CONVENTIONAL FINANCE, PROSPECT THEORY, AND MARKET EFFICIENCY

Transcription

1 CONVENTIONAL FINANCE, PROSPECT THEORY, AND MARKET EFFICIENCY PART ± I CHAPTER 1 CHAPTER 2 CHAPTER 3 Foundations of Finance I: Expected Utility Theory Foundations of Finance II: Asset Pricing, Market Efficiency, and Agency Relationships Prospect Theory, Framing, and Mental Accounting

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3 FOUNDATIONS OF FINANCE I: EXPECTED UTILITY THEORY 1 CHAPTER ± 1.1 INTRODUCTION The behavior of individuals, practitioners, markets, and managers is sometimes characterized as irrational, but what exactly does that mean? To answer this question, we must take several steps back and fully understand the foundations of modern finance, which are based on rational decision-making. The first two chapters of the book are designed to accomplish this goal, with the first presenting the standard theory of how individuals make decisions when confronted with uncertainty. As background, we first consider what standard (or neoclassical) economics argues about rational behavior when economic decisions are made and there is no uncertainty about the future. In the next section, it is argued that individuals should maximize utility (or happiness) based on their preferences, the constraints they face, and the information at their disposal. In Section 1.3, we note that decisions become complicated when there is uncertainty. An extension of utility theory has been developed for this purpose and is known as expected utility maximization. The basic procedure is to ascertain the utility level generated by varying levels of wealth, and then, when choosing among prospects, which are defined to be probability distributions of different wealth levels, to calculate the expected level of utility of each of these prospects. Finally, the decision-maker chooses the prospect with the highest expected utility. In the following section, we discuss the role of risk attitude. Specifically, since people prefer to avoid risk, it is necessary to compensate them for assuming it, and the degree to which they must be compensated depends on their risk aversion. Despite the elegance of expected utility, there are a number of occasions when many people act contrary to it. The best-known instance is the Allais paradox presented in Section 1.5. The final two sections of the chapter look ahead to the importance of how a problem is presented, that is,

4 4 CHAPTER 1 the decision frame, and to prospect theory, the principal alternative to expected utility maximization, which will be the major focus of Chapter NEOCLASSICAL ECONOMICS Traditional finance models have a basis in economics, and neoclassical economics is the dominant paradigm. In this representation, individuals and firms are selfinterested agents who attempt to optimize to the best of their ability in the face of constraints on resources. The value (or price) of an asset is determined in a market, subject to the influences of supply and demand. 1 In this chapter, we focus on individual decision-making, leaving markets to the following chapter. Neoclassical economics makes some fundamental assumptions about people: 2 1. People have rational preferences across possible outcomes or states of nature. 2. People maximize utility and firms maximize profits People make independent decisions based on all relevant information. These assumptions seem quite reasonable upon first consideration, but let s be sure we really understand what they mean. RATIONAL PREFERENCES What does it mean for individuals to have rational preferences? Certain conditions are commonly imposed on preferences. We will introduce some notation to understand these conditions. Suppose a person is confronted with the choice between two outcomes, x and y. The symbol means that one choice is strictly preferred to another, so that the relation x y means that x is always the preferred choice when x and y are offered to some individual. The symbol * indicates indifference, so that x * y indicates that the person values the two outcomes the same. Finally, the symbol indicates weak preference, so that x y means that the person prefers x or is indifferent between x and y. An important assumption is that people s preferences are complete. This means that a person can compare all possible choices and assess preference or indifference. Thus, for any pair of choices, x y or y x or both, which would mean that x * y. This assumption does not seem to cause too many problems. Surely most people know what they like and what they do not like. A second assumption, transitivity, does not seem to be too strong an assumption for most people. Suppose now that a person is confronted with a choice among three outcomes: x, y, and z. According to transitivity, if x y and y z, then x z. If I prefer vanilla ice cream to chocolate, and chocolate to strawberry, I should also prefer vanilla to strawberry. If transitivity does not hold, we cannot determine an optimal or best choice. So, rational choices are transitive. UTILITY MAXIMIZATION Utility theory is used to describe preferences. With a utility function, denoted as u( ), we assign numbers to possible outcomes so that preferred choices receive higher numbers. We can think of utility as the satisfaction received from a

5 FOUNDATIONS OF FINANCE I: EXPECTED UTILITY THEORY 5 particular outcome. Normally an outcome is characterized by a bundle of goods. For example, someone might have to choose between two loaves of bread plus one bottle of water and one loaf of bread plus two bottles of water. If this individual reveals a preference for the former, we would say that: 1.1 u(2 bread, 1 water) > u(1 bread, 2 water) Notice that we have not specified any numerical values for u( ). This is because, while the ordering of outcomes by a utility function is important, the actual number assigned is immaterial. The utility function is ordinal (i.e., order-preserving) but not cardinal (which would mean the exact utility value matters). To arrive at her optimal choice, an individual considers all possible bundles of goods that satisfy her budget constraint (based on wealth or income), and then chooses the bundle that maximizes her utility. 4 If there is a single good of interest, then ranking under certainty is trivial. This stems from the principle of non-satiation, which simply means the more the better. As an example of a single good, utility functions are often defined in relation to wealth. Though mathematically a utility function can be specified in different ways, we will use the example of a logarithmic function. In this case, the utility derived from wealth level w is u(w) = ln(w). Consider Table 1.1. In the table, wealth is defined in $10,000s, so that a wealth level of 1 translates to $10,000, a wealth level of 10 translates to $100,000, and so on. Figure 1.1 graphs this utility function. Notice that the slope gets flatter as wealth increases. For a person with this utility function, added wealth at low income levels increases utility more than added wealth at high income levels. We will return to this pattern later in the chapter. TABLE 1.1 LOGARITHMIC UTILITY OF WEALTH Wealth (in $10,000s) u(w) = ln(w)

6 6 CHAPTER 1 FIGURE 1.1 Logarithmic Utility Function Utility Wealth (in $10,000s) RELEVANT INFORMATION Neoclassical economics assumes that people maximize their utility using full information of the choice set. Of course, economists recognize that information is rarely free. Not only are there costs associated with acquiring information, but there are also costs of assimilating and understanding information that is already at hand. Students who spend many hours working toward success in a challenging course are keenly aware that information is not free. In the following chapter, we will return to this topic and consider how to define what information is relevant when making financial decisions. 1.3 EXPECTED UTILITY THEORY So far in this chapter, we have dodged the issue of uncertainty. In the real world, there is not much that we can count on with certainty. In financial decisionmaking, there is clearly a great deal of uncertainty about outcomes. Expected utility theory was developed by John von Neumann and Oskar Morgenstern in an attempt to define rational behavior when people face uncertainty. 5 This theory contends that individuals should act in a particular way when confronted with decision-making under uncertainty. In this sense, the theory is normative, which means that it describes how people should rationally behave. This is in contrast to a positive theory, which characterizes how people actually behave. Expected utility theory is really set up to deal with risk, not uncertainty. A risky situation is one in which you know what the outcomes could be and can assign a probability to each outcome. Uncertainty is when you cannot assign

7 FOUNDATIONS OF FINANCE I: EXPECTED UTILITY THEORY 7 probabilities or even come up with a list of possible outcomes. Frank Knight clarified the difference between risk and uncertainty: 6 But Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk, from which it has never been properly separated. The term risk, as loosely used in everyday speech and in economic discussion, really covers two things which, functionally at least, in their causal relations to the phenomena of economic organization, are categorically different. The essential fact is that risk means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating. It will appear that a measurable uncertainty, or risk proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. Risk is measurable using probability, but uncertainty is not. Whereas, conforming to common practice, we began by saying that we were going to address decisionmaking under uncertainty, the truth is that we will almost always focus on decision-making under risk. For almost all purposes, when considering decision-making under risk, it is sufficient to think in terms of just wealth. Let s suppose, for simplicity, that there are only two states of the world: low wealth and high wealth. When it is low, your wealth is $50,000, and when it is high, your wealth is $1,000,000. And further assume that you can assign probabilities to each of these outcomes. You are fairly optimistic about your future, so you assign a probability of 40% to low wealth and 60% to high wealth. Formally, a prospect is a series of wealth outcomes, each of which is associated with a probability. 7 If we call the latter prospect P1, we can represent this situation using the following convenient format: 1.2 P1(0.40, $50,000, $1,000,000) Note that with two outcomes the first number is the probability of the first outcome, and the next two numbers are the two outcomes. If only one dollar figure is given, as in P(.3, $100), the assumption is that the second outcome is 0. It can be shown that if one makes the assumptions previously discussed along with several others that most people consider to be reasonable, a procedure allowing us to make appropriate choices under risk results. In a nutshell, this procedure involves calculating the probability-weighted expected value of the different possible utility levels (that is, the expected utility). The Appendix to this chapter outlines a set of assumptions that allow us to rank outcomes based on expected utility maximization. In addition, a proof is sketched out, and certain characteristics of utility functions are described. Let us use the notation U(P) for the expected utility of a prospect. For P1, the expected utility, or U(P1), is: 1.3 U(P1) = 0.40u(50,000) u(1,000,000)

8 8 CHAPTER 1 With the logarithmic utility function previously presented, the expected utility of this prospect (using Table 1.1) is: 1.4 U(P1) = 0.40(1.6094) (4.6052) = Expected utility can be used to rank risky alternatives because it is order-preserving (i.e., ordinal). It can be shown that for a given individual it is also cardinal, in the sense that it is unique up to a positive linear transformation. This feature will come in handy in Chapter 3 for several demonstrations. If one prospect is preferred to another, its expected utility will be higher. Let us consider another prospect: 1.5 P2(.50, $100,000, $1,000,000) P2 is superior to P1 in that the low-wealth outcome is now higher ($100,000 vs. $50,000). On the other hand, it is inferior because the probability of the highwealth outcome is now lower (.50 vs..60). So it is not obvious which prospect would be preferred. Once again we take the expected utility of the prospect: 1.6 U(P2) = 0.50(2.3026) (4.6052) = Therefore, if someone has logarithmic utility, then they would prefer P2 to P1. Of course we could specify another functional form for utility such that P1 is preferred to P RISK ATTITUDE There is abundant evidence that most people avoid risk in most circumstances. People are, however, willing to assume risk if they are compensated for it. For example, when choosing between two stocks with the same expected return, if you are like most people, you would invest in the one with the lower risk. If you are going to take on a riskier investment, you will demand a higher return to compensate for the risk. In the following chapter, we will talk more about the trade-off between risk and return. Now we want to focus on what we mean by risk attitude. The utility function is useful in defining risk preferences. Returning to P1, the expected value of wealth is: 1.7 E(w) = 0.40($50,000) ($1,000,000) = $620,000 = E(P1) Note that the expected value of wealth is synonymous with the expected value of the prospect. The utility of this expected value of wealth is: 1.8 u(e(w)) = ln(62) =

9 On the other hand, as we saw before, the expected utility is , so u(e(w)) > U(P1). This means that a person, whose preferences can be described by a logarithmic utility function, prefers the expected value of a prospect to the prospect itself. In other words, if you have a logarithmic utility function, you would rather have wealth of $620,000 than a prospect in which you have a 40% chance of wealth of $50,000 and a 60% chance of wealth of $1,000,000. A person of this type dislikes risk, and we say this person is risk averse. Figure 1.2 illustrates the situation. In the figure, we see that the utility of the expected wealth (u(e(w)) = u(62) = ) is greater than the expected utility of the prospect (U(P1) = ). Someone who is risk averse has a concave utility function, which means that: 1.9 u(e(p)) > U(P) FOUNDATIONS OF FINANCE I: EXPECTED UTILITY THEORY 9 Such a person s preferences imply that the utility of the expected value of a prospect is greater than the expected utility of the prospect. This person would rather have the expected value of the prospect with certainty than actually take a gamble on the uncertain outcome. For our example, a risk-averse person would rather have wealth of $620,000 with certainty as compared to a prospect with a 40% chance of wealth of $50,000 and a 60% chance of wealth of $1,000,000. A risk-averse person is willing to sacrifice for certainty. The certainty equivalent is defined as that wealth level that leads the decision-maker to be indifferent between a particular prospect and a certain wealth level. In the case of P1 and logarithmic utility, the certainty equivalent is $301,700. This is because, as Figure 1.2 shows, wealth of (in $10,000s) leads to a utility level equal to the expected FIGURE 1.2 Utility Function for a Risk-averse Individual Utility Wealth (in $10,000s)

10 10 CHAPTER 1 utility of the prospect. The utility of the certainty equivalent is equal to the expected utility of the prospect or: 1.10 u(30.17) = u(w) = U(P1) = 0.40(1.6094) (4.6052) = You would give up $318,300 in expected value in order to exchange the prospect for certainty. We often assume that people are risk averse, but some people actually seem to prefer, at least at times, to take on risk. Such a person is called a risk seeker and has a convex utility function, as in: 1.11 u(e(p)) < U(P) For such an individual, the utility of the expected value of a prospect is less than the expected utility of the prospect. This person would rather gamble on the uncertain outcome than take the expected value of the prospect with certainty. Figure 1.3 shows the relationship between the utility of expected wealth and the expected utility of wealth for a risk seeker. For a risk seeker, the certainty equivalent level of wealth is greater than the expected value. Returning to our previous example, a risk seeker would rather have a prospect with a 40% chance of wealth of $50,000 and a 60% chance of wealth of $1,000,000 versus wealth of $620,000 with certainty. Finally, people who are risk neutral lie between risk averters and risk seekers. These people only care about expected values and risk does not matter at all. For someone who is risk neutral we have: 1.12 u(e(p)) = U(P) FIGURE 1.3 Utility Function for a Risk Seeker Utility U(P) u(e(w)) w1 E(w) Wealth w2

11 FOUNDATIONS OF FINANCE I: EXPECTED UTILITY THEORY 11 FIGURE 1.4 Utility Function for a Risk-neutral Individual Utility U(P) = u(e(w)) w1 E(w) Wealth w2 Thus the utility of the expected value of a prospect is equal to the expected utility of the prospect, as illustrated in Figure 1.4. Again, returning to our previous example, a risk-neutral individual would be indifferent between a prospect with a 40% chance of wealth of $50,000 and a 60% chance of wealth of $1,000,000 and wealth of $620,000 with certainty. For a risk-neutral person, the certainty equivalent level of wealth is equal to the expected value of the prospect. 1.5 ALLAIS PARADOX Throughout this book we will consider a number of observed behaviors that appear to be contrary to predictions generated by conventional finance models. Now we will look at one persistently documented contradiction of expected utility theory, the so-called Allais paradox. 8 Alternative approaches to decision-making under uncertainty have been developed because researchers have detected this and other departures from expected utility theory. The most famous is the prospect theory of Daniel Kahneman and Amos Tversky, which will be the major focus of Chapter 3. 9 Consider the prospect choices in Table 1.2. In the case of Question 1, people can choose between A and A*, while in the case of Question 2, people can choose between B and B*. Questions 1 and 2 have been presented to many people. Take a moment now and answer each question. For Question 1, would you prefer Prospect A or Prospect A*? For Question 2, would you prefer Prospect B or Prospect B*? Are you like many people? A large number of people choose A over A* and B* over B. We now show that this violates expected utility theory. If expected