the probability of a negative return = N(-.4167) = 1.55 x 10^-5

This implies that negative return is unlikely.