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 $2\mathrm{.}5\%$ over the coming month.

Beta R$-$Square Standard deviation of residuals

$10\mathrm{.}6511\%$ monthly

Now suppose that the manager misestimates the beta of Waterworks stock, believing it to be $0\mathrm{.}5$ instead of $1.$ The standard deviation of the monthly market rate of return is $10\%.$ If he holds a $$2,000,000$ portfolio of Waterworks stock. The S&P $500$ currently is at $2,000$ and the contract multiplier is $$50.$

a$.$ What is the standard deviation of the $($now improperly$)$ hedged portfolio? $($Round your answer to $3$ decimal places.$)$

b$.$ What is the probability of incurring a loss on improperly hedged portfolio over the next month if the monthly market return has an expected value of $1\%$ and a standard deviation of $10\%?$ The manager holds a $$2$ million portfolio of Waterworks stock, and wishes to hedge market exposure for the next month using $1-$month maturity S&P $500$ futures contracts. The S&P $500$ currently is at $2,000$ and the contract multiplier is $$50.$ Assume the risk$-$free rate is $0\mathrm{.}5\%$ per month. $($Enter your answer as percentages and not as a numbers, e$.$g$.,$ enter $"12\%"$ and not $"0\mathrm{.}12".)$

c$.$ What would be the probability of a loss if you hold an equally weighted portfolio of $100$ stocks with the same alpha, beta, and residual standard deviation as Waterworks and the manager similarly misestimated beta as $0\mathrm{.}5$ instead of $1?$ The manager holds a $$2$ million portfolio of Waterworks stock, and wishes to hedge market exposure for the next month using $1-$month maturity S&P $500$ futures contracts. The S&P $500$ currently is at $2,000$ and the contract multiplier is $$50.$ Assume the risk$-$free rate is $0\mathrm{.}5\%$ per month. $($Round your answer to $2$ decimal places. Enter your answer as percentages and not as a numbers. $($Eg: Enter $"12\%"$ and not $"0\mathrm{.}12".))$