. Provide solutions to the folly questions
State whether the following statements are true (T) or false (F). If a state- ment is true, brieyr explain why. If a statement is false, either briefly explain why it is false, or give a counter-example. (a) It is possible to have two events A and B for which P (AB) = [1.35 and P (A|B) = 0.5. (b) If X has variance 02, then the standard deviation of 5X + 3 is 50. (c) The function :2 :3 tn M{t):l+2f+3+4+ \":E'l-(-l'l} is a valid moment generating function. (Hint: what are the rst and second moments?) (d) If X is a discrete random variable with pf f); (:r), then it is possible that f); (0) = 2. (e) If the correlation between two random variables X and Y is 0, then X and Y must be independent. (f) The sum of two independent Gamma random variables has a lGamma distribution. (g) The following is a valid pdf fx (I) = Emm- (h) X and Y are independent random variables with nite variances. Their correlation is therefore 0. (i) If A and B are mutually exclusive, then P (AB) = P(A) P (B) . (j) If g (t) is the moment generating function of a random variable X, then e49 (t) is also a moment generating function. (Hint: Use the denition of the moment generating function). 2. Indifference curves. Assume that the consumer purchases only two goods, x and y. Based on the information in each of the following parts, sketch a plausible set of indifference curves (that is, draw at least two curves, and indicate the direction of higher utility). (a) Nomis enjoys muffins (x) and chai latte (y) if they are consumed together. (b) Tra likes chocolate (x), but he hates broccoli (y). (c) Aras likes steaks (x), but she doesn't care one way or the other about apples (y). (d) Fernando always buys five Polo-shirts (x) with every pair of jeans (y).Problem 1 (40 points) (Portfolios) There are five assets in Utopia, with the following mean and standard deviation of return. Since it's Utopia, all five assets are uncorrelated. Asset Risky 40% 30% High End 30% 25% Balanced 20% 15% Low End 15% 10% Safe 12% 8% (a) Minions residing Utopia have different investment preferences. Jerry wants to invest with the least risk possible. In what portfolio should Jerry invest? (b) Stuart, however, is not afraid of risk. He wants a portfolio with an average return of 25%, albeit with the least risk possible. Structure the right portfolio for Stuart. (c) Dave is middle of the road between Jerry and Stuart. He is seeking a 15% return with low risk. Structure the right portfolio for Dave. (d) Show the assets and the portfolios in (a)-(c) in the 7 - O plane. Sketch the efficient frontier on the same plot. (e) What are the correlation coefficients between the returns of (i) Dave and Jerry portfolios and (ii) Dave and Stewart Portfolios? Problem 2 (30 points) (AAPL Fun) Recent data, collected on a quarterly (3-month) basis suggests, that Apple stock price (AAPL) is equal likely to increase or decrease in every quarter. When AAPL increases at the end of a quarter, it does so by a factor of 1.105, with respect to its value at the beginning of the quarter. When AAPL decreases at the end of a quarter, it does so by a factor of 1/1.105 = 0.905, with respect to its value at the beginning of the quarter. AAPL closed at $143.17 today. In this exercise, we will do portfolio optimization over a period of six months. The current six-month U.S. Treasury bill yield closed today at 0.97%. The investor we consider, Aversy, has a logarithmic utility.four players, who gets the highest equilibrium utility? 3. Consider an ultimatum game where two players can share a "cake" of size one. Let (x,y) denote a proposal, where x is the proposer's share and y the responder's share. The cake is perfectly divisible, so a and y can be any non-negative real numbers such that a + y = 1. We will consider the effect of different preferences. In parts a and b, these preferences are commonly known: each player is sure about the other player's preferences. In part c, we consider the effect of uncertainty about the opponent's preferences. We always assume, for simplicity, that if the responder is indifferent between accepting and rejecting, then she accepts. Note that if the responder rejects, both players get utility 0. (a) Suppose they have altruistic preferences. Specifically, the proposer's utility function is up(x,y) = vi + vy and the responders utility function is up(r,y) = = + y. Use backward induction to find the subgame perfect Nash equilibrium. For example, suppose each player i keeps 7 and contributes 2; =3 dollars. The total contribution is c =4x3 = 12, so the public goods level is y =24. This $24 is distributed equally among the four players, i.e., $6 each. Thus, in this case each player i earns r, = # + (10 -3) = 13 dollars. For example, if on = 0.1 then player I's utility function is an + 0.1 (12 + 23 + 24). If each player i keeps 7 and contributes z, =3 dollars, then player I's utility is 13 + 0.1 x (13 + 13 + 13) = 16.9. (b) Again, the proposer's utility function is up(r, y) = vr + : vy. But now the responder is envious: her utility function is up(r, y) = 3/y - VI. Use backward induction to find the subgame perfect Nash equilibrium. (c) Now suppose the proposer is selfish, with the utility function up(x, y) = r. But he is uncertain about the responder's utility function. He thinks there is a probability q that the responder also is selfish, up(r,y) = y. But there is a probability 1 - q that the responder is envious, with utility function up(r, y) = 3 /y - vx (as in part (b)). The proposer is an expected utility maximizer. Explain exactly how his optimal proposal depends on q. Is there q such that the proposer is indifferent between two different proposals? If so, what
