A fund manager manages two portfolios, A and $B.$ The individual portfolios have expected returns of $E[r_A]=6.\%,E[r_B]=10\%($not excess return!$)$ and standard deviations of ${}_A=10.\%,{}_B=12.\%.$ T$-$bills earn a constant $4\%.$

a$)$ Find the Sharpe ratios of both portfolios and draw the Capital Allocation Line for each.

b$)$ The manager wishes to form one instead of two portfolios. They are constructing a new optimal risky portfolio from A and $B,$ but wants to know the weights. To achieve this, they ${}_B,$ and their excess returns of A and $B,E[R_A],E[R_B],$ their standard deviations ${}_A,{}_B,$ and their covariance Cov$(R_A,R_B)$ to assign weights according to:

$w_A=\frac{E[R_A]{}_B^2-E[R_B]Cov(R_A,R_B)}{E[R_A]{}_B^2+E[R_B]{}_A^2-(E[R_A]+E[R_B])Cov(R_A,R_B)}$

$w_B=1-w_A.$

Determine the weights $w_A$ and $w_B$ of the merged portfolio. The correlation $$ between portfolios A and $B$ is $0\mathrm{.}3.($Hint: Compute the covariance from the correlation first.$)$

c$)$ A client of the fund manager desires their investment to have a standard deviation of $12.\%.$ Determine their capital allocation into the updated optimal risky portfolio and T$-$bills.