Part A

Calculate the optimal risky portfolio for the following cases when short$-$sales are allowed. Compute its expected return and the standard deviation of its returns.

Three risky assets: $R_F=5,R^{\text{'}}=[6,10,15],$ and

$=\left[\begin{array}{ccc}12& 0& 0\\ 0& 21& 0\\ 0& 0& 39\end{array}\right]$

Part B

Mrs Z$\text{'}$s utility function is given by the following equation:

$U_Z((\frac{?}{\text{b}\text{a}\text{r}}(R)),{}^2)\frac{?}{b}=ar(R)-0\mathrm{.}4{}^2$

For each one of the above cases, find the overall optimal portfolio for Mrs Z and compute its expected return and the standard deviation of its returns.

Part A & B$-$ Further Guidance:

There are two parts in the coursework. In part A$,$ you are asked to solve for the optimal risky portfolio for $4$ different combinations of assets. Note that in each case, there is risk$-$free asset available. In part B$,$ you are given the utility function for an individual and required to find the overall optimal portfolio for each of the four cases in part A$.$ You can present your answer in the order of

Case $2$: Optimal Risky Portfolio

Case $2$: Overall Optimal Portfolio

OR in the order of

Case $2$: Optimal Risky Portfolio

Case $2$: Overall Optimal Portfolio

Please show all steps in your solution and explain them.