FINM8006 Advanced Investments Exercise Questions Short Answer Questions Suppose the CAPM holds and that there are no borrowing restrictions, so the mean- variance ecient frontier consists of combinations of Treasury bills and the market portfolio. Nonetheless, some households foolishly hold stock portfolios that are less than perfectly diversified. Show that the Sharpe ratio of an imperfectly diversified portfolio, divided by the Sharpe ratio of the market portfolio, equals the correlation of the imperfectly diversified portfolio with the market portfolio. solution SRi SRM = ERei i M EReM = M i SRi SRM = cov(ERei ,ER e M) 2M M i = iM 2M M i = Show graphically that small risk around reference point in prospect theory value function v(x) indicates risk aversion in the first order. Be specific about what features of v(x) results in this loss avoidance. Is the theory of first order stochastic dominance, i.e., w1 FOSD w2 i F1(w) F2(w), still valid under prospect theory Suppose the factor structure of two diversified portfolios a and b are given by Ra = 0.16 + 1.2f 1 + 0.4f 2 Rb = 0.26 + 0.8f 1 + 1.6f 2 0.16 = 0.04 + 1.21 + 0.42 0.26 = 0.04 + 0.81 + 1.62 We get 1 = 0.065, 2 = 0.105 If you try to run the OLS regression Rit Rf = i ◆+ it (1) where ◆ is a vector of 1 with dimension T , for each of the portfolio, that is, if you run time-series excess return on a single vector of 1s, what is the interpretation of your estimated i Why Recall the general OLS estimate = (X 0X )1X 0Y in exercise 1. Here we have = (◆0◆)1◆0(Ri Rf ) = PT 1 (Rit Rf ) T so the estimate is the mean value of excess return for each portfolio. Now if you plot your estimated i (on Y axis) against your estimated E (Ri Rf ) from CAPM model (on X axis), which portfolios (in terms of size and book-to-market) in the Fama-French portfolios do you expect to lie above the 45 degree line Value portfolios, especially small-value portfolio will lie above the line, that’s the so-called value premium. Consider N risky assets with mean vector Rˉ and variance covariance matrix . An arbitrary portfolio p with portfolio weights wp (N 1 vector ) on the mean-variance frontier has mean μp and variance 2p. find the zero-beta portfolio z for any mean-variance ecient portfolio min 1 2 w 0z wz s.t. w 0 z ◆ = 1, w 0 z wp = 0 where ◆ is a N 1 vector of 1s. For simplicity in expression, let define ◆0 1◆ C and Rˉ 0 1◆ B as in the lecture. Solve for the expected return of the zero-beta portfolio μz . min 1 2 w 0z wz s.t. w 0 z ◆ = 1w 0 z wp = 0 FOC wz ◆ wp = 0 wz = 1◆+ wp plug into the first constraint: ◆0 1◆+ w 0p◆ = 1 Notice w 0p◆ = ◆0wp = 1, so C + = 1 plug into second constraint: ◆0wp + w 0p wp = 0 + 2p = 0 solving the two equations we have = 1 1 C2p , = 2p 1 C2p μz = Rˉ 0wz = μp 2pB 1 C2p There is one risky asset with payo v that is normally distributed with zero mean and variance 2v . An informed investor who has zero endowment of the risky asset, observes v and places a market order of amount x at price p. The informed investor has constant absolute risk aversion a, so she maximizes expected CARA utility E [eaW ] , where W is terminal wealth and a is the absolute risk aversion. Risk neutral market makers observe the total order flow x + u, where u is the demand of noise traders and is normally distributed with mean zero and variance 2u. The informed trader is assumed to behave strategically; that is, in deciding on her optimal trade x , she takes the dependence of the price on the order flow into account. Assume risk free return is zero. The informed investor’s terminal wealth is given by W = (v p)x . Assume that the market makers uses a linear pricing rule p = (x + u). What is the conditional distribution of wealth, W (x , v), conditional on v and x W (x , v) = (v (x + u))x so W (x , v) is normally distributed with conditional expectation and variance: E [W (x , v)|x , v ] = (v x)x VAR[W (x , v)|x , v ] = 2x22u p = E [v |x + u] = E [v |v + u] By the projection theorem for jointly normal variables and the fact that E [v ] = E [u] = 0, we have p = cov(v ,v + u) Var(v + u) (v + u) = 2v 22v + 2 u (x + u) with a = 0 we also have = 12 and x = 1 2v . p = 1 2 2 v 1 42 2 v + 2 u (x + u) compare to pricing rule p = (x + u), we must have 1 2 2 v 1 42 2 v + 2 u = therefore, (0) = v2u and (0) = u v measures the sensitivity of price to order flows. It is increasing in standard deviation of stock value and decreasing in standard deviation of noise trading.
