Problem $20-17($Algo$)$

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 $1\%$ over the coming month.

$\backslash$table$[[$Beta$,$R$-$square,$\backslash$table$[[$Standard Deviation$],[$of Residuals$]]],[0\mathrm{.}60,0\mathrm{.}65,0\mathrm{.}06($i$.$e$.,6\%$ monthly$)]]$

Required:

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 $20\mathrm{.}1$ and $20\mathrm{.}2)$ on each of these stocks are independent of each other. What is the residual standard deviation of the portfolio? $($Round your percentage answer to $2$ decimal places.$)$

Residual standard deviation $\%$

b$.$ Recalculate the probability of a loss on a market$-$neutral stratey involving equally weighted, market$-$hedged positions in the $100$ stocks over the next month. Assume the risk$-$free rate is $1\mathrm{.}3\%$ per month. $($Do not round intermediate calculations. Round your percentage answer to $5$ decimal places.$)$

Probability of a loss $\%$