Chapter 1 Expected Utility and Risk Aversion Asset prices are determined by investors’ risk preferences and by the distrib- utions of assets’ risky future payments. Economists refer to these two bases of prices as investor "tastes" and the economy’s "technologies" for generating asset returns. A satisfactory theory of asset valuation must consider how in- dividuals allocate their wealth among assets having different future payments. This chapter explores the development of expected utility theory, the standard approach for modeling investor choices over risky assets. We first analyze the conditionsthatanindividual’spreferencesmustsatisfytobeconsistentwithan expected utility function. We then consider the link between utility and risk- aversion, and how risk-aversion leads to risk premia for particular assets. Our final topic examines how risk-aversion affects an individual’s choice between a risky and a risk-free asset. Modeling investor choices with expected utility functions is widely-used. However, significant empirical and experimental evidence has indicated that 3 4 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION individuals sometimes behave in ways inconsistent with standard forms of ex- pected utility. These findings have motivated a search for improved models of investor preferences. Theoretical innovations both within and outside the expected utility paradigm are being developed, and examples of such advances are presented in later chapters of this book. 1.1 Preferences when Returns are Uncertain Economists typically analyze the price of a good or service by modeling the nature of its supply and demand. A similar approach can be taken to price an asset. Asastartingpoint, let usconsider themodelingof an investor’sdemand foranasset. Incontrasttoagoodorservice,anassetdoesnotprovideacurrent consumptionbenefittoanindividual. Rather,anassetisavehicleforsaving. It is a component of an investor’s financial wealth representing a claim on future consumption or purchasing power. The main distinction between assets is the difference in their future payoffs. With the exception of assets that pay a risk-free return, assets’ payoffs are random. Thus, a theory of the demand for assets needs to specify investors’ preferences over different, uncertain payoffs. In other words, we need to model how investors choose between assets that have different probability distributions of returns. In this chapter we assume an environment where an individual chooses among assets that have random payoffs at a single future date. Later chapters will generalize the situation to consider an individual’s choices over multiple periods among assets paying returns at multiple future dates. Let us begin by considering potentially relevant criteria that individuals might use to rank their preferences for different risky assets. One possible measure of the attractiveness of an asset is the average or expected value of its payoff. Suppose an asset offers a single random payoff at a particular 1.1. PREFERENCES WHEN RETURNS ARE UNCERTAIN 5 futuredate,andthispayoffhasadiscretedistributionwithnpossibleoutcomes, n (x ,...,x ), and corresponding probabilities (p ,...,p ), where p = 1 and 1 n 1 n i i=1 p 0.1 Then the expected value of the payoff (or, more simplyP, the expected i ≥ n payoff) is x¯ E[x]= p x . i i ≡ i=1 P Is it logical toethink that individuals value risky assets based solely on the assets’ expected payoffs? This valuation concept was the prevailing wisdom until 1713 when Nicholas Bernoulli pointed out a major weakness. He showed that an asset’s expected payoff was unlikely to be the only criterion that in- dividuals use for valuation. He did it by posing the following problem that became known as the “St. Petersberg Paradox:” Petertossesacoinandcontinuestodosountilitshouldland"heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that on each additional throw the number of ducats he must pay is doubled.2 Suppose we seek to determine Paul’s expectation (of the payoff that he will receive). InterpretingPaul’sprizefromthiscoinflippinggameasthepayoffofarisky asset, how much would he be willing to pay for this asset if he valued it based onitsexpectedvalue? Ifthenumberofcoinflipstakentofirstarriveataheads is i, then p = 1 i and x =2i 1 so that the expected payoff equals i 2 i − ¡ ¢ 1As is the case in the following example, n, the number of possible outcomes, may be infinite. 2A ducatwas a3.5 gram gold coin usedthroughoutEurope. 6 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION ∞ x¯ = p x = 11+ 12+ 14+ 1 8+... (1.1) i i 2 4 8 16 i=1 X = 1(1+ 12+ 14+ 18+... 2 2 4 8 = 1(1+1+1+1+...= 2 ∞ The "paradox" is that the expected value of this asset is infinite, but, intu- itively,mostindividualswouldpayonlyamoderate,notinfinite,amounttoplay thisgame. In apaper published in1738, DanielBernoulli, acousinofNicholas, providedanexplanationfortheSt. PetersbergParadoxbyintroducingthecon- ceptof expected utility.3 Hisinsightwasthatanindividual’sutilityor"felicity" from receiving a payoff could differ from the size of the payoff and that people cared about the expected utility of an asset’s payoffs, not the expected value of its payoffs. Instead of valuing an asset as x = n p x , its value, V, would i=1 i i be P V E[U(x)]= n p U (1.2) ≡ i=1 i i P where U is the utility associated wieth payoff x . Moreover, he hypothesized i i that the "utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed." In other words, thegreateranindividual’swealth,thesmalleristheadded(ormarginal)utility received from an additional increase in wealth. In the St. Petersberg Paradox, prizes, x , go up at the same rate that the probabilities decline. To obtain a i finite valuation, the trick is to allow the utility of prizes, U , to increase slower i 3An English translation of Daniel Bernoulli’s original Latin paper is printed in Econo- metrica (Bernoulli 1954). Another Swiss mathematician, Gabriel Cramer, offered a similar solution in 1728. 1.1. PREFERENCES WHEN RETURNS ARE UNCERTAIN 7 thantheratethatprobabilitiesdecline. Hence,DanielBernoulliintroducedthe principle of a diminishing marginal utility of wealth (as expressed in his quote above) to resolve this paradox. The first completeaxiomatic development of expected utility isduetoJohn von Neumann and Oskar Morgenstern (von Neumann and Morgenstern 1944). Von Neumann, a renowned physicist and mathematician, initiated the field of gametheory,whichanalyzesstrategicdecisionmaking. Morgenstern,anecono- mist, recognized the field’s economic applications and, together, they provided arigorousbasisforindividualdecision-makingunderuncertainty. Wenowout- lineoneaspectof theirwork,namely, toprovideconditionsthatanindividual’s preferencesmust satisfy for these preferences to be consistent with an expected utility function. Define a lottery as an asset that has a risky payoff and consider an individ- ual’soptimalchoiceofalottery(riskyasset)fromagivensetofdifferentlotter- ies. All lotteries have possible payoffs that are contained in the set x ,...,x . 1 n { } In general, the elements of this set can be viewed as different, uncertain out- comes. Forexample, theycouldbeinterpretedasparticularconsumptionlevels (bundles of consumption goods) that the individual obtains in different states of nature or, more simply, different monetary payments received in different states of the world. A given lottery can be characterized as an ordered set n of probabilities P = p ,...,p , where, of course, p = 1 and p 0. A 1 n i i { } ≥ i=1 different lottery is characterized by another set of pProbabilities, for example, P = p ,...,p . Let , , and denote preference and indifference between ∗ { ∗1 ∗n} Â ≺ ∼ lotteries.4 We will show that if an individual’s preferences satisfy the following conditions (axioms), then these preferences can be represented by areal-valued 4Specifically,ifanindividualpreferslotteryP tolotteryP ,thiscanbedenotedasP P ∗ Â ∗ or P P. When the individual is indifferent between the two lotteries, this is written as ∗ ≺ P P . IfanindividualpreferslotteryP tolotteryP orsheisindifferentbetweenlotteries ∼ ∗ ∗ P and P ,thisiswritten as P P orP P. ∗ º ∗ ∗¹ 8 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION utility function defined over a given lottery’s probabilities, that is, an expected utility function V(p ,...,p ). 1 n Axioms: 1) Completeness For any two lotteries P and P, either P P, or P P, or P P. ∗ ∗ ∗ ∗ Â ≺ ∼ 2) Transitivity If P P and P P, then P P. ∗∗ ∗ ∗ ∗∗ º º º 3) Continuity IfP P P,thereexistssomeλ [0,1]suchthatP λP +(1 λ)P, ∗∗ ∗ ∗ ∗∗ º º ∈ ∼ − where λP +(1 λ)P denotes a“compound lottery,” namely with probability ∗∗ − λ one receives the lottery P and with probability (1 λ) one receives the ∗∗ − lottery P. These three axioms are analogous to those used to establish the existence of a real-valued utility function in standard consumer choice theory.5 The fourth axiom is unique to expected utility theory and, as we later discuss, has important implications for the theory’s predictions. 4) Independence For any two lotteries P and P , P P if for all λ (0,1] and all P : ∗ ∗ ∗∗ Â ∈ λP +(1 λ)P λP +(1 λ)P ∗ ∗∗ ∗∗ − Â − Moreover, for any two lotteries P and P , P P if for all λ (0,1] and all † † ∼ ∈ P : ∗∗ 5Aprimaryareaofmicroeconomicsanalyzesaconsumer’soptimalchoiceofmultiplegoods (and services) based on their prices and the consumer’s budget contraint. In that context, utility is a function of the quantities of multiple goods consumed. References on this topic include (Kreps 1990), (Mas-Colell, Whinston, and Green 1995), and (Varian 1992). In con- trast,theanalysisofthischapterexpressesutilityasafunctionoftheindividual’swealth. In future chapters, we introduce multi-period utility functions where utility becomes a function of the individual’s overall consumption at multiple future dates. Financial economics typi- cally bypasses the individual’s problem of choosing among different consumption goods and focuses on how the individual chooses a total quantity of consumption at different points in time and differentstates ofnature.