You are working as an investment advisor. The risk free rate is $0\mathrm{.}02(r_f=0\mathrm{.}02).$ Consider the following $4$ portfolios with expected returns $$ and standard deviations $$ :

$($a$)$ If you have clients with mean$-$variance preferences and they can only hold a portfolio with $100\%$ portfolio allocation among A$,$ B$,$ C$,$ or D$,$ show using indifference curves that some mean$-$variance investors would choose to hold portfolio $C.$

$($b$)$ If you have clients with mean$-$variance preferences and they can only hold a portfolio with $100\%$ portfolio allocation among $A,B,C,$ or $D,$ is there any portfolio that you should not recommend to anyone. Which one is it $?$ Explain.

$($c$)$ You can now offer your clients more investment options. In particular, they can invest in combinations of the risk$-$free asset and one of the risky portfolios. Which risky portfolio would you recommend?

$($d$)$ If your client has utility function given by $\frac{?}{\text{b}\text{a}\text{r}}(R{)}_c-(\frac{1}{2})RRA{}_c^2,$ with relative risk aversion, RRA$=4,$ how much portfolio weight should they put in the risk$-$free asset, and how much in the optimal risky portfolio.