Question $(1)($Minimum Variance Frontier, Efficient Portfolios and Optimal Portfolio

Selection$)(10$ points$)$

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

managed by Triad. Portfolio $C$ has an expected return of $E[R_c]=10\%$ and a standard deviation

of returns of $(R_c)=15\%.$ Portfolio B has an expected return of $E[R_b]=19\%$ and a standard

deviation of return of $(R_b)=25\%.$ The correlation coefficient between the returns of portfolios B

and $C$ is ${}_{b,c}=0\mathrm{.}2.$ Portfolio A consists entirely of riskfree securities$,$ and has a certain return of $4\%.$

Your client is leaning towards investing her money entirely in portfolio $C,$ since she is unwilling to

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

A$.$

$($a$)$ As a Triad investment advisor, you suggest to her an alternative portfolio $P($consisting of

a combination of only portfolios A and B$)$ that has the same expected return as portfolio C

but a lower standard deviation. If she has $$200,000$ to invest, how much should she invest in

$B$ and how much in A$?$ What is the standard deviation of the return on her investment in

this case? Sketch a mean standard deviation diagram that you would use to explain why the

portfolio you suggest is better.

$($b$)$ However, after your convincing presentation of the alternative portfolio $P,$ your client now

says that she is really comfortable with the level of risk in portfolio C$.$ So$,$ you suggest to her

another portfolio P$1($also consisting of a combination of only portfolios A and B$)$ that has

the same standard deviation as portfolio $C,$ but higher expected return. If she has $$200,000$

to invest, how much should she invest in B and how much in A$?$ What is the expected return

on her investment in this case? Sketch a mean standard deviation diagram that you would

use to explain why the portfolio you suggest is better.

$($c$)$ Your client also recalls reading a while ago in the $1$st edition of "Investments" by Sharpe and

Alexander $($Jeff Bailey at the time she went to school was not a coauthor of this book$),$ that

people should hold efficient portfolios, but she is unsure how to do it$.$ She asks you for advice

on how to optimally combine portfolios $C$ and $B$ in order to take advantage of diversification.

You suggest her to invest in a mix of the portfolio A and a so called mean$-$variance efficient

portfolio $($MVEP$)$ which is made up of portfolios B and C$.$ Find this MVEP portfolio $($i$.$e$.$

find the weights on portfolio B and C that generate the MVEP$).$ Use Excel's solver function to

do this $($feel free to use the Example Optimal CAL.xls that is posted on the course webpage$).$

What is the expected return, standard deviation and Sharpe ratio of the MVEP portfolio

that you found?

$($d$)$ Your client wants to keep the level of risk of her portfolio at the same level of portfolio $C$

$($i$.$e$.$ standard deviation of $15\%)$ but take advantage of diversification. If she invests $$200,000$

in A and MVEP, how much should she invest in portfolio A and how much in the MVEP $?$

How much in each of the portfolios B and C$?$ What is the expected return on her portfolio

in this case? Compare it to the expected return on P$1$ obtained in part $($B$)$ and explain the

difference, if any.

$($e$)$ Suppose that now portfolio $C$ has a standard deviation of $25\%($the same as portfolio B$,$ this

is not a mistake$),$ but the expected return on fund $C$ is the same as before, only $10\%.$ Would

you still advise your client to hold portfolio C$?$ If no$,$ explain why. If yes, also explain why.

Hint: find the new MVEP using Excel's solver and explain the result.