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.

Page $1$ of $2$

Problem Set $2$

Problem $2$ Suppose that you open your trading terminal on February $6,$ and see the

market parameters of four assets: The riskless asset $($f $),$ an individual stock $($stock X$),$ a

mutual fund $($fund Y $),$ and the market portfolio of risky assets $($M $).$ Based on the information

from the terminal, you build the following table:

E$($r$)\backslash$sigma $\backslash$beta

Riskless Asset $($f $)0\mathrm{.}05(5\%)0\mathrm{.}00$

Stock X $0\mathrm{.}120\mathrm{.}401\mathrm{.}6$

Market Portfolio $($M $)0\mathrm{.}100\mathrm{.}20$

Fund Y $0\mathrm{.}100\mathrm{.}2250\mathrm{.}2$

a$)$ What are the alphas $(\backslash$alpha $)$ of the stock X and of the fund Y $?$ According to CAPM,

are they underpriced, overpriced, or priced correctly?

b$)$ What are the Sharpe ratios of the stock X and of the fund Y $?$

c$)$ On February $7$ you learn that what you saw on February $7$ was due to a temporary

mispricing of some assets. Luckily, on February $7$ all assets are priced correctly.

Suppose that standard deviations and betas of the assets haven$$t changed. Also,

suppose that the expected return on the market portfolio and the riskless asset are

the same as on February $6.$

On February $7,$ your client asks you to build a portfolio that contains $40\%$ of fund

Y $,30\%$ of the market portfolio, and $50\%$ of stock X $($the client suggests to finance

the amount in excess of her budget by borrowing at the risk$-$free rate.$)$ What is the

beta $(\backslash$beta $)$ of this portfolio? What is its expected return? Is this portfolio efficient?