Problem $2$

The Triad family of mutual funds allows investors to split their money between several

portfolios managed by Triad $($none of the portfolios can be shorted$).$

Portfolio A consists entirely of risk$-$free securities$,$ and has a certain return of $4\%.$

Portfolio B has an expected return of $19\%$ and a standard deviation of $25\%.$

Portfolio C has an expected return of $10\%$ and a standard deviation of $15\%.$

Your client is leaning towards investing his money entirely in portfolio $C,$ since he is unwilling

to take the higher risk associated with portfolio B$,$ but wants a higher return than offered

by portfolio $A.$

$($a$)$ In your role as a Triad investment advisor, you suggest to him an alternative portfolio

$($consisting of a combination of A and B$)$ that has the same standard deviation as

portfolio C but a higher expected return. Assume he has $$200,000$ to invest.

How much should he invest in A and how much in B$?$

What is his expected return in this case?

$($Note that since you work for Triad you cannot recommend investing in a market index

fund, since this is not sold by Triad!$)$

$($b$)$ Sketch a risk$/$return diagram to help explain to him why his combination of risk and

return is improved by this option, and why his risk has not increased. Show all four

portfolios on the diagram $($the original three plus your suggestion$).$

$($c$)$ Now assume that the correlation between portfolios B and C is $0.$ Compute the weights

of the portfolio $T$ formed with B and $C$ that has the highest Sharpe ratio. Find also

the portfolio $G$ formed with $B$ and $C$ that has the smallest variance.