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 believes that Waterworks is underpriced, with an alpha of $2\mathrm{.}5\%$ over the coming month.

Beta R$-$square Standard Deviation of Residuals

$1\mathrm{.}250\mathrm{.}650\mathrm{.}14($i$.$e$.,14\%$ monthly$)$

Required:

If the fund holds a $$3$ million position in Waterworks stock and wishes to hedge market exposure for the next month using $1-$month maturity S&P $500$ futures contracts, how many contracts should it enter? Should it buy or sell contracts? The S&P $500$ currently is at $1,000$ and the contract multiplier is $$250.$

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

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 $0\mathrm{.}2\%$ per month.

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 on each of these stocks are independent of each other. What is the residual standard deviation of the portfolio?

Calculate the probability of a loss on a market$-$neutral strategy involving equally weighted, market$-$hedged positions in the $100$ stocks over the next month. Assume the risk$-$free rate is $0\mathrm{.}2\%$ per month.