$5.$ Suppose the following table represents the expected monthly return and the standard deviation of portfolio you currently hold, and a mutual fund you are thinking about investing in:

Expected returns $($in $\%)$ Standard deviation $($in $\%)$

Original portfolio $12\mathrm{.}5$

Mutual fund A $0\mathrm{.}92$

a$.$ How small does the correlation between both returns have to be in order to justify investing in mutual fund A$?$

b$.$ Suppose you decide to add the mutual fund to your portfolio. In your new portfolio, $90\%$ of your savings are invested in the original portfolio, and $10\%$ are invested in the mutual fund. Calculate the expected returns to your new portfolio.

c$.$ If the correlation of both returns is $0\mathrm{.}5,$ what is the covariance of the returns?

d$.$ Use your calculations from part $($c$)$ to calculate the standard deviation of the new portfolio.

e$.$ Suppose you want to substitute mutual fund A and replace it with mutual fund B with the same expected returns and standard deviation. You will also invest $10\%$ of your savings in mutual fund B $($and $90\%$ in the original portfolio$).$ Can you expect the returns and the standard deviation of this new portfolio to be the same as your response in parts $($b$)$ and $($d$)?$ Why or why not?