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Wriiten by George Pennachi, a very famous and classic material for asset pricing study. Theory of Asset Pricing unifies the central tenets and techniques of asset valuation into a single, comprehensive resource that is ideal for the first PhD course in asset pricing. Single-Period Portfolio Choice
1.1.PHE上上 RENCES WHEN RETURNS ARE UNCERTAIM 5 future date, and this payoff has a discrete distribution with n possible outcomes ) and corresponding probabilitics(pI,, Pn), where pi= 1 and Pi 20. Then the expected value of the payoff(or, more simply, the expected off)is =elI Is it logical to think 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 assets expected payoff was unlikely to be the only criterion that in dividuals usc for valuation. He did it by posing the following problcm that became known as the" St. Petersberg Paradox: Peter losses 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 if hc gots "heads"on the vcry first throw, two ducats if he gcts 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. Suppose we seek to determine Pauls expectation(of the payoff that he will receive Interpreting Paul,s prize from this coin Hipping game as the payoff of a risky asset. how much would he be willing to pay for this asset if he valued it based on its expected value? If the number of coin flips taken to first arrive at a heads is i, then p:=(2) and w;=2-so that the expected payoff equa As is the case in the following example, ole, n, the number of possible outcomes, may be nfinite A ducat was a 3.5 gram gold coin used throughout Europe 6 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION ∑na;=1+2+4+18 (1.1) (1+2+14+8+ (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 play this game. In a paper published in 1738, Daniel Bernoulli, a cousin of Nicholas provided an explanalion for the St. Petersberg Paradox by introducing the con- cept of expected utility. His insight was that an individual's utility or felicity l 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 Davo offs. Instead of valuing an asset as =21 pi Ti, its value, V, would V≡E[(x)= p (1 where Vi is the utility associated with payoff i. Moreover, he hypothesized that the utility resulting from any small increase in wealth will be inverse. proportionate to the quantity of goods previously possessed In other words the greater anl individuals wealth, the smaller is the added (or marginal)utility received from an additional increase in wealth. In the St. Petersberg paradox prizes, Ii, go up at the same rate that the probabilities decline. To obtain a finite valuation. the trick is to allow the utility of prizes, Ui, to increase slower An English translation of Daniel Bernoullis original Latin paper is printed in Econo metrica(Bernoulli 1954). Anot her Swiss ma thema.tician, Gabriel Cramer, offered a similar solution in 1728 1.1.PHE上上 RENCES WHEN RETURNS ARE UNCERTAIM than the rate that probabilities decline. Hence, Daniel Bernoulli introduced the principlc of a diminishing marginal atility of wealth(as cxpressed in his quote above) to resolve this paradox The first complete axiomatic development of expected utility is due to John von Neumann and Oskar Morgenstern(von Neumann and Morgenstern 1944) Von Neumann, a renowned physicist and mathematician, initiated the field of game theory which analyzes strategic decision making Morgenstern, an econo- mist, recognized the field's economic applications and, together, they provided a rigorous basis for individual decision-making under uncertainty. We now out- linc onc aspect of thcir work, namely, to providc conditions that an individual's preferences must 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 uals optimal choice of a lottery(risky asset) from a given set of different lotter ies. All lotteries have possible payoffs that are contained in the set 1,.., nj In general, the elements of this set can be viewed as different, uncertain out- comes. For example, they could be interpreted as particular consumption levels (bundles of consumption goods) that the individual obtains in diffcrent statos 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 of probabilities P={m1,…,mn}, where, of course,∑pz=1andp≥0.A different lottery is characterized by another set of probabilities, for example A, ,P*1. Let r, < and denote preference and indifference betweer lotteries. We will show that if an individual, s preferences satisfy the following conditions(axioms), then these preferences can be represented by a real-valued A Specifically, if an individual prefers lottery P to lottery P*, this can be denoted as P>P* or p* P. when the individual is indifferent between the two lotteries this is written as PN P*. If an individual prefers lottery P to lot tery P*or she is indifferent between lotteries P and Pt. this is written as p>p or p4< P. CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION utility function defined over a given lottery's probabilities, that is, an expected ility function V(pI Axioms 1) For any two lotteries P* and P, eit her P*> P, or P*< P, or P*N P 2)Transitiv If P**> P*and P* P, then P** P 3)Continuity IfP*∑P*P, thcrc exists somc入∈0,1 such that P*~AP米+(1-入P, where APw*+(1-A)p denotes a"compound lottery, namely with probability A one receives the lottery Pa and with probability(1-A)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.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 X E(0, 1 and all P AP*+(1-入PxAP+(1-入)P r for any two lotteries P and Pt, P Pf if for all A E(0, 1 and all A primary area of microeconomics analyzes a consumer's optimal choice of multiple good ility is a function of the quantities of multiple goods consumed. Refe Is topic include(Kreps 1990),(Mas-Colell, Whinston, and Green 1995), and(Varian 1992). In con- trast,the analysis of this chapter expresses utility as a function of the individuals wealth. Ir 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 by passes the individual's problem of choosing among different consumption goods and focuses on how the individual chooses a total quantity of consumpt ion at, different point s in time and different states of nature 1.1.PHE上上 RENCES WHEN RETURNS ARE UNCERTAIM 9 AP+(1-A)P~AP+(1-人)P水 To better understand the meaning of the independence axiom. note that P is preferred to P by assumption. Now the choice between AP++(1-A)P** and AP+(1-X)P* is equivalent, to a toss of a coin that has a probability (1-A)of landing "Lails', in which case both compound lotteries are equivalent to P**, and a probability A of landing "heads, in which case the first compound lottery is cquivalent to the singlc lottery P* and the sccond compound lottery is equivalent to the single lottery P. Thus, the choice between AP*+(1-A)P* x and AP+(1-A)P** is equivalent to being asked, prior to the coin toss, if one would prefer P* to P in the event the coin lands"heads It would seem reasonable that should the coin land "heads, we would go ahead with our original preference in choosing P over P. The independence axiom assumes that prcfcrcnccs ovcr the two lotteries arc independent of thc way in which we obtain them. G For this reason, the independence axiom 1s a iso known as the "no regret axiom. However, experimental evidence finds some systematic violations of this independence axiom, making it a questionable assumption for a theory of investor preferences. For example, the allais para- dox is a well-known choice of lotteries that. when offered to individuals, leads most to violate the independence axiom. 7 Machina(Machina 1987)summa- rizes violations of the independence axiom and reviews alternative approaches to modeling risk preferences. In spite of these deficiencies, von Neumann Morgenstern expected utility theory continues to be a useful and common ap b In the context of standard consumer choice theory, A would be interpreted as the amount (rather than proba bility) of a. particular good or bundle of goods consumed(say ()and (1-A)as the amount of another good or bundle of goods consumed (say C*%). In this case, it would not be reasonable to assume that the choice of these different bundles is independent This is due to some goods being substitutes or complements with other goods. Hence, the validity of the independence axiom is linked to outcomes being uncertain(risky), that is, the int, erpretation of X as a proba bility rat her than a deterministic amount A similar example is given as an exercise at the end of this chapter. CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION proach to modeling investor preferences, though research exploring alternative paradigms is growing The final axiom is similar to the independence and completeness axioms 5)Dominance Let pl be the compound lottery A1 Pi+(1-X1Pt and P2 be the compound lottery A2Pt+(1-A2)Pt. If Pf>Pt, then Pl>P2 if and only if A1> X2 Given preferences characterized by the above axions, we now show that the choice between any two(or more) arbitrary lotteries is that which has the higher Highest) expected utility The completeness axioms ordering on lotteries naturally induces an order- ing on the set of outcomes. To see this, define an elementaryorprimitive lottery, ei, which returns outcome Ti with probability 1 and all other outcomes with probability zero, that is, ei =p }={0,…,0,1,0,…0沿 where p ;= l and p;=0Vifi. Without loss of generality, suppose that the outcomes arc ordered such that en> en-l c1. This follows from the completeness axiom for this case of n elementary lotteries. Note that this or- dering of the elementary lot teries may not necessarily coincide with a ran king of the elements of strictly by the size of their monetary payoffs, as the state of nature for which a; is the outcome may differ from the state of nature for which c; is the outcome, and these states of nature may have different effects on how an individual values the same monetary outcome. For example, wi may be received in a state of nature when the economy is depressed, and monetary payoffs may be highly va lued in this state of nature. In contrast, i may be received in a state of nature characterized by high economic expansion, and monetary payments may not be as highly valued. Therefore, it may be that ei e, even if the monetary payment corresponding to i was less than that 8 This research includes "behavioral finance. a field that encompasses alternatives to both expected utility t heory and market, efficiency. An example of how a behaviora.I finance-type utility specification can impact asset prices will be presented in Chapter 15 1.1. PREFERENCES WHEN RETURNS ARE UNCERTAIN 11 corresponding to i From the continuity axiom, we know that for cach Ci, thcre exists a Ui E 0 such that ei N U;en +(1-Ui)er and for i=1, this implies U1=0 and for i=n, this implies Un=1. The values of the Ui weight the most and Icast preferred outcomes such that thc individual is just indifferent between a combination of these polar payoffs and the payoff of Ci. The Ui can adjust for both differences in monetary payoffs and differences in the states of nature during which the outcomes are received Now consider a given arbitrary lottery, P=p S. This can be cor sidered a compound lottery over the n elementary lotteries, where elementary lottery ci is obtaincd with probability Pi. By the independence axiom, and using equation(1.3), the individual is indifferent between the compound lottery, P, and the following lottery given on the right-hand-side of the equation below D1e1+….+mnCn~me1+…+p2-16-1+nUen+(1-U)el +P2+1e;+1+…+P where we have used the indifference relation in equation(1.3) to substitute for right hand side of(1.4). By repeating this substitution for all i i=1, .,n, we see that the individual will be indifferent between P, given by the left hand side of(1.4),and e1+…+ e1 12 CHAPTER 1. EXPECTED UTILITY AND RISK AVERSION Now define A=> PiUi. Thus, we see that lottery P is equivalent to a compound lottery consisting of aA probability of obtaining elementary lottery en and a(1-4) probability of obtaining elementary lottery el. In a similar manner,we can show that any other arbitrary lottery P*=pi, ..,*) is equiv- alent to a compound lottery consisting of a A' probability of obtaining en and a(1-A*)probability of obtaining e1, where A=2 P*Ui Thus we know from the dominance axiom that p*> P if and only ifA*>A which implies PiUi>pili. So defining an expected utility function as V(P1,…,pn,)=>n2 (16 will imply that P+> P if and only ifV(pi,.Pm)>V(p1,.,pn) The function given in equation(1.6)is known as von Neumann-Morgenstern expected utility. Note that it is linear in the probabilities and is unique up to a linear monotonic transformation. g This implies that the utility function has cardinal" properties, meaning that it does not preserve preference orderings for all strictly increasing transformations. 0 For example, if Vi= U(ai),an individual's choice over lotteries will be the same under the transformation al(ai)+b, but not a non-linear transformation that changes the "shape "of U(x2) The von Neumann-Morgenstern expected utility framework may only par tially explain the phenomenon illustrated by the St. Petersberg Paradox. Sup- pose an individual's utility is given by the square root of a monetary payoff, that is, Ui=U(,=va. This is a monotonically increasing, concave function of for wh cted utility is unie transformation can be traced to equation (1.3). The derivation chose to compare elementary lottery i in terms of the least and most preferred elementary lotteries. However, other bases for ranking a given lott. possible 10 An ordinal"utility function preserves preference orderings for any strictly increasing transformation, not just linear ones. The utility functions defined over multiple goods and used in standard consumer theory are ordinal measures

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