Assume the return on a market index represents the common factor and all stocks in the economy have a beta of $1.$ Firm$-$specific returns all have a standard deviation of $35\%.$

Suppose an analyst studies $20$ stocks and finds that one$-$half have an alpha of $4\mathrm{.}1\%,$ and one$-$half have an alpha of $4\mathrm{.}1\%.$ The analyst then buys $$1\mathrm{.}5$ million of an equally weighted portfolio of the positive$-$alpha stocks and sells short $$1\mathrm{.}5$ million of an equally weighted portfolio of the negative$-$alpha stocks.

Required:

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

b$-1.$ How does your answer for standard deviation change if the analyst examines $50$ stocks instead of $20?($Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest dollar amount.$)$

b$-2.$ How does your answer for standard deviation change if the analyst examines $100$ stocks instead of $20?($Enter your answer in dollars not in millions.$)$