Carla is considering investing in a portfolio of two stocks A and B$,$ and wants to minimise her risk. Based on her observations of their prices she finds that A has an expected return of $6\%$ per year with a standard deviation of $4\%,$ while B has an expected return of $7\%$ per year with a standard deviation of $2\%.$ The correlation coefficient of the two stocks is $0\mathrm{.}3.$

$($a$)$ Calculate the covariance matrix for the two stocks.

$($b$)$ Advise Carla on the best portfolio to choose.

$($c$)$ Calculate the expected return and the standard deviation of the portfolio that you recommended in part $($b$).$

$($d$)$ Comment on how the portfolio you recommended in part $($b$)$ compares with investing in the individual stocks, in terms of risk and return.

$($e$)$ Sketch a risk$-$return graph showing how the risks and returns of A$,$ B and the portfolio you recommended in part $($b$)$ are related.

$($f$)$ Is either of the two stocks dominated by the portfolio that you recommended in part $($b$)?$ Explain your answer.