Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of $2\mathrm{.}0$ on the market index. Firm$-$specific returns all have a standard deviation of $32\%.$

Suppose that an analyst studies $20$ stocks, and finds that half have an alpha of $+3\mathrm{.}0\%,$ and the other half an alpha of $3\mathrm{.}0\%.$ Suppose the analyst buys $$1\mathrm{.}0$ million of an equally weighted portfolio of the positive alpha stocks and shorts $$1\mathrm{.}0$ million of an equally weighted portfolio of the negative alpha stocks.

a$.$ What is the expected profit $($in dollars$)$ and standard deviation of the analyst$$s profit? $($Do not round intermediate calculations. Round your answers to the nearest whole dollar amount. Enter your answer in dollars, not millions. Omit the $"$$$"$ sign in your response.$)$

Expected profit $($in dollars$)$ $

Standard deviation $

b$.$ How does standard deviation change if the analyst examines $50$ stocks instead of $20$ stocks? $100$ stocks? $($Do not round intermediate calculations. Round your answers to the nearest whole dollar amount. Enter your answer in dollars, not millions. Omit the $"$$$"$ sign in your response.$)$

$50$ stocks $100$ stocks

Standard deviation $

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