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$)?$

Below is actual price and dividend data for three companies for each of seven months.

a$.$ Compute the rate of return for each company for each month

b$.$ Compute the average rate of return for each company

c$.$ Compute the standard deviation of the rate of return for each company

d$.$ Compute the correlation coefficient between all possible pairs of securities

e$.$ Compute the average return and standard deviation for the following portfolios

a$.\frac{1}{2}A+\frac{1}{2}B$

b$.\frac{1}{2}A+\frac{1}{2}C$

c$.\frac{1}{2}B+\frac{1}{2}C$

d$.\frac{1}{3}A+\frac{1}{3}B+\frac{1}{3}C$

Assume that the average variance of return for an individual security is $50$ and that the

average covariance is $10.$ What is the expected variance of an equally weighted portfolio

of $5,10,20,50$ and $100$ securities$?($Hint: use the risk reduction formula in the lecture

note.$)$