Do Problem$19$ in Chapter $6$ of Bodie, Kane, & Marcus, ISE Investments. Assume that the utility function is given by U $=$ E$($r$)0\mathrm{.}5$A${}^2$

Use these inputs for Problems $13$ through $19$: You manage a risky portfolio with an expected rate of return of $12\%$ and a standard deviation of $28\%.$ The T$-$bill rate is $2\%.$

$13.$ Your client chooses to invest $70\%$ of a portfolio in your fund and $30\%$ in an essentially risk$-$free money market fund. What are the expected value and standard deviation of the rate of return on his portfolio?

$14.$ Suppose that your risky portfolio includes the following investments in the given proportions:

Stock A $25\%$ Stock B $32\%$ Stock C $43\%$

What are the investment proportions of each asset in your client$$s overall portfolio, including the position in T$-$bills?

$15.$ What is the reward$-$to$-$volatility $($Sharpe$)$ ratio $($S$)$ of your risky portfolio? Your client$$s$?$

$16.$ Draw the CAL of your portfolio on an expected return$$standard deviation diagram. What is the

slope of the CAL? Show the position of your client on your fund$$s CAL.

$17.$ Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of $10\%.$

a$.$ What is the proportion y$?$

b$.$ What are your client$$s investment proportions in your three stocks and the T$-$bill fund?

c$.$ What is the standard deviation of the rate of return on your client$$s portfolio?

$18.$ 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 $12\%.$

a$.$ What is the investment proportion, y$?$

b$.$ What is the expected rate of return on the complete portfolio?

$19.$ Your client$$s degree of risk aversion is A $=3\mathrm{.}5.$

a$.$ What proportion, y$,$ of the total investment should be invested in your fund?

b$.$ What are the expected value and standard deviation of the rate of return on your client$$s optimized portfolio?