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1、Chapter 1Expected Utility and RiskAversionAsset prices are determined by investors risk preferences and by the distrib-utions of assets risky future payments.Economists refer to these two basesof prices as investor “tastes“ and the economys “technologies“ for generatingasset returns.A satisfactory t
2、heory 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 standardapproach for modeling investor choices over risky assets.We first analyze theconditions that an in
3、dividuals preferences must satisfy to be consistent with anexpected utility function.We then consider the link between utility and risk-aversion, and how risk-aversion leads to risk premia for particular assets. Ourfinal topic examines how risk-aversion affects an individuals choice between arisky a
4、nd a risk-free asset.Modeling investor choices with expected utility functions is widely-used.However, significant empirical and experimental evidence has indicated that34CHAPTER 1. EXPECTED UTILITY AND RISK AVERSIONindividuals sometimes behave in ways inconsistent with standard forms of ex-pected u
5、tility.These findings have motivated a search for improved modelsof investor preferences.Theoretical innovations both within and outside theexpected utility paradigm are being developed, and examples of such advancesare presented in later chapters of this book.1.1Preferences when Returns are Uncerta
6、inEconomists typically analyze the price of a good or service by modeling thenature of its supply and demand. A similar approach can be taken to price anasset. As a starting point, let us consider the modeling of an investors demandfor an asset. In contrast to a good or service, an asset does not pr
7、ovide a currentconsumption benefit to an individual. Rather, an asset is a vehicle for saving. Itis a component of an investors financial wealth representing a claim on futureconsumption or purchasing power.The main distinction between assets isthe difference in their future payoffs. With the except
8、ion of assets that pay arisk-free return, assets payoffs are random. Thus, a theory of the demand forassets needs to specify investors preferences over different, uncertain payoffs.In other words, we need to model how investors choose between assets thathave different probability distributions of re
9、turns.In this chapter we assumean environment where an individual chooses among assets that have randompayoffs at a single future date.Later chapters will generalize the situationto consider an individuals choices over multiple periods among assets payingreturns at multiple future dates.Let us begin
10、 by considering potentially relevant criteria that individualsmight use to rank their preferences for different risky assets.One possiblemeasure of the attractiveness of an asset is the average or expected value ofits payoff.Suppose an asset offers a single random payoffat a particular1.1. PREFERENC
11、ES WHEN RETURNS ARE UNCERTAIN5future date, and this payoffhas a discrete distribution with n possible outcomes,(x1,.,xn), and corresponding probabilities (p1,.,pn), wherenPi=1pi= 1 andpi 0.1Then the expected value of the payoff(or, more simply, the expectedpayoff) is x E e x =nPi=1pixi.Is it logical
12、 to think that individuals value risky assets based solely on theassets expected payoffs?This valuation concept was the prevailing wisdomuntil 1713 when Nicholas Bernoulli pointed out a major weakness. He showedthat an assets expected payoffwas unlikely to be the only criterion that in-dividuals use
13、 for valuation.He did it by posing the following problem thatbecame known as the “St. Petersberg Paradox:”Peter tosses a coin and continues to do so until it should land “heads“when it comes to the ground.He agrees to give Paul one ducat ifhe gets “heads“ on the very first throw, two ducats if he ge
14、ts it onthe second, four if on the third, eight if on the fourth, and so on, sothat on each additional throw the number of ducats he must pay isdoubled.2Suppose we seek to determine Pauls expectation (of thepayoffthat he will receive).Interpreting Pauls prize from this coin flipping game as the payo
15、ffof a riskyasset, how much would he be willing to pay for this asset if he valued it basedon its expected value? If the number of coin flips taken to first arrive at a headsis i, then pi=12iand xi= 2i1so that the expected payoffequals1As is the case in the following example, n, the number of possib
16、le outcomes, may beinfinite. 2A ducat was a 3.5 gram gold coin used throughout Europe.6CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION x=Xi=1pixi=1 21 +1 42 +1 84 +1 168 + .(1.1)=1 2(1 +1 22 +1 44 +1 88 + .=1 2(1 + 1 + 1 + 1 + . = The “paradox“ is that the expected value of this asset is infinite, but, intu-itively, most individuals would pay only a moderate, not infinite, amount to playthis game. In a paper published in 1738, Daniel Bernoulli, a cousin of Nich
