Plz answer part a to part c

Section B $(60\%)-$ FOUR Compulsory Short Questions

Short Question $($SQ$1)$

$(15\%)$

Hal and Stu have started a small fund, investing $$10$ thousands of their own capital. Their quantitative strategy has identified $10$ stocks to buy, and they plan to use their $$10$ thousands buy equal dollar amounts of each stock, that is$,$ $$1000$ in each stock. They would like to short sell a hedging portfolio constructed with the S&P $500$ portfolio and the risk$-$free asset to make their overall portfolio market$-$neutral. Assume there is no initial$/$maintenance margin requirement or any kind of restrictions on short$-$selling.

For each stock, the two partners have estimated the single market index model of the form,

$R_{it}^e={}_i+{}_i{R}_{mt}^e+lo{n}_{it}$

where $R_{it}^e$ is a stock's excess return in month $t,$ and $R_{Mt}^e$ is the excess return on the S&P $500$ index in month $t.$ Across the $10$ stocks that Hal and Stu want to buy, the partners estimate that the average ${}_i$ equals $0\mathrm{.}02,$ and the average ${}_i$ equals $0\mathrm{.}75.$

Based on historical estimates, they are also assuming that:

the standard deviation of $lo{n}_{it}$ equals $0\mathrm{.}10$ for the $10$ stocks that Hal and Stu plan to buy.

The correlation between $lo{n}_{it}$ and $lo{n}_{jt}$ equals $0$ for every possible pair for stocks that Hal and Stu want to buy.

the expected excess monthly return on the S&P $500$ equals $0\mathrm{.}005$

the standard deviation of the monthly market return is $0\mathrm{.}06$

Terminology:

A hedging portfolio here describes a position to be shorted to make the overall portfolio market neutral

Overall portfolio describes the overall position combining the buy position and the short$-$sell position

Market$-$neutral in the market index model indicates having zero beta on the market index

$($a$)$ What is the dollar amount of the hedging portfolio should they short sell?

$(5\%)$

$($b$)$ What is the expected dollar profit on the overall position over the next month? $(5\%)$

$($c$)$ What is the variance of the return on the overall position over the next month? $(5\%)$