Problem $20-18($Static$)$

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.

$\backslash$table$[[$Beta$,$R$-$square,$\backslash$table$[[$Standard Deviation$],[$of Residuals$]]],[0\mathrm{.}75,0\mathrm{.}65,0\mathrm{.}06($i$.$e$.,6\%$ monthly$)]]$

Now suppose that the manager misestimates the beta of Waterworks stock, believing it to be $0\mathrm{.}50$ instead of $0\mathrm{.}75.$ The standard deviation of the monthly market rate of return is $5\%.$

Required:

a$.$ If he holds a $$6$ million portfolio of Waterworks stock and wishes to hedge market exposure for the next month using one$-$month maturity S&P $500$ futures contracts, what is the standard deviation of the $($now improperly$)$ hedged portfolio? The S&P $500$ currently is at $3,000$ and the contract multiplier is $$50.($Do not round intermediate calculations. Round your percentage answer to $2$ decimal places.$)$

Standard deviation

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 $5\%?($Do not round intermediate calculations. Round your percentage answer to $2$ decimal places.$)$

$\backslash$table$[[$Probability of a negative retum,$42\mathrm{.}07]]$