$($POINTS: $30)$

There exists only two risky assets with returns $r_1$ and $r_2$ and a risk free rate $r_f.$ Let $in[-1,1]$

be the correlation between the two returns, ${}_1$ the standard deviation for asset $1$ and ${}_2$ the

standard deviation for asset $2.$ You are splitting your wealth only between the two risky assets.

Let $in[0,1]$ be the share of your wealth assigned to asset $1.$

$2.$I$)$ Write the expected excess return for an arbitrary portfolio as a function of $.($Points:

$2.$II$)$ Write the standard deviation of the return for an arbitrary portfolio as a function of $.$

$($Points: $5)$

$2.$III$)$ Compute the portfolio with the highest Sharpe ratio.

Attention: There are several steps to get to the solution.

Hint #$1$: The algebra is easier if you do this problem in terms of expected excess returns

$($ie$,$ risk$-$premium$)$ instead of in terms of expected returns minus the risk$-$free rate. That

is$,$ use $E[R_p]$ instead of $E[r_p]-r_f$ and break down $R_p$ into $R_1$ and $R_2$ as appropriate.

Hint #$2$: You will need to use the chain rule for derivatives. $($Points: $20)$