$1$ Treynor$-$Black model $[5$ points$]$ Consider the single$-$index model applied to a stock $($or a portfolio of stock$)$ S : R$\_($S$,$t$)=\backslash$alpha $\_($S$)+\backslash$beta $\_($S$)$R$\_($M$,$t$)+$e$\_($S$,$t$)\backslash$beta $\_($S$,$SML$)=0$ and $\backslash$beta $\_($S$,$HML$)=0\backslash$alpha $\_($S$)=2\%($and statistically different from zero$),\backslash$beta $\_($S$)=0\mathrm{.}6,$V$($e$\_($S$,$t$))=0\mathrm{.}09.$ In addition, E$($R$\_($S$,$t$))=6\mathrm{.}8\%,$E$($R$\_($M$,$t$))=8\%,$V$($R$\_($M$,$t$))=0\mathrm{.}0625.$ The risk$-$free rate is r$\_($f$)=1\%$ and let's assume that r$\_($f$)$ is constant over time. Build the portable alpha portfolio Z$.$ That is$,$ compute the weights w$\_($S$)$ and w$\_($M$)$ such that the portfolio Z$0\mathrm{.}5$ pointsP that combines Z and M$.$ When we build the portfolio P$,$ we use the weights w$\_($Z$)$ on Z and $1-$w$\_($Z$)$ on M$.$ If w$\_($Z$)=0\mathrm{.}08,$ what are the expected return and standard deviation of P $?[0\mathrm{.}5$ point$]$ Assume that you have used Excel and you've found out that the best risky portfolio P$^(*)$ has weight wZ$=0\mathrm{.}08.$ Compute the Sharpe ratios of Z$,$M$,$ and P$^(*)$ and check that the Sharpe ratio of P$^(*)$ is the highest among the three. $($When you compute the Sharpe ratio, use four decimal digits.$)[1$ point$]$ Draw a diagram with $\backslash$sigma on the horizontal axis and E$($r$)$ on the vertical axis. Identify: $[1$ point$]$ the market portfolio M the portable alpha portfolio Z the efficient frontier built using M and ZCov$($R$\_($Z$,$t$),$R$\_($M$,$t$))=0$P$^(*)$ and the risk$-$free asset the CAL associated to P$^(*)$ Label the axes and the coordinates of any points.