Assume the risk$-$free rate is $4\%(r_f=4\%),$ the expected return on the market portfolio is

$12\%(E[rM]=12\%)$ and the standard deviation of the return on the market portfolio is

$16\%({}_M=16\%).($All numbers are annual.$)$ Assume the CAPM holds.

a$.$ What are the expected returns on securities with the following betas: $($i$)=1\mathrm{.}0,$

$($ii$)=1\mathrm{.}5,($iii$)=0\mathrm{.}5,($iv$)=0\mathrm{.}0,($v$)=-0\mathrm{.}5?$

b$.$ What are the betas of securities with the following expect returns: $($i$)12\%,($ii$)$

$20\%,($iii$)-4\%?$

c$.$ What are the portfolio weights $($in the risk$-$free asset and the market portfolio$)$ for

efficient portfolios $($portfolios on the efficient frontier$)$ with expected returns of $($i$)$

$8\%,($ii$)10\%,($iii$)4\%,($iv$)24\%.$

d$.$ What are the portfolio weights $($in the risk$-$free asset and the market portfolio$)$ for

efficient portfolios $($portfolios on the efficient frontier$)$ with standard deviations of

$($i$)4\%,($ii$)20\%,($iii$)16\%.$

e$.$ What are the correlations between the portfolios in $($i$)$ Q$\mathrm{.}1$c$($i$)$ and Q$\mathrm{.}1$c$($iv$),($ii$)$

Q$.1d(i)$ and Q$.1d(ii)?$

f$.$ Can securities or portfolios with the following characteristics exist in equilibrium,

assuming the CAPM holds $($yes or no$)$: $($i$)$ expected return $0\%,$ standard deviation

$40\%,($ii$)$ expected return $9\%,$ standard deviation $9\%,($iii$)$ expected return $34\%,$

standard deviation $70\%.$

g$.$ A stock with a beta of $1(=1\mathrm{.}0)$ has a current price of $$\frac{40}{?}$ share. Assuming it

pays no dividends, what is the expected price in $1$ year? If it is expected to pay a

dividend of $$\frac{4}{?}$ share at the end of the year, what is the expected price in $1$ year

$($after the payment of the dividend$)?$ If the beta of the stock is $2(=2\mathrm{.}0),$ what are

the expected prices under these $2$ scenarios, i$.$e$.,$ no dividends or a dividend of $$4?$

h$.$ For a moment $($but just a moment$)$ assume that the CAPM may not hold. In other

words, alpha $()$ is non$-$zero. If a non$-$dividend paying stock with a beta of

$1\mathrm{.}0)$ has a current price of $$\frac{50}{?}$ share and an expected price in $1$ year of $$\frac{60}{?}$ share

$($based on your personal analysis of the companies prospects$),$ what is the alpha

$()$ of this stock? What if the beta is $2(=2\mathrm{.}0)?$ What if the beta is $3(=3\mathrm{.}0)?$