Problem $1$ You are given the following information about stock X and the market port$-$

folio, M :

E$($r$)\backslash$sigma

Riskless Asset $($f $)0\mathrm{.}03(3\%)0\mathrm{.}00$

Stock X $0\mathrm{.}10\mathrm{.}60$

Market Portfolio $($M $)?0\mathrm{.}30$

The market portfolio includes all stocks in the economy $($including X$).$ You are not given the

expected return of stock X$.$ The correlation between the returns on the stock X and the

market portfolio is equal to $0\mathrm{.}5.$ Assume that CAPM holds.

a$)$ What is the expected return on the market portfolio M $?$

b$)$ Construct a portfolio with $\backslash$beta $=1\mathrm{.}5$ that is efficient$\mathrm{.}1$ What is the standard deviation

and expected return of this portfolio?

c$)$ Consider a portfolio that has correlation of $0\mathrm{.}6$ with the market portfolio. What

is the idiosyncratic risk of this portfolio if its standard deviation is $50\%?$ Use the

standard deviation of the non$-$systematic component of returns as the measure of

idiosyncratic risk.

d$)$ You have $$1000$ to invest in a combination of the risk$-$free asset, stock X$,$ and the

market portfolio. You are thinking about investing $$300$ in the riskless asset, $$400$

in stock X$,$ and $$300$ in the market portfolio. What are the overall expected return,

standard deviation, and beta of this portfolio?

e$)$ You seem to dislike the portfolio obtained in $($d$).$ You understand that you can

tolerate the overall risk of your portfolio up to $\backslash$sigma P $=30\%.$ Thus, you are willing

to invest your $$1000$ in any combination of the risk$-$free asset, the stock X$,$ and

the market portfolio that gives you the highest expected return, given a standard

deviation of $30\%.$ How much money do you invest in each of the three securities$,$

and what expected return and beta does your portfolio have?

$1$To construct $($or to find$)$ a portfolio, it$$s sufficient to describe its weights on all assets that are used to

build this portfolio.

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