$(5$ points$)$ You manage a risky portfolio with expected rate of return of $18\%$ and standard deviation

of $24\%.$ The T$-$bill rate is $6\%.$

a$)$ Suppose that your client prefers to invest in your fund a proportion $y$ that maximizes the expected

return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation

will not exceed $19\%.$ What is the investment proportion, $y?$ What is the expected rate of return on the

complete portfolio?

b$)$ Suppose you have two risky asset, Asset A and Asset B$.$ Asset A has an expected return of $12\%$ and

a standard deviation of $20\%.$ Asset B has an expected return of $22\%$ and a standard deviation of $50\%.$

The correlation coefficient between Asset A and Asset $B$ is $-0\mathrm{.}3.$ The T$-$bill rate is $6\%.$ Write the

optimization problem whose solution will give the optimal risky portfolio if short selling is not allowed

in this market? $($Do not solve the optimization problem!$).$

c$)$ Suppose your client's degree of risk aversion is $A=6,$ what proportion, $y,$ of the total investment

should be invested in your fund? What is the expected return and standard deviation of this new

portfolio? The utility function is $U=E[R]-\frac{1}{2}A{}^2.$