$(45$points$)$ We have two stocks. The random variables of return for those two stocks are $x_1$ and

$x_2,$ respectively. Suppose the random vector $(x_1,x_2)$ follows the $2-$dimensional normal

distribution $N_{{?}_{?}}(,)$ with $=(0\mathrm{.}04,0\mathrm{.}04)$ and $=\left[\begin{array}{cc}0\mathrm{.}2& -0\mathrm{.}05\\ -0\mathrm{.}05& 0\mathrm{.}1\end{array}\right].$

We have two portfolios as follows:

Portfolio$1(P_1)$ : capital allocation weight is $W_{P1}-=(w_1,w_2)=(0\mathrm{.}6,0\mathrm{.}4).$

Portfolio return for $P_1$ is $r_{P1}=0\mathrm{.}6x_1+0\mathrm{.}4x_2$

Portfolio$2(P_2)$ : capital allocation weight is $W_{P2}=(w_1,w_2)=(0\mathrm{.}2,0\mathrm{.}8).$

Portfolio return for $P_2$ is $r_{P2}=0\mathrm{.}2x_1+0\mathrm{.}8x_2$

Suppose that the risk free rate of return is $r_f=0\mathrm{.}01.$

a$.$ Calculate mean for the return of Portfolio$1.$

b$.$ Calculate mean for the return of Portfolio$2.$

c$.$ Calculate standard deviation for the return of Portfoliol

d$.$ Calculate standard deviation for the return of Portfolio$2.$

e$.$ Calculate Sharpe Ratio of Portfolio $1$

f$.$ Calculate Sharpe Ratio of Portfolio$2.$

g$.$ Calculate $99\%-$Value at Risk $(=Va{R}_1\%(r_{P1}))$ of Portfolio $1.$

h$.$ Calculate $99\%-$Value at Risk $(=$VaR $1\%(rP2)$ of Portfolio $2.$

Note that $99\%$ quantile value of the standard normal distribution is $q99\%=2\mathrm{.}326.$

i$.$ Which portfolio is more efficient? Explain the reason of your selection in the sense of Mean$-$

Variance Optimization.