ECON Financial Economics

Transcription

1 ECON 8 - Financial Economics Michael Bar August, 0 San Francisco State University, department of economics.

2 ii

3 Contents Decision Theory under Uncertainty. Introduction Expected Utility Theory Axiomatic foundations of Expected Utility Theory Risk Aversion Measuring risk aversion and applications Mean-Variance Theory Critique of the mean-variance theory Validity of the mean-variance theory Prospect Theory Motivation PT v.s. CPT Two-Period Model: Mean-Variance Approach 4. Mean-Variance Portfolio Analysis Portfolios with two risky assets Portfolios with n risky assets Adding a risk-free asset Capital Asset Pricing Model (CAPM) Deriving the CAPM CAPM in practice General Principles of Asset Pricing 89. Assets and Portfolios in a Two-Period Model Financial markets Redundant assets Complete markets Law of One Price iii

4 iv CONTENTS.. State prices Arbitrage Fundamental Theorem of Asset Pricing State prices and utility maximization Pricing using risk-neutral probabilities Option Pricing Options terminology Binomial model (-states) Black-Scholes formula (in nitely many states)

5 Chapter Decision Theory under Uncertainty. Introduction Almost every decision we ever make in our lives, involves uncertainty, i.e. the outcome of our choices cannot be predicted with absolute certainty. These decisions include not only nancial investment choices, but career choices, marriage, and college major. Decision theory under uncertainty makes the foundations of all nance and portfolio theories, therefore it must be the starting point of this course. We assume that decision maker faces a choice among a number of risky alternatives. Each risky alternative may result in one of a number of possible outcomes, but which outcome will actually occur is uncertain at the time of decision making. We represent these risky alternatives by lotteries. De nition A lottery is a probability distribution de ned on a set of payo s. A lottery can be discrete, in which case it is described by the list of payo s x ; x ; :::; x N and the probabilities of these payo s p ; p ; :::; p N. The number of outcomes in a discrete lottery can be nite N <, or in nite. A lottery can also be continuous, in which case the set of payo s is usually a subset of real numbers (x R) and the distribution is described with a probability density function (pdf) f (x) or the cumulative distribution function (cdf) F (x) R f (t) dt. Lotteries can also be a mix of discrete and continuous, but we will x not deal with such lotteries in this course. We use the notation L to denote the space of all possible lotteries. Just like in the case of any probability distribution of a random variable, we require that

6 CHAPTER. DECISION THEORY UNDER UNCERTAINTY probabilities (or densities) will be non-negative, and add (or integrate) to : [Discrete] : p i 0 8i, X i [Continuos] : f (x) 0 8x, p i Z f (x) dx Just like with any random variables, we might want to compute the mean and variance of a lottery, and also the covariance between any two lotteries. In general, lotteries can have any kind of abstract outcomes, not only monetary payo s. For example, when you play basketball, you can win, loose, end with an overtime or have an injury. These outcomes are not stated in monetary terms. For simplicity of analysis, we assume that these outcomes can be represented in terms of money, so a win is for example equivalent to +$00, and a loss is equivalent to -$400. Moreover, money lotteries are all we need for the study of nancial decisions, where outcomes are naturally represented with monetary returns. Also note that sure outcomes can also be viewed as lotteries, that have one outcome occurring with probability. Example A discrete lottery A, with two outcomes, $000 and $00, each achieved with equal probabilities. ( $000 w.p. A $00 w.p. The expected value and the variance of this lottery are: E (A) 000 0: : $0 V ar (A) (000 0) 0: + (00 0) 0: ; 00 De nition Let A; B L be lotteries and [0; ]. Then C A + ( ) B denotes a compound lottery where with probability the lottery A is played and with probability the lottery B is played. If the lotteries are discrete, then the probability of outcome i in C is p C i p A i + ( ) p B i, and when the lotteries are continuous, the probability density of C is f C (x) f A (x) + ( ) f B (x). Exercise Consider two lotteries A ( 8 $000 w.p. >< $00 w.p., B >: $000 w.p. $400 w.p. 0 w.p.

7 .. INTRODUCTION Describe the payo s and the associated probabilities of the compound lottery C 0:A+0:B. Solution The compound lottery C 0:A + 0:B is 8 >< C >: $000 w.p. + $00 w.p. $400 w.p. 0 w.p. For example, the payo of $000 is achieved if lottery A is played (which occurs with probability 0:) and the outcome of $000 is realized (which occurs with probability ), or if lottery B is played (which occurs with probability ) and then $000 is realized (with probability ), i.e. Pr (000) Pr (A) Pr (000jA) + Pr (B) Pr (000jB). The main challenge of decision theory is to de ne preference over lotteries that will allow comparison of di erent lotteries. For example, consider a lottery D, which pays $0 with probability. Which lottery is better, A or D? This question is equivalent to asking "would you be willing to pay $0 for lottery A? More generally, how much are you willing to pay for lottery A? This is the fundamental question of asset pricing. Most people will say that they would pay less than $0 for lottery A. In general, most people are not willing to pay the mean of a lottery to buy that lottery. This suggests that things other than the mean of a lottery are also important. Nevertheless, until recently (mid 0th century), the only theory of preferences over risky alternatives was the mean theory - i.e. the value of a lottery is given by its mean payo. The next example illustrates the rst challange to the mean theory. It it is knows as the "St. Petersburg Paradox", analyzed by the Swiss mathematician Daniel Bernoulli in 8. Example (St. Petersburg Paradox). Consider the lottery A, based on the following gamble. A fair coin is tossed repeatedly, until "tails" rst appears. This ends the game. Let the number of times the coin is tossed until "tails" appears be k. The lottery pays $ k. Thus, if you toss the coin once, and "tails" appear, then you are paid $. If it takes tosses until the coin shows "tails", then you are paid 4 $. Thus, the possible payo s k, of this lottery are k, k ; ; :::, and the probabilities are k ; ; :::. The expected value of this lottery is: E (A) X k k k X k k0 k X k0

8 4 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Once again, the question is, how much are you willing to pay for this lottery? Despite the fact that the expected payo of this lottery is, there is not a single person who would pay $ to play this lottery. In fact, most people are willing to pay no more than a few dollars to play this lottery. Indeed, this game can give you a high payo of more than a million dollars, if you toss the coin times before "tails" rst appears (your payo in this case is 0 $; 048; ). But this happens with very low probability: 0: , or once in more than million plays. Example (St. Petersburg Paradox) demonstrates once again that, in general, the value of a lottery is not equal to the expected value of its payo. Nevertheless, until middle of the 0th century, the expected value was the well-accepted theory of decisions under risk. In the next sections we develop alternative theories for choices under risk: (i) Expected Utility Theory, (ii) Mean-Variance Theory, and (iii) Prospect Theory. We start with the mainstream theory in economics - the Expected Utility Theory (EUT). The advantage of EUT is that it is derived from reasonable axioms about rational behavior in risky environment. The EUT is therefore prescriptive, in that it tells us how rational agents should behave. Next we proceed to a simpler Mean-Variance Theory, which is popular exactly because of its simplicity. Under MVT, all lotteries (and in particular all nancial assets and all portfolios of assets) can be represented with two numbers: mean and variance (; ). The MVT violates some of the axioms of rational behavior, but nevertheless serves as the foundation of modern portfolio theory. MVT is therefore the most practical theory in nance, exactly due to its simplicity. Finally, the prospect theory (PT) and its variants, are a result of recent developments in behavioral and experimental economics, and it attempts to understand actual behavior. PT is therefore a descriptive theory, and it tries to describe how people actually behave. PT is by far the most complicated theory of the three, because it was developed as a generalization to EUT with the goal to " x" some problems with it. While PT is successful in resolving some paradoxes which the EUT fails to explain, due to its complexity the PT is di cult to apply to real world nancial problems of choosing portfolios and pricing nancial assets.. Expected Utility Theory The motivation for the development of EUT is Example (St. Petersburg Paradox). Bernoulli realized that twice the money is not always "twice as good". If a person has only a small amount of money, say $000, then doubling it increases his utility by more, than say doubling the wealth of someone who has 0 million dollars. Put in another way, the marginal utility from money is diminishing - the more money you have, the smaller is the

9 .. EXPECTED UTILITY THEORY gain from additional $. This idea of diminishing marginal utility from money is equivalent to risk aversion in EUT, and will be formalized later. For now, Bernoulli s intuition is that instead of computing the expected payo of a lottery, we need to compute the expected utility of a lottery. The strength of EUT is that it is not only intuitively appealing, but can be derived from more fundamental axioms about preferences... Axiomatic foundations of Expected Utility Theory The starting point for any decision theory under risk is lotteries, and the assumption that people have some preferences over the space of all lotteries. De nition Let the space of all lotteries be L. We assume that there exists a weak preference relation % on L, such that for two lotteries A; B L, the notation A % B means that "lottery A is at least as good as lottery B". From the weak preference relation, we can derive the strict preference relation and the indi erence relation as follows. The strict preference relation on L means that A B if A % B but not B % A. We read A B as "lottery A is strictly better than lottery B". The indi erence relation on L means A B if A % B and B % A. We read A B as "lottery A is as good as (or equivalent to) lottery B". Next, we make some assumptions (axioms) about the preference relation %, that will allow representing it with a utility functional and enable practical usage. De nition 4 A preference relation % on L is called rational if it satis es the following two axioms: A. Completeness. A preference relation % on L is complete if for any two lotteries A; B L, either A % B or B % A or both. Completeness means that the decision maker is able to choose among risky alternatives. A. Transitivity. A preference relation % on L is transitive if for any three lotteries A; B; C L, we have A % B and B % C ) A % C The transitivity assumption is a natural consistency of preferences. We can show that if transitivity is violated for some individual, i.e his preferences are A % B and B % C and C A, then we can easily extract all his wealth by o ering him to trade B for C, A for B, C for A, and repeat many times, and each time he gets C for A he pays some amount (because C is strictly better than A).

10 CHAPTER. DECISION THEORY UNDER UNCERTAINTY The next assumption is technical, and is needed in order to ensure representation of preferences with utility. De nition A. Continuity. The weak preference relation % on L is continuous if for any lotteries A; B; C L with A % B % C, there exists a probability p [0; ] such that B pa + ( p) C The right hand side is the compound lottery, where with probability p lottery A is played and with probability p lottery C is played. Continuity means that any lottery "in between" two other lotteries (here B is in between A and C) is equivalent to some mixture of the two lotteries. What it also means is that there are no "jumps" in preferences due to small changes in probabilities. For example, suppose that A is "basketball game with friends", and B is "staying at home", and C is a "knee injury", and I prefer playing basketball over staying at home: A B. Then, when I add a small enough probability of a knee injury to the basketball game, I would not suddenly change my mind and decide to stay at home. That is, B does not become better than pa + ( p) C for small enough ( p). This axiom is reasonable because most people do go out to work, shopping, movies, despite the small risk of getting into accident. The next theorem states that preferences which satisfy completeness, transitivity and continuity assumptions, can be represented with a continuous utility functional. Theorem (Utility Representation). Let % on L be a weak preference relation on the space of lotteries. If % satis es the axioms A, A and A, then there exists a continuos utility functional U : L! R such that U (A) U (B) if and only if A % B for any lotteries A; B L. This also implies that, for any lotteries, U (A) > U (B) if and only if A B and U (A) U (B) if and only if A B. Proof. Omitted. The concept of functional is needed here to emphasize that U maps lotteries (probability distributions) into real numbers, while a regular function maps real numbers (or vectors) into real numbers. Also note that theorem does not state that U must have any speci c form; only that U is continuous. In particular, theorem does not state that U must have expected utility form. De nition We say that a utility functional U : L! R has expected utility form if there exists a utility function u : R! R, such that for every lottery L L, U (L) E [u (x)]

11 .. EXPECTED UTILITY THEORY In words, the utility of a lottery is equal to the expected utility of its payo s. Speci cally, for discrete lottery U (L) E [u (x)] X i u (x i ) p i and for continuous lottery (with pdf f (x)) U (L) E [u (x)] Z u (x) f (x) dx Once again, the utility function u maps sure payo s (real numbers) into real numbers, while the utility functional U maps lotteries (probability distributions) into real numbers. The utility function u is sometimes called "von Neumann-Morgenstern" (vnm) utility function, after the mathematician John von Neumann (90 9) and the economist Oskar Morgenstern (90 9) who provided the axiomatic foundations for the EUT. These two are also the founders of Game Theory. Observe that a key advantage of the EUT is that in order to know the preferences over risky alternatives, all we need to know is the preferences over sure outcomes, given by the vnm utility function u : R! R. In all applications, we make the standard assumption that u is increasing, which means that more money is better (formally monotonicity assumption). An important property of expected utility form is linearity. Proposition (Linearity of Expected Utility Functional). Suppose that U : L! R has the expected utility form. Then U is linear, i.e. for any ( ; ; :::; K ) > 0, P k k, and lotteries L ; L ; :::; L K L, U! KX k L k k KX k U (L k ) k In words, the utility of a compound lottery is the weighted average of utilities of individual lotteries. Observe that in a two-lottery case, A; B L, linearity means: U (A + ( ) B) U (A) + ( ) U (B) It is recommended that you rst prove the proposition for the two-lottery case, before proving the K-lottery case. Usually in math, the concept of functional is used to describe a map from a space of functions into real numbers, and it is appropriate here because lotteries are probability distributions, which are functions themselves.

12 8 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Proof. Suppose that the lotteries are discrete, and all the possible payo s are indexed by i ; ; :::; N. Then the probability of payo x i, in the compound lottery, is Pr (x i ) p i; + p i; + ::: + K p i;k KX k p i;k ; k where p i;k is the probability of payo i in lottery k. Then! " KX NX K # X U k L k u (x i ) k p i;k k i k " KX N # X k u (x i ) p i;k k i KX k U (L k ) k The proof for continuous case is similar, with probabilities replaced with pdfs, and sums with integrals. Our last axiom will allow us to represent preferences over lotteries not just with any U, but with U that has expected utility form. De nition A4. Independence of irrelevant alternatives. Let A; B L two lotteries with A B, and let (0; ]. Then for any lottery C L, it must be A + ( ) C B + ( ) C The independence axiom means that the ranking of two lotteries does not change if you mix each of them with a third lottery. Preferences that satisfy the independence axiom, in addition to completeness, transitivity and continuity, can be represented with expected utility form. Theorem (Expected Utility Theorem). Let % on L be a weak preference relation on the space of lotteries. (i) If % satis es the axioms A, A, A and A4, then % can be represented with expected utility form. That is, there exists a (continuous) von Neumann-Morgenstern utility function u : R! R, such that for every lottery L L, U (L) E [u (x)]. (ii) Any expected utility functional satis es axioms A, A, A, and A4. Proof. Part (i) is very technical and is omitted. We will only prove part (ii). Thus, we assume that U (L) E [u (x)] represents % on L, and prove that U () satis es the axioms.

13 .. EXPECTED UTILITY THEORY 9 A, Completeness. Since by de nition U : L! R, which means that any lottery is assigned a real number, then for any two lotteries A; B L, the expected utility U (A), U (B) are two numbers. Thus, just like for any two real numbers, we must have either U (A) U (B) or U (B) U (A) or both. In other words, the completeness of % on L follows from the completeness of "" on R. A, Transitivity. For any three lotteries A; B; C L with U (A) U (B) and U (B) U (C), we must have U (A) U (C) because U (A), U (B) and U (C) are just three real numbers. In other words, the transitivity of % on L follows from the transitivity of "" on R. A, Continuity. Let A; B; C L with U (A) U (B) U (C). We need to show that there exists a probability p [0; ] such that U (B) U (pa + ( p) C) Using the linearity of expected utility functional (proposition ), U (pa + ( p) C) pu (A) + ( p) U (C) If U (A) U (C) U (B), then any p R satis es U (B) pu (A) + ( U (A) > U (C), then we can solve for p directly: p) U (C). If U (B) pu (A) + ( p) U (C) U (B) U (C) ) p [0; ] U (A) U (C) A4, Independence. Let A; B L two lotteries with U (A) > U (B), and let (0; ]. We need to show that for any lottery C L, we must have U (A + ( ) C) > U (B + ( ) C). Using the linearity of expected utility functional (proposition ), U (A + ( ) C) U (A) + ( ) U (C) > U (B) + ( ) U (C) U (B + ( ) C) The next exercise demonstrates how the Expected Utility Theory resolves the St. Petersburg Paradox, and explains why people are willing to pay very little for the gamble in example, despite in nite expected payo.

14 0 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Exercise Suppose that your preferences over lotteries can be represented with expected utility, and that your vnm utility function on sure payo s is u (x) ln (x). How much are you willing to pay for the gamble in St. Petersburg Paradox in example? Solution The expected utility from the gamble is: E [u (x)] X u (x k ) p k k " X ln () k k The second term in the brackets is X k X ln k k X ln () (k ) k X # k ln () k k k k k To compute the rst term in the brackets, we use the rule P k kak a ( a) : X k k k ( ) Therefore, I will pay at most $ for this gamble, since u () ln (), which means that the utility from $ is the same as the utility from this gamble. We therefore say that $ is the certainty equivalent of this lottery. De nition 8 The certainty equivalent of a lottery L is the non-random payo CE which is equivalent to playing the lottery: CE L. If preferences can be represented by expected utility, then certainty equivalent is de ned by u (CE) E [u (x)]. Thus, in the last example, with vnm utility function u (x) ln (x), we say that the certainty equivalent to the St. Petersburg gamble is CE, since ln () E [ln (x)], i.e. utility from the sure payo of $ is equal to the expected utility from the gamble. This exercise is our rst example of asset pricing. St. Petersburg gamble is a risky asset that has a payo k with probability k. We found that the value of this risky asset, according to EUT, to an investor with vnm utility function u (x) ln (x) is $. Exercise Suppose that your preferences over lotteries can be represented with expected utility, and that your vnm utility function on sure payo s is u (x) p x. How much are you willing to pay for the gamble in St. Petersburg Paradox in example?

15 .. EXPECTED UTILITY THEORY Solution Do it yourself. You should get CE :94, which means that according to EUT, an investor with vnm utility function u (x) p x is willing to pay about $:9 for the St. Petersburg gamble (risky asset). The above examples illustrate that the value of any lottery depends on investor s vnm utility function. Investors with u (x) p x are willing to pay $:9, while investors with u (x) ln (x) are willing to pay only $. It would be nice if there was a way to estimate the vnm utility function. We end this section by pointing out that the vnm utility function u is not unique, and any increasing linear transformation of u represents the same preferences as u. This property helps us estimate the vnm of individual investors. Proposition (Invariance of Expected Utility). Let a R and b > 0. Suppose that the preference relation % on L has expected utility representation with vnm utility function u. Then, v (x) a + bu (x) is another vnm utility function representing the same preferences as u. Proof. We need to show that for any two lotteries A; B L, E A [u (x)] E B [u (x)] () E A [v (x)] E B [v (x)] where the subscripts A and B indicate that we are computing the expected values using the probability distributions A and B. Note that we can write: E A [v (x)] E A [a + bu (x)] a + be A [u (x)] E B [v (x)] E B [a + bu (x)] a + be B [u (x)] Subtracting the second from the rst: E A [v (x)] E B [v (x)] b fe A [u (x)] E B [u (x)]g All expectations are numbers, and since b > 0 we have: E A [v (x)] E B [v (x)] 0 () E A [u (x)] E B [u (x)] 0 Example The above result implies that if u (x) p x is vnm utility function corresponding to preference relation % on L, then v (x) + p x or v (x) +0:8 p x also represent

16 CHAPTER. DECISION THEORY UNDER UNCERTAINTY the same preferences as u. This means that for any lotteries A; B L, A % B () E A [u (x)] E B [u (x)] () E A [v (x)] E B [v (x)] Exercise 4 Suppose some vnm utility function u has the following values at two points: u (0) 0 and u (000). Show that there exists another utility function v, such that v (0) 0, and v (000), that represents the same preferences as u. Solution 4 By proposition, we know that v represents the same preferences as u if we can nd two numbers, a R and b > 0 such that v (x) a + bu (x). Since the values of the utility functions are given at two points, we have two equations that can be solved for a and b, v (0) a + bu (0), and v (000) a + bu (000), i.e. 0 a + b 0 a + b The solution is a, and b 0:. Thus, v (x) + 0:u (x) represents the same preferences as u, and also v (0) 0, v (000) as required. From the previous exercise we learned that we can assign any arbitrary to a vnm utility function at any two points x and x. This property allows us to estimate the preferences of a decision maker by estimating one of her equivalent vnm utility functions. Suppose that Steve has vnm utility function u, which is known at two points: u (0) 0 and u (000). We o er him a lottery with payo s at these two points, and ask him what is the most he is willing to pay it: ( $000 w.p. L 0 w.p. Suppose Steve tells us that his certainty equivalent for this lottery is CE 40. We can nd Steve s utility from 40, using the de nition of certainty equivalent: u (CE ) E [u (x)] u (40) 0:u (0) + 0:u (000) 0: Now, can obtain another point on the his utility function by o ering him a lottery: ( $000 w.p. L $40 w.p.

17 .. EXPECTED UTILITY THEORY Suppose tells us that his certainty equivalent for this lottery is CE 00. We can nd Steve s utility from 00, using the de nition of certainty equivalent: u (CE ) E [u (x)] u (00) 0:u (40) + 0:u (000) 0: Notice that we started from two points u (0) 0 and u (000), and found two more points u (40) 0: and u (00) 0:. We can nd more popints on Steve s vnm utility function, by o ering him lotteries based on any two points previously found, and asking him for the certainty equivalent of these lotteries... Risk Aversion We have seen in example a lottery that pays $000 or $00 with equal probabilities, ( $000 w.p. A $00 w.p. The expected value of this lottery was found to be E (A) $0, and we mentioned that most people would not pay $0 for this lottery. In general, most people prefer the mean of a lottery for sure, over the lottery itself. This behavior is called risk aversion. De nition 9 We call a person risk-averse if he prefers the expected value of every lottery L L over the lottery itself, i.e. E (L) L. If preferences over lotteries can be represented with expected utility form, then risk aversion means u [E (x)] > E [u (x)]. A person is riskseeking if he prefers every lottery over its expected value, L E (L). A person is riskneutral if he is indi erent between lotteries and their expected return, L E (L). Once again, if preferences over lotteries can be represented with expected utility form, then riskseeking means E [u (x)] > u [E (x)], and risk-neutral means u [E (x)] E [u (x)]. The concept of risk aversion in EUT is closely related to the shape of the vnm utility function, as the next theorem states. Theorem (Risk aversion and concavity - Jensen s inequality). A person described by EUT is (i) risk-averse if and only if his vnm utility function u is strictly concave, (ii) risk-seeking if and only if his vnm utility function u is strictly convex, (iii) risk-neutral if and only if his vnm utility function u is linear.

18 4 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Proof. We only prove that if the vnm utility function u is strictly concave, then the person described by EUT is risk averse. This result is known as Jensen s inequality. We also assume that u is once di erentiable. Suppose that vnm utility function u : R! R is strictly concave. We need to prove that u [E (x)] > E [u (x)] for any lottery. Any tangent line to the graph of a strictly concave function, lies above the graph of the function. Figure. illustrates this graphically. In particular, the gure shows a tangent line g (x) at the Figure.: Strictly concave u and a tangent line at (; u ()). point (; u ()), where E (x) is the mean of a lottery. The equation of the tangent line is g (x) u () + u 0 () (x ), and since it lies above u (x) for all x with the exception of x, we have 8x g (x) > u (x) u () + u 0 () (x ) > u (x)

19 .. EXPECTED UTILITY THEORY Taking expectations: u () + u 0 () [E (x) ] > E [u (x)] u () + u 0 () [ ] > E [u (x)] u [E (x)] > E [u (x)] The proof of the converse, u [E (x)] > E [u (x)] for any lottery ) u is strictly concave, is omitted. One can prove this by contradiction, i.e. assuming that for any lottery u [E (x)] > E [u (x)], but u is not strictly concave and has a convex area. Then construct a lottery for that convex area, so that the lottery is better than its mean, which will lead to the contradiction of u [E (x)] > E [u (x)]. Part (ii) about risk seeking is proved in a similar way, but now the tangent line will be below the graph of u. Part (iii) is the simplest, and is left as an exercise. Theorem established the equivalence between risk aversion and concavity of the vnm utility function. Intuitively, risk aversion is a result of the diminishing marginal utility of concave functions, where a gain of a certain amount increases utility by less than the drop in utility resulting from losing that same amount. Consider again example, where lottery A pays $000 or $00 with equal probabilities. This lottery can be seen as initial wealth of $0 together with an equal chances of winning or losing $0. Looking at gure. with $0, we can see that the increase in utility from additional $0 is smaller than the loss in utility from -$0. As a numerical example, consider the vnm utility function u (x) p x, which is strictly concave. With this utility function, starting from $0, the gain in utility from winning additional $0 is p p :, and the loss in utility from -$0 is p p 0 00 :0. Thus, this risk averse individual prefers $0 with certainty over the risk of winning or losing $0 with equal probabilities. Exercise Let A and B be two lotteries, with the same mean payo. Suppose that lottery B has higher variance than lottery A. Prove that any risk neutral individual will be indi erent between these two lotteries. Solution Do it yourself... Measuring risk aversion and applications We have established that risk averse individuals, whose preferences can be represented with expected utility functional, must have concave vnm utility function u. The question is, which utility functions are suitable? Di erent individuals might have di erent degree of

20 CHAPTER. DECISION THEORY UNDER UNCERTAINTY risk aversion, so how do we capture these di erences with the utility function? The next de nition introduces two ways of measuring the degree of risk aversion. De nition 0 Given a twice di erentiable vnm utility function u : R! R, the Arrow-Pratt coe cient of absolute risk aversion at x is de ned ARA (x) u00 (x) u 0 (x) De nition Given a twice di erentiable vnm utility function u : R! R, the Arrow-Pratt coe cient of relative risk aversion at x is de ned RRA (x) u00 (x) u 0 (x) x Intuitively, the "more concave" the utility function is, the greater should be the degree of risk aversion. Therefore, both measures have the second derivative, which is supposed to capture the degree of concavity. The minus in front makes both measures positive numbers (since the second derivative is negative for strictly concave functions). Recall from proposition that multiplying u by a positive constant, will not change the preferences it represents, and therefore the degree of risk aversion should not change as well. This is the reason why both have u 0 (x) in the denominator - to eliminate the e ect of multiplication by a positive constant. The absolute risk aversion is relevant for choices involving absolute gains and losses from current wealth, while the relative risk aversion is relevant for choices involving percentage (or fractional) gains or losses of current wealth. The Arrow-Pratt measures of degree of risk aversion allow us to establish whether one individual is more risk-averse than another individual. That is, individual is more risk averse than individual if for every x we have ARA (x) > ARA (x) or RRA (x) > RRA (x) for x > 0. There are several other, equivalent ways, of making the same comparison across individuals. Recall from de nition 8 that the Certainty Equivalent (CE) of a lottery is the non-random payo that gives the same utility as playing the lottery, CE L When preferences are represented by expected utility, the certainty equivalent is de ned by the following equation: u (CE) E [u (x)]

21 .. EXPECTED UTILITY THEORY We can think of certainty equivalent of a lottery as the maximal amount that an individual is willing to pay for that lottery. We de ned a risk averse individual as someone who prefers the expected payo of a lottery over the lottery itself: u [E (x)] > E [u (x)]. Since u is increasing function, this implies that for risk averse individuals, the certainty equivalent of any lottery is smaller than the mean of the lottery: CE < E (x). We can now say that individual is more risk averse than individual if CE < CE for any lottery. Intuitively, a more risk averse individual is willing to pay less for the same lottery. A closely related concept to certainty equivalent is risk premium. De nition A risk premium RP for a given lottery is the amount that an individual is willing to pay out of the expected payo in order to avoid playing the lottery, and instead receive its mean with certainty: u [E (x) RP ] E [u (x)] Notice that on the left side we have utility from certainty equivalent. That is, the risk premium is just the di erence between the mean of a lottery and its certainty equivalent, i.e. RP E (x) CE. Therefore, an individual is more risk-averse if the risk premium he is willing to pay for avoiding any lottery is higher. Yet another equivalent way to compare individuals according to their degree of risk aversion, is by letting the more risk averse vnm utility function be "more concave". In other words, the vnm utility function u is more risk averse than u if u is obtained through an increasing and concave transformation of u. In other words, u (x) (u (x)) for some increasing and concave function (). It turns out that all the above ways of de ning more risk aversion are equivalent, as the next theorem states. Theorem 4 (More Risk Aversion Equivalence Theorem). The following de nitions of "more risk aversion" are equivalent. An individual with vnm utility function u is more risk averse than individual with vnm utility function u, if (i) ARA (x) > ARA (x) or RRA (x) > RRA (x) for every x > 0, (ii) There exists an increasing and concave function () such that u (x) (u (x)), (iii) CE < CE for any lottery, (iv) RP > RP for any lottery. Proof. First, we will prove that (i) and (ii) are equivalent. For any increasing utility functions u and u, we can always nd some increasing function () such that u (x) (u (x))

22 8 CHAPTER. DECISION THEORY UNDER UNCERTAINTY In particular, we can de ne (u) u u (u), since an inverse of increasing function always exists. The rst and second derivatives of u are: u 0 (x) 0 (u (x)) u 0 (x) u 00 (x) 00 (u (x)) (u 0 (x)) + 0 (u (x)) u 00 (x) Dividing both sides by u 0 (x) > 0 (remember that all vnm utility functions are assumed increasing), and then multiplying by : u 00 (x) u 0 (x) u 00 (x) u 0 (x) ARA (x) 00 (u (x)) (u 0 (x)) + 0 (u (x)) u 00 (x) 0 (u (x)) u 0 (x) 00 (u (x)) 0 (u (x)) u0 (x) u 00 (x) u 0 (x) 00 (u (x)) 0 (u (x)) u0 (x) + ARA (x) Notice that the rst expression on the right is positive whenever () is concave function. Thus, ARA (x) > ARA (x) if and only if () is concave. Next we prove that (ii) and (iii) are equivalent. Suppose that u (x) (u (x)) for some increasing and concave function (), and we prove that CE < CE. Then E [u (x)] E [ (u (x))] < [E (u (x))] The inequality above is the familiar Jensen s inequality (see theorem ), which applies because is concave. Next, using the de nition of certainty equivalents u (CE ) E [u (x)] and u (CE ) E [u (x)] in the above inequality, gives u (CE ) < [u (CE )] u (CE ) Since u is increasing, we must have CE < CE. To prove the converse, suppose that CE < CE, and we will prove that () must be concave. Since u () is increasing, we have u (CE ) < u (CE ) For example, suppose that u u (x) p x. Then x u (u) u.

23 .. EXPECTED UTILITY THEORY 9 By de nition of certainty equivalent, the above inequality is E [u (x)] < u (CE ) As before let u (x) (u (x)), for some increasing function (). above inequality, gives E [ (u (x))] < (u (CE )) Substituting in the Again, by de nition of CE, we have u (CE ) E [u (x)]. Plug this in the last inequality E [ (u (x))] < [E (u (x))] Thus, by Jensen s inequality (theorem ), the function () must be concave. Finally, one can see immediately that (iii) and (iv) are equivalent CE E (x) RP CE E (x) RP Subtracting the above CE CE (RP RP ) which implies that CE < CE () RP > RP Exercise Based on the above theorem, show that individual with vnm utility function u (x) ln (x) is more risk averse than individual with u (x) p x. You should be able to answer this question in more than one way. Solution Do it yourself. Exercise Find the certainty equivalents and risk premia of the lottery, ( $000 w.p. L $00 w.p. for the following vnm utility functions: (i) u (x) p x, (ii) u (x) ln (x).

24 0 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Solution Do it yourself. The di erent de nitions of risk aversion and of more risk aversion help us prove some testable predictions about the behavior of risk averse individuals. For example, even risk averse individuals will invest some of their wealth in risky assets, provided that the return of the risky assets is high enough. However, the amount invested in risky assets is smaller for more risk averse individuals. Investment in risky asset Suppose that an individual, with preferences represented by expected utility, has wealth w and can invest an amount x [0; w] in a risky asset with random net return r (return in addition to risk-free return). Then, after one period, the wealth of the investor is w x + x ( + r) w + rx. The optimal investment problem is: max E [u (w + rx)] 0xw The rst order condition for interior optimum, x, is E [u 0 (w + rx ) r] 0 Second order condition for interior optimum E u 00 (w + rx ) r < 0 The second order condition is satis ed if u 00 () < 0, i.e. if the investor is risk averse. This condition guarantees a unique global maximum of the objective function (expected utility). Proposition Any risk-averse investor will invest positive amount in the risky asset, if the expected return on the asset is positive. Proof. Figure. illustrates the optimal investment in risky asset, for a risk-averse investor. We know that the investor is risk averse because the shape of the objective function (expected utility) is strictly concave. In the gure, the optimal investment is positive. The optimal investment is strictly positive if and only if the objective function at x 0 is increasing, which is like saying that the rst derivative is positive at x 0: E [u 0 (w + rx) r] jx0 > 0 E [u 0 (w) r] > 0 u 0 (w) E (r) > 0

25 .. EXPECTED UTILITY THEORY Figure.: Investment in risky asset. Since the vnm utility function is increasing, we must have u 0 (w) > 0, and therefore the expected return must be positive. Example 4 We can modify the above example, by adding a risk-free asset with sure return of r f, and then we will see that a risk-averse investor will invest positive amount in the risky asset if its expected return is higher than the return on the risk-free asset. The wealth of the investor after one period is (w x) ( + r f ) + x ( + r) w ( + r f ) + x (r r f ) Here we assumed that any amount not invested in the risky asset is invested in the risk-free asset. The optimal investment problem is: max E [u (w ( + r f) + x (r r f ))] 0xw The rst order condition for interior optimum, x, is E [u 0 (w ( + r f ) + x (r r f )) (r r f )] 0

26 CHAPTER. DECISION THEORY UNDER UNCERTAINTY And the condition of positive investment in risky asset, x > 0, becomes: E [u 0 (w ( + r f )) (r r f )] > 0 u 0 (w ( + r f )) E (r r f ) > 0 Which holds if and only if E (r) > r f In other words, any risk averse investor will invest some positive amount in the risky asset, as long as the expected return is greater than the return on the risk-free asset. In the examples above, we established that even risk-averse investor will invest some positive amount in a risky asset, provided that the expected return on the risky asset is su ciently high. We now show that investors who are more risk averse, will invest less in the risky asset. Proposition 4 Optimal investment in risky asset is decreasing with the degree of risk aversion. In particular, suppose individual with vnm utility function u is more risk averse than individual with u. Suppose that the individuals are identical in other respects, i.e. they both have the same initial wealth and consider investment in the same risky asset. We will show that their optimal investments are x < x, i.e. the more risk averse person, will invest less in a risky asset (ceteris paribus). Without loss of generality, the analysis below is for the case r f 0. Proof. Let f (r) be the pdf or the risky return r. We write the rst order conditions for interior optimum for both individuals explicitly, as integrals: Z E [u 0 (w + rx ) r] u 0 (w + rx ) rf (r) dr 0 (.) Z E [u 0 (w + rx ) r] u 0 (w + rx ) rf (r) dr 0 Since u is more risk averse than u, theorem 4 established that there exists an increasing and concave function () such that u (x) (u (x)). Now we evaluate the slope of the

27 .. EXPECTED UTILITY THEORY objective function of individual, if he invested x, and we claim that this slope is negative: Z E [u 0 (w + rx ) r] 0 (u (w + rx )) u 0 (w + rx ) rf (r) dr Z 0 Z (u (w + rx )) u 0 (w + rx ) rf (r) dr 0 (u (w + rx )) u 0 (w + rx ) rf (r) dr < 0 The last step breaks the integral into two parts, where the rst integrates over negative returns r < 0, and the second integrates over the positive returns. The integrand in the last inequality is the same as in (.), except that it is multiplied by 0 (u (w + rx )). Since () is concave, 0 () is a decreasing function, so the term 0 (u (w + rx )) is larger when r < 0 then when r > 0. Since the integral in (.) is equal to zero, it must now be negative when we weigh the negative terms more than the positive ones. Demand for insurance Suppose that an individual, with preferences represented by expected utility, has wealth w and faces a risk that with probability his wealth will sustain damage d [0; w], and with probability his wealth remains intact. The individual can buy insurance with premium p per unit of wealth insured, and he needs to choose the amount of coverage q to purchase, where 0 q d. Assuming that the utility over certain outcomes is represented by u (), which is strictly increasing and strictly concave, the individual s expected utility is E [u] u (w d pq + q) + ( ) u (w pq) Notice that in the case of damage occurring, he still needs to pay the premium, but he receives a compensation at the amount of his coverage q. His problem is therefore max u (w d pq + q) + ( ) u (w pq) 0qd The rst order condition for interior solution is: u 0 (w d + q ( p)) ( p) ( ) u 0 (w pq ) p 0 There must be some chance of negative returns here because otherwise both individuals would invest all their wealth in the risky asset. To see this, note that if r > r f, the objective function is always increasing in x, i.e. x w (corner solution).

28 4 CHAPTER. DECISION THEORY UNDER UNCERTAINTY The second order condition is u 00 (w d + q ( p)) ( p) + ( ) u 00 (w pq ) p < 0 Thus, since the individual is risk averse (utility is strictly concave i.e. u 00 < 0), the second order condition is satis ed. For interior solution, we must have u 0 (w d + q ( p)) ( p) ( ) u 0 (w pq ) p u 0 (w d + q ( p)) ( ) p u 0 (w pq ) ( p) An important general result is that if insurance is actuarially fair, then a risk averse person will buy full coverage, i.e. q d. An actuarially fair insurance requires that p, in which case the insurance company makes zero expected pro t: E () pq u 0 (w d + q ( p)) u 0 (w pq ) w d + q ( p) w pq q d q 0. Thus If p >, the insurance company makes positive pro t and risk averse individuals will not buy full coverage, and possibly will not buy any insurance at all if the premium is too high 4. In order to nd exactly how much insurance is purchased, we need to know the function u (). Proposition Prove that only risk averse individuals will ever buy insurance. Proof. We consider only the cheapest possible insurance - actuarially fair (i.e. p ). Individuals who won t buy actuarially fair insurance, will de nitely not buy a more expensive one. With coverage q [0; d], and with p, the wealth is equal to the following lottery: L ( w d q + q w.p. w q w.p. We will show that the objective function (expected utility) is decreasing in q for risk seeking individuals, as illustrated in gure.. 4 This might explain why many Americans do not purchase health insurance. Cheaper than actuarially fair insurance leads to negative pro t of insurance companies, and therefore is not sustainable.

29 .. EXPECTED UTILITY THEORY Figure.: Demand for insurance: Risk Seeking Individual. The slope of the objective function (expected utility) is u 0 (w d q + q) ( ) ( ) u 0 (w q) ( ) [u 0 (w d q + q) u 0 (w q)] For risk-seeking individuals, the vnm utility function is convex, and thus the marginal utility is increasing: u 0 (x ) < u 0 (x ) () x < x. Thus, u 0 (w d q + q) u 0 (w q) < 0 () w d q + q < w q q < d At q d (full coverage), the expected utility attains its global minimum, as shown in gure.. Thus, the optimal coverage for any risk averse individual is q 0, i.e. no insurance. For risk-neutral individuals, the vnm utility function is linear, i.e. u (x) a + bx, for a R, and some b > 0. The expected utility is then: E [u] (a + b (w d pq + q)) + ( ) (a + b (w pq)) a + bw bd + b ( p) q

30 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Notice that with actuarially fair insurance, p, the expected utility does not depend on coverage: E [u] a + bw bd, and risk-neutral individual is indi erent between all levels of coverage q [0; d]. With realistic insurance, with p >, notice that expected utility is decreasing in q, and therefore the optimal coverage of a risk-neutral individual is q 0, i.e. no insurance. Exercise 8 Consider two lotteries: X U [0; ] and Y ( :4 w.p. 0:8 0:0 w.p. 0: This means that X has a continuous uniform distribution on the interval [0; ]. (i) Calculate the mean and variance of both lotteries. (ii) Which lottery is preferred by an individual whose vnm utility function is u (x) ln (x)? (iii) Based on your answers to (i) and (ii), is it true that any risk averse individual, when comparing two lotteries with the same mean, would always prefer the one with the lower variance? Solution 8 Do it yourself.. Mean-Variance Theory Expected utility theory has several advantages. First, it is derived from precise axioms about human behavior, so users of the theory know exactly what assumptions about preferences make the EUT valid. Second, the expected utility theory helped us understand why people are usually unwilling to pay for a lottery the expected value of its payo s - a behavior known as risk aversion. EUT explains why people buy insurance, while at the same time invest in risky assets. We have seen applications of EUT to optimal investment and demand for insurance. Notice however that in choosing optimal investment in risky asset, we assumed that the entire distribution of returns is known. In a more realistic setting, of choosing optimal portfolio with many risky assets, we would need to know (or estimate) the joint distribution of all asset returns - a monumental task. This is why in 9 Harry Markowitz introduced a theory of portfolio selection based on the simplifying assumption that all risky alternatives can be summarized by two numbers - mean and variance - the Mean- Variance Theory (MVT). In the next chapter we will learn the details of Markowitz portfolio selection theory, but in this section we discuss the assumptions behind the mean-variance

31 .. MEAN-VARIANCE THEORY analysis. In particular, how good is the key assumption of the MVT, that in evaluating risky alternatives people only care about the mean and variance of the returns? We also ask, under what assumptions the MVT is as valid as the EUT. De nition We say that a utility functional U : L! R has mean-variance form if there exists a utility function u : R R +! R, such that for every lottery L L U (L) u ; where E (L) and V ar (L), and such u is called the mean-variance utility function. Equivalently, the mean-variance utility function can be written as u (; ), i.e. as a function of mean and standard deviation, instead of variance. In other words, the utility derived from any lottery depends only on the mean and variance of that lottery. Notice the notation u : R R +! R, which means that the mean-variance utility function u maps elements from the two dimensional Euclidian space into real numbers. The second dimension is restricted to non-negative real numbers because variance cannot be negative. Thus, the mean-variance utility function maps vectors (; ) into numbers. The mean-variance utility function is therefore very di erent from the vnm utility function that maps single numbers (quantities of money or payo s) into real numbers. But there are some similarities between the mean-variance utility functions and the vnm utility functions. Similar to monotonicity (more is better) of the vnm utility function u, which requires that it is an increasing function, the mean-variance utility is assumed to be increasing in the mean. De nition 4 A mean-variance utility function u : R R +! R is called monotone if u ( ; ) u ( ; ) for all and all >. It is called strictly monotone if > ) u ( ; ) > u ( ; ). In other words, lotteries with higher mean are better, ceteris paribus (for the same variance). We will always assume that u is strictly monotone. Using mean-variance utility, one can de ne risk aversion in the standard way, just like in EUT, where risk averse individual is one who prefers the mean of any lottery over the lottery itself. De nition A mean-variance utility function u : R R +! R is called risk-averse if u (; 0) > u (; ) for all and all > 0. Similarly, an individual is risk-seeking if > 0 ) u (; 0) < u (; ), while risk-neutrality means that u (; 0) u (; ) 8.

32 8 CHAPTER. DECISION THEORY UNDER UNCERTAINTY The next de nition is a bit stronger assumption than risk aversion. De nition A mean-variance utility function u : RR +! R is called variance-averse if u (; ) u (; ) for all and all <. It is strictly variance-averse if < ) u (; ) > u (; ). Notice that risk aversion is a special case of variance aversion, when 0. Example An example of a typical mean variance utility function is u (; ), > 0 The parameter represents the degree of variance aversion. Notice that in this example u (; ) is strictly variance averse. We can de ne indi erence curves as the set of all and that give the same utility: u, u + Figure.4 illustrates indi erence curves in the (; ) space. Higher indi erence curve represents higher utility level. Also notice that indi erence curves are increasing because higher variance (a negative feature) needs to be compensated with a higher mean (a positive feature). The applications of the mean-variance theory are vast, especially to the portfolio analysis. We now turn to the critique of the mean-variance theory... Critique of the mean-variance theory One problem that is very obvious is that the mean-variance theory cannot always be applied to all lotteries. For example, recall the St. Petersburg Paradox example. The lottery in that example has in nite mean and in nite variance, and there is absolutely no meanvariance utility function which can evaluate such lottery. The axioms in theorem guarantee that for any preferences that satisfy completeness, transitivity, continuity and independence, there exists a vnm utility function that can represent these preferences. Mean-variance representation is not grounded in such axioms, and it is not immediately clear what kind of preferences can be represented with mean-variance utility. The main problem with mean-variance theory is that it does not satisfy First Order Stochastic Dominance.

33 .. MEAN-VARIANCE THEORY 9 Figure.4: Mean-variance indi erence curves. De nition (First Order Stochastic Dominance). Let F () and G () be two cumulative distribution functions describing the distribution of monetary payo s. Distribution F () rst-order stochastically dominates distribution G () if F (x) G (x) for all x Recall that cdf gives the probability that a lottery (or random variable) yields a payo no greater than x, that is F (x) Pr (payo x). This can be written as F (x) G (x), i.e. under F the probability of gettig pay s higher than x is greater than under G, for any possible payo x. Intuitively, every decision maker, regardless of how we represent their preferences, should prefer lottery with F () over the lottery with G (). Example Consider two lotteries: F ( 8 $000 w.p. >< $00 w.p. ; G >: $000 w.p. $400 w.p. 0 w.p.

34 0 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Lottery F rst-order stochastically dominates lottery G. To see this, one needs to check that for any payo x, the probability of getting a higher payo under F is no smaller than under G. x F (x) comparison G (x) x x < 000 < 400 x < 00 0 < 0 x < < x < or F (x) comparison G (x) 0 0 > > > Thus, F (x) G (x) 8x or F (x) G (x) 8x, i.e. F F OSD G. Our intuition tells us that any decision maker, regardless of his attitude towards risk, should prefer a rst-order stochastically dominant lottery, over the dominated one. Meanvariance utility however often violates rst-order stochastic dominance. Example Consider the two lotteries: F ( $ w.p. 0: 0 w.p. 0:9, G n 0 w.p. Notice that F () rst-order stochastically dominates G (). The lottery G pays $0 with certainty, while in lottery F there is a positive chance of getting $. Any decision maker should prefer F over G. Consider a very typical mean-variance utility function u (; ). This utility function prefers G over F. The utility from G is u (0; 0) 0. The mean of F is 0: 0:, and the variance of F is 0: ( 0:) + 0:9 (0 0:) 0:. Thus, the utility from F is u (; ) 0: 0: 0:. Therefore, G is preferred to F. The above example is not out of the ordinary. In fact, for any mean variance utility function one can construct lotteries such that lottery F rst-order stochastically dominates lottery G, but the mean-variance utility chooses G over F. The expected utility theory on the other hand, always satis es rst-order stochastic dominance. Theorem (Expected Utility Theory satis es FOSD). For any two lotteries F and G, such that F rst-order stochastically dominates G, i.e. F (x) G (x) 8x, and for any increasing vnm utility function u (whether concave or convex), we have Z u (x) df (x) Z u (x) dg (x)

35 .. MEAN-VARIANCE THEORY Proof. Integrating by parts (recall R b a u (x) dv (x) [u (x) v (x)]b a R b v (x) du (x) ): a Z Z u (x) df (x) [u (x) F (x)] u (x) dg (x) [u (x) G (x)] Z Z u 0 (x) F (x) dx u 0 (x) G (x) dx Subtracting the above Z u (x) df (x) Z u (x) dg (x) [u (x) (F (x) G (x))] +Z u 0 (x) [G (x) F (x)] dx Assuming that lim x! u (x) (F (x) G (x)) 0 (since lim x! F (x) lim x! G (x), this requires that u (x) does not increase "too fast"), the above reduces to Z u (x) df (x) Z u (x) dg (x) Z u 0 (x) [G (x) F (x)] dx Since F () FOSD G (), G (x) F (x) 0 8x, and since utility is increasing, u 0 (x) > 0 8x. Thus, the right hand side is positive, which means that the left hand side is positive as well. Corollary Suppose a random variable X with cdf F () rst order stochastically dominates a random variable Y with cdf G (). Then, E (X) E (Y ). This is a special case of the above theorem, with u (x) x. Thus, we have shown that the mean-variance theory violates rst-order stochastic dominance, while the expected utility theory always satis es FOSD. As a result, one can show that every strictly monotone and risk-averse Mean-Variance utility function also violates the Independence Axiom. This result however is not very important, because to begin with, we did not have much trust in the Independence Axiom, and experimental work showed that it is violated on many occasions. The violation of the FOSD is a much bigger problem. However, there are some cases, and additional assumptions, under which the MVT arises naturally and is just as valid as the EUT. We discuss these cases in the next section... Validity of the mean-variance theory We discuss three cases when the mean-variance theory arises as a special case of the expected utility theory:. The vnm utility function u is quadratic, i.e. u (x) x bx, b > 0,

36 CHAPTER. DECISION THEORY UNDER UNCERTAINTY. The payo s are normally distributed, i.e. x N (; ),. The risks are small, so the vnm utility function can be locally approximated by a quadratic function. Quadratic vnm utility function Theorem Let % be a preference relation on L which can be described by expected utility with vnm utility function u. If u is quadratic, then there exists a mean-variance utility function v (; ) which also describes %. Proof. Suppose u (x) x bx, b > 0. Then expected utility from any lottery is given by E [u (x)] E x bx E (x) be x b + v (; ) Here we used the fact that V ar (x) E (x ) E (x). Exercise 9 Prove that the above mean-variance utility function, v (; ) b [ + ], is (i) Not monotone, i.e. it is not always increasing in, (ii) Decreasing in, i.e. variance-averse. Solution 9 v (; b This is positive when <, and negative when >. Monotonicity requires that v (; ) b b is always increasing in. v (; ) b < There are drawbacks to assuming that vnm utility function is quadratic. For starters, it is not monotone, and becomes decreasing for payo s that exceed certain amount. As we have seen, this results in a non-monotone mean-variance utility function as well. Moreover, experimental work nds very little support for quadratic vnm utility function. Exercise 0 Let the vnm utility function be quadratic u (x) x increasing absolute and relative risk aversion. bx. Show that it exhibits

37 .. MEAN-VARIANCE THEORY Solution 0 The rst and second derivatives Thus, u 0 (x) bx u 00 (x) b ARA (x) RRA (x) u 00 (x) u 0 (x) b bx b bx bx bx b x b We see that ARA (x) is increasing since x is in the denominator with negative sign. Similarly, after rewriting RRA (x) we can also see right away that x is decreasing the denominator, and so RRA (x) is increasing in x. The last exercise gives us another argument against quadratic vnm utility, because evidence shows that risk aversion, at least the absolute, is decreasing with wealth. Normally distributed payo s Theorem Let % be a preference relation on L which can be described by expected utility with vnm utility function u. Suppose that payo s are normally distributed, i.e. x N (; ), with pdf " f (x) p exp x #, < x < Then, there exists a mean-variance utility function v (; ) which describes % for all normal lotteries. Proof. The expected utility from any lottery with normal distribution is E [u (x)] Z Z u (x) f (x) dx u (x) " p exp x # dx v (; ) which is a function of and only. The above result extends to distributions that are "relatives" of normal distribution, for example, the LogNormal distribution, or any distribution that is completely determined by

38 4 CHAPTER. DECISION THEORY UNDER UNCERTAINTY its mean and variance. Exercise Prove that if payo s are normally distributed and if vnm utility function u is strictly increasing and strictly concave, then the mean-variance utility function, v (; ), derived from expected utility is (i) Increasing in, i.e. monotone, (ii) Decreasing in, i.e. variance-averse. Solution We can change the variable x to standardized variable z x N (0; ) The inverse transform is then x + z. Then, the mean-variance utility function becomes v (; ) Z u ( + z) p exp z dz The integral is computed with respect to the standard normal distribution. Now, v (; ) u 0 ( + z) p z dz > 0 The sign is positive because u 0 (x) > 0 for all x, since u is strictly increasing. Next, Now the sign is not obvious because v (; ) u 0 ( + z) z p z dz < z <, so half of the values of z are negative. Note that if we replace u 0 ( + z) by a constant c > 0, then the integral is zero: Z c z p exp z dz 0 This is because the mean of N (0; ) is zero. The function u 0 ( + z) puts more weight on < z < 0 than on 0 < z < because u 0 () is decreasing function since it is given that u () is strictly concave. Thus, u 0 ( + z) is larger for < z < 0 than for 0 < z <, and the integral has to be negative. While sometimes the assumption of normal returns is valid, there are many important cases in nance in which the returns are clearly not normally distributed. The most common We say that Y has LogNormal distribution, Y LN ;, if ln (Y ) N ;. In other words, a LogNormal random variable is obtained through exponentiation of a normal random variable: Y exp (X), where X N ;.

39 .. MEAN-VARIANCE THEORY case is options. Small risks Suppose that you have initial wealth w, to which we add a small risk z, with E (z) 0 and V ar (z), so the total wealth is random: w + z. We can always make the assumption that z 0, because if the risk was y with y 0, we could write the resulting wealth as w + y + y y, and then rede ne the initial wealth as w + y and the mean-zero risk z y y. Therefore, the mean and variance of the nal wealth are: E (w + z) w + E (z) w V ar (w + z) V ar(z) Suppose that preferences are represented with expected utility, and with vnm utility function u (). We show that the expected utility when z is small, can be closely approximated with mean variance utility function v (; ). The vnm utility function u (w + z) can be closely approximated with second-order Taylor approximation around w, when z is small: Taking expectations: u (w + z) u (w) + zu 0 (w) + z u00 (w) E [u (w + z)] u (w) + E (z) u 0 (w) + E (z ) u 00 (w) Since E (z) 0, we have E (z ) V ar (z). Thus, the above becomes E [u (w + z)] u (w) + u00 (w) u () + u00 () v (; ) Thus, the expected utility functional E [u (w + z)] can be approximated with a mean-variance utility function, provided that the risk is small (so that w + z is not too-far from w). Exercise Prove that if vnm utility function u is strictly increasing and strictly concave, then the mean-variance utility function, v (; ), derived from approximating the expected utility functional is variance-averse. Is it also monotone? Solution It is straightforward to verify that v (; ) v (; ) u00 () < 0

40 CHAPTER. DECISION THEORY UNDER UNCERTAINTY Here we use the given that u () is strictly concave, u 00 (x) < 0 8x. It is less clear that v (; ) is monotone, i.e. increasing v (; ) u0 () + u000 ()? > 0 We always assume that u () is strictly increasing, u 0 (x) > 0 8x, but the sign of the derivative of v (; ) with respect to depends on the third derivative of u. If we assume that u 000 (x) 0, which is the case for most vnm utility functions encountered in practice, then indeed v (; ) is monotone. Exercise Suppose that preferences have EUT form, with vnm uitlity function u (x). A small risk z with E (z) 0 is added to wealth w, so the wealth with the risk is w + z. Here the risk leads to gains and losses of absolute amounts. Show that the risk premium is approximately Solution u (CE) u (w RP ARA (w) risk premium. First order Taylor expansion of u (w RP ) E [u (w + z)] by de nition of certainty equivalent and u (w RP ) u (w) RP u 0 (w) RP ) around RP 0, gives We have shown above that u (w + z) u (w) + zu 0 (w) + z u00 (w) Plugging these approximations into the de nition of certainty equivalent, gives: u (w RP ) E [u (w + z)] u (w) RP u 0 (w) u (w) + u00 (w) RP u 0 (w) u00 (w) RP u 00 (w) u 0 (w) ARA (w) Exercise 4 Suppose that preferences have EUT form, with vnm uitlity function u (x). A small risk z with E (z) 0 is multiplicative, i.e. the wealth with the risk is given by w ( + z). In this example the risk leads to gains and losses that are a fraction z of wealth. Show that the relative risk premium, rp (which is a fraction of w the individual is willing to

41 .. MEAN-VARIANCE THEORY pay to avoid the risk z) is approximately rp RRA (x) Solution 4 u (CE) u (w rp w) E [u (w ( + z))] by de nition of certainty equivalent. First order Taylor expansion of u (w rp w) around rp 0, gives u (w rp w) u (w) rp wu 0 (w) The vnm utility function u (w ( + z)) can be closely approximated with second-order Taylor approximation around w (when z is small): Taking expectations: u (w ( + z)) u (w) + zwu 0 (w) + z w u00 (w) E [u (w ( + z))] u (w) + E (z) wu 0 (w) + E (z ) w u 00 (w) Since E (z) 0, we have E (z ) V ar (z). Thus, the above becomes E [u (w + z)] u (w) + w u 00 (w) Plugging the approximations into the de nition of certainty equivalent, gives: u (w rp w) E [u (w ( + z))] u (w) rp wu 0 (w) u (w) + w u 00 (w) rp u 00 (w) u 0 (w) w RRA (w) To summarize, in this section we introduced the widely used in practice Mean-Variance Theory. Its main advantage is simplicity, and the main drawback is that it violates the First Order Stochastic Dominance. However, we have shown that under certain special cases, the MVT is equivalent to the EUT: (i) when vnm is quadratic, (ii) when payo s have normal distribution or any distribution completely determined by mean and variance, (iii) when risks are small.

42 8 CHAPTER. DECISION THEORY UNDER UNCERTAINTY.4 Prospect Theory Recall that Expected Utility Theory is grounded in axioms of what is considered rational behavior. In that sense, the EUT is prescriptive - it prescribes how people should (or supposed to) behave when choosing among risky alternatives. But does the EUT do a good job describing how people actually behave? In many situations the EUT does describe actual behavior pretty well. Nevertheless, there are plenty of experimental and real world examples where the EUT fails. The development of Prospect Theory (Daniel Kahneman and Amos Tversky 99), and the Cumulative Prospect Theory (Daniel Kahneman and Amos Tversky 99) is an attempt to improve on the EUT in order to explain actual behavior with risky alternatives. Thus, Prospect Theory is descriptive attempts to describe actual behavior..4. Motivation The key ingredients of the di erent version of Prospect Theory are: (i) framing, and (ii) probability weighting. Framing refers to the way the particular choices are formulated (framed), and the reference for gains and losses. Probability weighting is the observed tendency of people to consistently in ate low probabilities, and de ate high probabilities. The next example illustrates that people s choices among risky alternatives may depend on the framing of the choices. Example 8 (Framing). Imagine that your country is preparing for the outbreak of a deadly virus, which is expected to kill 00 if no cure is found. You need to choose between two programs: A,B. With program A, 00 people will be saved. With program B there is a / probability of saving all 00 people, and / probability of not saving anyone. Thus, when the choice is framed in terms of numbers of people saved, the two alternatives are: ( n +00 w.p. A +00 w.p., B 0 w.p. The majority (%) of doctors who participated in this experiment chose A over B. This is indeed what the EUT predicts that a risk-averse individuals should choose. The mean of B is 00, so risk-averse individuals should prefer the mean of this lottery (i.e. program A) over the lottery itself. In a di erent experiment, with another group of doctors, the participants had to choose between programs C and D. If program C is adopted, 400 people will die with certainty, and The word prospect has many meanings, but one of them is "probability of success", and in the work of Kahneman and Tversky a prospect has the same meaning as a lottery, or a risky alternative.

43 .4. PROSPECT THEORY 9 if program D is adopted, there is / probability that nobody will die, and / probability that 00 will die. Thus, framed in terms of people dead, the two alternatives are: ( n 0 w.p. C 400 w.p., D 00 w.p. The majority of doctors who participated in this experiment (8%) chose D over C. Here the EUT predicts that risk-averse individuals should again choose the mean of a lottery over the lottery. The mean of D is 400, so the EUT predicts that risk-averse individuals will choose C over D, which is the opposite from what the majority of participants chose. Thus, it seemed like the participants in this experiment were risk-seeking. Notice that A is the same as C, while B is exactly the same program as D. The di erence is in the way these programs are framed - A and B are framed in terms of lives saved (gains) and C and D are framed in terms of numbers of dead (losses). Thus, Kahneman and Tversky modi ed the original vnm utility function so that it is consistent with risk-averse behavior for gains (concave), and risk-seeking behavior for losses (convex). Figure. describes such utility function, which we call value function, and denote by v : R! R. As in EUT, we will always assume that the value function v is monotone increasing. Figure.: Prospect theory value function. Notice that the vnm utility function u : R! R in EUT, was not restricted to be concave or convex, and it potentially could have the shape of the Prospect Theory value function in gure.. However, the EUT is usually applied to nal wealth, which is usually positive. For example, recall the investment in risky asset, example.., with initial wealth w, uncertain return r and amount invested is x. The expected utility theory was applied

44 40 CHAPTER. DECISION THEORY UNDER UNCERTAINTY to the nal wealth, and thus, we maximized E [u (w + rx)]. The prospect theory however abstracts from wealth, and focuses on gains and losses. But nevertheless, the above value function seems more like a re nement of the EUT, rather than an altogether new theory. In addition, Kahneman and Tversky estimated the shape of the value function in lab experiments, and obtained a shape like this: v (x) ( x x 0 ( x) x < 0 with ; ; > 0. Their estimates are 0:88, and :. Despite the experimental evidence that supports risk-averse behavior for gains and riskseeking behavior for losses, there is plenty of real world data that suggests exactly the opposite behavior! For example, millions of people buy state lottery tickets, like "Mega Millions" California lottery, despite the fact that the price of the lottery is much greater than the expected return. A risk-averse individual will never invest in an asset with negative net return (see again the example..). Kahneman and Tversky s explanation for this is that people can be risk-seeking with gains, when the probability of a gain is very small (like Mega Millions lottery). People tend to "in ate" the tiny probability of winning the lottery, and behave as if it was much larger than it actually is. Another real life example, that seems at odds with the shape of the prospect theory value function in gure. is insurance. Real world people spend large fractions of their incomes on health insurance, life insurance, old-age insurance (pensions and retirement plans), property insurance, and other types of insurance. Insurance is related to losses, and we showed that only risk-averse individuals will buy any insurance. Therefore, the enormous demand for insurance suggests that people are risk averse when it comes to losses, which again contradicts the shape of the prospect theory value function in gure.. Kahneman and Tversky s explanation for this is again that people tend to overweight small probabilities, and insurance is usually purchased against small probability events. Thus, if people overweight small probabilities, they can still buy insurance against a small probability event, despite being risk-seeking for losses. In general, Kahneman and Tversky proposed a weighting function w (p) which overweights small probabilities and underweights large probabilities. Figure. illustrates a typical shape of the weighting function. In their experimental work from 99, Kahneman and Tversky estimated the following weighting functions: w + (p) p [p + ( p) ], w (p) p hp + ( p) i

45 .4. PROSPECT THEORY 4 Figure.: Weighting function: 0: where w + (p) and w (p) are weighting functions for gains and for losses. The weighting functions are always assumed to be monotone increasing. Kahneman and Tversky allowed the probability weighting functions to be di erent from gains and losses. Their estimates are 0:, and 0:9. Since these two estimates are similar, we will assume for simplicity that there is only one weighting function, the same for gains and for losses..4. PT v.s. CPT There are many versions of Prospect Theory today, but they all share the properties of (i) framing, and (ii) probability weighting. De nition 8 (Original Prospect Theory 99). For any lottery A L, with outcomes x < x < ::: < x n and probabilities p ; p ; :::; p n, and given the individual s value function v : R! R and his weighting function w : [0; ]! [0; ], the Prospect Theory utility from the lottery A is given by nx P T (A) w (p i ) v (x i ) i This looks similar to EUT, except that the vnm utility function u is replaced by the prospect theory value function v, and instead of the actual probabilities of outcomes p i, the individual uses the weights w (p i ).

46 4 CHAPTER. DECISION THEORY UNDER UNCERTAINTY It turns out that the original Prospect Theory can violate First Order Stochastic Dominance, which is the main drawback of the Mean-Variance Theory. Recall that if lottery A FOSD lottery B, then for any payo x, the probability of getting at least x under A is greater or equal than under B, and we feel that any decision theory should choose dominant lotteries over the dominated. Therefore, the original prospect theory was modi ed, and in 99 Kahneman and Tversky developed the so called Cumulative Prospect Theory (CPT). The new theory assumes that instead of weighting the actual probabilities of payo s, the weighting function applies to the cumulative distribution function F i Pr (payo x i ) P i j p j, when the payo s are ordered in increasing order: x < x < ::: < x n. De nition 9 (Cumulative Prospect Theory 99). For any lottery A L, with outcomes x < x < ::: < x n and probabilities p ; p ; :::; p n, and given the individual value function v : R! R and his weighting function w : [0; ]! [0; ], the Cumulative Prospect Theory utility from the lottery A is given by CP T (A) nx [w (F i ) w (F i )] v (x i ) i where F is the cumulative distribution function of the lottery A, and F 0 0. This seems like a very similar de nition to the original Prospect Theory, and indeed in many cases the resulting utility from a lottery is very similar (P T (A) is similar to CP T (A)). However, the main advantage of the CPT is that it satis es FOSD. Proposition CPT satis es First Order Stochastic Dominance, i.e. for any non-identical lotteries A; B L, with A F OSD B, we must have CP T (A) > CP T (B) Proof. Let F A and F B denote the cumulative distribution functions of lotteries A and B. Both lotteries are assumed discrete, with payo s ordered in increasing order: x < x < ::: <

47 .4. PROSPECT THEORY 4 x n. CP T (A) nx i nx i w F A n i w F A i w Fi A v (xi ) w F A i v (xi ) Xn w Fi A v (xi+ ) i0 n X v (xn ) + i w Fi A v (xi ) Xn w Fi A v (xi+ ) w F0 A v (x ) Xn [v (x i ) v (x i+ )] w Fi A + w () v (xn ) w (0) v (x ) The last step follows from the general property of cumulative distribution functions, i.e. F A (x n ) F B (x n ), and the de nition F A (x 0 ) F B (x 0 ) 0. Due to the monotonicity of the value function v, we have x i+ > x i ) v (x i+ ) > v (x i ), so all the terms in the squared brackets are negative. Thus, CP T (A) > Xn [v (x i ) v (x i+ )] w Fi A + w () v (xn ) w (0) v (x ) i n X i CP T (B) i [v (x i ) v (x i+ )] w Fi B + w () v (xn ) w (0) v (x ) The inequality arises from the stochastic dominance of A over B: Fi B Fi A for all i, and monotonicity of the weighting function w (), which implies that w Fi B w F A i 8i. Since the lotteries being compared are not identical, at least one the the inequalities must be strict. To summarize, the prospect theory (PT or CPT) is more general than the EUT, and seem to be able to resolve some inconsistencies of EUT with empirical and experimental evidence. The disadvantage of the CPT is its complexity, which makes it di cult to apply in practice. Of the three theories we presented in this chapter, the CPT is by far the most complicated, while the MVT is the simplest. The next chapter presents the modern portfolio theory, which is based on the assumption that people choose among risky alternative according to the MVT, i.e. all we need to compare lotteries or nancial assets is the mean and variance of their returns.

48 44 CHAPTER. DECISION THEORY UNDER UNCERTAINTY

49 Chapter Two-Period Model: Mean-Variance Approach We begin our analysis of nancial markets with simplifying assumptions: (i) there are only two periods, and (ii) preferences over risky returns are represented with Mean-Variance Theory (MVT). That is, we assume that the mean-variance utility function is v (; ), where is the mean return and is the variance of the return. Throughout this chapter we assume that the mean-variance utility function is monotone (increasing in ) and variance-averse (decreasing in ). For example, a typical mean-variance utility function is v (; ), which is monotone and variance-averse. We allow investors to have di erent mean variance utility functions. We shall see that the main results regarding optimal portfolio of risky assets do not depend on the spici c utility functions of investors. In later chapters we relax these assumptions, and extend our analysis to many periods and more general preferences.. Mean-Variance Portfolio Analysis Suppose that there are n assets indexed by i ; ; :::; n. The price of an asset i in the rst period is q i and in the second period the asset pays dividend Di 0 and has the price qi. 0 Thus, the total value of the asset in the second period is A 0 i Di 0 + qi. 0 The gross return on the asset is R i A 0 iq i, and the net return is r i R i. In our analysis we focus primarily on the net returns, since most of the examples we will encounter present data on net returns. Mathematically however, the analysis of gross and net returns is identical. From the point of view of the investor, who makes portfolio decisions in the rst period, the return on a 4

50 4 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH given asset, r i, is a random variable, with mean (expected value) and variance: [mean] : i E (r i ) [variance] : i V ar (r i ) [standard deviation] : i p V ar (r i ) The standard deviation (or variance) tells us how volatile the asset returns are. An asset that guarantees a particular return r f with certainty, is called a risk-free asset, and we have E (r f ) r f, and V ar (r f ) 0. For any two assets, i and j, we will also be interested in the degree of comovement of their returns (r i and r j ), which is measured with covariance or correlation: [covariance] : ij Cov(r i ; r j ) [correlation] : ij Cov(r i; r j ) i j We will see that the correlation between assets is key determinant of the degree of risk reduction due to diversi cation (i.e. combining several assets in one portfolio). Speci cally, we will see that the variance of the portfolio depends crucially on the correlations of the asset returns. Since preferences are assumed to be MVT, the mean and variance (or standard deviation) are the only two features of any asset (or portfolio of assets) that investors care about. Thus, any asset can be represented as a point in the mean-variance space, as gure. illustrates. The gure shows assets: a risk-free asset with mean return of % and Figure.: Mean-variance space.

51 .. MEAN-VARIANCE PORTFOLIO ANALYSIS 4 variance of zero, a risky asset with mean return of % and standard deviation of %, and another risky asset with mean return of % and standard deviation of %. The mean returns, variances and covariances are usually estimated based on historical data. Although gure. is called the mean-variance space, we prefer to use (standard deviation) on the x-axis instead of (variance), because standard deviations have the same units as the data, while the units of variance are the original units suqared. For example, if returns are measured as percentages, the units of standard deviation are percentage points. That is, if the standard deviation of asset return is %, then it means that the returns are on average distanced % away from the mean. As another example, if the data represents prices of cars in dollars, then a standard deviation of $0,000 means that on average, the prices of cars are $0,000 away (above or below) from the mean price of a car... Portfolios with two risky assets Suppose an investor divides some amount of his wealth, w, between two assets i and j, such that a fraction [0; ] is invested in i and the rest is invested in j. The net return on a portfolio is (you should prove this): r r i + ( ) r j Since the returns r i and r j are random variables, the return on the portfolio r is also a random variable. Thus, any portfolio composed of risky assets can be viewed as just another risky asset, with random return r, and with mean and variance as follows: E [r i + ( ) r j ] i + ( ) j V ar [r i + ( ) r j ] i + ( ) j + ( ) Cov (r i ; r j ) i + ( ) j + ( ) ij i j q V ar [r i + ( ) r j ] Notice that the mean return on the portfolio depends on the mean returns of the composing assets ( i and j ) as well as the asset shares and. Thus, the mean return is a weighted average of the mean returns on individual assets. For example, if i % and j %, the mean return on any portfolio consisting of these two assets, is between these two returns ( i r j ). By combining several assets in a portfolio, the mean return can never exceed the highest mean return of the assets in the portfolio, and also it cannot fall below the minimal mean return of the assets in the portfolio.

52 48 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH The variance of the portfolio return depends on the variances of returns of the composing assets ( i and j), on the asset shares and, and on the covariance (or correlation) of the asset returns. Therefore, even an investor who cares only about mean and variance of his portfolio (MVT investor), will still need information about the correlations of assets in his portfolio. We assume that asset returns are exogenous to investors, and investors can only choose the portfolio share (or portfolio weight) of each asset, [0; ] (and ) in this case. By varying [0; ], the investors can create in nitely many portfolios, even if there are only two assets available. The set of all possible portfolios can be presented graphically, in the mean-variance space, as the mean-variance opportunity set. De nition 0 For any two assets with returns r i and r j, the mean-variance opportunity set, is the set of all mean and standard deviations of returns on portfolios created from investing a share [0; ] in asset i and share in asset j. Mathematically, the mean-variance opportunity set is de ned as follows: OS ij q ( ; ) R R + j [0; ] ; i + ( ) j ; V ar (r i + ( ) r j ) Figure. describes the mean-variance opportunity set for two assets with: Table.: Two assets Asset i Asset j % % % % ij 0 Each point on the graph represents a portfolio created with a particular. For example, the point (%, %) is created from, which means that the portfolio consists entirely of asset i. Similarly, the point (%, %) is the portfolio corresponding to 0, which means that the entire investment is in asset j. All the other points on the graph represent portfolios with 0 < <, i.e. portfolios containing positive shares of both assets. Notice the very interesting feature of the mean-variance opportunity set. The standard deviation ("risk") of asset i is %, and when combined with a more risky asset j, with standard deviation of %, we are able to create a portfolio with around 4% standard deviation! This is a stunning feature, which is called the diversi cation e ect.

53 .. MEAN-VARIANCE PORTFOLIO ANALYSIS 49 Figure.: Mean-variance opportinity set for two assets. De nition The diversi cation e ect is the reduction in portfolio risk (variance) that results from combining assets with certain statistical (probabilistic) features. Exercise Consider two assets i and j, with mean, standard deviation and correlation of returns from table.. Compute the mean return and standard deviation of return on a portfolio with 0:. Solution The mean return on the portfolio is: i + ( ) j The standard deviation of the portfolio is: 0: % + ( 0:) % 4:% q i q + ( ) j + ( ) ij i j 0: 0:0 + ( 0:) 0:0 4:0% Notice once again that we mixed a lower risk asset i with a higher risk asset j, and the resulting portfolio has even lower risk than asset i. This diversi cation e ect was achieved despite the fact that the assets are not correlated. diversi cation e ect by the following example: Many texts in nance motivate the

54 0 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH "You are invested equally in a company that produces suntan lotion and a company that produces umbrellas. If the summer turns out to be sunny, the rst company does well and the second poorly. In contrast, if the summer turns out to be rainy, the rst company does poorly and the second does well. In other words, their returns are negatively correlated. By investing in both of them, instead of only one of them, obviously you reduce your portfolio risk a lot." This popular example, although intuitive, is very misleading, because it makes students think that negative correlation of asset returns in necessary for risk reduction of portfolios. In the example of assets in table., the returns on two assets are uncorrelated, ij 0. If we change this example to positive, but not too large correlation, say ij 0:, we can still obtain some diversi cation e ect. Although, if the correlation between the two asset returns is positive and large, the diversi cation e ect disappears. Figure. plots the meanvariance opportunity sets, based on two assets in table., for di erent values of correlation of returns: ij 0:, 0, 0:. Notice that indeed, as the correlation between the two assets Figure.: Correlation and mean-variance opportunity set. gets smaller, the diversi cation e ect increases. This means that we can reduce the variance of a portfolio more if we combine assets with more negatively correlated returns. So far, we did not discuss at all about which portfolios an investor might want to choose. We merely described the opportunity set of portfolios that are available to investors. At this point however it is instructive to think about which portfolio would you choose if you had

55 .. MEAN-VARIANCE PORTFOLIO ANALYSIS a mean-variance opportunity set like the one in gure., or opportunity sets like the ones in gure.. While it is premature to decide on the optimal portfolio, it should be obvious that some of the portfolios are clearly inferior - the ones located on a decreasing section of the opportunity set. Assuming that higher return and lower risk are desirable, for any portfolio on a decreasing section of the mean-variance opportunity set, there are portfolios to the left of it which have higher mean return and lower variance. So any mean-variance investor, with monotone and variance-averse preferences, will never choose a portfolio on a decreasing part of the opportunity set. Diversi cation e ect - a closer look Let us take a closer look at diversi cation e ect in portfolios with two assets. The mean and variance of portfolio returns were found to be: i + ( ) j (.) i + ( ) j + ( ) ij i j (.) We discuss three cases of possible correlation between asset returns: (i) perfect positive correlation, (ii) perfect negative correlation, and (iii) imperfect correlation. Case : ij Suppose that assets are perfectly positively correlated. In this case, the portfolio variance in equation (.) becomes i + ( ) j + ( ) i j Recognize the quadratic form a + b + ab (a + b), where a i and b ( ) j. Thus, the variance and standard deviation of the portfolio can be written as: [ i + ( ) j ] i + ( ) j Thus, the standard deviation of portfolio return is a weighted average of the standard deviation of assets i and j, just like the mean return on the portfolio. In this case cannot be smaller than min f i ; j g when short sells are prohibited (i.e. when [0; ]). The mean-variance opportunity set in this case is a straight line connecting the two assets in the mean-variance space (see gure.4, the line labeled ). To prove this

56 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH Figure.4: Range of diverci cation with two assets. claim, i j i j This is a constant slope, which is positive whenever asset with higher return also entails greater risk ( i > j ) i > j 8i; j). Case : ij Now suppose that the two assets are perfectly negatively correlated. In this case, the portfolio variance in equation (.) becomes i + ( ) j ( ) i j Recognize the quadratic form a + b ab (a b) (b a). Thus, the variance is [ i ( ) j ] [( ) j i ]. But the standard deviation must always be nonnegative, so depending on, the standard deviation is the term in the squared brackets,

57 .. MEAN-VARIANCE PORTFOLIO ANALYSIS which turns out to be non-negative. The rst term is non-negative if i ( ) j 0 i j + j 0 j i + j Thus, the standard deviation of the portfolio has two sections: ( i ( ) j if j i + j < [ i ( ) j ] if 0 < j i + j These two pieces of give rise to the portfolio opportunity set labeled in gure.4, which is a broken line. Notice that with j i + j, the portfolio variance is 0. Thus, with two risky assets, that have perfectly negative correlation, we can always nd a portfolio of the two assets, which has zero variance. In the real world, there are of course no such assets that have perfectly negative correlation. Exercise Consider portfolios of two risky assets, with random returns r i and r j, with crenellation ij. (i) Suppose that i %, i %, j % and j %. Calculate the portfolio share invested in asset i, which eliminates the portfolio risk altogether (i.e., the portfolio has zero variance: 0). (ii) Find the slopes of the two linear segments of the portfolio opportunity set. Solution ( i j i + j i j [ i + j ] j i + j + + if j i + j [+] if 0 j i + j Case : < ij < Finally, consider a more realistic case, where the assets are not perfectly correlated. We can nd the global minimum-variance portfolio mathematically i ( ) j + ( ) ij i j 0

58 4 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH This gives the portfolio weight on asset i that minimizes the variance of the portfolio. j i + j i j ij i j ij One also needs to check the second order > 0, which guarantees that indeed have found the minimal variance. In general, this portfolio share does not result in zero variance, i.e. it is in general impossible to eliminate the risk of any portfolio. In gure.4, the curve labeled 0, illustrates the portfolio opportunity set for two uncorrelated assets. The global minimum variance portfolio in this case is the leftmost point on the portfolio opportunity set, and it has a standard deviation of about 4%. In general, the portfolio opportunity set for non-perfectly correlated assets, will be in between the two curves labeled and. Now that we have investigated the portfolio opportunity sets, it is useful to pause and think what portfolios would an investor choose. Visit gure., and think about what portfolio would you choose from the set of all possible portfolios. Notice that the increasing section of the opportunity set represents a tradeo between mean and variance. On the increasing section, you can increase the mean return only at the cost of higher variance. The decreasing section of the opportunity set, on the other hand, depicts portfolios where it is possible to increase the mean return and decrease the risk of the portfolio at the same time - no tradeo. These are dominated portfolios, because if investors want a higher return and lower risk, than for each portfolio on the decreasing section of the opportunity set, there is another portfolio that achieves exactly that. So no mean-variance investor, with monotone and variance averse preferences, will ever choose a portfolio on the decreasing part of the opportunity set. You should think of the opportunity set as conceptually equivalent to the budget constraint from consumer s choice theory in microeconomics. Consumers with monotone preferences would never choose a consumption bundle under the budget line, because there are other bundles on the budget line (frontier), which contain higher quantities of both goods, and "more-is-better". therefore dominated by bundles on the budget line. Bundles underneath the budget line are Exercise Suppose an investor has mean-variance utility function v (; ). She can invest in any portfolio consisting of two risky assets, with random returns r i and r j. (i) Write the asset allocation problem of this investor. (ii) Derive the optimal portfolio for the investor. Make sure that you also check that the second order conditions are satis ed.

59 .. MEAN-VARIANCE PORTFOLIO ANALYSIS Solution (i) max u ( ; ) 0 s:t: i + ( ) j i + ( ) j + ( ) ij i j Substituting the constraints into the objective function, gives max u ( ; ) i + ( ) j i + ( ) j + ( ) 0 ij i j (ii) Do it yourself. Notice that the objective function is quadratic in the choice variable. This means that the rst order condition is a linear function of, and can be easily dolved. Exercise 8 Suppose an investor has vnm utility function u (). She has initial wealth of w, and she can invest in any portfolio consisting of two risky assets, with random returns r i and r j. The joint distribution of the returns is described by the pdf f (x; y). (i) Write the asset allocation problem of this investor. (ii) Discuss the di culties with actually solving such problem. Solution 8 Do it yourself... Portfolios with n risky assets Suppose there are n > risky assets with random returns r ; r ; :::; r n. Let ; ; :::; n be the portfolio shares (weights) of some fund allocated to these assets. The shares must add up to, which is like a "budget constraint": nx i The return on a portfolio with given shares is ; ; :::; n i r r + r + ::: + n r n nx i r i i

60 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH As before, the return on a portfolio is random, and has mean and variance + + ::: + n n nx nx i j Cov (r i ; r j ) i j nx i i i Recall that variance of a random variable can be de ned as the covariance of that random variable with itself. Therefore, the double summation terms for i j captures the individual variance terms of every return r i, multiplied by i. The notation becomes much easier if we express the above in matrix form. Moreover, matrix notation makes programming with Matlab straightforward. The means of all asset returns can be expressed with the n-by- vector : 4. n n The covariance matrix of all individual assets is n-by-n matrix : n n n n n nn Thus, the ijth element is ij Cov (r i ; r j ), and the elements on the diagonal are the individual variances. We introduce two more notations: the vector of portfolio weights, and a vector of -s: 4., n 4. n n With these notations, we can write the budget constraint as: n [BC] : 0 n (.)

61 .. MEAN-VARIANCE PORTFOLIO ANALYSIS The portfolio mean is: and the portfolio variance as: 0 (.4) 0 (.) With more than assets, the mean-variance opportunity set is not a curve, but rather an elliptic shape as in gure.. We have seen that combining two assets in a portfolio, creates Efficient frontier Figure.: Mean-variance opportunity set, n > assets. an opportunity set which is a curve. Even with assets, one can create portfolios with pairs of assets, and that will give three curves. But in addition to pairs, any portfolio consisting of two assets can be considered as another asset, which can be combined with individual assets in yet new portfolios. This is why the opportunity set is connected, i.e. does not have "holes". Imagine that you choose portfolios that give you minimum variance of return for any level of mean return. These portfolios would be on the left boundary of the opportunity set, and are called the minimum-variance frontier. However, only the increasing part of the minimum-variance frontier constitutes the e cient frontier, because any portfolio below the

62 8 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH e cient frontier is dominated by some portfolio on the e cient frontier. In other words, for any portfolio under the e cient frontier, there is another portfolio on the e cient frontier which has higher return and lower variance. Thus, with the opportunity set as in gure., any investor should choose portfolios on the e cient frontier only. The minimum-variance frontier is found by minimizing the variance of portfolio return for any given level of portfolio mean, say c. Using the matrix notation, the minimumvariance, for any required level of returns, is found by solving the following problem: min 0 s:t: [BC] : 0 n [portfolio mean] : 0 c The solution to this problem is c, and it gives the portfolio weights that minimize variance for the required mean c. By varying c, we obtain additional portfolios on the minimumvariance frontier. Notice that min i f i g c < max i f i g. To nd the global minimumvariance portfolio, we solve the above problem without the second constraint (i.e. minimizing the variance with only the budget constraint). This is the portfolio located at the tip on the left bound of the opportunity set. If there are short-sale restrictions, additional constraints might be added to the above problem. For example, if short sales are not allowed (you cannot sell a stock that you don t hold), then 0 is another constraint added. Also notice that the minimum variance problem does not require any speci c mean-variance utility function u (; ). There are e cient algorithms developed for solving such optimization problems, which are called quadratic programming (optimizing a quadratic objective, subject to linear constraints). Matlab s function quadprog.m (in Optimization toolbox), is perfect for most portfolio problems. With nonlinear constraints, one needs to use fmincon.m, which solves more general optimization problems. Even Excel s solver can be used for optimal portfolio problems, if there are not too many assets (it is limited to 0 unknowns). Diversi cation e ect with n assets Although general results regarding diversi cation e ect with n assets are hard to obtain, our intuition says that with more assets available, there are more possibilities to reduce the portfolio risk, than with only two assets. The next exercise illustrates that even with uncorrelated assets, it is possible to eliminate the portfolio risk (variance) entirely, if we have many assets (precisely, n! ). The second part of the exercise demonstrates that with

63 .. MEAN-VARIANCE PORTFOLIO ANALYSIS 9 positively correlated assets, the portfolio risk cannot be eliminated completely, even if we have access to in nitely many assets. Exercise 9 Consider a portfolio with n assets, with equal shares of each asset ( i n 8i). (i) Suppose the assets returns are uncorrelated ( ij 0 8 i j) and all assets have the same variance of return i 8i. Find the portfolio variance, and show that it goes to zero as the number of assets increases to in nity. That is, prove that lim n! 0. That is, all the risk can be eliminated if we have many assets. (ii) Still under the assumption that all assets have the same variance of return i 8i, suppose that asset returns are correlated, and all pairs of assets have the same positive correlation, ij > 0 8 i j. Find the portfolio variance, and show that lim n! > 0. Thus, no matter how many assets we have in our portfolio, there is no way to eliminate the risk entirely. Solution 9 (i) The portfolio variance is V ar n r + n r + ::: + n r n n V ar (r + r + ::: + r n ) n n n The last step used the property that for uncorrelated random variables, the variance of a sum is sum of the variances. As n gets large, we have lim n! lim n! n 0 (ii) The portfolio variance is nx nx Cov n r i; n r j nx nx Cov (r n i ; r j ) i j i j " nx # " nx nx nx V ar (r n i ) + Cov (r i ; r j ) + n i i ji i n + n (n ) n n n + n # nx nx i ji

64 0 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH As n gets large, we have lim n! lim n! n + lim n! n n > 0 Thus, with positive correlations, diversi cation cannot eliminate all the risk, no matter how many assets we include in the portfolio. However, diversi cation can signi cantly reduce the risk. For example, suppose that 0:, and we have a symmetric portfolio of n 00 assets. Then, using our result from above, the variance of such portfolio is n + n n : 0:0 Thus, the variance of the portfolio is about half of the variance of individual assets, even when we assume positive correlation of 0:, and even though we used symmetric weights instead of optimal weights! To summarize, in this section we illustrated the shape of the mean-variance opportunity set, and concluded that any MVT investor, with monotone and variance averse utility function, will choose a portfolio on the e cient frontier only. However, we will show in the next section, that investors can in general do much better than the e cient frontier, if there exists a risk-free asset. In reality, there are such assets, for example government bonds or treasury bills, that have a guaranteed return over one period... Adding a risk-free asset Suppose that in addition to the n risky assets with random returns r ; r ; :::; r n, we also have a risk-free asset with guaranteed return r f. Let the portfolio share in the risk-free asset be 0, and the shares in other risky assets be as before [ ; ; :::; n ] 0. The "budget constraint" on the weights is 0 + nx i i Notice that the sum of weights on risky assets is P n i i 0. For example, the portfolio 0 0:4, [0:; 0:; 0:] 0 consists of 40% investment in risk-free asset, and 0%, 0% and 0% investment in risky assets, and. Thus, 0% of the portfolio is invested in risky assets. If we refer to the portfolio of the risky assets as a separate portfolio, then the weights have to add up to 00%, and this is achieved by dividing by the sum of its weights: P n i i 0: 0 0: ; 0: 0: ; 0: 0 0:

65 .. MEAN-VARIANCE PORTFOLIO ANALYSIS Consider a portfolio which combines a fraction 0 invested in the risk-free asset with 0 invested in some portfolio p consisting of the other n risky assets only. Notice that if 0 < 0, the investor borrows money at the risk-free return. The return on this new portfolio is: r 0 r f + ( 0 ) r p with mean and variance: r 0 r f + ( 0 ) p r ( 0 ) p, r ( 0 ) p Thus, combinations of the risk-free asset with any other portfolio of risky assets are located on the line that connects the risk free asset and this other portfolio p (you should prove it). In gure. we see that the best investment opportunities are created when we combine the risk-free asset with the tangent portfolio on the e ciency frontier (portfolio T). The line CML T Figure.: Capital Market Line (CML) that connects the risk-free portfolio with the tangent portfolio is called the Capital Market

66 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH Line (CML). The slope of the capital market line is called the Sharpe ratio, and is equal to r f SR T (.) T The Sharpe ratio gives the tradeo between risk and return, i.e. the excess return that investors can get over the risk-free asset, for every additional unit of risk (the standard deviation). Example 9 Suppose that T %, r f %, T 0%. Then, the Sharpe ratio is SR T r f T 0 0: This means that an increase in risk of a portfolio (standard deviation) by % is compensated with a 0.% higher expected return. Notice that with the risk-free asset, investors have better set of portfolios to choose from (CML) than the e cient frontier, because the CML lies above the e cient frontier, and coincides with the e cient frontier only at the tangent portfolio. All investors will therefore choose some portfolios on the CML. Those who desire less risk, will invest a greater proportion of their wealth in the risk-free asset ( 0 is greater). These conservative investors will choose portfolios on the CML that are close to r f. Others, more aggressive investors will choose smaller 0 or even 0 < 0 (i.e. borrow at the risk-free rate, if they can). These, more aggressive portfolios, are located higher on CML, and if 0 < 0, will be above the tangent portfolio. However, regardless of investors preferences, all of them will invest in the same portfolio of risky assets - the tangent portfolio. This result is called the Two-Fund Separation Theorem. Theorem 8 (Two-Fund Separation). If investors preferences are Mean-Variance, and if a risk-free asset exists, then all optimal investments combine some fractions of two funds only: (i) the risk-free asset, and (ii) a single optimal portfolio of risky assets - the tangent portfolio. In other words, the theorem says that all investors should choose portfolios on the Capital Market Line. The theorem also implies that all investors will hold the same portfolio of risky assets - the tangent portfolio. In order to make optimal investment choices, we now need to nd the tangent portfolio (which is the same as nding the Capital Market Line). Harry M. Markowitz, Merton H. Miller, William F. Sharpe were awarded Nobel Memorial prize in economics in 990, "for their pioneering work in the theory of nancial economics". For more information, visit:

67 .. MEAN-VARIANCE PORTFOLIO ANALYSIS The easiest way of nding the tangent portfolio is solving any utility maximization prob- lem, for example, maximizing u ( ; ). It doesn t matter which utility function we pick, because according to the Two-Fund Separation Theorem, all mean-variance investors will buy the same portfolio of risky assets - the tangent portfolio T. Thus, the mean and variance of return of an investor who allocates 0 of his wealth to risk-free asset (and the rest [ ; ; :::; n ] 0 are invested in the risky assets of the tangent portfolio) are: 0 r f Notice that the risk-free asset does not contribute to the portfolio variance. maximization problem is therefore: The utility max 0 ; u ( ; ) 0 r f n s:t: 0 We can plug the budget constraint into the objective function, to get: max [ 0 n ] r f + 0 max r f 0 n r f + 0 max r f + 0 ( n r f ) The constant r f does not a ect the maximization, so it can be dropped from the objective function. Also, since 0 ( n r f ) ( r f n ) 0 (true for any two n vectors), we write the above problem as: max ( r f n ) 0 0 (.) The above problem has almost the same format as required by the Matlab function quadprog. In particular, quadprog solves problems of the form:

68 4 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH min 0.*x *H*x + f *x x subject to: A*x < b Aeq*x beq lb < x < ub (.8) Notice that quadprog solves minimization problems, while the problem we want to solve in (.) is maximization. However, maximizing any function f(x) is equivalent to minimizing f (x) - the optimal x is always the same. So we can write the problem in (.) as min 0 ( r f n ) 0 Then, by letting x, H, and f ( r f n ), we get exactly the same problem as in (.8) min 0 H + f 0 Without any constraints, the solution to a quadratic problem (such as Ordinary Least Squares) is a system of linear equations, and is usually expressed in matrix form. still needs to solve the linear system of n equations in order to obtain the portfolio shares [ ; ; :::; n ] 0. The other optional constraints in (.8) allow accommodating various practical constraints on traders in nancial markets. For example, if a traders are not allowed to short-sale assets, then setting lb 0 (lower bound on shares), will accommodate this restriction. If there is an upper bound restriction of certain assets, say no more that % in asset, then setting ub() 0. will accommodate this restriction. Borrowing constraint at the risk-free rate can be accommodated by setting utilizing the constraint A*x < b in (.8). For example, suppose that investors cannot borrow more than 40% of the funds at the risk-free rate, which means the cannot incest more that 40% of their own wealth in risky assets (i.e. cannot leverage more than.4). Thus, the 0 0:4 ) 0 P n i i :4. This can be programmed by setting A [; ; :::; ] and b :4. Matlab s quadprog can easily solve the problem in (.8) numerically, and almost any conceivable practical constraint can be imposed. After the problem in (.) of in (.8) is solved, let the solution be opt. Notice however that the sum of all the shares in opt adds up to One 0, not to, and therefore opt are not the weights of the tangent portfolio T. We solved an arbitrary utility maximization problem, and we know that the solution is on the CML, but not necessarily at the tangent portfolio.

69 .. MEAN-VARIANCE PORTFOLIO ANALYSIS The tangent portfolio T consists of only risky assets and therefore the weights of all the risky assets must add up to. Therefore, the weights of the tangent portfolio are obtained by normalizing the weights as follows: T i opt i P n j opt j opt i (.9) 0 Also observe that opt depends on the parameter, which describes the degree of variance aversion of this particular investor. However, when we normalize by the sum of opt, the parameter must cancel out. Thus, the tangent portfolio, should not depend on any preference parameter of particular investors, which is the main point of the Two-Fund Separation Theorem. Exercise 0 Consider asset market with risk-free return r f % and two risky assets with returns r ; r, and mean returns and covariance matrix as follows: " # " # " %, :% # " # % % % 4% Let the share invested in the risk free asset be 0 and shares [ ; ] 0 invested in the risky assets. (i) Calculate the mean and variance of the portfolio with shares 0 ;. (ii) Calculate the Global Minimum-Variance portfolio of risky assets. (iii) Calculate the Tangent Portfolio. (iv) Find the Capital Market Line (CML). Solution 0 (i) The return on a portfolio with shares 0 ; is: r 0 r f + r + r The mean and variance of the portfolio: 0 r f % + % + :% + + Cov (r ; r ) % + 4% + ( %) (ii) You are asked to nd the global minimum variance portfolio of risky assets since the risk-free asset has variance of zero, so the portfolio 0, 0 is the global minimum-variance portfolio if risky-free asset is included. The global minimum-variance

70 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH portfolio of risky assets is found by solving: min ; + + Cov (r ; r ) s:t: + Notice that the budget constraint requires that all investment is in the two risky assets. Substituting the budget constraint, gives The rst order condition min + ( ) + ( ) ( ) + ( ) 0 Solving for : Plugging the numbers: 4 ( ) + 4 ( ) 8 8 The mean return, variance and standard deviation of this portfolio can be calculated. + 8 % + :% :9% Cov (r ; r ) % + 4% + ( %) 0: p p 0:8 0:9 4 Notice that the variance of this portfolio is smaller than the smallest variance of individual

71 .. MEAN-VARIANCE PORTFOLIO ANALYSIS assets in this portfolio - diversi cation e ect. (iii) The easiest way to nd the tangent portfolio is to solve a utility maximization of some MVT investor, say with mean-variance utility function u (; ), by choosing optimal portfolio weights 0 ; ;. The budget constraint now is 0 + +, and plugging in the mean return, gives Thus, to maximize utility, we need to solve: ( ) r f + + r f + ( r f ) + ( r f ) max ( r f ) + ( r f ) ; + + The rst order conditions are: Solving the second condition for : [ ] : r f + 0 [ ] : r f + 0 r f 0 r f r f Plugging into the rst order condition for : r f r f r f r f Solving for optimal : opt r f ( ) ( r f )

72 8 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH Similar steps yield the optimal : opt r f ( ) ( r f ) Plugging the numerical values of means and covariances, gives: Solving for 0 : opt opt " 4 # (: ) : ( ) 4 # ( ) 4 ( ) " : opt 0 opt opt : 4: Notice that the greater is, the more variance-averse the investor is, and the greater is the share in the risk-free asset 0. The tangent portfolio is the same for all investors, so the shares of each risky asset out of the total fraction invested in risky assets 0, must be the same for all investors. In other words, opt T ( 0 ), and opt T ( 0 ). Therefore, to nd the tangent portfolio weights, we use the normalization in equation (.9): T opt 0 T opt 0 : 4: 4: : 4: 9 4: 4 9 Notice that the shares in the market portfolio do not depend on the variance aversion parameter. This is a consequence of the Two-Fund Separation Theorem: all investors hold the same portfolio of risky assets - the tangent portfolio. Thus, under the assumption which led to this theorem, the tangent portfolio is the same for every investor. ratio: (iv) To nd the Capital Market Line (CML), we need to calculate its slope, i.e. the Sharpe SR T T r f

73 .. CAPITAL ASSET PRICING MODEL (CAPM) 9 This requires calculating the mean and standard deviation of return on the tangent portfolio: The Sharpe ratio is then: T T + T : 9 :% T T + T + T T ( ) 0:98% 9 q T T p 0:0098 9:8% SR T r f : T 9:8 0:4 The Capital Market Line is: r f + SR + 0:4 This means that along the CML, an increase in risk of a portfolio (standard deviation) by % is compensated with a 0.4% higher expected return. As you can see, nding the optimal portfolios analytically (the CML) is a tedious task, even with only two risky assets. With more that risky assets, it is a mission impossible, and requires the use of computer optimization software. All the calculations in the last example can be performed with the Matlab script CML.m. Use this Matlab program to verify the answers in the above exercise.. Capital Asset Pricing Model (CAPM) In the previous section we derived optimal portfolios for investors whose preferences are described by Mean-Variance Theory (MVT). Without a risk-free asset, we showed that investors will choose portfolios from the e cient frontier of the opportunity set. Under the assumption that there is a risk-free asset, and that investors can borrow and lend at the risk-free rate r f, we showed that all MVT investors will choose portfolios from the Capital Market Line (CML), gure., which is the highest slope line connecting the risk-free asset with the mean-variance opportunity set. Thus, the CML is the set of optimal portfolios such that any investor with MVT preferences will choose from. The implication is that all investors hold the same portfolio of risky assets, called the tangent portfolio T, and the only

74 0 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH di erence between investors is the fraction 0 invested in risk-free asset versus the fraction invested in the tangent portfolio T. Since all investors hold the same portfolio of risky assets, shares of risky assets ; ; :::; n in the tangent portfolio are common to all investors. For example, if IBM stock represents 0.% of investor A portfolio of risky assets, it also represents 0.% of investor B portfolio of risky assets. Suppose there are I investors. The market share of asset i, out of the total market value of all risky assets is: P I j iw j P I j P n i iw j P I i j w j! P I nx j w j i i {z } iw W i where w j is investor j wealth (money) invested in the market for risky assets (say the stock market), and P I j w j W is the total value of all the risky-assets. Therefore, since all the investors hold all the risky assets, the individual share of asset i in the tangent portfolio is also the market share of asset i, and the tangent portfolio T in equilibrium is also the Market portfolio M. Thus, in this section we will refer to the tangent portfolio as the market portfolio. Notice that this discussion implies that the market portfolio contains all assets from all markets. If i 0 for some asset i for some investor j, then by the Two-Fund Separation Theorem all investors don t hold asset i, and therefore asset i will not exist in the market. The Capital Asset Pricing Model, developed by Jack Treynor, William Sharpe, John Lintner and Jan Mossin in the early 0s, builds on the Mean-Variance portfolio analysis in the last section (by Markowitz), and shows its implications for individual assets returns. The key assumption is that the tangent portfolio is the same as the market portfolio, and therefore must contain all assets from all markets. With this assumption, we will show that the expected return on any asset (here also called security) i (or portfolio of assets) is a linear function of that asset s market risk, i, and the market portfolio risk premium. Obviously, not all investors in reality hold exactly the same portfolio of risky asset, for a number of reasons to be discussed later. However, one should see this model as a benchmark, similar to the perfectly competitive equilibrium, which is the starting point of principles of microeconomics... Deriving the CAPM The starting point is the result that all investors hold the same market portfolio M of risky assets (which is the same as the tangent portfolio T). Moreover, the market portfolio contains

75 .. CAPITAL ASSET PRICING MODEL (CAPM) positive shares of all existing assets. Next we examine the e ect of slightly changing the share of some security i. Consider mixing a small fraction! of some security i with! of the market portfolio. Such portfolios are similar to the market portfolio M, with the share of security i is changed slightly. The combinations of asset i with the market portfolio, for di erent!, are depicted on the mean-variance space in gure. as the curve labeled i. When CML M i Figure.: Market portfolio mixed with asset i.! 0, the i-curve coincides with the market portfolio. For 0 <!, the combination is between M and i. Values of! < 0 mean that we are selling some of the existing holdings of asset i in the market portfolio (it is a good exercise to plot such a curve in Matlab). Notice that the i-curve cannot go outside the e cient frontier since then there would be points on this curve that dominate the e cient frontier, which is impossible. Thus, the slope of the i-curve and the capital market line must coincide at point M. The return on portfolios that combine! of asset i and! of the market portfolio is: r p!r i + (!) r M

76 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH With mean and variance: p! i + (!) M p! i + (!) M +! (!) Cov (r i ; r M ) p! i + (!) M +! (!) Cov (r i ; r M ) The slope of the i-curve in the space, @ i i + (!) M +! (!) im! i (!) M + (!) im As!! 0, portfolio p approaches the market portfolio M, we M Thus, the slope of the i-curve at point M is M + im im M p i M!0 ( im M ) M This slope must be equal to the slope of the CML (Sharpe Ratio, equation (.)): i M ( im M ) M M r f M

77 .. CAPITAL ASSET PRICING MODEL (CAPM) We can now solve for i, the expected return on asset i: i M ( im M ) M M r f M i M im M ( M r f ) M im i M ( M r f ) M i M M + r f + im ( M r f ) M i r f + im ( M r f ) M The above equation is known as the Capital Asset Pricing Model (CAPM), or the Security Market Line (SML): where i r f + i ( M r f ) (.0) i Cov (r i; r M ) M im i M (.) Figure.8 plots the Security Market Line (SML) from equation (.0). Equation (.0) must hold (according to the model) for all securities (or portfolios of securities) in the market. It says that the expected return on any security i (or portfolio of securities) must be the sum of the risk-free rate r f and a risk premium i ( M r f ). The term M r f is known as the market risk premium, i.e. the return on the market portfolio in excess of the risk-free return. The term i is called the Beta risk of security i. If i > 0, then security i moves with the market portfolio (its return is positively correlated with the market return), and therefore it adds to the market risk. In this case, investors should be compensated with higher expected return ( i > r f ). If i < 0, then security i is negatively correlated with the market portfolio, and therefore reduces the market risk. The expected return on such security will be smaller than the risk-free rate. Finally, if i 0, the expected return on asset i is i r f. Thus, according to the model, expected returns on securities di er because they have di erent, i.e. according to their contribution to the risk of the market portfolio. Only assets with positive should have a positive excess return over the risk-free return r f. Also observe that i i im M increases in the standard deviation of asset i relative to the standard deviation of the market portfolio. Thus, more risky assets are supposed to have higher risk premium, according to this theory. Notice that the slope of the SML is the market risk premium m r f. Thus, in times of nancial crisis, when market excess return is negative, people estimate a downward slopping SML curve.

78 4 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH Figure.8: Security Market Line. Example 0 Suppose the market portfolio has an expected excess return of 8% (which happens to be close to the value for the US market return since WWII). An asset with a Beta of 0:8 should then have an expected excess return of :4%, and an asset with a Beta of : should have an expected excess return of 9:%. The Security Market Line (SML) should not be confused with the Capital Market Line (CML), despite the vertical axes measuring the mean return in both graphs. While the CML plots all the optimal portfolios that investors with MVT preferences might choose, the SML on the other hand plots all assets and all portfolios (e cient or not). Any portfolio p can be viewed as a security, and therefore it has it own p. Exercise We mentioned that according to the CAPM theory, all assets and all portfolios are points on the SML, and every asset and every portfolio has some. Consider a portfolio p consisting of 0 share in risk-free asset and a share 0 in the market portfolio (note that such portfolio is located on CML). Find this portfolio s p, and show its location on the SML.

79 .. CAPITAL ASSET PRICING MODEL (CAPM) Solution The return on portfolio p is: r p 0 r f + ( 0 ) r M The Beta of this portfolio, using the de nition of Beta in equation (.), is: p Cov (r p; r M ) V ar (r M ) Cov ( 0r f + ( 0 ) r M ; r M ) V ar (r M ) 0 z } { 0Cov (r f ; r M ) + ( 0 ) Cov (r M ; r M ) V ar (r M ) 0 Thus, the Beta of any portfolio on the Capital Market Line is equal to the share invested in the market portfolio, 0. For example, if the investor borrows at risk-free rate, i.e. 0 < 0, then p >, and the expected excess return on such portfolio will be greater than the market excess return. From the SML, equation (.0), we have: p r f {z} p ( M r f ) > Exercise Consider asset market with risk-free return r f % and two risky assets with returns r ; r, and mean returns and covariance matrix as follows: " # " % :% #, " # " % % % 4% We found in exercise 0 that the market portfolio is M and 9 M 4. Calculate the 9 Betas of the two risky assets, ;, and the Beta of the risk-free asset f. Solution There are two ways of calculating the Betas. One way uses the de nition of Beta in equation (.), we have: Cov (r ; r M ) V ar (r M ) Cov r ; M r + M r V ar M r + M r # M 9 M + M + M + M M + 4 ( ) ( ) 0:9 Recall that we already calculated the variance of the tangent (market) portfolio in exercise

80 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH 0, but I repeat the calculations here for clarity. Similarly, Cov (r ; r M ) V ar (r M ) Cov r ; M r + M r V ar M r + M r M 9 M + M + M + M M ( ) ( ) 99 4 :8 And for the risk-free asset 0 Cov (r f; r M ) V ar (r M ) Another way to calculate the betas is from the SML (since all assets are points on SML, according to the CAPM theory): 0 i r f i ( M r f ) ) i i r f M r f Recall that we calculated the expected market return in exercise 0: Plugging M ; i and r f, we nd: M M + M : 9 :% r f M r f : 0:9 r f M r f : : :8 0 r f r f M r f 0 Thus, if the mean return of asset i is known, and if we know the mean return on the market portfolio and the risk free asset, we can calculate asset i s Beta from the SML. A useful property of Betas is that it is linear. That is, the Beta of a linear combination of assets (or portfolios) is a linear combination of the Betas of the individual assets. We prove this formally in the next proposition. It can help calculating Betas of portfolios, if you have already calculated the Betas of individual assets in that portfolio. Proposition (Linearity of Beta). Let r r ; r ; :::; r n be asset returns, and the corre-

81 .. CAPITAL ASSET PRICING MODEL (CAPM) sponding Betas are ; ; :::; n. Consider a protfolio with wieghts [ ; ; :::; n ] 0, and return r r + r + ::: + n r n. Let be the Beta risk of this portfolio. Then, + + ::: + n n Proof. Applying the de nition of Beta on the portfolio: Cov (r ; r M ) M Cov ( r + r + ::: + n r n ; r M ) M Cov (r ; r M ) + Cov (r ; r M ) + ::: + n Cov (r n ; r M ) M Cov (r ; r M ) Cov (r ; r M ) + M M + + ::: + n n + ::: + n Cov (r n ; r M ) M The above result can be used to calculate Betas of portfolios, when the individual Betas are known, as illustrated in the next example. Example Suppose the betas on three assets are 0:, 0:8, and :4. Calculate the beta on a portfolio, which combines these assets with weights 0:, 0:, 0:. Then, the Beta of the portfolio is + + 0: 0: + 0: 0:8 + 0: :4 0: + 0: + 0: :.. CAPM in practice In this chapter we studied optimal portfolio selection under the assumption that all investors care about is the mean and variance of the return to their investment. In other words, we assumed that preferences are described by the Mean-Variance Theory (MVT). This is not the same as assuming that there is only one investor, or that all investors are identical. In our examples we assumed that the mean-variance utility of individual i is u i (; ) i Thus, there could be unlimited number of investors, which di er by their variance-aversion (or risk aversion) parameter i.

82 8 CHAPTER. TWO-PERIOD MODEL: MEAN-VARIANCE APPROACH Despite the di erences in preferences among individual investors, we arrived at a striking conclusion - the Two-Fund Separation Theorem, which implies that all investors will invest in two funds only: (i) the risk-free asset and (ii) market portfolio. In other words, all investors will hold the same portfolio of risky assets, which we called the tangent portfolio T, and in equilibrium is also the market portfolio M. All optimal portfolios then are located on the Capital Market Line (CML), which connects the risk-free asset with the market portfolio. The only di erence between investors portfolios is the fraction of their nancial wealth invested in the risk-free asset, 0. We have seen in exercise 0 that 0 is increasing in the investor s risk-aversion parameter, i.e. more variance-averse investors will hold a larger portion of their nancial wealth in the risk-free asset. But nevertheless, all investors will hold the same market portfolio of risky assets - M. Moreover, we showed that all assets (and all portfolios for that matter), must lie on the same Security Market Line (SML, or the CAPM equation), given in equation (.0): i r f i ( M r f ) The expected excess return on asset i (or any portfolio) is proportional to its Beta i Cov (r i; r M ) V ar (r M ) im i M In reality, investors di er substantially in the composition of their risky-assets portfolio. Thus in reality, the Two-Fund Separation Theorem does not hold. In addition, some investors (e.g. some hedge funds ) report returns that are above (or below) the expected returns predicted by the SML. These deviations of actual returns from those predicted by the SML are called the asset s Alpha: i i r f i ( M r f ) (.) The term i r f is the expected excess return on asset i, and i ( M r f ) is the predicted excess return by the CAMP (SML). Figure.9 shows two such assets (or portfolios), which deviate from the SML. Asset i has mean return higher than the one predicted by the SML (positive Alpha), while asset j has mean return lower than the predicted by SML (negative A hedge fund is an investment fund that can undertake a wider range of investment and trading activities than other funds, but which is only open for investment from particular types of investors speci ed by regulators. These investors are typically institutions, such as pension funds, university endowments and foundations, or high net worth individuals. As a class, hedge funds invest in a diverse range of assets, but they most commonly trade liquid securities on public markets. They also employ a wide variety of investment strategies, and make use of techniques such as short selling and leverage.

83 .. CAPITAL ASSET PRICING MODEL (CAPM) 9 Alpha). SML M Figure.9: The Alpha of an asset (or portfolio). Our rst instinct tells us that positive Alpha of an asset (or portfolio) is an indicator that the asset outperformed the market, and had average return above what the theory (CAPM) predicts. Similarly, a negative Alpha indicates that an asset (or portfolio) underperforming, and delivering returns below the levels required by its Beta risk. A very appealing (but dangerous!) investment strategy of buying assets with positive Alpha and short selling assets with negative Alpha. In fact, many hedge funds emerged (and sunk), promising to generate Alpha - i.e. higher returns and lower risks than the market. Goldman Sachs used to have Global Alpha, which closed in 0 after large losses. Before adopting a strategy based on the "quest for Alpha", we need to understand the reasons behind observing non-zero Alphas. One explanation for observing non-zero Alphas is that di erent investors tend to specialize in di erent subsets of assets. If a hedge fund specializes in particular industries, its tangent portfolio will consist of that industry s securities. The derivation of the CAPM shows that if the tangent portfolio consists of a group of assets, then all the assets that make up the tangent portfolio, or any combination of these assets, must have Alpha of zero. However,