Problem $3(40$ points$).$ Suppose there are two uncorrelated risky assets A and $B.$

Asset A has volatility $15\%$ and return $15\%.$ Asset B has volatility $25\%$ and return

$25\%.$ You are a mean$-$variance investor with risk$-$aversion parameter a:

$U(,{}^2)=-{}^2$

a$)$ Suppose you are an investor with parameter of risk$-$aversion $=1.$ What is

the optimal portfolio consisting of only assets A and B$?$b$)$ Suppose you are investor with parameter of risk$-$aversion $=2.$ What is the

optimal portfolio consisting of only assets A and B$?$c$)$ Suppose the risk$-$free rate is $5\%.$ Do portfolios computed in parts a$)$ and b$)$

have different Sharpe Ratios? Provide intuition.d$)$ Compute optimal portfolios for investors a$)$ and b$)$ if they also have access

to the risk free rate. Do the optimal portfolios still have different Sharpe

Ratios? Provide intuition.