Problem $7$ This question deals with the optimal portfolio choice for an investor with mean$-$

variance preferences in a world with two risky securities$,$ A and B $,$ and a risk free asset, F$.$ Security

A offers an expected return of $2\%$ and has a standard deviation of return of $2\%.$ Security B has

expected return and standard deviation of $9\%$ and $7\%,$ respectively. The correlation coefficient

between securities A and B is $-0\mathrm{.}5.$ The investor can also borrow and lend at a risk free rate of $1\%$

per month.

a$.$ Compute and plot the expected return and standard deviation combinations that the investor

can achieve by combining securities A and B into a portfolio.

b$.$ What is the optimal portfolio of risky assets for the investor in such a world?

c$.$ Does the optimal portfolio of risky assets in $($b$)$ depend on the risk preferences of the investor?

Why or why not?

d$.$ Suppose that the investor's preferences are represented by the utility function $U=E(r)-$

$0\mathrm{.}5A{}^2,$ where $A$ is an index of the investor's risk aversion. Furthermore, suppose that $A=5.$

How should the investor allocate his wealth between the optimal portfolio of risky assets

computed in $($b$)$ and the risk free asset?

e$.$ What is the Sharpe ratio $($reward$-$to$-$volatility$)$ of the investor's complete portfolio obtained

in $(d)?$

f$.$ How does the answer to $($d$)$ change if $A=1?$

g$.$ Interpret the difference in results between $($d$)$ and $($f$).$