An investor with the well know mean$-$variance preference function:

$U(P)=E[r_P]-\frac{A{}_P^2}{2}$

gets equal utility if he adds $5\%$ to his expected returns or if he reduces his

standard deviation by $10\%.$

a$)$ What is the investor's degree of risk aversion, $A?$

$(2$ marks$)$

b$)$ If this investor's optimal portfolio consists of $50\%-50\%$ in risk free T$-$bills and

optimal market portfolio, what is the ratio

$\frac{E(r_M)-r_F}{{}_M^2}$

between the optimal market portfolio excess returns and the market

portfolio's variance equal to$?$ Interpret. If ${}_M=0\mathrm{.}2,$ what is the Sharpe ratio S$?$

What is the excess return of the optimal portfolio, $E(r_M)-r_F?$

$(5$ marks$)$

c$)$ Suppose our investor meets a friend, who says he has the same optimal

portfolio $(50\%-50\%$ in risk free T$-$bills and optimal market portfolio$).$ However,

the friend is more optimistic about the market and thinks the Sharpe ratio,

denoted by $S_f($where the index $f$ refers to the friend$),$ is twice as large as the

one you derived in part b$)$ above, that is$,S_f=2**S.$

If the friend also knows that ${}_M=0\mathrm{.}2,$ then what is the friend's degree of risk

aversion, A$?$ According to this friend, what is the excess return of the optimal

portfolio, $(E[r_M]-r_F)?$

$(5$ marks$)$

d$)$ In the previous question, we saw that our investor and his friend have

different opinions about what the Sharpe ratio is$.$ Show that if both the

investor and his friend receive the same news that the true Sharpe ratio is

$S_t=0\mathrm{.}15,$ then they adjust their portfolio holdings in opposite directions $($i$.$e$.$

show that one may increase their allocation to the risky asset and the other

may decrease it$).$

$(3$ marks$)$