$($b$)$ Seth is not sure of how much to invest in BP and Shell shares. However, he just want to maximise the return of the portfolio, while keeping the risk of the portfolio very low. The information for the $2$ shares is as tabulated below:

Expected Return E$($r$)$ Standard Deviation $,\backslash$sigma

BP $2\%1\%$

Shell $4\%2\%$

$($i$).$ Compute the covariance between BP and Shell

$($ii$).$ Compute the optimal weights for the two shares

$($iii$).$ What is the expected return of the optimal portfolio

$($iv$).$ What is the standard deviation of the optimal portfolio