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 $2\mathrm{.}3\%$ over the coming month.

$\backslash$table$[[$Beta$,\backslash$table$[[$R$-],[$square$]],\backslash$table$[[$Standard Deviation$],[$of Residuals$]]],[1\mathrm{.}5,0\mathrm{.}65,0\mathrm{.}12($i$.$e$.,12\%$ monthly$)]]$

Now suppose that the manager misestimates the beta of Waterworks stock, believing it to be $0\mathrm{.}5$ instead of $1\mathrm{.}5.$ The standard deviation of the monthly market rate of return is $11\%.$ If he holds a $$4,000,000$ portfolio of Waterworks stock. The S&P $500$ currently is at $2,000$ and the contract multiplier is $$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 $11\%?$ The manager holds a $$4$ million portfolio of Waterworks stock, and wishes to hedge market exposure for the pext month using $1-$month maturity S&P $500$ futures contracts. The S&P $500$ currently is at $2,000$ and the contract multiplier is $$50.$ Assume the risk$-$free rate is $0\mathrm{.}2\%$ per month. $($Enter your answer as percentages and not as a numbers, e$.$g$.,$ enter $"12\%"$ and not $"0\mathrm{.}12".)$

Probability of a negative return