Midterm Exam:Answer Sheet


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1 Econ 497 Barry W. Ickes Spring 2007 Midterm Exam:Answer Sheet 1. (25%) Consider a portfolio, c, comprised of a riskfree and risky asset, with returns given by r f and E(r p ), respectively. Let y be the proportion of the portfolio invested in the risky asset. Let the utility function of the agent be given by where A is a constant. U = E(r) 1 2 Aσ2 (1) (a) Why does (1) make sense? What does it mean if A<0? IfA =0?Drawtheindifference curves of an agent who has the preferences described by (1) for each of the three cases: A<0,A=0,A>0. Which case is the most sensible? brief answer Utility depends on positively on expected return and negatively on the risk. This makes sense if people are risk averse or risk loving. Indeed, this equation can even represent preferences for riskneutral agents (A = 0). In the latter case indifference curves will have zero slope: the agent cares only about expected return, and is indifferent to risk. For A>0indifference curves will be positively sloped: the agent is risk averse, implying that higher expected return is required to get her to hold more risk. If A<0 the indifference curves are negatively sloped as the agent is arisklover. A>0 makes more sense given that agents typically purchase insurance. (b) Denote the standard deviation of the risky asset as σ p. What will the standard deviation of the portfolio, σ c, be equal to? What will the variance of the portfolio, σc, 2 be equal to? brief answer The riskfree asset must have zero variance. So σ c = yσ P. Then σc 2 = y 2 σc. 2 (c) What will the expected return on the complete portfolio, E(r c ) be equal to? brief answer E(r c )=ye(r P )+(1 y)r f (d) How will the optimal choice of y vary with (i) the variance of the risky asset; (ii) the excess return on the risky asset; (iii) the value of A? brief answer If σp 2 increases, then y will decrease given A>0; this is the case of the capital allocation line becoming flatter. If E(r P ) r f increases then y will increase; this is equivalent to the capital allocation line becoming steeper. If A rises then the agent is more risk averse, so he will hold less y. Think of it as the indifference curve becoming steeper. 2. (20%) When Kendall first showed that prices in financial markets evolved randomly he took this to be disturbing new for economists. It seemed to imply that stock markets are erratic and irrational. Today this conclusion seems exactly backwards. Why might we expect price changes in a wellfunctioning market to evolve randomly? Explain. 1
2 brief answer If markets function efficiently then any information agents have will be incorporated in the price. If you know the price will rise tomorrow, you bid more today. Hence, arbitrage causes the current price to reveal all available information. If the current price reveals all available information, then prices change only on new information. New information by definition is not yet known. So prices are just as likely to rise or fall. (a) If asset prices are determined by some theory (CAPM, APT, etc.) how can price changes be random? Explain. brief answer Asset pricing theories explain the current price. They are based on available current information. New information causes prices to change. So it makes sense that prices will evolve randomly. Think about p t = E t [m t+1 x t+1 ]. If we obtain new information about future payoffs this will change the price. Since we do not know what the new information will reveal, we cannot know how the price will change. (b) Suppose changes in stock returns were evolving randomly as in an efficient market. At some date, t 1, the risk premium rises (people become more risk averse). What would happen to stock prices at t 1?Aftert 1 what will happen to stock prices? Does this mean that the market is inefficient? Explain. brief answer If the risk premium rises stock prices will fall. This is necessary so that expected returns can rise. (You can think of this as a higher value of A in problem 1. If A rises agents hold less of the risky asset, so its price must fall today given the supply). With a higher risk premium assets require higher expected returns to be held. But if prices fall today to reflect the higher risk premium, prices will be expected to rise in the future returning to where they were. Indeed, if prices were not expected to rise in the future returns would not be higher and people would not hold the assets (this phenomenon is called overshooting). So it seems that prices are predictable. But this is not a violation of efficient markets. Although the next period s price will be a function of the current price, this is because it is the only way for markets to clear. It is fully compatible with fully informed agents and full arbitrage. Indeed, there is no way to make excess returns, since the predictable price is precisely what is required to match the higher risk premium that agents now demand. Returns are only truly random if agents are risk neutral, or if the risk premium is uncorrelated with past prices. In this case they are not. (c) Samuelson convincingly argued that in properly functioning markets it is the change in the log of the price (e.g., the percentage change in prices), rather than the change in the absolute price ( P t ) itself that evolves randomly. Why would a random walk in the level (as opposed to the log) of the price of a security be incompatible with economics? brief answer If P were random then prices could be negative, but asset prices are bounded below by zero. Negative asset prices make no sense for securities. But if the growth rate of prices is random then prices no longer can become negative. 3. (30%) Assume the agent is a price taker in the asset (i.e., she can purchase or sell as much of the payoff x t+1 as she wants at the price p t ).Letx t+1 be the payoff of this asset in period 2
3 t +1. The solution to the consumer s optimal consumption problem yields p t = E t [m t+1 x t+1 ] (2) where m t+1 = β u0 (c t+1 ) u 0 (c t ), β is the discount factor, and u0 (c i ) is the marginal utility of consumption in period i. (a) Explain the logic behind the expression (2). Why does this represent optimal behavior? brief answer h It is useful to substitute the definition of m in expression (2): p t = E t β u 0 (c t+1 ) x i u 0 (c t ) t+1.iknowu 0 (c t ) today, so I can write this as u 0 (c t )p t = E t [βu 0 (c t+1 )x t+1 ]. The lefthand side is the cost of buying a unit of the asset today. It reduces my consumption by p t and the utility cost of that is u 0 (c t )p t. The righthand side is the expected gain from buying more of the asset. I get the payoff x t+1 and the utility of that is u 0 (c t+1 )x t+1, but it is in the future, so I discount it by β. Ifthisequationdid not hold true I could raise my utility by rearranging my consumption. One can see this at point C in figure 1. If we were at point E, condition (2) is not satisfied, and utility can be raised by reducing current consumption and having higher expected future consumption. Figure 1: Optimal Consumption (b) Why is expression (2) useful (indeed cool)? brief answer The stochastic discount factor, m t+1, can be used to price any asset. Indeed, the same discount factor can be used to price any asset. That is cool. (c) Suppose an asset s payoff, x t+1, positively covaries with m t+1. What does (2) imply about the price of this asset (about your desire to hold it in your portfolio)? brief answer If the payoff covaries with m t+1 you would like to hold more of it, so its price will be higher. A higher m means lower future consumption, since marginal utility decreases with consumption. You would like an asset that pays off higher when consumption is lower. That is insurance. This is obvious from expression (2) but it is nice to know why. 3
4 (d) Suppose the asset s payoff has zero correlation with m t+1. What does this imply about the rate of return that this asset will bear? Explain. brief answer It must pay the riskfree rate. Only risk correlated with market risk bears higher return. One could show this formally from the basic price equation again, p = E(mx). Recall the definition of covariance: cov(m, x) = E(mx) E(m)E(x).So p = E(m)E(x)+ cov(m, x) Now we can divide through by p to obtain 1 = E t [m t+1 R t+1 ]or R f = 1 is a riskfree asset. Thus we have: E(m), since it p = E(x) R f + cov(m, x) (3) the first term on RHS is discounted present value, second term is risk adjustment. Now if the covariance is zero, as in this question last term on the RHS drops off. So E(x) = E(R p t+1 )=R f. We could let σ 2 (x) equal a billion, but if cov(m, x) =0,it will still yield only the riskfree rate. Even if people are totally risk averse. (e) What relationship, if any, does (2) have with the CAPM? Explain. brief answer Everything! Suppose that m is inversely related to the return on the market portfolio, R P (This makes sense: when consumption is high, returns are less valuabletoyouthanwhenitislow.). Specifically, let m = a br P.Thenm and R P are perfectly negatively correlated. So an asset s payoff depends on its correlation with the market portfolio. So the CAPM follows from the assumption that investor s marginal utility declines linearly when the market goes up. The CAPM says that the risk premium we require to add asset A to our portfolio is proportional to its β. If an asset s return is uncorrelated with the market return its beta is zero. Such a risky asset is riskless in the market portfolio so its expected return is the riskfree interest rate. 4. (25%) Let X U be the future operating income (payoff) oftheunleveredfirm, and X L be the same for the levered firm. Assume that they are of the same risk class, i.e., X = X U = X L. The value of the unlevered firm is equal to the value of the firms equity: V U S U. For the levered firm the value is equal to debt plus equity, V L S L + D L,whereS is the current value oftheequityandd L is the current value of the debt. Let r be the rate of return on this debt. (a) Consider the portfolio that consists of holding α% of the shares of the unlevered firm. What is the current cost of this portfolio? What is the future payoff equal to? brief answer It costs you αs U. Its future payoff is αx. (b) Consider an alternative portfolio that consists of α% of the bonds of the levered firm, and α% of the equity of the levered firm. What is the current cost of this portfolio? What is the future payoff of this portfolio equal to? brief answer The bonds cost you αd L and the equity costs you αs L,orα(S L +D L ). The payoff from the bonds is the αmin[x, rd L ], since bondholders receive interest unless the payoff is insufficient to cover this. The equity holders receive the rest, so their 4
5 payoff is αmax[x rd L, 0], since there is limited liability. Adding these two terms we have αmin[x, rd L ]+αmax[x rd L, 0] = α [Min[X, rd L ]+Max[X rd L, 0]] = αx. (c) Compare the payoffs from the two portfolios. What can you conclude about the current costs? What does this imply about the relationship between V U and V L? brief answer The payoffs from the two portfolios are the same, so the law of one price says they must cost the same. So α(s L +D L )=αs U.SoS L +D L = S U = V L = V U. In words, the value of the firm is independent of its capital structure. (d) Why is this an important result? Explain. brief answer We have shown that capital structure is irrelevant. This is the MM theorem. This theorem is important because it shows that what matters for firm value are the firm s prospects, X, nothowitisfinanced. If leverage does matter for valuation some assumption of the theorem must be violated. So the theorem makes us think about what must be true if leverage matters. It also shows us the power of the LOP. It tells us that if leverage does matter, then the LOP must not hold, or there must be tax considerations, or costs of bankruptcy. We use 4 assumptions: (i) No arbitrage (equalsized bites of the pie have the same taste ). (ii) Operating income (from assets) is not affected by capital structure. (iii) The proportion of operating income that is jointly allocated to stocks and bonds is not affected by the firm s capital structure ( only stockholders and bondholders eat the pie ). (iv) The present value function (economywide state prices) is not affected by capital structure( tasteperbiteofthepieisfixed ). Notice that assumption (ii) rules out bankruptcy costs, differential transaction costs, peculiar managerial incentive schemes based on capital structure. Assumption (iii) rules out differential taxes for income from stocks and bonds. Assumption (iv) rules out the possibility of creating or destroying desired patterns of returns not otherwise existing in the market by changing capital structure. 5
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