The $($annual$)$ expected return and standard deviation of returns for $2$ assets are as follows:

Asset A Asset $B$

$E[r],10\%,20\%$

$SD[r],30\%,50\%$

The correlation between the returns is $0\mathrm{.}15.$

a$.$ Calculate the expected returns and standard deviations of the following portfolios:

$($i$)80\%$ in $A,20\%$ in $B$

$($ii$)50\%$ in $A,50\%$ in $B$

$($iii$)20\%$ in $A,80\%$ in $B$

b$.$ Find the weights for a portfolio with an expected return of $25\%?$ What is the

standard deviation of this portfolio?

c$.($Difficult$)$ Find the weights for a portfolio with the same standard deviation as

asset A but a higher expected return? $($Trial and error may be a viable strategy;

however, there is an analytical solution.$)$ What is the expected return of this

portfolio?

d$.($Optional$)$ What is the correlation between the returns on the portfolios in parts

a$($i$)$ and a$($iii$)?($Recall that the correlation is the covariance divided by the product

of the standard deviations.$)$

In addition to the information in Q$\mathrm{.}3,$ assume that the $($annual$)$ risk$-$free $($T$-$bill$)$ rate is

$4\%.$

a$.$ Calculate the expected returns and standard deviations of the following portfolios:

$($i$)75\%$ in the risk$-$free asset, $25\%$ in $B$

$($ii$)25\%$ in the risk$-$free asset, $75\%$ in $B$

$($iii$)50\%$ in the risk$-$free asset, $50\%$ in the portfolio in Q$\mathrm{.}3$a$($ii$)$

b$.$ Calculate the Sharpe ratios of

$($i$)$ asset $A$

$($ii$)$ asset $B$

$($iii$)$ the portfolio in Q$\mathrm{.}3$a$($i$)$

$($iv$)$ the portfolio in Q$\mathrm{.}3$a$($ii$)$

$($v$)$ the portfolio in Q$\mathrm{.}3$a$($iii$)$

$($vi$)$ the portfolio in Q$\mathrm{.}4$a$($iii$)$

c$.$ Find the weights $($T$-$bill, asset A$,$ asset B$)$ for a portfolio with the same expected

return as asset $B,$ using only a combination of the risk$-$free rate and the portfolio

in Q$\mathrm{.}3$a$($ii$)?$ What is the standard deviation of this portfolio? What is the

correlation of this portfolio with the portfolio in Q$\mathrm{.}4$a$($iii$)?$