<br><h2>CHAPTER 1</h2><p><b>THE EXPECTED UTILITY MAXIM</p><br><p>INTRODUCTION</b></p><p>As explained in the preceding preface to the four parts planned for the presentbook, the fundamental assumptions of Markowitz (1959) are in its Part 4,Chapters 10 through 13. These fundamental assumptions are at the back ratherthan the front of Markowitz (1959) because Markowitz feared that no practitionerwould read a book that began with an axiomatic treatment of the theory ofrational decision making under uncertainty. But now, clearly, these matters havebecome urgent. They bear directly on controversies such as:</p><p>• Under what conditions one should use mean-variance (MV) analysis</p><p>• What should be used in its stead when mean-variance analysis is not applicable</p><p>• How a single-period risk-return analysis relates to its many-period context</p><p>• How parameter uncertainty should be handled</p><br><p>It may seem frivolous to carefully weigh questions such as:</p><p>If an investor prefers Portfolio A to Portfolio B, should that investor preferto go with Portfolio A, or would it be as good or better to toss a coin tochoose between Portfolio A and Portfolio B?</p><p>Or again, if the investor prefers Portfolio A to Portfolio B, should he or sheprefer Portfolio B for sure or a 50-50 chance of A or B?</p><br><p>Perhaps surprisingly, the answers to such questions imply how one should judgealternative measures of risk.</p><p>For example, two measures of risk that have been proposed are (1) standarddeviation and (2) maximum loss. But maximum loss is a risk measure that violatesthe principle that if one prefers Portfolio A to Portfolio B, one should prefera chance of Portfolio A to the certainty of Portfolio B. For example, supposethat Portfolio A has a 50-50 chance of a 30 percent gain or a 10 percent loss,whereas Portfolio B has a 50-50 chance of a 40 percent gain or a 20 percentloss. Both have an expected (a.k.a. average or "mean") return of 10 percent.Portfolio A has a standard deviation of 20 percent and a maximum loss of 10percent, whereas Portfolio B has a standard deviation of 30 percent and amaximum loss of 20 percent. Thus, by either criterion of risk—standard deviationor maximum loss—Portfolio A is preferable to Portfolio B. If one flips a coin todecide between Portfolio A and Portfolio B, the whole process (flip a coin, thenchoose one portfolio or the other accordingly) has a lower standard deviationthan just choosing Portfolio B, but the process has the <i>same maximum possibleloss</i> as just choosing B. The most you can lose by either choice is 20 percent.Thus the maximum loss criterion violates the desideratum to prefer a chance of abetter thing to the certainty of a worse thing. If one accepts the latter, thenmaximum loss is not even permitted into the contest between alternative riskmeasures.</p><p>This chapter generalizes this discussion. In roughly the following order, it:</p><p>• Defines certain concepts, including the expected utility maxim</p><p>• Describes the properties of expected utility maximization, including whetherpreferences determine utility numbers uniquely and what shape utility functionencourages portfolio diversification</p><p>• Contrasts the HDM (human decision maker) with the RDM (rational decisionmaker), the latter being the topic of this book</p><p>• Discusses objections that have been raised to the expected utility rule andhow these confuse the behavior observed in an HDM with that to be expected froman RDM</p><p>• Presents three decision-choice "axioms" that we believe it is reasonable toexpect of an RDM</p><p>• Refers the reader to a proof [in Markowitz (1959)] that a decision maker whoacts consistently with the aforementioned axioms must necessarily act accordingto the expected utility rule</p><p>• Ties up an important loose end</p><br><p>Subsequent chapters of this part of the book consider the merits of variousrisk-return criteria as approximations to the expected utility rule. As noted inthe preface, subsequent parts are planned that will consider rational decisionmaking over time with known odds, rational decision making when odds are notknown, and certain implementation considerations, especially the division oflabor among computers, data, algorithms, and judgment.</p><p>This chapter and the three that follow may all seem very academic, but thetopics covered are, in fact, of central importance in practice. Few, if any,decisions are more important in the actual use of risk-return analysis than thechoice of risk mmeasure. A false statement on the subject—such as, "The use ofvariance as a risk measure assumes that return distributions are normallydistributed"—ccan be stated in a fraction of a sentence and then left as self-evident, but an accurate, nuanced account of the topic—including a descriptionof the boundaaaaries where mean-variance (MV) approximations begin to break down—requiresmore space.</p><br><p><b>DEFINITIONS</b></p><p>In this section, we define the terminology used in our discussion of the maximumexpected utility rule. Imagine an RDM who must choose between alternativeprobability distributions such as</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><p>After the RDM selects one such distribution, a wheel is spun and—eventually—theoutcome is announced. The RDM can make no relevant decision between the choiceof a probability distribution and the announcement of the outcome. This absenceof possibly relevant intervening decisions characterizes this as a single-periodchoice situation. See Markowitz (1959), Chapter 11, the section entitled"Intermediate Decisions, Incomplete Outcomes" for an elaboration of this point.</p><p>We will speak of "outcomes," "probability distributions of outcomes,""preferences among probability distributions of outcomes," and "the expectedutility maxim." By definition, <i>one and only one outcome</i> can occur. If it ispossible, for example, for the individual to win both $1,000 and a car, thenthis combination is defined as a single outcome. If it is possible for theindividual to neither win nor lose, this is defined as another outcome. We canimagine a situation whose outcomes, thus defined, are:</p><p>• Win a car</p><p>• Win $1,000</p><p>• Win both a car and $1,000</p><p>• Win nothing</p><br><p>These would be the four possible "outcomes" of the particular single-periodchoice situation.</p><p>It will be convenient to assume that there is only a finite number (<i>n</i>) ofoutcomes. This is not a serious practical limitation because <i>n</i> can equal thenumber of microseconds since the Big Bang. (We will cite literature thatgeneralizes the results presented here to nonfinite probability distributions.)One may represent a probability distribution among the n possible outcomes by avector:</p><p><i>P = (p<sub>1</sub>, p<sub>2</sub>, ..., p<sub>i</sub>, ..., p<sub>n</sub></i></p><br><p>where <i>p<sub>i</sub></i> is the probability that the ith outcome will be the one to occur. Sinceone and only one outcome will occur, the <i>p<sub>i</sub></i> sum to 1:</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1a)</p><br><p>and, of course, cannot be negative:</p><p><i>p<sub>i</sub></i> ≤ 0 <i>i</i> = 1, ..., <i>n</i> (1b)</p><br><p>One may think of two probability distributions,</p><p><i>P = (p<sub>1</sub>, ..., p<sub>n</sub>)</i> and <i>Q = (q<sub>1</sub>, ..., q<sub>n</sub>)</i></p><br><p>as "lottery tickets" offering different probabilities of outcomes. One canimagine flipping a coin and then engaging in lottery <i>P</i> if heads appears orlottery <i>Q</i> if tails appears. If this is done, the probability of obtaining theith outcome is</p><p>(the probability of engaging in lottery <i>P</i>) times</p><p>(the probability of obtaining outcome <i>i</i> if <i>P</i> is engaged in)</p><p>plus</p><p>(the probability of engaging in lottery <i>Q</i>) times</p><p>(the probability of obtaining outcome <i>i</i> if lottery <i>Q</i> is engaged in)</p><br><p>If the coin is fair, this equals</p><p>(1/2)<i>p<sub>i</sub></i> + (1/2)<i>q<sub>i</sub></i></p><br><p>In general, if there is a probability (<i>a</i>) of engaging in <i>P</i> and a probability (11 - <i>a</i>) of engaging in <i>Q</i>, then the probability of obtaining the ith outcome is</p><p><i>ap<sub>i</sub></i> + (1 - <i>a)q<sub>i</sub></i></p><br><p>The probability distribution of ultimate outcomes associated with a probability(<i>a</i>) of <i>P</i> and (1 - <i>a</i>) of <i>Q</i> is therefore the probability distribution representedby the vector</p><p>[<i>ap</i><sub>1</sub> + (1 - <i>a</i>)<i>q<sub>1</sub>, ap<sub>2</sub></i> +(1 - <i>a</i>)<i>q<sub>2</sub>, ..., ap<sub>n</sub></i> + (1 - <i>a</i>)<i>p<sub>n</sub></i>]</p><br><p>In matrix notation, this is the vector denoted as <i>aP</i> + (1 - <i>a</i>)<i>Q</i>. Thus the lattermay be interpreted either as a chance of <i>P</i> or <i>Q</i> or as a new vector ofprobabilities obtained from vectors <i>P</i> and <i>Q</i> by the rules of matrix algebra.</p><p>An implicit assumption of an analysis such as ours—which is frequentlyoverlooked or underappreciated—is that a situation to be analyzed is set up sothat the RDM's decisions do (and the HDM's decisions should) depend only on theprobability distributions of outcomes <i>and not on how these probabilities aregenerated</i>. In particular, we assume that outcomes have been "suitably defined,"that is, so that the decision maker is indifferent between (1) having an outcomegenerated by a single distribution with probabilities equal to <i>aP</i> + (1 - <i>a</i>)<i>Q</i> and (2) thepreviously described two-stage process. If the decision maker enjoys the processitself, such as the watching of a horse race, then this must be accounted for indefining "outcomes." See Markowitz (1959 and 1997) for further elaboration ofthis point.</p><p>We consider an RDM who has preferences among probability distributions of the noutcomes. Specifically, if <i>P</i> and <i>Q</i> are any two such probability distributions,the RDM either prefers <i>P</i> to <i>Q</i>, prefers <i>Q</i> to <i>P</i>, or considers both equally good. Aset of preferences may or may not be in accordance with the expected utilitymaxim. If preferences are in accordance with the expected utility maxim, thereare numbers</p><p><i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, <i>u</i><sub>3</sub>, ..., <i>u<sub>n</sub></i></p><p>such that the probability distribution</p><p>(<i>p</i><sub>1</sub>, ..., <i>p<sub>n</sub></i></p><p>is preferred to the probability distribution</p><p><i>q</i><sub>1</sub>, ..., <i>q<sub>n</sub></i></p><p>if and only if</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)</p><br><p>We call <i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ... the utilities assigned to each outcome. The average of theseutilities, weighted by probabilities <i>p</i><sub>1</sub>, <i>p</i><sub>2</sub>, ..., is the distribution's expectedutility. Thus the expected utility rule assumes that some <i>linear</i> function[summation]<i>u<sub>i</sub>p<sub>i</sub></i> describes the RDM's preferences amongprobability distributions.</p><br><p><b>UNIQUENESS</b></p><p>If <i>u</i><sub>1</sub>, <i>u</i><sub>2</sub>, ..., <i>u<sub>n</sub></i> are utility numbers that describe an RDM's preferences, and onecomputes new utility numbers <i>u'</i><sub>1</sub>, ..., <i>u'<sub>n</sub></i> by multiplying the old utilities by a positivenumber and perhaps adding a constant:</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)</p><br><br><p>then <i>u' = (u'<sub>1</sub>, ..., u'<sub>n</sub>)</i> ranks probability distributions exactly as does<i>u = (u<sub>1</sub>, ..., u<sub>n</sub>)</i>. That is,</p><p><i>[summation]u'<sub>i</sub>p<sub>i</sub> ≥ [summation]u'<sub>i</sub>q<sub>i</sub></i> (4a)</p><p>if and only if</p><p><i>[summation]u<sub>i</sub>p<sub>i</sub> ≥ [summation]u<sub>i</sub>q<sub>i</sub></i> (4b)</p><p>Proof</p><p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]</p><br><p>But since <i>b > 0, E<sub>p</sub>U' ≥ E<sub>Q</sub>U'</i> if and only if <i>E<sub>p</sub>U ≥ E<sub>Q</sub>U</i>.</p><br><p>If a decision maker had preferences among probability distributions of threeoutcomes with utilities <i>u</i><sub>1</sub> = 3, <i>u</i><sub>2</sub> = 4, and <i>u</i><sub>3</sub> = 8, and it were desirable todescribe these preferences in such a way that the utility attached to the firstoutcome is 0 and that attached to the third is 1, we may divide each utility by5 and then subtract 3/5 to obtain equivalent utilities:</p><p><i>u'</i><sub>1</sub> = 0, <i>u'</i><sub>2</sub> = 0.2, and <i>u'</i><sub>3</sub> = 1</p><br><p>The <i>u'</i> describe the same set of preferences as the <i>u</i> and have the specifiedoutcomes as the "origin" and "unit" for their utility scale.</p><p>In general, any outcome can be assigned zero utility, and any preferred outcomecan be assigned unit utility. Beyond that, the decision maker's preferencesamong probability distributions uniquely determine the utility number that mustbe assigned to each outcome. For example, suppose that an Outcome 1 is assignedzero utility and an Outcome 2 is assigned unit utility. What utility number ornumbers may be assigned to an Outcome 3 that is preferred to Outcome 1 but notto Outcome 2? In a subsequent section, we will see that if the decision makeracts according to the expected utility maxim, there is a probability <i>p</i> such thatthe probability <i>p</i> of getting Outcome 1 and the probability 1 - <i>p</i> of gettingOutcome 2 are exactly as good as getting Outcome 3 with certainty. Hence</p><p><i>u<sub>3</sub> = pu<sub>1</sub> + (1 - p)u<sub>2</sub>= p(0) + (1 - p)(1)= 1 - p</i></p><br><p>This value of <i>u</i><sub>3</sub> must be assigned to Outcome 3 to represent the RDM'spreferences once <i>u</i><sub>1</sub> = 0 and <i>u</i><sub>2</sub> = 1 have been assigned. Other cases are dealtwith in an endnote.</p><br><p><b>CHARACTERISTICS OF EXPECTED UTILITY MAXIMIZATION</b></p><p>In the particular case in which random outcomes are the returns on a portfolio,the utility associated with various levels of return can be represented by acurve such as <i>ll'</i> or <i>ss'</i> in <b>Figure 1.1</b>. (To be consistent with our assumptionthat there are only a finite number of possible outcomes, we can imagine thatthe curves in <b>Figure 1.1</b> connect the utilities of various discrete levels ofreturn, for example, from 100 percent loss to some huge gain, by increments of0.0001 percent.)</p><p>A curve that is shaped like <i>ll'</i> in <b>Figure 1.1</b> is said to be <i>concave</i>; one that isshaped like <i>ss'</i> is called convex. The property that identifies a concave curveis that a straight line drawn between any two different points on the curve lieseverywhere on or below the curve. If it actually lies below the curveeverywhere, as it would in the case of <i>ll'</i>, we say that the curve is <i>strictly</i>concave. With a convex curve, a straight line drawn between two points of thecurve lies everywhere on or above it. In the case of a <i>strictly</i> convex curve,the line lies above the curve everywhere.</p><p>Suppose that an individual who acts according to the strictly concave utilitycurve <i>ll'</i> must choose between:</p><p>1. A 50-50 chance of a <i>gain</i> of (<i>a</i>) percent or a <i>loss</i> of (<i>a</i>) percent</p><p>2. The certainty of no change</p><br><p>The expected return in both cases is zero. <b>Figure 1.1</b> is drawn so that theexpected utility of option 2 is also zero. The utility attached to option 1 is</p><p>(1/2)<i>U(a)</i> + (1/2)<i>U(-a)</i></p><br><p>This equals the utility level of the point (<i>b</i> on <i>ll', c</i> on <i>ss'</i>) halfway between[<i>-a, U(-a)</i>] and [<i>a, U(a)</i>] of the respective curves. These lie on the straight linesconnecting the points. Because <i>ll'</i> is strictly concave, the utility at <i>b</i> is lessthan zero: the certainty is preferred to the gamble. Thus an RDM with a concaveutility function is risk-averse and will tend to prefer diversified portfolios.The implications of a convex utility curve such as <i>ss'</i> are the opposite. Theexpected utility at point c is greater than that of no bet. An RDM with a convexutility function would be a risk seeker and would not diversify his or herportfolio.</p><p>Unless otherwise specified, throughout this book we will assume that thedecision maker under discussion has a strictly concave (and therefore risk-averse) utility function.</p><br><p><b>RDMs VERSUS HDMs</b></p><p>In a later section, we postulate, as "axioms," certain properties that we wouldexpect of an RDM in making choices among risky alternatives. In some ways, anRDM is like an HDM; in other ways, the RDM and the HDM are quite different. Twoways in which the RDM and the HDM are alike are:</p><p>• The RDM cannot predict the outcome of a flip of a coin.</p><p>• Like an HDM, an RDM has objectives. Specifically, he or she would like a"good" probability distribution of the possible outcomes of any risky actionthat he or she takes.</p><br><p>A major difference between an RDM and an HDM is that the former computesinstantly with no errors of logic or arithmetic, whereas most HDMs cannot—intheir head, without a computer—compute the probabilities of relatively simpleprobabilistic outcomes, partly because they do not know the applicable formulas.