Suppose we are constructing a complete portfolio from a combination of a

risky portfolio, P$,$ and a risk free asset. A long$-$time acquaintance calls you to

let you in on a new investment opportunity from the firm Rockquity with a

very low risk level, given by $\backslash$sigma R$.$ The distributions of these returns are such

that

E$($rP $)=0\mathrm{.}12$

E$($rR$)=0\mathrm{.}18$

$\backslash$sigma P $=0\mathrm{.}2$

$\backslash$sigma R $=0\mathrm{.}25$

Cov$($rP $,$ rR$)=0\mathrm{.}30$

rf $=0\mathrm{.}03$

$($a$)$ If you do not use the new investment from Rockquity, and your risk

aversion parameter is A $=3,$ what fraction will you hold in the risky

portfolio?

$($b$)$ Your friend demands that you switch to Rockquity $($and abandon your

other risky asset$)$ as part of your complete portfolio. What fraction of

your complete portfolio will be held in Rockquity?

$($c$)$ Which of the above situations results in higher utility for you, using just

P or Roquity as part of a complete portfolio.

$1$

$2$ Investment Theory and Applications

$($d$)$ You suggest to your friend that you want to use a bit of Rockquity in

your portfolio and the original portfolio P$.$ Find the optimal fractions in

P and R to maximize Sharpe ratio, and find the expected return and

standard deviation of this new portfolio.

$($e$)$ Which risky portfolio should you use as part of the complete portfolio

$($which would include some part risk free$),$ P$,$ Rockquity, or the combina$-$

tion of the two as computed in d$)?$ Be sure to justify your answer with

numbers