I have a problem with this question 2d.
Everything else is done.
I really need help asap, as the assignment is for tomorrow.
Please please pelase, help me out
Thank you so much,
1. An investor with mean-variance preferences 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, Ai?
b) If this investor’s optimal portfolio consists of 50% - 50% in risk free T-bills and optimal market portfolio, what is the ratio (E[rM]-rF)/σM2 between the optimal market portfolio excess returns and the market portfolio’s variance equal to? If σM=0.2, what is the Sharpe ratio S? What is the excess return of the optimal portfolio, E[rM]-rF?
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 Sf (where the index f refers to the friend), is twice as large as the one you derived in b), that is, Sf=2*S. If the friend also knows that σM=0.2, then what is the friend’s degree of risk aversion, Af? According to this friend, what is the excess return of the optimal portfolio, (E[rM]-rF)f?
d) In the previous question, we saw that our investor and his friend have different opinions about what the Sharpe ratio is (if you did not compute the Sharpe ratios above, assume that S=0.1 and Sf=0.2). Show that if both the investor and his friend receive the same news that the true Sharpe ratio is St=0.15, then they adjust their portfolio holdings in opposite directions (i.e. show that one may increase its allocation to the risky asset and the other may decrease it). Comment in light of the Efficient Market Hypothesis.
2. Consider two portfolios, A and B, which are assumed to be priced correctly according to CAPM. In this question we will study how much we can find out about the market from the fact that the correlation structure between these assets is -1.
a) What is the risk free return, rF, assuming we know the equilibrium expected returns of the portfolios are E[rA]= 5% and E[rB] = 10%? Assume that σA=1/2*σB.
b) What is the relative sensitivity of these assets to the market excess returns, namely what is βA / βB ?
If βB = 1.5, what is the expected market return E[rM]? What is the market excess return E[rM]-rF?
c) Suppose that the true excess return E[rM]-rF turns out to be one-half the value you found in question b), but all other parameters (E[rA], E[rB], βA , βB ) are as above. Suppose that the values of βA and βB are correct. Then are assets A and B under-priced or over-priced? For which asset is the mispricing largest, that is, which asset has the largest absolute difference between the expected return given in a) and the correct expected return according to CAPM?
d) Suppose asset A is now correctly priced, so that E[rA] satisfies the CAPM equilibrium condition. Is there an arbitrage opportunity? If so, how would you explore it, and is it risk free(recall that if stochastic variable s1 is perfectly anticorrelated with stochastic variable s2 then s1=-a*s2+b).