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

folio, $M$ :

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 $=1\mathrm{.}5$ that is efficient. ${?}^1$ 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 ${}_P=30\%.$ 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. Note: Even though it says we are not given the E$($R$)$ x$,$ we actually are and it is $10\%.$ It is supposed to say we are not given the E$($R$)$ of the market.