Suppose that the risk$-$free rate is $3\%$ and that the corresponding optimal risky portfolio has an

expected return of $15\%$ and a standard deviation of $20\%.$ Consider an investor with preferences

represented by the utility function $U=E(r)-0\mathrm{.}5A{}_2,$ where $A=2.$

$($a$)$ What fraction of her wealth should she invest in the risky portfolio?

$($b$)$ Should she invest more or less fraction of her wealth in the optimal risky portfolio if she is

more risk$-$averse? Why?

$($c$)$ Should she invest more or less fraction of her wealth in the optimal risky portfolio if the

optimal risky portfolio offers a higher expected return $($but the same standard deviation$)?$ Why?

$($d$)$ If the risk$-$free rate is higher, what would you expect the optimal risky portfolio to differ from

the current one $($for $3\%$ risk$-$free rate$)$ in terms of expected return and standard deviation?

$($e$)$ Now suppose that the investor faces a higher risk$-$free rate when borrowing $($because you are

facing more borrowing constraints than the government, or people won't allow you to borrow at

the same rate as the government$).$ Specifically, the investor may lend at a risk$-$free rate of $3\%$ but

has to borrow at a risk$-$free rate of $5\%.$ Assume for simplicity that the optimal risky portfolio is

the same for either $3\%$ risk$-$free rate or $5\%$ risk$-$free rate, i$.$e$.,$ with an expected return of $15\%$ and

a standard deviation of $20\%($this should not be the case in reality, as you will find out in

question $($d$)).$ What fraction of her wealth should the investor now put in the risky portfolio?

Hint: The risk$-$free rate of $3\%$ will imply a fraction $y_1$ of wealth invested in the optimal risky

portfolio. The risk$-$free rate of $5\%$ will imply another fraction $y_2$ of wealth invested in the

optimal risky portfolio. However, $3\%$ is only available for the investor to lend $($so she cannot

borrow to invest more than her wealth to invest in the optimal risky portfolio$).$ If she wants to

borrow to invest more than her wealth to invest in the optimal risky portfolio, she will have to go

with $5\%$ risk$-$free rate $($but she won't be able to lend at $5\%$ rate$).$

The key here is: you would need to understand what weights in the optimal risky portfolio mean

she will lend, and what weights mean she will borrow.

$($f$)$ What is the expected return and standard deviation of the portfolio that you found in $($e$)?$