Asset Pricing. Teaching Notes. João Pedro Pereira

Asset Pricing Teaching Notes João Pedro Pereira Nova School of Business and Economics Universidade Nova de Lisboa joao.pereira@novasbe.pt http://docentes.fe.unl.pt/ jpereira/ June 18, 2015

Contents 1 Introduction 5 2 Choice theory 7 2.1 Motivation................................... 7 2.2 The utility function............................... 8 2.2.1 Choice under certainty......................... 8 2.2.2 Choice under uncertainty....................... 9 2.2.3 Interpretation of utility numbers................... 11 2.3 Risk aversion.................................. 12 2.3.1 Concepts................................ 12 2.3.2 Measures of risk aversion....................... 13 2.3.3 Risk neutrality............................. 15 2.4 Important utility functions.......................... 15 2.5 Certainty Equivalent.............................. 16 2.6 Stochastic dominance............................. 18 2.6.1 First Order Stochastic Dominance.................. 18 2.6.2 Second Order Stochastic Dominance................. 19 2.7 Exercises.................................... 20 3 Portfolio choice 24 3.1 Canonical portfolio problem.......................... 24 3.2 Analysis of the optimal portfolio choice................... 26 3.2.1 Risk aversion.............................. 26 3.2.2 Wealth................................. 27 3.3 Canonical portfolio problem for N > 1.................... 31 3.4 Exercises.................................... 31 4 Portfolio choice for Mean-Variance investors 35 4.1 Mean-Variance preferences........................... 35 4.1.1 Quadratic utility............................ 36 4.1.2 Normal returns............................. 36 4.1.3 Conclusion............................... 38 4.2 Review: Mean-Variance frontier with 2 stocks................ 39 2

Contents 3 4.3 Setup for general case............................. 41 4.3.1 Notation................................. 41 4.3.2 Brief notions of matrix calculus.................... 41 4.4 Frontier with N risky assets.......................... 42 4.4.1 Efficient portfolio............................ 42 4.4.2 Frontier equation............................ 44 4.4.3 Global minimum variance portfolio.................. 45 4.5 Frontier with N risky assets and 1 risk-free asset.............. 45 4.5.1 Efficient portfolio............................ 45 4.5.2 Frontier equation............................ 46 4.5.3 Tangency portfolio........................... 47 4.6 Optimal portfolio................................ 48 4.7 Additional properties of frontier portfolios.................. 50 4.8 Exercises.................................... 50 5 Capital Asset Pricing Model 54 5.1 Introduction................................... 54 5.2 Derivation.................................... 54 5.3 Important results................................ 55 5.3.1 Capital Market Line.......................... 56 5.3.2 Security Market Line.......................... 56 5.4 Other remarks................................. 59 5.5 Exercises.................................... 60 6 Arbitrage Pricing Theory and Factor Models 61 6.1 Factor Structure................................ 61 6.2 Example of simple factor structure: Market Model............. 63 6.2.1 Return generating process....................... 63 6.2.2 Application: the Covariance matrix is simplified.......... 63 6.2.3 Implication: Diversification eliminates Specific risk......... 64 6.2.4 Another interpretation of the CAPM β............... 65 6.3 Pricing equation................................ 67 6.3.1 Exact factor pricing with one factor................. 67 6.3.2 Exact factor pricing with more than one factor........... 68 6.3.3 Approximate factor pricing...................... 70 6.4 How to identify the factors.......................... 70 6.4.1 Overview................................ 70 6.4.2 Fama and French model........................ 71 6.5 Applications................................... 72 6.5.1 Fund performance........................... 72 6.5.2 Market neutral strategy........................ 74 6.6 Exercises.................................... 75

Contents 4 7 Pricing in Complete Markets 77 7.1 Basic and Complex securities......................... 77 7.2 Computing AD prices............................. 78 7.3 Complete Markets............................... 79 7.3.1 Price of complex securities....................... 79 7.3.2 Quick test for market completeness.................. 80 7.4 Risk-Neutral Pricing.............................. 81 7.4.1 Price of complex securities....................... 81 7.4.2 Fundamental theorems......................... 83 7.5 Conclusion................................... 84 7.6 Exercises.................................... 84 8 Consumption-Based Asset Pricing 86 8.1 The investor s problem............................. 86 8.2 Fundamental Asset Pricing Equation..................... 88 8.3 Relation to Arrow-Debreu Securities..................... 89 8.4 Relation to the Risk-Neutral measure.................... 90 8.5 Risk Premiums................................. 91 8.6 Consumption CAPM (CCAPM)....................... 92 8.7 The CAPM reloaded.............................. 94 8.8 Conclusion................................... 96 8.9 Exercises.................................... 96 9 Conclusion 99 Bibliography 100 A Background Review 102 A.1 Math Review.................................. 102 A.1.1 Logarithm and Exponential...................... 102 A.1.2 Derivatives............................... 103 A.1.3 Optimization.............................. 105 A.1.4 Means and Variances.......................... 106 A.2 Undergraduate Finance Review........................ 107 A.2.1 Financial Markets and Instruments.................. 107 A.2.2 Time value of money.......................... 109 A.2.3 Risk and Return............................ 110 A.2.4 Equilibrium and No Arbitrage.................... 111 B Solutions to Problems 112

Chapter 1 Introduction These notes follow Danthine and Donaldson (2005) closely, though we will use other sources as needed. We will start by analyzing individual choices and portfolio decisions. Then, we will study the prices that result from the interaction of many individuals in the market. To motivate the work to come, consider the following question: What is the role of financial markets? Answer: allowing the desynchronization of agents income and consumption. Example: buy a house now and pay for it during the next 20 years. This is achieved by trading financial securities with financial institutions. Preference for smooth consumption Financial economists see the world in two dimensions. It is useful to understand why agents want to dissociate consumption and income across these two dimensions. 1. Time Dimension. Most people prefer to smooth their consumption through their life cycle. Usually, consumption is higher than income during early years of life (buy the house), then people save during active life (y > c), finally people consume their savings after retirement (y = 0, c > 0). 2. Risk Dimension. The future is uncertain. At any point in the future, one of many states of nature will be realized. 1 Most people want to smooth consumption 1 A state of nature is a complete description of a possible scenario for the future across all the dimensions relevant for the problem at hand. 5

6 across the different possibilities that may arise. That s why people buy health insurance (to be able to consume even if they stop working) or fire insurance for the new house (avoid low consumption in the burned to the ground state of nature). Financial assets serve precisely to move consumption through time and across states of nature. Modelling the preference for smoothness Financial economics builds on the fact that people have a preference for smoothness, as just mentioned. How to model this preference for smoothness, also called risk aversion? Consider two assets that offer two different consumption plans: asset 1 asset 2 time/state 1 4 3 time/state 2 4 5 Since investors like smoothness, they must prefer asset 1. 2 Let U(c) be the utility function, i.e., it tells us how much the investor likes consumption c. The utility function must thus satisfy U(4) + U(4) > U(3) + U(5) U(4) > 1 2 U(3) + 1 2 U(5) What shape must U(.) have to satisfy this condition? 3 Plot it: U(c) 2 Suppose your employer offers you the following salary scheme: under scheme 1, you get $4,000 per month; under scheme 2, you get $3,000 if it rains or $5,000 if it is sunny. Which scheme would you take? 3 Answer: It must be strictly concave c

Chapter 2 Choice theory 1. Under certain conditions, investors preferences can be represented by a utility function, 2. Typical utility functions: x y E[U(x)] E[U(y)] U(w) = ln(w) U(w) = w 1 γ /(1 γ) U(w) = exp( αw) U(w) = aw bw 2 [CRRA] [CRRA] [CARA] 2.1 Motivation We want to find a method to choose between risky assets. Consider the following simple example: Example 2.1.1. There are 3 assets and 2 equally likely possible states of nature in the future: t = 0 t = 1 state θ = 1 state θ = 2 asset 1-100 100 120 asset 2-100 91 131 asset 3-100 100 140 7

2.2. The utility function 8 Which asset would you rather have? In this case, the choice is easy. Asset 3 clearly dominates the other assets, since it pays at least as much in all states of nature, and strictly more in some states. This is an example of state-by-state dominance. State-by-state dominance is the strongest possible form of dominance. We can safely assume that all rational agents will always prefer asset 3. 1 However, the world is not that simple and we will not usually be able to use this concept to make choices. (Is it likely we will observe a market like in this example? Why not?) Suppose now that asset 3 does not exist. Do you prefer asset 1 or asset 2? The choice is not obvious... To understand the choices people make in the real world we need a better machinery utility theory. 2.2 The utility function To be able to represent agents preferences by a formal mathematical object like a function, we need to make precise assumptions about how people make choices. 2 2.2.1 Choice under certainty We start by postulating the existence of a preference relation. For two consumption bundles a and b (two vectors with the amount of consumption of each good), we either say that a b a is strictly preferred to b a b a is indifferent to b a b a is strictly preferred or indifferent to b (a not worse than b) We make the following economic rationality assumptions: A1: Every investor possesses a complete preference relation. I.e., he must be able to state a preference for all a and b. 1 More precisely, we are assuming agents to be nonsatiated in consumption (always like more consumption) 2 People have wasted time thinking about reformulating the canonical portfolio problem just because they were not aware of the axioms that lead to an expected utility representation.

2.2. The utility function 9 A2: The preference relation satisfies the property of transitivity: a, b, c, a b and b c a c A3: The preference relation is continuous. 3 Under these circumstances, we can now state the following useful theorem: Theorem 2.2.1. Assumptions A1 3 are sufficient to guarantee the existence of a continuous function u : R N R such that, for any consumption bundles a and b, a b u(a) u(b) This real-valued function u is called a utility function. Note that the notion of consumption bundle used in the theorem is quite general. Different elements of the bundle may represent the consumption of the same good in different time periods or in different states of nature. 2.2.2 Choice under uncertainty Even thought the previous thm is quite general, we want to extend it in a way that captures uncertainty explicitly and separates utility from probabilities. Definition (Lottery). The simple lottery (x, y, π) is a gamble that offers payoff x with probability π and payoff y with probability 1 π. This notion of lottery is quite general. The payoffs x and y can represent monetary or consumption amounts. If there is no uncertainty, we can write (x, y, 1) = x The payoffs can themselves be other lotteries, leading to compound lotteries. For example, if y = (y 1, y 2, τ), we will have (x, y, π) = (x, (y 1, y 2, τ), π) We assume that the agent is able to work out the probability tree and only cares about the final outcomes. 4 Assume the following axioms: 3 Technical assumption. See Danthine and Donaldson (2005) for details on this and Huang and Litzenberger (1988) for further technical details. 4 A lottery is the simplest example of a random variable. Stock prices are random variables, so you can see where we are going.

2.2. The utility function 10 B1: There exists a preference relation, defined on lotteries, which is complete, transitive, and continuous. Since the consumption bundles in theorem 2.2.1 where general enough to include consumption in different states of nature, it can be applied here to ensure that there exists a utility function U() defined on lotteries. To get an expected utility representation of preferences, we need the following crucial axiom: B2: Independence of irrelevant alternatives. Let (x, y, π) and (x, z, π) be any two lotteries. Then, y z (x, y, π) (x, z, π) In other words, x is irrelevant; including it does not change the investor s preferences about y and z. This axiom is not trivial and has been strongly contested. One well know violation is the Allais Paradox. 5 This and other violations have lead to the exploration of alternatives to the expected utility framework, namely to the growing field of Behavioral Finance. Despite this, recall that the goal of financial economics is to understand the aggregate market behavior and not individual behavior. At this point, expected utility is the most useful framework. We now get to the punchline: Theorem 2.2.2 (Expected Utility Theorem). If axioms B1 2 hold, then there exists a real-valued function U, defined on the space of lotteries, such that the preference relation can be represented as an expected utility, that is, for any lotteries x and y, x y E[U(x)] E[U(y)] The function U(), defined over lotteries, is called a von Neumann-Morgenstern (vnm) utility function. 6 5 Allais Paradox. Given the four lotteries defined below, most people show the following preferences: and L1 = ($10000, $0, 0.10) L2 = ($15000, $0, 0.09) L3 = ($10000, $0, 1.00) L4 = ($15000, $0, 0.90) However, given that L1 = (L3, $0, 0.1) and L2 = (L4, $0, 0.1), with $0 the irrelevant alternative, the independence axiom would imply L3 L4 L1 L2! 6 This designation is sometimes confusing. Some people define U := E[U()] and call this U the vnm utility function, while others call vnm to the u() defined on sure things. Nonetheless, it is always used in the context of preferences that have an expected utility representation theorem 2.2.2

2.2. The utility function 11 Note that x and y can be lotteries with multiple outcomes. outcome in state s that occurs with probability π s, 7 we have Denoting by x s the { s E[U(x)] = U(x s)π s x is a discrete r.v. s U(x s)π s ds x is a continuous r.v. Example 2.2.1. 2.1.1. Let U(x) = x. Choose between assets 1 and 2 in example Example 2.2.2. Now consider another investor with U(x) = x 1 2 /(1 2) = 1/x. (It will soon become clear that this investor is very similar to the previous one, though a little bit more risk averse). Check that this investor prefers the other asset. 2.2.3 Interpretation of utility numbers The numbers returned by the utility function do not have any meaning per se, as the following proposition makes clear. Proposition 2.2.1. If U(x) is a vnm utility function for a given preference relation, then V (x) = au(x) + b, a > 0, is also a vnm utility function for the same preference relation, that is, E[U(x)] E[U(y)] E[V (x)] E[V (y)] Proof. E[U(x)] E[U(y)] ae[u(x)] + b ae[u(y)] + b, since a > 0 E[aU(x) + b] E[aU(y) + b] E[V (x)] E[V (y)] Example 2.2.3. Suppose a different investor has utility V (x) = 1+2 x. His choice between assets 1 and 2 (from example 2.1.1) will be the same as the choice of the investor with U(x) = x. (Check it!) Hence, the utility function serves only to rank the choices under consideration. The precise magnitude of the number does not have any meaning. 7 More often, especially in probability classes, the state of nature is denoted by ω Ω, and the probability measure by P (ω).

2.3. Risk aversion 12 2.3 Risk aversion 2.3.1 Concepts Consider an investor with wealth Y. Consider also the fair gamble, or lottery, L = (+h, h, 1/2). Definition (Risk aversion). An investor displays risk aversion if he wishes to avoid a fair gamble, i.e., Y Y + L. This implies that the utility function of a risk-averse agent must satisfy E[U(Y )] > E[U(Y + L)] U(Y ) > 1 2 U(Y + h) + 1 U(Y h) 2 This inequality is satisfied for all wealth levels if the utility function is strictly concave. 8 Plot it: U(Y ) Y For twice differentiable utility functions, the sufficient condition for concavity is that U (Y ) < 0. This means that U (Y ) is decreasing in wealth. This important economic concept is called decreasing marginal utility. As wealth increases, the utility from additional consumption decreases. When I am starving, a sandwich tastes great, while when I am almost satiated I don t care about another sandwich. 8 This is formally justified by Jensen s inequality: E[g(X)] g(e[x]), for concave g. If g is strictly concave, the inequality is strict. For the utility function in particular, E[U(Y + L)] < U(E[Y + L]) = U(E[Y ] + E[L]) = U(Y + 0) = U(Y )

2.3. Risk aversion 13 2.3.2 Measures of risk aversion We would like to compare utility functions and say which one is more risk averse. Toward this end, we define the following measures of risk aversion: Absolute Risk Aversion: ARA(Y ) U (Y ) U (Y ) Relative Risk Aversion: RRA(Y ) Y U (Y ) U (Y ) Interpretation of ARA. Let π(y, h) be the probability of the favorable outcome at which the investor with wealth Y is indifferent between accepting or rejecting the lottery L = (+h, h, π()). Note that h is an amount of money. It can be shown that π(y, h) = 1 2 + 1 h ARA(Y ) (2.1) 4 The favorable odds requested increase with the amount at stake h. More importantly, the higher the ARA, the more favorable odds the investor demands to accept the lottery. Example 2.3.1. A commonly used utility function is U(Y ) = exp( γy ), which is known for having constant ARA, ie, ARA = γ. 9 For this investor, π(y, h) = 1 2 + 1 4 hγ The higher the degree of ARA (parameter γ), the higher the favorable odds requested (π). However, π does not depend on the level of wealth Y. Is this particular utility function U(Y ) = exp( γy ) a good description of human behavior? We now derive equation (2.1). Proof. π(y, h) must be such that π : Y Y + L E[U(Y )] = E[U(Y + L)] U(Y ) = πu(y + h) + (1 π)u(y h) 9 This is the only utility function with constant ARA. To see this, write U (Y ) U (Y ) = γ U (Y )+γu (Y ) = 0, which is a homogeneous linear differential equation of the second order with constant coefficients. The two special solutions are U 1 = 1 and U 2 = exp( γy ) and the general solution is thus U(Y ) = c 1 + c 2 exp( γy ). This is a linear transformation of U(Y ) = exp( γy ), therefore representing the same preferences. Thanks to Diogo Bessam for pointing this out.

2.3. Risk aversion 14 Expanding U(Y + h) and U(Y h) in Taylor series around Y, we get 10 U(Y + h) = U(Y ) + hu (Y ) + 1 2 h2 U (Y ) + O(h 2 ) U(Y h) =... Ignoring terms of higher order, replacing both these approximations in the previous equation, and canceling terms, we get equation (2.1). Interpretation of RRA. Now we define a gamble in terms of a proportion of the investor s initial wealth. Specifically, we set h = θy, and the lottery becomes L = (θy, θy, π()). π(y, θ) is the probability of the favorable outcome at which the investor is indifferent between accepting or rejecting the lottery. It can be shown that π(y, θ) = 1 2 + 1 θ RRA(Y ) (2.2) 4 The favorable odds requested increase with the proportion of wealth at stake θ. More importantly, the higher the RRA, the more favorable odds the investor demands to accept the lottery. Example 2.3.2. An important utility function is U(Y ) = Y 1 γ /(1 γ), which is known for having constant RRA, ie, RRA = γ. 11 For this investor, π(y, θ) = 1 2 + 1 θγ (2.3) 4 The higher the degree of RRA (parameter γ), the higher the favorable odds requested (π). Again, π does not depend on the level of wealth Y. It depends only on the proportion of wealth θ at stake. We do like this! Historically, stock returns look stationary (same mean through time), while aggregate wealth has been increasing. Thus, investors must require an expected return that cannot depend on the amount of wealth at risk. (Note that the expected return is determined by π.) The utility function with constant 10 Taylor series: f(x) = f(a) + f (a)(x a) + 1 2 f (a)(x a) 2 + + 1 n! f (n) (a)(x a) n +... 11 This is the only utility function with constant RRA. To see this, write Y U (Y ) U (Y ) = γ U (Y ) + γ Y U (Y ) = 0, which is a homogeneous linear differential equation of the second order. One specific solution is U 1 = Y 1 γ /(1 γ) (check that it satisfies the equation). The second exp{ γ/y dy } linearly independent solution is given by U 2 = U 1 (U 1) dy = 1. The general 2 solution is thus U(Y ) = c 1 Y 1 γ /(1 γ) c 2, a linear transformation of U(Y ) = Y 1 γ /(1 γ), therefore representing the same preferences. Again, thanks to Diogo Bessam for pointing this out.

2.4. Important utility functions 15 RRA (RRA = γ only, Y does not show up) is consistent with these empirical facts. 12 The proof of equation (2.2) is left as an exercise. 2.3.3 Risk neutrality Risk-neutral investors don t care about risk. Their utility function is linear: U(Y ) = a + by, b > 0 Check that ARA = 0 and RRA = 0, which implies π(y, h) = π(y, θ) = 1/2. Hence, risk neutral investors are indifferent to fair games (i.e., symmetrical games with 50 50 chances). They will always choose the asset with highest expected payoff, regardless of its risk. 2.4 Important utility functions The most common utility functions are the following: Name U(Y ) = Restrictions ARA RRA on parameters Log ln(y ) na Power Y 1 γ /(1 γ) Exponential exp( αy ) Quadratic ay by 2

2.5. Certainty Equivalent 16 Complete the table. In particular, define the restrictions on parameters s.t. the functions are proper utility functions, i.e., U > 0 and U < 0. Note that the quadratic utility function also needs a restriction on the domain (Y <... ). Also, compute the ARA and RRA functions, and classify the corresponding utility as increasing, decreasing, or constant ARA/RRA. As mentioned above, the power (and log) utility are considered good utility functions. Typical values for the degree of risk-aversion are γ = 1, 2, 3, 5. The other two utility functions are not so good descriptors of human behavior (as you can see by the ARA and RRA functions you got). As we will see in later sections, the exponential utility is used because it simplifies the calculations when asset returns are normally distributed, and the quadratic utility simplifies them even further for any distribution. 2.5 Certainty Equivalent Consider an investor with initial wealth Y. Consider a gamble Z = (Z 1, Z 2, π). How much is this risky asset worth? Definition (Certainty Equivalent). CE(Y, Z), the certainty equivalent of the risky investment Z, is the certain amount of money which provides the same utility as the gamble, i.e., E[U(Y + Z)] = U(Y + CE) The investor is indifferent between receiving CE(Y, Z) for sure and playing the gamble Z. In other words, if the investor owns the asset, he is willing to sell it at a price equal to the certainty equivalent. The CE is useful to compare different assets in more intuitive terms (money, instead of utility numbers). Note that a risk-averse agent will always value an asset at something less than its expected payoff: CE < E[Z]. 13 12 Thinking about the cross section of assets, note that (2.3) allows different assets to have different expected returns: π increases with θ, and thus the expected return also increases with θ. Does this make sense? Think about risk! 13 Let Z be any random variable. Since U is strictly concave (U < 0), from Jensen s inequality, Hence, from the definition of CE, Since U is increasing (U > 0), we must have E[U(Y + Z)] < U(E[Y + Z]) = U(Y + E[Z]) U(Y + CE) < U(Y + E[Z]) Y + CE < Y + E[Z] CE < E[Z]

2.5. Certainty Equivalent 17 Example 2.5.1. The investor has log utility and initial wealth Y = 1000. The risky investment is Z = (200, 0, 0.5). Compute the CE: E[U(Y + Z)] = U(Y + CE)... CE = 95.45 Why is the investor willing to accept less than the expected value of the gamble, ie, why is CE = 95.45 < E[Z] = 100? Risk aversion. Plot the utility function, marking the points Y + Z 1, Y + Z 2, Y + EZ, Y + CE. U(Y ) Y Consider now a fair gamble: Example 2.5.2. The investor has log utility and initial wealth Y = 100. The risky prospect is Z = (20, 20, 0.5). We get: E[U(Y + Z)] = U(Y + CE) 1/2 ln(120) + 1/2 ln(80) = ln(100 + CE) CE = 2.02 What does it mean the CE to be negative? Plot the utility function, marking the points Y + Z 1, Y + Z 2, Y + EZ, Y + CE.

2.6. Stochastic dominance 18 2.6 Stochastic dominance We now reverse gears and look for circumstances where the ranking among random variables is preference free, that is, where we do not need to specify a utility function. We will develop two concepts of dominance that are weaker, thus more broadly applicable, than state-by-state dominance. 2.6.1 First Order Stochastic Dominance Consider two assets, X 1, X 2, with the following payoffs: Payoff State (s) Prob(s) X 1 X 2 1 0.4 10 10 2 0.4 100 100 3 0.2 100 2000 Clearly, all rational investors prefer X 2 : probability of exceeding it. it at least matches X 1 and has a positive To formalize this intuition, let F i (x) denote the cumulative distribution function of X i, that is, F i (x) = Prob[X i x]. Definition (1SD). F a (x) 1SD F b (x) F a (x) F b (x), x Plot the two distribution functions in the example and check that F 2 (x) F 1 (x), x. Note that if the distribution of X 2 is always below X 1, then the probability of X 2 exceeding a given payoff is always larger, that is, F 2 (x) F 1 (x) 1 F 2 (x) 1 F 1 (x) Prob[X 2 x] Prob[X 1 x], x The usefulness of this concept comes from the following theorem: Theorem 2.6.1. F a (x) 1SD F b (x) E a [U(x)] E b [U(x)] for all nondecreasing U where E i is the expectation under the distribution of i, E i [U(x)] = U(x) df i (x) = U(x)fi (x) dx. Hence, all nonsatiable investors prefer asset X 2. Note that 1SD is not the same as state-by-state dominance. Danthine and Donaldson (2005). See exercise 4.8 in

2.6. Stochastic dominance 19 2.6.2 Second Order Stochastic Dominance 1SD is still a very strong condition, thus not applicable to most situations. If we add the assumption of risk aversion, we get the much more useful concept of Second Order Stochastic Dominance (2SD). Consider the following investments: X 3 X 4 Payoff Prob Payoff Prob 4 0.25 1 0.33 5 0.50 6 0.33 9 0.25 8 0.33 Plot the two distribution functions. Even though no investment 1SD the other, intuitively X 3 looks better. To make this precise: Definition (2SD). F a (x) 2SD F b (x) x F a(s) ds x F b(s) ds, x x [F b(s) F a (s)] ds 0, x That is, at any point the accumulated difference between F b and F a must be positive. Note that 1SD implies 2SD, but the converse is not true. In the plot of the previous example, this basically means that the area of the difference where F 3 > F 4 is small. To make this a bit more precise, we can compute the integrals at all relevant jump points. x x F 3 (x) 0 F x 3(s)ds F 4 (x) 0 F 4(s)ds x 0 F 4(s)ds x 0 F 3(s)ds 1 0.00 0 1/3 0 0 0 4 0.25 0 1/3 1 1 0 5 0.75 0.25 1/3 4/3 13/12 0 6 0.75 1.00 2/3 5/3 2/3 0 8 0.75 2.50 3/3 3 0.50 0 9 1.00 3.25 3/3 4 0.75 0 The last columns shows that x [F 4(s) F 3 (s)] ds 0, x. (After x = 9, the difference between the two integrals will always be 0.75 0.) All risk averse investors will prefer X 3, as the following theorem shows. Theorem 2.6.2. F a (x) 2SD F b (x) E a [U(x)] E b [U(x)] for all nondecreasing and concave U Note that risk aversion is enough, i.e., we do not have to assume a specific utility function.

2.7. Exercises 20 Mean preserving spread. The concept of 2SD is even more useful to understand the tradeoff between risk and return. Definition. Suppose there exists a random variable Z s.t. X b = X a +Z, with E[Z X a ] = 0 for all values of X a. Then, we say that X b is a mean preserving spread of X a. (Or F b or f b is a m.p.s. of F a or f a ). Note that X b has the same mean as X a, but it is more noisy, i.e., risky. Intuitively, all risk averse investors should prefer the payoff with less risk, X a. The following theorem justifies this intuition: Theorem 2.6.3. Let F a (x) and F b (x) be two distribution functions with identical means. Then, F a (x) 2SD F b (x) F b is a mean preserving spread of F a Mean-Variance criterion. This popular investment criterion states that: (i) for two investments with the same mean, investors prefer the one with smaller variance; (ii) for two investments with the same variance, investors prefer the one with higher mean. We will discuss later the exact conditions for this criterion to be true. For now, note that theorem 2.6.3 helps to explain part (i). 2.7 Exercises Ex. 1 (This is problem 3.1. in Danthine and Donaldson (2005)) Utility function. Under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this statement to linear transformations if we are to keep the same preference representation. Check it with this example. Assume an initial utility function attributes the following values to 3 perspectives: B u(b) = 100 M u(m) = 10 P u(p) = 50 a. Check that with this initial utility function, the lottery L = (B, M, 0.50) P. b. The proposed transformations are f(x) = a + bx, a 0, b > 0 and g(x) = ln(x). Check that under f, L P, but that under g, P L.

2.7. Exercises 21 Ex. 2 (This is problem 3.3. in Danthine and Donaldson (2005)) Inter-temporal consumption. Consider a two-date economy and an agent with utility function over consumption: U(c) = c1 γ 1 γ, γ > 0 at each period. Define the inter-temporal utility function as V (c 1, c 2 ) = U(c 1 ) + U(c 2 ). Show that the agent will always prefer a smooth consumption stream to a more variable one with the same mean, that is, U( c) + U( c) > U(c 1 ) + U(c 2 ), if c = c 1 + c 2 2 1. Start by showing that the utility function U is concave. 2. Then, show the required relation geometrically. 3. Finally, do the proof formally. Hint: use the following definition of a concave function. A function f : R N R 1 is concave if f(ax + (1 a)y) af(x) + (1 a)f(y), x, y R N and a [0, 1] Ex. 3 An agent with wealth = 100 is faced with the following game: with probability 1/2 his wealth will increase to 200; with probability 1/2 it will decrease to 0. Complete the following sentence: If the agent is a risk- he is willing to pay some money to play this game, whereas if he is risk- he is willing to pay some money to avoid the game. Ex. 4 The ARA and RRA measures have the first derivative of the utility function in the denominator. Why? Hint: read Danthine and Donaldson (2005) Ex. 5 Prove equation (2.2). Ex. 6 Ex. 7 Complete the table in section 2.4 and plot the utility functions. The CRRA utility function is usually presented as { ln(w ), γ = 1 U(W ) = W 1 γ /(1 γ), γ > 1 because ln(w ) is almost the limiting case as γ 1. More precisely, the true limit is lim γ 1 W 1 γ 1 1 γ = ln(w ). 1. Explain why U 1 (W ) = W 1 γ 1 γ preferences. and U 2 (W ) = W 1 γ 1 1 γ represent exactly the same

2.7. Exercises 22 2. Prove that Hint: L Hôpital s rule. W 1 γ 1 lim = ln(w ) γ 1 1 γ Ex. 8 Consider the utility function U(Y ) = 5 + 10Y 2. What does it imply in terms of risk-taking behavior? Would it be economically reasonable to model an investor s behavior with this utility function? Ex. 9 An investor has an initial wealth of Y = 10. To play a game where he could win or loose 5% of his wealth, he demands π = 0.6, where π is the probability of the favorable outcome (winning 5%). Nonetheless, if his wealth were Y = 1000, he would still demand the same π = 0.6 to play the game. 1. What can you say about the risk characteristics of this investor? (One sentence answer). 2. Give an example of an utility function consistent with this behavior. Ex. 10 The risk-aversion characteristics of an investor can be described by two functions: ARA and RRA. 1. Give a very brief definition in words of these two measures. 2. What does it mean to say that an investor has increasing ARA? Does it make intuitive sense? Give an example of an utility function with this characteristic. 3. Give an example of an utility function with constant RRA (compute the actual coefficient of RRA). Ex. 11 An investor with initial wealth Y 0 = 100 is faced with the following lottery: win 20 with 0.3 probability; loose 20 with 0.7 probability. The utility function is U(W ) = ln(w ). What is the Certainty Equivalent of this lottery? What does this number mean? Ex. 12 Consider the following risky investment: Z = (100, 0, 0.5). The investor has log utility, U = ln(y ). 1. If the initial wealth is Y = 100, what is the certainty equivalent of the gamble? 2. If the initial wealth is Y = 1, what is the certainty equivalent of the gamble? 3. Explain in simple terms the change in CE. Ex. 13 Exercise 4.5 in Danthine and Donaldson (2005, p.354) Ex. 14 Exercise 4.7 in Danthine and Donaldson (2005, p.355). They meant to refer to table 4.2. Ex. 15 Exercise 4.8 in Danthine and Donaldson (2005, p.355). Be careful in distinguishing between states of nature and distributions defined over payoffs.

2.7. Exercises 23 Ex. 16 Consider two assets with returns r a N(0.1, 0.2) and r b N(0.1, 0.3). An investor has the utility function U(W ) = exp( γw ). Which asset does the investor prefer?

Chapter 3 Portfolio choice 1. The investor s typical problem is maximize a E[U(Y )] 2. It can be solved explicitly if we assume either: 1. Quadratic utility, or 2. CARA utility and normal returns. 3.1 Canonical portfolio problem This section analyzes the problem of an investor that must decide how much to invest in a risky asset. Consider the following notation 1 a amount (in $) to invest in a risky portfolio r uncertain rate of return on the risky portfolio r f risk-free (certain) rate of return Y 0 initial wealth Ỹ 1 terminal wealth = a(1 + r) + (Y 0 a)(1 + r f ) = Y 0 (1 + r f ) + a( r r f ) The investor s problem is maximize a E[U(Ỹ1)] (3.1) 1 Tildes denote random variables. We ll drop them when it is clear which variables are random. 24

3.1. Canonical portfolio problem 25 The (necessary) first order condition for a maximum is [ ] d du(.) foc: da E[U(Ỹ1)] = 0 E ( r r f ) = 0 dỹ1 and the (sufficient) second order condition is [ d 2 soc: da 2 E[U(Ỹ1)] < 0 E which is true if the investor is risk averse (U < 0). ] d 2 U(.) ( r r f ) 2 < 0 dỹ 2 1 Example 3.1.1. Assume U = 11Y 5Y 2, with Y 0 = $1. Let r f = 0, E[r] = 0.1, Var[r] = 0.2 2. Recall Var[x] = E[x 2 ] E[x] 2. Use the foc to get the optimal amount invested in the risky asset: foc:... a = $0.2 Use the soc to check that this is indeed a maximum: soc: The analysis of the optimality conditions produces the following important theorem: Theorem 3.1.1. Let â denote the solution to problem (3.1) and assume the investor is nonsatiable (U > 0) and risk-averse (U < 0). Then â > 0 E[r] > r f â = 0 E[r] = r f â < 0 E[r] < r f

3.2. Analysis of the optimal portfolio choice 26 The theorem says that a risk-averse investor will only invest in the risky asset (stocks) if its expected return is higher than the risk-free rate. Conversely, if this is the case ( E[r] > r f ), then the investor will always participate in the stock market (even if with just a tiny amount of money). Example 3.1.2. Suppose U(Y ) = ln(y ). For simplicity, assume the risky return is the simple lottery (r 2, r 1, π). Further assume r 2 > r f > r 1 (why?). The problem is thus maximize E[ln(Ỹ1)] a The foc is [ ] r r f E = 0 Y 0 (1 + r f ) + a(r r f ) or, given the two possible states, r 2 r f r 1 r f π + (1 π) Y 0 (1 + r f ) + a(r 2 r f ) Y 0 (1 + r f ) + a(r 1 r f ) = 0 which after some algebra is a = (1 + r f )( E[r] r f ) Y 0 (r 1 r f )(r 2 r f ) Check that the sign of the rhs depends on the sign of E[r] r f. In particular, if E[r] r f > 0, we get a/y 0 > 0, as in theorem 3.1.1. Note also the following intuitive results: 1) The fraction of wealth invested in the risky asset (a/y 0 ) increases with the return premium ( E[r] r f ); 2) The fraction of wealth invested in the risky asset (a/y 0 ) decreases with the return dispersion around r f, ( (r 1 r f )(r 2 r f )). Lastly, note that the fraction of wealth invested in the risky asset (a/y 0 ) does not depend on the level of wealth (there is no Y 0 on the rhs). This result is specific to the CRRA utility function as described in a theorem below. 2 3.2 Analysis of the optimal portfolio choice 3.2.1 Risk aversion We now relate the portfolio decision to the risk aversion of the investor. The follwoing theorem states, quite intuitively, that a more risk averse individual will invest less in the stock market: 2 See the numerical examples in Danthine and Donaldson (2005) for further interpretation.

3.2. Analysis of the optimal portfolio choice 27 Theorem 3.2.1. Let â denote the solution to problem (3.1). Y > 0, ARA inv1 (Y ) > ARA inv2 (Y ) = â inv1 < â inv2 Furthermore, since ARA inv1 (Y ) > ARA inv2 (Y ) RRA inv1 (Y ) > RRA inv2 (Y ), we also have Y > 0, RRA inv1 (Y ) > RRA inv2 (Y ) = â inv1 < â inv2 Lets check this result: Example 3.2.1. Assume r f = 0.05 and r = (r 2 = 0.4, r 1 = 0.2, 1/2). For U(Y ) = ln(y ), we can use the results in the last example to get â/y 0 = 0.6 Now consider the power utility function U(Y ) = Y 1 γ /(1 γ), with γ = 3. Note that it has both higher RRA (3 > 1) and ARA (3/Y > 1/Y ). Check (end-of-chapter exercise 18) that the optimal portfolio decision for this utility function is â/y 0 = 0.198 Hence, this more risk-averse agent invests a smaller percentage of his wealth in the risky asset. The initial wealth (Y 0 ) is the same for both investors, so the money invested (â) is also smaller, as the theorem stated. 3.2.2 Wealth We now analyze the portfolio decision as the initial wealth changes. We might expect wealthier investors to put more money in the stock market. However, the result is not so simple; it depends on the characteristics of the specific utility function. Absolute Risk Aversion Theorem 3.2.2. Let â = â(y 0 ) denote the solution to problem (3.1). Then, (Decreasing ARA) ARA (Y ) < 0 â (Y 0 ) > 0 (Constant ARA) ARA (Y ) = 0 â (Y 0 ) = 0 (Increasing ARA) ARA (Y ) > 0 â (Y 0 ) < 0

3.2. Analysis of the optimal portfolio choice 28 DARA. If the investor has decreasing absolute risk aversion (DARA), he is willing to put more money at risk as he becomes wealthier. Recall that power utility has DARA (ARA(Y ) = γ/y ). (Is this reasonable behavior?) CARA. The second case, constant absolute risk aversion (CARA) is also important because the exponential utility satisfies this condition. Recall that U(Y ) = exp( αy ) ARA(Y ) = α ARA (Y ) = 0 The theorem states that this investor will put the same amount of money in the risky asset regardless of how much wealth he has. (Is this a reasonable description of investors behavior?) Illustration: solving the problem for CARA Lets verify the CARA case of the theorem. The portfolio problem is with Y 1 = Y 0 (1 + r f ) + a(r r f ). The foc is maximize {E[ exp( αy 1 )]} (3.2) a E [α(r r f ) exp( αy 1 )] = 0 (3.3) which cannot be solved explicitly for a without further assumptions! To proceed, we consider two alternatives. 1. Implicit Function Theorem Even though we cannot explicitly solve the problem, we can still describe the optimal solution using a very useful trick in economics: the Implicity Function Theorem. 3 Intuitively, this theorem says the following. Suppose the (implicity) function y = y(x) is the solution to some equation, that is, f(x, y) = 0. More 3 Implicit Function Theorem. Consider the equation f(y, x 1,..., x m ) = 0 and the solution (ȳ, x 1,..., x m ). If f(ȳ, x)/ y 0, then there exists an implicit function y = y(x 1,..., x m ) that satisfies the equation for every (x 1,..., x m ) in the neighborhood of ( x 1,..., x m ), i.e., f(y(x 1,..., x m ), x 1,..., x m ) = 0. Furthermore, the partial derivatives are given by y( x 1,..., x m ) x i = f(ȳ, x 1,..., x m )/ x i f(ȳ, x 1,..., x m )/ y

3.2. Analysis of the optimal portfolio choice 29 precisely, as we change x, y(x) adjusts to keep f at 0, f(x, y) 0. We can thus conclude that f does not change, ie, its total differential is zero. Therefore, df(x, y) = 0 f f dx + x y dy = 0 dy dx = f/ x f/ y Going back to the maximization problem, â = â(y 0 ) is the implicit function that guarantees that the lhs of (3.3) is always zero. We can thus take the total differential of the foc and get dâ(y 0 ) = E[... ]/ Y 0 dy 0 E[... ]/ a =0 (foc) {}}{ = (1 + r f )α E[α(r r f )e αy 1 ] E[α 2 (r r f ) 2 e αy 1] }{{} >0 = 0 Hence, the amount invested in the risky asset does not change with the investor s wealth, as the theorem claimed. Furthermore, the implicit function theorem allowed us to check this without solving the maximization problem explicitly. 2. Normal returns To get an explicit closed-form solution to problem (3.2) we need an additional assumption. It is this assumption that justifies the wide use of exponential utility. Assume the return on the risky asset is normally distributed, r N(µ, σ 2 ). Then, next period s wealth is also normally distributed, Y 1 N(Y 0 (1 + r f ) + a(µ r f ), a 2 σ 2 ). Using the moment generating function for the normal distribution 4, we can simplify the portfolio problem: max a { ( {E[ exp( αy 1 )]} = max exp α[y0 (1 + r f ) + a(µ r f )] + 1/2α 2 a 2 σ 2)} a that is, the rhs does not have E[.]. We can thus solve the maximization problem and get a closed-form solution for a. Exercise 24 asks you to do these final steps. Check that the final expression for a does not depend on Y 0, as the theorem stated. To summarize, even though the exponential utility is not the best intuitive description of human behavior, it is very useful if we assume that returns are normally distributed. 4 If X N(m, s 2 ), then E [ e γx] = exp ( γm + 1 2 γ2 s 2), for any γ.

3.2. Analysis of the optimal portfolio choice 30 Relative Risk Aversion We can also characterize the optimal portfolio choice in terms of the relative risk aversion measure, RRA. Define ŵ â/y 0, the optimal proportion of wealth invested in the risky asset, or the optimal portfolio weight in the risky asset. Theorem 3.2.3. Express the solution to problem (3.1) as a fraction of wealth, ŵ(y 0 ) â(y 0 )/Y 0. Then, (Decreasing RRA) RRA (Y ) < 0 ŵ (Y 0 ) > 0 (Constant RRA) RRA (Y ) = 0 ŵ (Y 0 ) = 0 (Increasing RRA) RRA (Y ) > 0 ŵ (Y 0 ) < 0 For example, if the investor has decreasing RRA, he will invest a higher proportion of wealth in the risk asset as he becomes wealthier. The most interesting case is perhaps the constant relative risk aversion (CRRA) case, as it characterizes the power and log utility functions. These investors always invest the same fraction of their wealth in the stock market, regardless of their initial wealth. 5 Example 3.2.2. Consider U = ln(y ). Define w a/y 0, the fraction of wealth invested in the risky asset. The investor s problem is to maximize w E[ln(Y 1 )] with Y 1 = Y 0 (1 + r f ) + wy 0 (r r f ). Writing the foc and using the implicit function theorem, we can show that (end-of-chapter exercise 19) dŵ dy 0 = 0 That is, the optimal fraction does not change with wealth. 5 This theorem can also be expressed in terms of η dâ/â dy 0/Y 0, the wealth elasticity of the investment in the risky asset: (Decreasing RRA) RRA (Y ) < 0 η > 1 (Constant RRA) RRA (Y ) = 0 η = 1 (Increasing RRA) RRA (Y ) > 0 η < 1 To see that increasing ŵ(y 0 ) â(y 0 )/Y 0 is the same as η > 1, note d [ŵ(y 0 )] = d [â(y0 ] ) > 0 dâ 1 â/y0 2 > 0 dâ/ dy 0 > â/y 0 dâ/â > 1 dy 0 dy 0 Y 0 dy 0 Y 0 dy 0 /Y 0 and similarly for the other cases.

3.3. Canonical portfolio problem for N > 1 31 3.3 Canonical portfolio problem for N > 1 Now we generalize the portfolio choice problem. There are N risky assets and 1 risk-free asset. Terminal wealth is The investor s problem is thus maximize {a 1,...,a N } E Ỹ 1 = Y 0 (1 + r f ) + [ U ( N a i ( r i r f ) i=1 Y 0 (1 + r f ) + )] N a i ( r i r f ) i=1 It will be convenient to choose weights instead of $ values. We thus define w i a i /Y 0 and write Y 1 = Y 0 (1 + r f ) + N i=1 w iy 0 ( r i r f ). The investor s problem can thus be rewritten as [ [ ])] N maximize E U (Y 0 (1 + r f ) + w i ( r i r f ) {w 1,...,w N } Define r p to be the return on the portfolio: r p := w f r f + i=1 N w i r i Imposing the constraint that the weights must add up to one, we have that ( ) N N N r p = 1 w i r f + w i r i = r f + w i ( r i r f ) i=1 i=1 i=1 Hence, the portfolio problem can also be written as i=1 maximize E [ U (Y 0(1 + r p ))] {w 1,...,w N } Unfortunately, this problem is hard to solve without some simplifying assumptions. 3.4 Exercises Ex. 17 State the investor s problem (expression 3.1) in words.

3.4. Exercises 32 Ex. 18 Check the results in example 3.2.1. The final expression is in the book; you just need to do the intermediate calculations. Caution: the expression in the book is correct, but the number is not (at least I get a different answer: a/y = 0.198 instead of 0.24). Ex. 19 Check the results in example 3.2.2, ie, do the intermediate computations. Ex. 20 Consider the standard portfolio choice between a risk-free asset and a risky stock. An investor with initial wealth $1000 makes an optimal choice to allocate $400 to the stock. We know that if the same investor had an initial wealth larger than $1000, he would allocate more than $400 to the stock. 1. This investor has (decreasing / constant / increasing) ARA. 2. Give an example of a utility function consistent with this behavior. Consider the utility function U(Y ) = e gy, where g is a constant param- Ex. 21 eter. 1. Compute the ARA and RRA coefficients. 2. Interpret in words the result obtained for ARA (relate it to a simple lottery and to the portfolio choice problem). Ex. 22 Consider the canonical portfolio choice problem with 1 risky asset (with random return r) and 1 risk-free asset (with return r f ). The investor chooses the amount of money (a) to invest in the risky asset. 1. Write the problem explicitly for an investor with U(Y ) = exp( αy ), where Y is the wealth. 2. If the risk-free rate increases, what should happen to the amount invested in the risky asset? Explain intuitively (5 lines). 3. Show it explicitly. Hint: compute da dr f and determine its sign. Ex. 23 There is a risk-free and a risky asset. The investor chooses the amount invested in the risky asset, a, to maximize a EU(Y 1 ), where Y 1 is next period s wealth. Assume a regular utility function (U > 0, U < 0). 1. In general, what can you say about the sign of da/dy 0? 2. Assume U(Y ) = e αy. Compute da/dy 0. Ex. 24 Consider the standard portfolio choice problem maximize E[ exp( γy 1 )] a where next-period s wealth is Y 1 = Y 0 (1 + r f ) + a(r r f ), and the return on the risky asset is normally distributed, r N(µ, σ 2 ). Compute the explicit optimal amount to

3.4. Exercises 33 invest in the risky asset (a). Hint. Use the following property of the normal distribution (called moment generating function): If X N(m, s 2 ), then E [ e γx] = exp ( γm + 1 2 γ2 s 2), for any γ. Ex. 25 Computing returns with dividends. Consider the following daily closing prices and dividends (D) for two stocks (in $): Stock A Stock B day t P t D t P t D t fri 0 10 10 mon 1 11 11 tue 2 10 10 wed 3 11 11 1.1 thu 4 9 9 fri 5 12 12 Note that when a stock pays dividends, the return should be computed as r t = Pt+Dt P t 1 1. 1. Compute daily returns for these two stocks. Compute also the weekly returns assuming that the dividends are reinvested in the stock. This is a standard assumption, so use the standard formula, 1+r 0,T = (1+r 0,1 )(1+r 1,2 )... (1+r T 1,T ). Note: this is usually called Holding Period Return in databases such as CRSP or DataStream. 2. Suppose you invested $4,000 in A and $6,000 in B in the beginning of the week. Compute the portfolio return over this week. (Use the weekly returns already computed and apply the standard formula for the portfolio return). 3. Since we assume that dividends are reinvested in the stock, we may end up with more shares than we started with. How many shares of each stock do you have at the beginning of the week? How many shares do you have at the end of the week? Note: to check that you have the right answer, compute the terminal value of the portfolio by doing V 5 = P A,5 N A,5 + P B,5 N B,5, where N is the number of shares that you got. It should imply the same weekly return as in the previous question. 4. Again, the way weekly returns were computed assumes that dividends are reinvested in the stock. Hence, while for the stock without dividends (A) we have r A,week = P 5 /P 0 1 0.2 = 12/10 1 the same is no longer true for the dividend-paying stock (stock B) r B,week P 5 /P 0 1 0.32 12/10 1

3.4. Exercises 34 Hence, databases usually also show an adjusted price, P a, that can be used to compute returns without having to know the dividends. The true return from market closes plus dividends must equal the return with adjusted closes: P t + D t P t 1 1 = P t a Pt 1 a 1 Fix the last price P a 5 = P 5 = 12. Compute the adjusted prices for the previous days for both stocks. (Check my website for an exercise with data from finance.yahoo.com)

Chapter 4 Portfolio choice for Mean-Variance investors 1. Quadratic utility or Normal returns imply mean-variance preferences, E[U] = f(µ p, σ 2 p). 2. The optimal investment opportunities are described by the meanvariance frontier. 3. The investor s portfolio choice problem with N > 1 risky assets can be solved explicitly. These concepts were developed by Harry Markowitz in 1952 and they are still the benchmark for optimal portfolio allocation. 4.1 Mean-Variance preferences The general portfolio problem (N > 1) is hard to solve unless we make one of the simplifying assumptions below. Either one of these assumptions will lead to meanvariance preferences, that is, to investors that care only about the first two moments of Y 1 or r p. 1 Expand U(Ỹ1) around E(Ỹ1). To simplify the notation, let Y Y 1. U(Y ) = U( EY ) + U ( EY ) (Y EY ) + 1/2 U ( EY ) (Y EY ) 2 + remainder 1 Note that the two are related: E[Y 1 ] = Y 0 (1 + E[r p ]) and Var[Y 1 ] = Y 2 0 Var[r p ]. 35