Diversification Preferences in the Theory of Choice
DDiversification Preferences in the Theory of Choice
Enrico G. De Giorgi ∗ Ola Mahmoud † October 7, 2016
Abstract
Diversification represents the idea of choosing variety over uniformity. Withinthe theory of choice, desirability of diversification is axiomatized as preference for aconvex combination of choices that are equivalently ranked. This corresponds to thenotion of risk aversion when one assumes the von-Neumann-Morgenstern expectedutility model, but the equivalence fails to hold in other models. This paper analyzesaxiomatizations of the concept of diversification and their relationship to the re-lated notions of risk aversion and convex preferences within different choice theoreticmodels. Implications of these notions on portfolio choice are discussed. We covermodel-independent diversification preferences, preferences within models of choiceunder risk, including expected utility theory and the more general rank-dependentexpected utility theory, as well as models of choice under uncertainty axiomatizedvia Choquet expected utility theory. Remarks on interpretations of diversificationpreferences within models of behavioral choice are given in the conclusion.
Keywords: diversification, risk aversion, convex preferences, portfoliochoice.JEL Classification: D81, G11. ∗ Department of Economics, School of Economics and Political Science, University of St. Gallen,Bodanstrasse 6, 9000 St. Gallen, Switzerland, Tel. +41 +71 2242430, Fax. +41 +71 224 28 94, email:[email protected]. † Faculty of Mathematics and Statistics, School of Economics and Political Science, University of St.Gallen, Bodanstrasse 6, 9000 St. Gallen, Switzerland and Center for Risk Management Research, Univer-sity of California, Berkeley, Evans Hall, CA 94720-3880, USA, email: [email protected] a r X i v : . [ q -f i n . E C ] O c t Introduction
Another rule which may prove useful can be derived from our theory. This isthe rule that it is advisable to divide goods which are exposed to some dangerinto several portions rather than to risk them all together. – Daniel Bernoulli, 1738The term diversification conveys the idea of introducing variety to a set of objects.Conceptually, Bernoulli (1738) may have been the first to appreciate the benefits of diver-sification in an economic context. In his fundamental 1738 article on the St. Petersburgparadox, he argues by example that risk averse investors will want to diversify.In finance, diversification is perhaps the most important of investment principles. Here,it is roughly understood as the mitigation of overall portfolio risk by investing in a widevariety of assets. The seminal work of Markowitz (1952) on portfolio theory laid the firstmathematical foundations for what we understand today under investment diversificationin finance. Markowitz’s portfolio theory provides a crucial formalization of the link betweenthe inseparable notions of diversification and risk; it postulates that an investor shouldmaximize portfolio return while minimizing portfolio risk, given by the return variance.Hence, diversification in finance is equivalent to the reduction of overall risk (but notgenerally its elimination). Interestingly, assuming that markets are populated by investorsas in Markowitz (1952), only non-diversifiable risk is priced at equilibrium, as shown bythe well-known Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner(1965), and Mossin (1966). Briefly, under the CAPM, investors are only rewarded for non-diversifiable or systematic risk.Diversification plays an equally important role in economic theory. The way in whichan individual economic agent makes a decision in a given choice theoretic model formsthe basis for how diversification is viewed. An economic agent who chooses to diversifyis understood to prefer variety over similarity. Axiomatically, a preference relation (cid:37) ona set of choices X exhibits preference for diversification if for any x , . . . , x n ∈ X and α , . . . , α n ∈ [0 ,
1] for which (cid:80) ni =1 α i = 1, x ∼ · · · ∼ x n ⇒ n (cid:88) i =1 α i x i (cid:37) x j for all j = 1 , . . . , n. An individual will hence want to diversify among a collection of choices all of which areranked equivalently. This notion of diversification is equivalent to that of convexity ofpreferences, which states that α x + (1 − α ) y (cid:37) y , for all α ∈ [0 , x (cid:37) y .The most common example of diversification in the above choice theoretic sense iswithin the universe of asset markets, where an investor faces a choice amongst risky posi-tions, such as equities, derivatives, portfolios, etc. Such risky positions are usually modeledas random variables on some state space Ω under a given objective reference probability2 . Diversification across two equivalently ranked risky assets x and y is then expressedby the state-wise convex combination α x ( ω ) + (1 − α ) y ( ω ) for P -almost all ω ∈ Ω and α ∈ [0 , α to asset x and a fraction 1 − α to asset y rather than fullyinvest in either one of the assets. Within traditional decision theory under risk, wherepreferences are formed over lotteries, that is probability measures p : Z → [0 ,
1] over aset of prizes Z , diversification has yet another interpretation. Here, a convex combination α p + (1 − α ) q of equally desirable lotteries p and q is defined by taking the convex com-bination of each prize z separately, that is, ( α p + (1 − α ) q )( z ) = α p ( z ) + (1 − α ) q ( z ).This convex combination of lotteries can be interpreted as some additional randomization,since it corresponds to the sampling of either p or q depending on the outcome of a binarylottery with probability α or 1 − α .The concepts of diversification and risk aversion are closely intertwined. In axiomaticchoice theory, risk aversion is roughly the preference for a certain outcome with a possiblylower payoff over an uncertain outcome with equal or higher expected value. More pre-cisely, a decision maker is said to be risk averse if the expected value of a random variablewith certainty is preferred to the random variable itself. Informally, one might say thatthe goal behind introducing variety through diversification is the reduction of “risk” or“uncertainty”, and so one might identify a diversifying decision maker with a risk averseone. This is indeed the case in expected utility theory (EUT), where risk aversion andpreference for diversification are exactly captured by the concavity of the utility function.However, this equivalence fails to hold in other models of choice.Even though the desirability for diversification is a cornerstone of a broad range ofportfolio choice models in finance and economics, the precise formal definition differs frommodel to model. Analogously, the way in which the notion of diversification is interpretedand implemented in the investment management community varies greatly. Diversifyingstrategies thus span a vast range, both in theory and in practice, from the classical ap-proaches of Markowitz’s variance minimization and von-Neumann-Morgenstern’s expectedutility maximization, to the more naive approaches of equal weighting or increasing thenumber of assets.Diversification, in its essence, can be regarded as a choice heuristic that comes indifferent forms. This article provides the first comprehensive overview of the variousexisting formalizations of the notion of diversification from a choice theoretic perspective.Different axiomatizations of the concept of diversification and their relationship to relatednotions of risk aversion are reviewed within some of the most common decision theoreticframeworks. Motivated by Bernoulli’s realization of its financial and economic benefits,we discuss the implications of each of the major definitions of diversification on portfoliochoice.We start by setting up the theoretical choice theoretic framework in Section 2. Section3 examines various choice theoretic axiomatizations of the concept of diversification and3heir relationship to convexity of preferences and concavity of utility. Given the intrinsiclink between risk aversion and diversification, Section 4 reviews common definitions of riskaversion, including weak, strong and monotone risk aversion, and their inter-relationship.Section 5 studies the connection between diversification preferences, convex preferencesand risk aversion under no particular model assumptions. Section 6 reviews the classicresults within the framework of expected utility theory, where all definitions of diversifi-cation preferences as well as all notions of risk aversion coincide with the concavity of thevon-Neumann-Morgenstern utility representation. Section 7 considers the more generalrank-dependent expected utility model of Quiggin (1982), where the equivalence betweenweak and strong risk aversion does not carry over from the expected utility model. Sim-ilarly, the correspondence between risk aversion and preference for diversification doesnot hold any longer. Section 8 extends the discussion from models of choice under riskto models of choice under uncertainty. This covers decision models where there is nogiven objective probability distribution on the set of states of the world, the axiomati-zation of which is given by Choquet expected utility theory. Notions of diversificationand uncertainty aversion, rather than risk aversion, are discussed within such models ofexpected utility under a non-additive subjective probability measure. Under uncertainty,the phenomenon of ambiguity aversion roughly captures the preference for known risksover unknown risks. In Section 9, we recall the most prominent definitions of ambiguityaversion, link them to the notion of diversification and discuss the implications on portfoliochoice. Section 10 concludes with remarks on interpretations of diversification preferenceswithin models of behavioral choice. We adopt the classical setup for risk assessment used in mathematical finance and portfoliochoice. We consider a decision maker who chooses from the vector space X = L ∞ (Ω , F , P )of essentially bounded real-valued random variables on a probability space (Ω , F , P ), whereΩ is the set of states of nature, F is a σ -algebra of events, and P is a σ -additive prob-ability measure on (Ω , F ). Note that the decision maker is also able to form compoundchoices represented by the state-wise convex combination α x + (1 − α ) y for x, y ∈ X and α ∈ [0 , α x ( ω ) + (1 − α ) y ( ω ) for P -almost all ω ∈ Ω. The space X is endowedwith the order x ≥ y ⇔ x ( ω ) ≥ y ( ω ) for P -almost all ω ∈ Ω. Our assumption that theoutcome space is the set R of real numbers, which comes with an intrinsic ordering andmixing operations, enables a natural monetary interpretation of outcomes.A weak preference relation on X is a binary relation (cid:37) satisfying:(i) Completeness : For all x, y ∈ X , x (cid:37) y ∨ y (cid:37) x .(ii) Transitivity : For all x, y, z ∈ X , x (cid:37) y ∧ y (cid:37) z ⇒ x (cid:37) z .4very weak preference relation (cid:37) on X induces an indifference relation ∼ on X defined by x ∼ y ⇔ ( x (cid:37) y ) ∧ ( y (cid:37) x ). The corresponding strict preference relation (cid:31) on X is definedby x (cid:31) y ⇐⇒ x (cid:37) y ∧ ¬ ( x ∼ y ). A numerical or utility representation of the preferencerelation (cid:37) is a real-valued function u : X → R for which x (cid:37) y ⇐⇒ u ( x ) ≥ u ( y ).For x ∈ X , F x denotes the cumulative distribution function of x , defined by F x ( c ) = P [ x ≤ c ] for c ∈ R , and e ( x ) is the expectation of x , that is, e ( x ) = (cid:82) c dF x ( c ). For c ∈ R , δ c denotes the degenerated random variable with δ c ( ω ) = c for P -almost all ω ∈ Ω.The certainty equivalent of x ∈ X is the value c ( x ) ∈ R such that x ∼ δ c ( x ) , i.e., c ( x )is the certain value which the decision maker views as equally desirable as a choice x with uncertain outcome. The risk premium π ( x ) of x ∈ X is the amount by which theexpected return of a choice x ∈ X must exceed the value of the guaranteed outcome inorder to make the uncertain and certain choices equally attractive. Formally, it is definedas π ( x ) = e ( x ) − c ( x ). Monotonicity.
Emulating the majority of frameworks of economic theory, it seems rea-sonable to assume that decision makers prefer more to less. In particular, in view of themonetary interpretation of the space X , a natural assumption on the preference relation (cid:37) is monotonicity .(iii) Monotonicity : For all x, y ∈ X , x ≥ y = ⇒ x (cid:37) y .Monotonicity of preferences is equivalent to having a strictly increasing utility function u . Indeed, for x ≥ y , we have x (cid:37) y and thus u ( x ) ≥ u ( y ). Monotonicity of the utilityfunction simply implies that an agent believes that “more is better”; a larger outcomeyields greater utility, and for risky bets the agent would prefer a bet which is first-orderstochastically dominant over an alternative bet. Continuity.
Continuity of preferences is often assumed for technical reasons, as it canbe used as a sufficient condition for showing that preferences on infinite sets can haveutility representations. It intuitively states that if x (cid:31) y , then small deviations from x orfrom y will not reverse the ordering.(iv) Continuity : For every x (cid:31) y , there exist neighborhoods B x , B y ⊆ X around x and y , respectively, such that for every x (cid:48) ∈ B x and y (cid:48) ∈ B y , x (cid:48) (cid:31) y (cid:48) .Throughout this article, unless otherwise stated, we assume that preferences are bothmonotonic and continuous. Debreu’s theorem (Debreu 1964) states that there exists acontinuous monotonic utility representation u of a monotonic and continuous preferencerelation (cid:37) . 5 Rudiments of convexity, diversification, and risk
The notions of convex preferences, preferences for diversification, and risk are inherentlylinked, both conceptually and mathematically. We recall the formal definitions and theirrelationships.
We begin with the mathematically more familiar concept of convexity. The notion of con-vexity of preferences inherently relates to the classic ideal of diversification, as introducedby Bernoulli (1738). To be able to express convexity of a preference relation, one assumesa choice-mixing operation on X that allows agents to combine (that is diversify across)several choices. By combining two choices, the decision maker is ensured under convexitythat he is never “worse off” than the least preferred of these two choices. Definition 1 (Convex preferences) . A preference relation (cid:37) on X is convex if for all x, y ∈ X and for all α ∈ [0 , , x (cid:37) y = ⇒ α x + (1 − α ) y (cid:37) y. In mathematics and economic theory, convexity is an immensely useful property, par-ticularly within optimization. The role it plays in the theory of choice leads to someconvenient results, because of what it says about the corresponding utility representation.We recall some well-known properties of utility functions representing convex preferences.
Proposition 1.
A preference relation (cid:37) on X is convex if and only if its utility represen-tation u : X → R is quasi-concave. This means that convexity of preferences and quasi-concavity of utility are equivalent.A direct corollary to this result is that concavity of utility implies convexity of preferences.However, convex preferences may have numerical representations that are not concave. More strongly even, some convex preferences can be constructed in a way that does notadmit any concave utility representation.
An important property within the theory of choice is that of diversification. An economicagent who chooses to diversify is understood to prefer variety over similarity. Axiomati-cally, preference for diversification is formalized as follows; see Dekel (1989). To see this, suppose u is a concave utility function representing a convex preference relation (cid:37) . Then ifa function f : R → R is strictly increasing, the composite function f ◦ u is another utility representation of (cid:37) . However, for a given concave utility function u , one can relatively easily construct a strictly increasingfunction f such that f ◦ u is not concave. efinition 2 (Preference for diversification) . A preference relation (cid:37) exhibits preferencefor diversification if for any x , . . . , x n ∈ X and α , . . . , α n ∈ [0 , for which (cid:80) ni =1 α i = 1 , x ∼ · · · ∼ x n = ⇒ n (cid:88) i =1 α i x i (cid:37) x j for all j = 1 , . . . , n. This definition states that an individual will want to diversify among a collection ofchoices all of which are ranked equivalently. The most common example of such diversi-fication is within the universe of asset markets, where an investor faces a choice amongstrisky assets. We recall that this notion of diversification is, in our setup, equivalent tothat of convexity of preferences.
Proposition 2.
A monotonic and continuous preference relation (cid:37) on X is convex if andonly if it exhibits preference for diversification. Various other definitions of diversification exist in the literature. Chateauneuf andTallon (2002) introduce the stronger notion of sure diversification . Roughly, sure diversi-fication stipulates that if the decision maker is indifferent between a collection of choicesand can attain certainty by a convex combination of these choices, he should prefer thatcertain combination to any of the uncertain choices used in the combination.
Definition 3 (Preference for sure diversification) . A preference relation (cid:37) exhibits pref-erence for sure diversification if for any x , . . . , x n ∈ X and α , . . . , α n ≥ satisfying (cid:80) ni =1 α i = 1 , and c, β ∈ R , (cid:34) x ∼ · · · ∼ x n ∧ n (cid:88) i =1 α i x i = βδ c (cid:35) = ⇒ βδ c (cid:37) x i , ∀ i = 1 , . . . , n. Chateauneuf and Lakhnati (2007) introduce a weakening of the concept of preferencefor diversification, which is referred to as preference for strong diversification . Preferencefor strong diversification means that the decision maker will want to diversify between twochoices that are identically distributed.
Definition 4 (Preference for strong diversification) . A preference relation (cid:37) exhibits preference for strong diversification if for all x, y ∈ X with F x = F y and α ∈ [0 , , αx + (1 − α ) y (cid:37) y . Preference for diversification implies preference for sure diversification, but the conversedoes not hold. One can also show that preferences for strong diversification are equivalentto requiring that preferences respect second-order stochastic dominance (see Chateauneufand Lakhnati (2007) for an outline of the proof). Moreover, Chateauneuf and Lakhnati(2007) also provide a counterexample showing that preferences for strong diversificationdo not imply convex preferences.Yet another notion of diversification was introduced by Chateauneuf and Tallon (2002),namely that of comonotone diversification . Two random variables x, y ∈ X are comono-tonic if they yield the same ordering of the state space from best to worst; more formallyif for every ω, ω (cid:48) ∈ S , ( x ( ω ) − x ( ω (cid:48) ))( y ( ω ) − y ( ω (cid:48) )) ≥
0. Comonotonic diversification isdefined as follows: 7 efinition 5 (Comonotone diversification) . A decision maker exhibits preference for comonotone diversification if for all comonotonic x and y for which x ∼ y , αx + (1 − α ) y (cid:37) x for all α ∈ (0 , . Comonotone diversification is essentially convexity of preferences restricted to comono-tonic random variables, just as Schmeidler (1989) restricted independence to comonotonicacts (see Section 8 for a more detailed discussion of Schmeidler’s model). Under this morerestrictive type of diversification, any hedging in the sense of Wakker (1990) is prohibited.
In mathematical finance, diversification is often understood to be a technique for reducingoverall risk , where, here, one may follow the Knightian (Knight 1921) identification of thenotion of risk as measurable uncertainty . In classical risk assessment within mathematicalfinance, uncertain portfolio outcomes over a fixed time horizon are represented as randomvariables on a probability space. A risk measure maps each random variable to a realnumber summarizing the overall position in risky assets of a portfolio. In his seminalpaper, Markowitz (1952), even though he proposed variance as a risk measure, emphasizedthe importance for a risk measure to encourage diversification. Over the past two decades,a number of academic efforts have more formally proposed properties that a risk measureshould satisfy, for example the work of F¨ollmer and Schied (2010) and F¨ollmer and Schied(2011) who echo Markowitz in that a “good” risk measure needs to promote diversification.The key property is once again that of convexity, which, if satisfied, does not allow thediversified risk to exceed the individual standalone risks. It thus reflects the key principleof economics and finance, as well as the key normative statement in the theory of choice,namely that diversification should not increase risk .We review this diversification paradigm in the context of risk measures within thetheory of choice. We emulate the formal setup of Drapeau and Kupper (2013), wherethe risk perception of choices is modeled via a binary relation ˆ (cid:37) , the “risk order”, on X satisfying some appropriate normative properties. A risk order represents a decisionmaker’s individual risk perception, where x ˆ (cid:37) y is interpreted as x being less risky than y .Risk measures are then quasiconvex monotone functions, which play the role of numericalrepresentation of the risk order.The two main properties of risk captured by a risk order are those of convexity andmonotonicity. The convexity axiom reflects that diversification across two choices keepsthe overall risk below the worse one; the monotonicity axiom states that the risk order iscompatible with the vector preorder. A formal definition follows. “ Diversification is both observed and sensible; a rule of behavior which does not imply the superiorityof diversification must be rejected. ” efinition 6 (Risk order) . A risk order on the set X is a reflexive weak preference relationsatisfying the convexity and monotonicity axioms. Numerical representations of risk orders inherit the two key properties of convexityand monotonicity of a decision maker’s risk perception and are called risk measures.
Definition 7 (Risk measure) . A real-valued mapping ρ : X → R is a risk measure if it is:(i) quasiconvex: for all x, y ∈ X and λ ∈ [0 , , ρ ( λx + (1 − λ ) y ) ≤ max { ρ ( x ) , ρ ( y ) } (ii) monotone: for all x, y ∈ X , x ≥ y = ⇒ ρ ( x ) ≤ ρ ( y )The following theorem states the bijective correspondence between risk orders and theirrepresentation via risk measures. Theorem 1 (Drapeau and Kupper (2013)) . Any numerical representation ρ ˆ (cid:37) : X → R ofa risk order ˆ (cid:37) on X is a risk measure. Conversely, any risk measure ρ : X → R definesthe risk order ˆ (cid:37) ρ on X by x ˆ (cid:37) ρ y ⇐⇒ ρ ( x ) ≤ ρ ( y ) . Risk orders and risk measures are bijectively equivalent in the sense that ˆ (cid:37) = ˆ (cid:37) ρ ˆ (cid:37) and ρ ˆ (cid:37) ρ = h ◦ ρ for some increasing transformation h : R → R . Note that, in the context of theories of choice, the risk measure ρ ˆ (cid:37) corresponding to agiven risk order ˆ (cid:37) is in fact the negative of the quasiconcave utility representation of theconvex and monotonic total preorder ˆ (cid:37) . In our setup of choice amongst risky positions L ∞ (Ω , F , P ), a commonly used example of such a risk measure is the tail mean (Acerbiand Tasche 2002b, Acerbi and Tasche 2002a), defined by TM α = E [ x | x > q α ( x )], where α ∈ (0 ,
1) is the confidence level and q α ( x ) = inf { x (cid:48) ∈ R : P ( x ≤ x (cid:48) ) ≥ α } is the lower α -quantile of the random variable x . Based on the previous discussion, a risk order exhibits preference for diversification(through the equivalent convexity axiom) if and only if the corresponding risk measurerepresenting it is quasiconvex. This is a weakening of the general understanding of di-versification within the theory of quantitative risk measurement, where diversification isencouraged when considering convex risk measures (F¨ollmer and Schied 2010, F¨ollmerand Schied 2011) or, even more strongly, subadditive risk measures (Artzner, Delbaen,Eber, and Heath 1999). Drapeau and Kupper (2013) refer to the convexity property as quasiconvexity , which we believe isa mathematically more appropriate nomenclature. However, we stick to the more widely used convexity terminology for consistency. This definition of tail mean holds only under the assumption of continuous distributions, that is forintegrable x . A risk measure ρ : X → R is convex if for all x, y ∈ X and λ ∈ [0 , ρ ( λx + (1 − λ ) y ) ≤ λρ ( x ) + (1 − λ ) ρ ( y ). A risk measure ρ : X → R is subadditive if for all x, y ∈ X , ρ ( x + y ) ≤ ρ ( x ) + ρ ( y ).
9n mathematical finance, the passage from convexity to quasiconvexity is conceptuallysubtle but significant; see, for example, Cerreia-Vioglio, Maccheroni, Marinacci, and Mon-trucchio (2011a). While convexity is generally regarded as the mathematical formalizationof the notion of diversification, it is in fact equivalent to the notion of quasiconvexity un-der a translation invariance assumption . By considering the weaker notion of quasiconvexrisk, one disentangles the diversification principle from the assumption of liquidity of theriskless asset – an abstract simplification encapsulated through the translation invarianceaxiom. As we have seen above, the economic counterpart of quasiconvexity of risk measuresis quasiconcavity of utility functions, which is equivalent to convexity of preferences. The concepts of diversification and risk aversion are closely intertwined. Informally, onemight say that the goal behind introducing variety through diversification is the reductionof “risk” or “uncertainty”, and so one might identify a diversifying decision maker with arisk averse one. We will later see that this is generally not the case.In axiomatic choice theory, risk aversion is roughly the preference for a certain outcomewith a possibly lower expected payoff over an uncertain outcome with equal or higherexpected value. In the economics literature, risk aversion is often exactly captured by theconcavity of the utility function, and this is based on the underlying implicit framework ofexpected utility theory. In other models, however, risk aversion no longer goes along with aconcave utility function, unless perhaps the very definition of risk aversion is reconsidered.In this section, when relating risk aversion to diversification or convexity of preferences,we look at intrinsic notions of risk aversion rather than model-dependent definitions. Tothis end, we use the three most frequently used definitions of weak, strong, and monotonerisk aversion. In expected utility theory, all of these notions coincide and are characterizedby the concavity of the utility function. We stress, however, the model-independency ofthe following definitions.
The first, most common notion of risk aversion is based on the comparison between arandom variable and its expected value. A decision maker is weakly risk averse if healways prefers the expected value of a random variable with certainty to the randomvariable itself.
Definition 8 (Weak risk aversion) . The preference relation (cid:37) on X is weakly risk averse if δ e ( x ) (cid:37) x for every x ∈ X , where e ( x ) denotes the expected value of the random variable x . A decision maker is weakly risk seeking if he always prefers any random variable toits expected value with certainty; formally if for all x ∈ X , x (cid:37) δ e ( x ) . A decision maker is A risk measure ρ : X → R is tranlsation invariant (or cash-additive) if for all x ∈ X and m ∈ R , ρ ( x + m ) = ρ ( x ) − m . if he is always indifferent between any random variable and its expectedvalue with certainty; formally if for all x ∈ X , x ∼ δ e ( x ) . A straightforward characterization of weak risk aversion can be given in terms of therisk premium. Indeed, a decision maker is weakly risk averse if and only if the risk premium π ( x ) associated to any x ∈ X is always nonnegative. Using this, one obtains a relationbetween decision makers ranking their level of risk aversion. Decision maker D is saidto be more risk averse than decision maker D if and only if for every x ∈ X , the riskpremium π ( x ) associated to x is at least as great for D as it is for D . The second notion of risk aversion is based on the definition of increasing risk of Hadar andRussell (1969) and Rothchild and Stiglitz (1970) (see also Landsberger and Meilijson 1993)who define it in terms of the mean preserving spread.
Definition 9 (Mean preserving spread) . For two random variables x, y ∈ X , y is a mean preserving spread of x if and only if e ( x ) = e ( y ) and x second-order stochasticallydominates y , written as x (cid:37) SSD y , that is if for any C ∈ R , (cid:90) C −∞ F x ( c ) dc ≤ (cid:90) C −∞ F y ( c ) dc. The mean preserving spread is intuitively a change from one probability distribution toanother probability distribution, where the latter is formed by spreading out one or moreportions of the probability density function or probability mass function of the formerdistribution while leaving the expected value unchanged. As such, the concept of meanpreserving spreads provides a stochastic ordering of equal-mean choices according to theirdegree of risk. From the definition, we see that ordering choices by mean preservingspreads is a special case of ordering them by second-order stochastic dominance when theexpected values coincide. Moreover, this ordering has the following properties.
Lemma 1.
The stochastic ordering induced by the mean preserving spread on x, y ∈ X (i)depends only on the probability distributions of x and y ; and (ii) implies a non-decreasingvariance, but a non-decreasing variance does not imply a mean preserving spread. The notion of strong risk aversion can be viewed as aversion to any increase in risk,formalized next in terms of the mean preserving spread.
Definition 10 (Strong risk aversion) . The preference relation (cid:37) is strongly risk averse if and only if for any x, y ∈ X such that y is a mean preserving spread of x , x (cid:37) y . Itis strongly risk seeking if for any x, y ∈ X such that y is a mean-preserving spread of x , y (cid:37) x . The preference relation (cid:37) is strongly risk neutral if for any x, y ∈ X such that y is a mean-preserving spread of x , x ∼ y . Preferences that are strongly risk averse (respectively strongly risk seeking) are alsoweakly risk averse (respectively weakly risk seeking). This is because for any x ∈ X , x is11lways a mean preserving spread of δ e ( x ) . Note also that strong and weak risk neutralityare equivalent, because weak risk neutrality implies that any x is indifferent to δ e ( x ) sothat if y is a mean preserving spread of x , they are both indifferent to δ e ( x ) . The notion of strong risk aversion can be considered as too strong by some decision makers.This lead Quiggin (1992) (see also Quiggin (1991)) to define a new way for measuringincreasing risk and, as a consequence, a new weaker notion of risk aversion, called monotonerisk aversion .The definition of monotone risk aversion involves comonotonic random variables. Recallthat two random variables x, y ∈ X are comonotonic if they yield the same ordering of thestate space from best to worst. Clearly, every constant random variable is comonotonicwith every other random variable. One can then define a measure of increasing risk ofcomonotonic random variables as follows.
Definition 11 (Mean preserving monotone spread I) . Suppose x and y are two comono-tonic random variables. Then y is a mean preserving monotone spread of x if e ( x ) = e ( y ) and z = y − x is comonotonic with x and y . This definition ensures that, since x and z are comonotonic, there is no hedging between x and z , and thus that y can be viewed as more risky than x . It can be extended to randomvariables that are not necessarily comonotonic, as follows. Definition 12 (Mean preserving monotone spread II) . For two random variables x, y ∈ X , y is a mean preserving monotone spread of x if there exists a random variable θ ∈ X suchthat y has the same probability distribution as x + θ , where e ( θ ) = 0 and x and θ arecomonotonic. The notion of monotone risk aversion can be viewed as aversion to monotone increasingrisk and is based on the definition of mean preserving monotone spread. We will later showthat this concept of risk aversion is consistent with the rank-dependent expected utilitytheory of Quiggin (1982), one of the most well-known generalizations of expected utilitytheory, in which comonotonicity plays a fundamental part at the axiomatic level.
Definition 13 (Monotone risk aversion) . The preference relation (cid:37) on X is monotonerisk averse if for any x, y ∈ X where y is a mean preserving monotone spread of x , x (cid:37) y .It is monotone risk seeking if for any x, y ∈ X where y is a mean preserving monotonespread of x , y (cid:37) x , and it is monotone risk neutral if for any x, y ∈ X where y is a meanpreserving monotone spread of x , x ∼ y . Finally, the following relationship between the three notions of weak, strong, and mono-tone risk aversion holds.
Proposition 3 (Cohen (1995)) . Strong risk aversion implies monotone risk aversion;monotone risk aversion implies weak risk aversion; weak risk neutrality, strong risk neu-trality, and monotone risk neutrality are identical. Model-independent diversification preferences
We study the relationship between diversification preferences, convex preferences and riskaversion when preferences are not assumed to fit a specific choice theoretic model. Asbefore, we simply assume that preferences are monotonic and continuous.
Because the conventional definition of diversification is too strong to yield an equivalenceto weak risk aversion when we move outside the assumptions of expected utility theory, theweaker concept of sure diversification was introduced. This weaker notion of diversificationis indeed equivalent to weak risk aversion, independent of any model.
Proposition 4 (Chateauneuf and Lakhnati (2007)) . A monotonic and compact contin-uous preference relation (cid:37) exhibits preference for sure diversification if and only if it isweakly risk averse. In the space of probability distributions rather than random variables, Dekel (1989) showsthat – assuming no particular choice model – preference for diversification is usuallystronger that strong risk aversion. Indeed, he shows that preference for diversificationimplies risk aversion, but the converse is false. Proposition 5 (Dekel (1989)) . A strongly risk averse preference relation (cid:37) does notnecessarily exhibit preference for diversification.
This means that preference for diversification is generally stronger than strong riskaversion. However, a complete characterization of strong risk aversion can be achievedthrough a weakening of preference for diversification obtained through the notion of strongdiversification.
Theorem 2 (Chateauneuf and Lakhnati (2007)) . A monotonic and compact continuouspreference relation (cid:37) exhibits preference for strong diversification if and only if it respectssecond-order stochastic dominance.
An immediate consequence of this result is that convexity of preferences, or equivalentlypreference for diversification, implies strong risk aversion. We point out that Dekel (1989)proves the same result in the framework of probability distributions.
Corollary 1.
A preference relation (cid:37) exhibiting preference for diversification is stronglyrisk averse. A preference relation (cid:37) is compact continuous if x (cid:37) y whenever a bounded sequence ( x n ) n ∈ N con-verges in distribution to x and x n (cid:37) y for each n . As noted by Chew and Mao (1995), many widelyused examples of expected utility preferences are in fact compact continuous and not continuous whenthe corresponding utility function is discontinuous or unbounded. The theoretical setup of Dekel (1989) used to derive the results reviewed in this section is a veryparticular one, and we encourage the reader to read his article for the details. Choice under risk I: expected utility theory
Since the publication of the seminal
Theory of Games and Economic Behavior of vonNeumann and Morgenstern (1944), expected utility theory (EUT) has dominated theanalysis of decision-making under risk and has generally been accepted as a normativemodel of rational choice. A wide range of economic phenomena that previously laybeyond the scope of economic formalization were successfully modelled under expectedutility theory.This Section illustrates that, in the axiomatic setup of expected utility theory, (i) allnotions of risk aversion coincide with concavity of utility, (ii) all notions of diversifica-tion coincide, and (iii) preference for diversification (that is convexity of preferences) isequivalent to risk aversion.
A utility representation of the preference relation (cid:37) on X under EUT takes the form u ( x ) = (cid:90) u ( c ) dF x ( c ) , x ∈ X , c ∈ R , u : R → R . Traditionally, the objects of choice in the von Neumann–Morgenstern setup are lotteries rather than random variables, which are formalized via probability distributions. Theagent’s choice set is hence the set M of all probability measures on a separable metricspace ( S, B ), with B the σ -field of Borel sets. A utility representation u of a preferencerelation (cid:37) on M takes the form u ( µ ) = (cid:90) u ( x ) µ ( dx ) , µ ∈ M , u : S → R . A compound lottery that is represented by the distribution αµ + (1 − α ) ν ∈ M , for α ∈ [0 , µ with probability α and ν with probability (1 − α ), and so the probabilityof an outcome x under the compound lottery is given by( αµ + (1 − α ) ν ) ( x ) = αµ ( x ) + (1 − α ) ν ( x ) . The crucial additional axiom that identifies expected utility theory is the independenceaxiom . It has also proven to be the most controversial. The independence axiom statesthat, when comparing the two compound lotteries αµ + (1 − α ) ξ and αν + (1 − α ) ξ ,the decision maker should focus on the distinction between µ and ν and hold the samepreference independently of both α and ξ . The key idea is that substituting ξ for part of µ and part of ν should not change the initial preference ranking. The formal statement ofthe axiom follows. This is despite the evidence supporting alternative descriptive models showing that people’s actualbehavior deviates significantly from this normative model. See Stanovich (2009) and Hastie and Dawes(2009) for a discussion. ndependence : For all µ, ν, ξ ∈ M and α ∈ [0 , µ (cid:37) ν = ⇒ αµ + (1 − α ) ξ (cid:37) αν + (1 − α ) ξ . Remark 1.
In this paper we adopted the classical setup in mathematical finance, whichis the space of random variables. However, as mentioned above, the von Neumann–Morgenstern setup is the space of lotteries. In view of our discussion on preference fordiversification, we refer to Dekel (1989, Proposition 3), which states that convexity ofpreferences over the space of lotteries together with risk aversion implies preference fordiversification over the space of random variables. This clarifies the link between convexityof preferences over the space of lotteries and convexity of preferences over the space ofrandom variables.
The power of the analysis of concepts of risk aversion and of the corresponding interpre-tation of increasing risk in terms of stochastic dominance contributed to a large degree tothe success of EUT in studying problems relating to risk. Indeed, in EUT, risk aversioncorresponds to a simple condition on the utility function: within the class of preferencerelations which admit a von-Neumann-Morgenstern representation, a decision maker ischaracterized via his concave utility function.
Proposition 6.
Suppose the preference relation (cid:37) satisfies expected utility theory andadmits a von Neumann-Morgenstern utility representation u . Then: (i) (cid:37) is weakly riskaverse if and only if u is concave; and (ii) (cid:37) is strongly risk averse if and only if u isconcave. An immediate consequence of Proposition 6 and Proposition 3 is that, under the ex-pected utility framework, all three notions of weak, strong, and monotone risk aversioncoincide.
Corollary 2.
Under the assumptions of expected utility theory, the definitions of weak,strong, and monotone risk aversion are equivalent.
In its essence, expected utility theory imposes restrictions on choice patterns. Indeed, itis impossible to be weakly risk averse without being strongly risk averse. As a consequence,in EUT one simply speaks of “risk aversion” without any need to specify the particularnotion. Because it is characterized by concavity of utility, one can characterize the level ofrisk aversion through the curvature of the utility function. We recall the most commonlyused such measure of risk aversion, introduced by Arrow (1965) and Pratt (1964), and itscharacterization in the expected utility model.
Definition 14 (Arrow-Pratt measure of risk aversion) . For an expected utility theorydecision maker with utility function u , the Arrow-Pratt coefficient of absolute risk aversion is defined for any outcome c ∈ R by A ( c ) = − u (cid:48)(cid:48) ( c ) u (cid:48) ( c ) . roposition 7. Suppose that u and u are two strictly increasing and twice continuouslydifferentiable functions on R representing expected utility preferences with correspondingArrow-Pratt coefficients A and A and risk premiums π and π , respectively. Thenthe following conditions are equivalent: (i) A ( c ) ≥ A ( c ) for all outcomes c ∈ R ; (ii) u = g ◦ u for some strictly increasing concave function g ; (iii) π ( x ) ≥ π ( x ) for all x ∈ X . This essentially characterizes the relation of being more risk averse through the Arrow-Pratt coefficient.
Under the assumptions of expected utility theory, the two forms of sure and comonotonediversification are both represented by concavity of the utility index and consequentlycannot be distinguished. Furthermore, they cannot be distinguished from the traditionalnotion of diversification, which corresponds to convexity of preferences.
Proposition 8 (Chateauneuf and Tallon (2002)) . Suppose (cid:37) is a preference relation inthe expected utility theory framework with utility index u . Then the following statementsare equivalent:(i) (cid:37) exhibits preference for diversification(ii) (cid:37) exhibits preference for sure diversification(iii) (cid:37) exhibits preference for comonotone diversification(iv) u is concave Moreover, recall that the equivalence between diversification and risk aversion estab-lished in the expected utility framework does not hold in more general frameworks. Inparticular, the notion of strong diversification, that is preference for diversification amongtwo identically distributed assets, was shown to be equivalent to strong risk aversion. Un-der the assumptions of EUT, this simply means that strong diversification coincides withall forms of risk aversion and, therefore, with concavity of utility.
Corollary 3.
Suppose (cid:37) is a preference relation in the expected utility theory frameworkwith utility index u . Then the following statements are equivalent:(i) (cid:37) exhibits preference for strong diversification(ii) (cid:37) is risk averse(iii) u is concave In summary, all notions of risk aversion (weak, strong, monotone) and of diversification(sure, strong, comonotone, and convex preferences) introduced in this article coincide withconcavity of utility in the framework of expected utility theory.As an illustration of a concrete portfolio choice implication of this correspondence,consider the problem of investing in one risk-free asset with return r and one risky asset16aying a random return z . Under a full investment assumption, suppose the decisionmaker invests w in the risky asset and the remaining 1 − w in the safe asset, implying anoverall portfolio return of wz + (1 − w ) r . Maximizing utility u formally meansmax w (cid:90) u ( wz + (1 − w ) r ) dF ( z ) , where F is the cumulative distribution function (cdf) of the risky asset. An investor who isrisk neutral will only care about the expected return, and hence will put all of the availablewealth into the asset with the highest expected return. If the investor is risk averse, theutility is concave, and hence the second order condition of the maximization problemcompletely characterizes the solution. More importantly, this means that if the risky assethas a postitve rate of return above the risk-free rate (that is z > r ), the risk averse investorwill still choose to invest some amount w > u is more risk-aversethan a second expected utility investor with utility v , then in the portfolio choice problem,the former investor will optimally invest less in the risky asset than the latter for any initiallevel of wealth. However, Ross (1981) has pointed out that this result depends cruciallyon the fact that the less risky asset has a certain return. If both assets are risky, with oneless risky than the other, a much stronger condition is needed to guarantee that the lessrisk averse investor will invest more in the riskier asset than the more risk averse investorfor equal levels of wealth. Rank-dependent expected utility theory (RDEU) is a generalization of expected utilitytheory accommodating the observation that economic agents both purchase lottery tickets17implying risk-seeking preferences) and insure against losses (implying risk aversion). Inparticular, RDEU explains the behaviour observed in the Allais paradox by weakening theindependence axiom. RDEU was first axiomatized by Quiggin (1982) as anticipated utilitytheory , and was further studied by Yaari (1987), Segal (1989), Allais (1987), and Wakker(1990), amongst others.After reviewing the basics of RDEU, we show that the equivalence between weak andstrong risk aversion does not carry over from the expected utility model; see Machina(1982) and Machina (2008). Similarly, the correspondence of preference for diversificationand risk aversion fails in the RDEU framework.
The development of rank-dependent expected utility theory was motivated by the idea thatequally probable events should not necessarily receive the same decision weights. Such aprobability weighting scheme is meant to incorporate the apparent feature of overweightingof low probability events with extreme consequences that has been observed in violationsof EUT models.The RDEU model has a simple formalization in which outcomes are transformed by avon-Neumann-Morgenstern utility function and probabilities are transformed by a weight-ing function. The utility function being the same as in EUT implies that standard toolsof analysis developed for EUT may be applied, with some modifications, to the RDEUframework. The probability weighting scheme first arranges states of the world so that theoutcomes they yield are ordered from worst to best, then gives each state a weight thatdepends on its ranking as well as its probability. These ideas are formalized as follows.
Definition 15 (Preferences under rank-dependent expected utility theory) . A decisionmaker satisfies rank-dependent expected utility (RDEU) theory if and only if his preferencerelation (cid:37) can be represented by a real-valued function V such that for every x, y ∈ X , x (cid:37) y ⇐⇒ V f,u ( x ) ≥ V f,u ( y ) where V f,u is defined for every z ∈ X by V f,u ( z ) = (cid:90) −∞ ( f ( P [ u ( z ) > t ]) − dt + (cid:90) ∞ f ( P [ u ( z ) > t ]) dt , where u : R → R , the utility function representing (cid:37) , is assumed to be continuous, strictlyincreasing and unique up to positive affine transformations, and f : [0 , → [0 , is aunique, continuous and strictly increasing function satisfying f (0) = 0 and f (1) = 1 .When X has a finite number of outcomes x ≤ x ≤ · · · ≤ x n , this representationreduces to V f,u ( z ) = u ( x ) + n (cid:88) i =2 f (cid:32) n (cid:88) j = i p j (cid:33) ( u ( x i ) − u ( x i − )) . u and f ; theutility function u is interpreted as the utility level under certainty, and the transformationfunction f is interpreted as the perception of probabilities.Note that in the case that f ( p ) = p for all p ∈ [0 , f ( p ) ≤ p , which means that the decisonmaker’s perception of probability is less than the actual probability. For finite X , thiscondition implies that the decision maker, having at least utility u ( x ), systematicallyunderweights the additional utilities u ( x i ) − u ( x i − ). Such a decision maker is referred toas being (weakly) pessimistic . Definition 16.
A RDEU preference relation (cid:37) is weakly pessimistic if and only if f ( p ) ≤ p for all p ∈ [0 , , and weakly optimistic if and only if f ( p ) ≥ p for all p ∈ [0 , . RDEU models suggest an approach to risk aversion that differs from EUT. By definition,RDEU theory can be viewed as embodying a fundamental distinction between attitudesto outcomes and attitudes to probabilities. Risk aversion within the RDEU frameworkshould then encompass two different phenomena. The first is the standard notion ofrisk aversion within EUT associated with preferences over outcomes in terms of decliningutility of wealth. The second relates to preferences over probabilities, that is to thetransformation function f . We next review characterizations of both the utility function u and the probability weighting function f in terms of the various notions of risk aversion. We begin with weak risk aversion. Chateauneuf and Cohen (1994) give necessary andsufficient conditions under which preferences are weakly risk averse. We only mention oneparticular case in which their conditions are both necessary and sufficient.
Proposition 9 (Chateauneuf and Cohen (1994)) . A RDEU decision maker whose utilityfunction u is concave and differentiable is weakly risk averse if and only if he is weaklypessimistic. Strong risk aversion in rank-dependent utility theory has been characterized by Chew,Karni, and Safra (1987) as follows.
Proposition 10 (Chew, Karni, and Safra (1987)) . A RDEU preference relation (cid:37) isstrongly risk averse if and only if its utility u is concave and probability weighting f isconvex. This is a rather strong characterization, since under RDEU, a decision maker cannotbe strongly risk averse without having concave utility. On the other hand, under the dualtheory of Yaari (1987), strong risk aversion corresponds to the convexity of f (see Yaari(1987)). For a more complete review of the notions of risk aversion within the theory of choice under risk, werefer the reader to Cohen (1995).
Proposition 11 (Quiggin (1992)) . (i) A RDEU decision maker who is monotone riskaverse and whose utility u is concave is weakly pessimistic. (ii) A RDEU decision makerwho is weakly pessimistic and has concave utility u is monotone risk averse. Moreover, one obtains a characterization of the mean-preserving monotone spread.
Proposition 12 (Quiggin (1992)) . For x, y ∈ X for which e ( x ) = e ( y ) , y is a monotonemean-preserving spread of x if and only if for the preference relation (cid:37) under RDEU, x (cid:37) y . In expected utility theory, risk averse decision makers will always prefer a diversifiedportfolio over a concentrated one. A similar result holds for RDEU decision makers whoare strongly risk averse in the sense of second-order stochastic dominance.
Proposition 13 (Quiggin (1993)) . A RDEU decision maker exhibits preference for di-versification if and only if he is strongly risk averse, that is if and only if the utility u isconcave and probability weighting f is convex. The weaker notion of sure diversification, where the decision maker prefers diversifica-tion if he can attain certainty by a convex combination of choices (in addition to beingindifferent between these choices), is actually equivalent to that of weak risk aversion inthe RDEU framework.
Proposition 14 (Chateauneuf and Lakhnati (2007)) . A RDEU decision maker exhibitspreference for sure diversification if and only if he is weakly risk averse.
Assuming additionally concavity (and differentiability of utility), we can immediatelyrelate sure diversification and the notion of weak pessimism under RDEU as follows.
Corollary 4.
A RDEU decision maker with differentiable and concave utility u exhibitspreference for sure diversification if and only if he is weakly pessimistic. Finally, consider the case of comonotonic random variables and the associated notionof comonotone diversification, which essentially restricts preference for diversification tocomonotonic choices. Given comonotonic random variables, one can show that a RDEUdecision maker will in fact be indifferent regarding diversification across such comonotonicprospects. This was proven by R¨oell (1987).
Remark 2 (Convexity of preferences in RDEU) . Quiggin (1993) discusses the relation-ship between diversification as a linear mixture of random variables and the concepts ofconvexity and concavity of preferences. Recall that we identify convex preferences with thetraditional definition of diversification. However, in RDEU theory, where preference fordiversification over outcome mixtures arises without convexity, this analogy is misleading,even though the correspondence between strong risk aversion and diversification carriesover to this model. We refer the reader to Quiggin (1993) for a detailed discussion andfurther references on this topic. emark 3 (Implications on portfolio choice) . Rank-dependent expected utility contains thefollowing two components: a concave utility function and a reversed S-shaped probabilitydistortion function. The first component captures the observation that individuals dislikea mean-preserving spread of the distribution of a random outcome. The second componentcaptures the tendency to overweight tail events — a principle that can explain why peoplebuy both insurance and lotteries. Indeed, Quiggin (1991) points out that the behavior of anindividual whose preferences are described by a RDEU functional with a concave outcomeutility function and a (reversed) S-shaped probability weighting function will display riskaversion except when confronted with probability distributions that are skewed to the right. Models of choice under risk were originally formulated to be used with pre-specified orobjective probabilities. One of their main limitations is that uncertainty is treated asobjective risk. Not all uncertainty, however, can be described by such an objective prob-ability. Following the fundamental works of Keynes (1921), Knight (1921), and Ramsey(1926), we now draw a distinction between uncertainty and risk — risk is used when thegamble in question has objectively agreed upon known odds, while uncertainty roughlyrefers to situations where the odds are unknown. Notions of diversification and uncertainty aversion, rather than risk aversion, are hencediscussed within models of expected utility under a (not necessarily additive) subjectiveprobabilty measure, which seeks to distinguish between quantifiable risks and unknown uncertainty . Such choice theoretic models originated with the seminal works of Savage(1954) and Anscombe and Aumann (1963). Our framework will be the axiomatic treat-ment of such models as developed by Schmeidler (1989); see also Gilboa (1987). Theseaxiomatizations involve the use of the Choquet integral to derive the corresponding rep-resentation results for nonadditive probabilities — we will hence refer to such models ofuncertainty as
Choquet expected utility (CEU) models.
Before formally setting up the axiomatic framework of Schmeidler (1989), we quicklyreview the basics of subjective expected utility theory under additive and nonadditiveprobability measures. In this subsection we depart from the theoretical setting of Section 2and adopt the classical decisional theoretical setup where decision makers choose from aset of acts. A number of papers have studied portfolio theory, risk-sharing and insurance contracting in the RDEUframework; see Bernard, He, Yan, and Zhou (2013) for a detailed review. Objective risk is typically available in games of chance, such as a series of coin flips where the proba-bilities are objectively known. In practice, the notion of risk also encompasses situations, in which reliablestatistical information is available, and from which objective probabilities are inferred. Uncertainty, onthe other hand, can arise in practice from situations of complete ignorance or when insufficient statisticaldata is available, for example. .1.1 Subjective expected utility In the standard model of (subjective) uncertainty pioneered by Savage (1954), the decisionmaker chooses from a set of acts. The formal model consists of a set of prizes or conse-quences X = R and a state space S endowed with an algebra Σ. The set of acts A is theset of all finite measurable functions (the inverse of each interval is an element of Σ fromΩ to R ). Preference relations (cid:37) are defined over acts in A . A subjective expected utilityrepresentation of (cid:37) is given by a subjective probability measure P on the states S and autility function u : X → R on the consequences X satisfying f (cid:37) g ⇐⇒ (cid:90) S u ( f ( s )) d P ≥ (cid:90) S u ( g ( s )) d P . The expectation operation here is carried out with respect to a prior probability de-rived uniquely from the decision maker’s preferences over acts. Both Savage (1954) andAnscombe and Aumann (1963) provide axiomatizations of preferences leading to the cri-terion of maximization of subjective expected utility, making this representation hold, thelatter being the simpler development, since a special structure is imposed on the set X ofconsequences. The classic development of Savage (1954) is more general yet more complex,a thorough review of which is beyond the scope of the present article. We do, however,briefly mention one key axiom of Savage’s, known as the Sure-Thing Principle. Axiom 1 (Savage’s Sure-Thing Principle) . Suppose f, f (cid:48) , g, g (cid:48) ∈ A are four acts and T ⊆ S is a subset of the state space such that f ( s ) = f (cid:48) ( s ) and g ( s ) = g (cid:48) ( s ) for all s ∈ T , and f ( s ) = g ( s ) and f (cid:48) ( s ) = g (cid:48) ( s ) for all s ∈ T C . Then f (cid:37) g if and only if f (cid:48) (cid:37) g (cid:48) . This axiom can be interpreted in terms of preferences being separable — if the decisionmaker prefers act f to act g for all possible states in T C (for example knowing a certainevent will happen), and act f is still preferred to act g for all states in T (for exampleif that certain event does not happen), then the decision maker should prefer act f toact g independent of the state. Using the example of an event occurring, this meansthat f is preferred to g despite having no knowledge of whether or not that certain eventwill happen. The axiom essentially states that outcomes which occur regardless of whichactions are chosen, “sure things”, should not affect one’s preferences. Schmeidler’s axiomatization of choice under uncertainty without additivity formally sepa-rates the notion of individual perception of uncertainty from valuation of outcomes. Underthe key axiom of comonotonic independence, preferences under uncertainty are character-ized by means of a functional that turns out to be a Choquet integral.22llsberg’s paradox (Ellsberg 1961) was the main reason motivating the developmentof a more general theory of subjective probabilities, in which the probabilities need not beadditive. Definition 17 (Nonadditive probability measure) . A real-valued set function v : Σ → [0 , on an algebra Σ of subsets of a set of states S is a nonadditive probability measure if it satisfies the normalization conditions v ( ∅ ) = 0 and v ( S ) = 1 , and the monotonicitycondition, that is for all A, B ∈ Σ , if A ⊆ B , then v ( A ) ≤ v ( B ) . We now formally set up Schmeidler’s CEU model. The state space S is endowed withan algebra Σ of subsets of Ω. The set of consequences is assumed to be the positive realline, i.e., X = R + . The set of acts is the set of L nonnegative measurable functions on S . Preferences (cid:37) of the CEU decision maker are defined over the set of acts L and areassumed to be monotonic and continuous. In order to weaken the independence axiom, thenotion of comonotonic independence was introduced by Schmeidler (1989). Very roughly,it requires that the usual independence axiom holds only when hedging effects (in thesense of Wakker (1990)) are absent: Axiom 2 (Comonotonic Independence) . For all pairwise comonotonic acts f, g, h ∈ L and for all α ∈ (0 , , f (cid:37) g implies α f + (1 − α ) h (cid:37) α g + (1 − α ) h . A nonadditive probability measure is referred to as a capacity v : Σ → [0 , A ∈ Σ such that v ( A ) ∈ (0 , core of a capacity v is defined bycore( v ) = { π : Σ → R + | ∀ A ∈ Σ , π ( A ) ≥ v ( A ); π ( S ) = 1 } . Schmeidler (1989) proved that the preference relation (cid:37) on L satisfying comonotonicindependence (and the usual monotonicity and continuity axioms) is represented througha Choquet integral with respect to a unique capacity v rather than a unique additiveprobability measure. Definition 18 (Choquet integral) . The
Choquet integral (Choquet (1954)) of a real-valuedfunction u : L → R on a set of states S with respect to a capacity v is defined by (cid:90) u ( f ( · )) dv = (cid:90) −∞ ( v ( u ( f ) ≥ t ) − dt + (cid:90) ∞ v ( u ( f ) ≥ t ) dt . Note that if the capacity v is in fact an additive probability measure p , then theChoquet integral reduces to the mathematical expectation with respect to p . Ellsberg (1961) proposed experiments where choices violate the postulates of subjective expected util-ity, more specifically the Sure-Thing Principle. The basic idea is that a decision maker will always choosea known probability of winning over an unknown probability of winning even if the known probability islow and the unknown probability could be a guarantee of winning. His paradox holds independent of theutility function and risk aversion characteristics of the decision maker, and implies a notion of uncertaintyaversion , which is an attitude of preference for known risks over unknown risks. Recall that two acts f, g ∈ L are said to be comonotonic if for no s, t ∈ S , f ( s ) > f ( t ) and g ( s ) < g ( t ). heorem 3 (Schmeidler’s Representation Theorem (Schmeidler (1989))) . Suppose thatthe preference relation (cid:37) on L satisfies the comonotonic independence, continuity, andmonotonicity axioms. Then there exists a unique capacity v on Σ and an affine utility-on-wealth function u : R → R such that for all f, g ∈ L , f (cid:37) g ⇐⇒ (cid:90) S u ( f ( · )) dv ≥ (cid:90) S u ( g ( · )) dv . Conversely, if there exist u and v as above with u nonconstant, then the preference relation (cid:37) they induce on L satisfies the comonotonic independence, continuity, and monotonicityaxioms. Finally, the function u is unique up to positive linear transformations. We say that a function V : L → R represents the preference relation (cid:37) if for all acts f, g ∈ L , f (cid:37) g ⇐⇒ V ( f ) ≥ V ( g ), where, under the axiomatization of CEU, V ( f ) for f ∈ L , is given by the Choquet integral (cid:82) S u ( f ( · )) dv , where u and v satisfy the propertiesof the previous Theorem. In the Choquet expected utility framework, preference for diversification is equivalentto the utility index being concave and capacity being convex. The following Theoremprovides this characterization:
Theorem 4 (Chateauneuf and Tallon (2002)) . Assume that u : R + → R is continuous,differentiable on R ++ and strictly increasing, and let V be the functional representing thepreference relation (cid:37) under the CEU model. Then the following statements are equivalent:(i) (cid:37) exhibits preference for diversification.(ii) V is concave.(iii) u is concave and v is convex. Recall that the weaker notion of sure diversification introduced by Chateauneuf andTallon (2002) can be interpreted as an axiom of uncertainty aversion at large; if thedecision maker can attain certainty by a convex combination of equally desirable randomvariables, then he prefers certainty to any of these random variables. Sure diversificationhence embodies a notion of aversion to ambiguity (in the sense of imprecise probability)as well as a notion of aversion to risk.In the context of Choquet expected utility, sure diversification does not have a fullcharacterization in terms of the utility index u and capacity v , as the following Theoremstates. Theorem 5 (Chateauneuf and Tallon (2002)) . Let v be the capacity functional in the Cho-quet expected utility framework, and suppose that the utility index u is continuous, strictlyincreasing and differentiable on R ++ . If the preference relation (cid:37) exhibits preference forsure diversification, then core( v ) (cid:54) = ∅ . On the other hand, if core( v ) (cid:54) = ∅ and u is concave,then (cid:37) exhibits preference for sure diversification. However,the concavity of the utility index can be shown to be equivalent to a different form ofdiversification, namely that of comonotone diversification, which is a restriction of convex-ity of preferences to comonotone random variables. Indeed, comonotone diversification ischaracterized completely by the concavity of the utility index in the CEU framework:
Theorem 6 (Chateauneuf and Tallon (2002)) . Let v be the capacity functional in theChoquet expected utility framework, and suppose that the utility index u is continuous,strictly increasing and differentiable on R ++ . Then the preference relation (cid:37) exhibitspreference for comonotone diversification if and only if u is concave. Corollary 5 (Chateauneuf and Tallon (2002)) . Let v be the capacity functional in theChoquet expected utility framework, and suppose that the utility index u is continuous,strictly increasing and differentiable on R ++ . Then the preference relation (cid:37) exhibitspreference for comonotone and sure diversification if and only if u is concave and core( v ) is non-empty. Cerreia-Vioglio, Maccheroni, Marinacci, and Montrucchio (2011b) provide a characteriza-tion for a general class of preferences that are complete, transitive, convex and monotone,which they refer to as uncertainty averse preferences . They establish a representation foruncertainty averse preferences in an Anscombe-Aumann setting which is general yet richin structure. Let F be the set of all uncertain acts f : S → X , where S is the state space and X is a convex outcome space, and let ∆ be the set of all probability measures on S .Cerreira-Vioglio et al. show that a preference relation (cid:37) is uncertainty averse (and satisfiesadditional technical conditions) if and only if there is a utility index u : X → R and aquasi-convex function G : u ( X ) × ∆ → ( −∞ , ∞ ], increasing in the first variable, such thatthe preference functional V ( f ) = min p ∈ ∆ G (cid:18)(cid:90) u ( f ) dp, p (cid:19) , ∀ f ∈ F represents (cid:37) , where both u and G are essentially unique.In their representation, decision makers consider all possible probabilities p and theassociated expected utilities u ( f ) dp of act f . They then summarize all these evaluations See Example 1 in Chateauneuf and Tallon (2002). Uncertainty averse preferences are a general class of preferences. Special cases that can be obtainedby suitably specifying the uncertainty aversion index G defined below include, amongst others, varia-tional preferences and smooth ambiguity preferences. See Cerreia-Vioglio, Maccheroni, Marinacci, andMontrucchio (2011b) for more details.
25y taking their minimum. The function G can be interpreted as an index of uncertaintyaversion; higher degrees of uncertainty aversion correspond to pointwise smaller indices G .It is shown that the quasiconvexity of G and the cautious attitude reflected by theminimum derive from the convexity of preferences. Uncertainty aversion is hence closelyrelated to convexity of preferences. Under the formalization of Cerreiro-Vioglio et al.,convexity reflects a basic negative attitude of decision makers towards the presence ofuncertainty in their choices. In the theory of choice under risk, the notion of risk aversion comes from a situation wherea probability can be assigned to each possible outcome of a situation. In models of choiceunder uncertainty, however, probabilities of outcomes are unknown, and thus the idea ofa decision maker being risk averse makes little sense when such risks cannot be quantifiedobjectively. Under uncertainty, the phenomenon of ambiguity aversion roughly capturesthe preference for known risks over unknown risks. Whereas risk aversion is defined bythe preference between a risky alternative and its expected value, an ambiguity averseindividual would rather choose an alternative where the probability distribution of theoutcomes is known over one where the probabilities are unknown.The use of the term ambiguity to describe a particular type of uncertainty is due toDaniel Ellsberg (Ellsberg 1961). As his primary examples, Ellsberg offered two experi-mental decision problems, which continue to be the primary motivating factors of researchon ambiguity aversion. Unlike the economic concept of risk aversion, but similar to thenotion of diversification, there is not unanimous agreement on what ambiguity aversion,also often referred to as uncertainty aversion, formally is. However several models anddefinitions have been proposed. We recall the most prominent such definitions and linkthem to the notion of diversification and discuss the implication of ambiguity aversion onportfolio choice.
Schmeidler (1989) introduced a similar notion of aversion towards the unknown and calledit uncertainty aversion . Uncertainty aversion roughly describes an attitude of preferencefor known risks over unknown risks. Formally, it is defined through convexity of prefer-ences:
Definition 19 (Schmeidler’s uncertainty aversion) . A preference relation (cid:37) on L is said “The nature of one’s information concerning the relative likelihood of events... a quality depending onthe amount, type, reliability and ‘unanimity’ of information, and giving rise to one’s degree of ‘confidence’in an estimation of relative likelihoods.” (Ellsberg 1961) They are referred to as the
Two-Urn Paradox and the
Three-Color Paradox – see Ellsberg (1961). o exhibit uncertainty aversion if for any two acts f, g ∈ L and any α ∈ [0 , , we have f (cid:37) g = ⇒ α f + (1 − α ) g (cid:37) g . To present an intuition for this definition, Schmeidler (1989) explains that uncertaintyaversion encapsulates the idea that “ smoothing or averaging utility distributions makesthe decision maker better off .” This quote once again represents the general notion ofdiversification as discussed throughout this paper, namely that “if f and g are preferredto h , so is any convex mixture λf + (1 − λ ) g with λ ∈ (0 , Theorem 7 (Schmeidler (1989)) . A preference relation (cid:37) exhibits uncertainty aversion ifand only if the capacity v is convex. The uncertainty aversion notion of Schmeidler (1989) represents the first attempt toformalize the notion that individuals dislike ambiguity. The intuition is that, by mixingtwo acts, the individual may be able to hedge against variation in utilities, much like, byforming a portfolio consisting of two or more assets, one can hedge against variation inmonetary payoffs.
Remark 4 (Alternative notions of ambiguity aversion) . Other attempts to characterize adislike for ambiguity have been proposed in the literature. Epstein (1999) proposed an al-ternative definition of uncertainty aversion that is more suited to applications in a Savagedomain. His motivation is the weak connection between convexity of capacity and behaviorthat is intuitively uncertainty averse (see Epstein (1999) and Zhang (2002) for examples).Yet another approach was proposed by Ghirardato and Marinacci (2002). They considerboth the Savage setting, with acts mapping to prizes, and the horse-roulette act framework,with acts mapping to objective probability distributions over prizes, and restrict their atten-tion to preferences that admit a Choquet Expected Utility representation on binary acts, butare otherwise arbitrary. For a comprehensive review of the notion of ambiguity aversion,we refer the reader to Machina and Siniscalchi (2014).
Keynes (1921) was perhaps the first economist to grasp the full significance of uncertaintyfor economic analysis and portfolio choice. Whereas conventional models of choice underrisk promote diversification, Keynes expressed the view that one should allocate wealth inthe few stocks about which one feels most favorably. Indeed, even though Markowitz’s “As time goes on I get more and more convinced that the right method in investment is to put fairlylarge sums into enterprises which one thinks one knows something about and in the management of whichone thoroughly believes. It is a mistake to think that one limits one’s risk by spreading too much betweenenterprises about which one knows little and has no reason for special confidence. [...] One’s knowledgeand experience are definitely limited and there are seldom more than two or three enterprises at any giventime in which I personally feel myself entitled to put full confidence.” See (Keynes 1983). familiar . Aversion to ambiguity essentially captures tilting away from theunknown and preference for the familiar. Boyle, Garlappi, Uppal, and Wang (2012) showthat if investors are familiar about a particular asset, they tilt their portfolio toward thatasset, while continuing to invest in other assets; that is, there is both concentration in themore familiar asset and diversification in other assets. If investors are familiar about aparticular asset and sufficiently ambiguous about all other assets, then they will hold onlythe familiar asset, as Keynes would have advocated. Moreover, if investors are sufficientlyambiguous about all risky assets, then they will not participate at all in the equity market.Their model also shows that when the level of average ambiguity across all assets is low,then the relative weight in the familiar asset decreases as its volatility increases; but thereverse is true when the level of average ambiguity is high. An increase in correlationbetween familiar assets and the rest of the market leads to a reduction in the investmentin the market. And, even when the number of assets available for investment is very large,investors continue to hold familiar assets.Dimmock, Kouwenberg, Mitchell, and Peijnenburg (2014) provide empirical evidencethat ambiguity aversion relates to five household portfolio choice puzzles: non-participationin equity markets, low portfolio fractions allocated to equity, home-bias, own-companystock ownership, and portfolio under-diversification. Consistent with the theory, ambi-guity aversion is negatively associated with stock market participation, the fraction offinancial assets allocated to stocks, and foreign stock ownership, but ambiguity aversionis positively related to owncompany stock ownership. Conditional on stock ownership,ambiguity aversion also helps to explain portfolio under-diversification. Under the definition of uncertainty aversion of Schmeidler (1989), Dow and da Costa Wer-lang (1992) derive the nonadditive analog of Arrow’s local risk neutrality theorem. Witha nonadditive probability distribution over returns on a risky asset, there is an intervalof prices within which the economic agent has no position in the asset. At prices belowthe lower limit of this interval, the agent is willing to buy this asset. At prices above theupper end of the interval, the agent is willing to sell the asset short. The highest price atwhich the agent will buy the asset is the expected value of the asset under the nonadditiveprobability measure. The lowest price at which the agent sells the asset is the expectedvalue of selling the asset short. This reservation price is larger than the other one if thebeliefs reflect uncertainty aversion `a-la-Schmeidler, that is, with a nonadditive probabilitymeasure, the expectation of a random variable is less than the negative of the expectation See (De Giorgi and Mahmoud 2014) for a review of empirical evidence suggesting underdiversification. A number of other research efforts empirically studying the effect of ambiguity aversion on portfoliochoice reach the conclusion of under-diversification in some form, including the works of Uppal and Wang(2003), Maenhout (2004), Maenhout (2006), Garlappi, Uppal, and Wang (2007), Liu (2010), Campanale(2011), and Chen, Ju, and Miao (2014).
28f the negative of the random variable. These two reservation prices, hence, depend onlyon the beliefs and aversion to uncertainty incorporated in the agent’s prior, and not onattitudes or aversion towards risk.
10 Some Concluding Behavioral Remarks
We have surveyed axiomatizations of the concept of diversification and their relationship tothe related notions of risk aversion and convex preferences within different choice theoreticmodels. Our survey covered model-independent diversification preferences, preferenceswithin models of choice under risk, including expected utility theory and the more generalrank-dependent expected utility theory, as well as models of choice under uncertaintyaxiomatized via Choquet expected utility theory.The traditional diversification paradigms of expected utility theory and Markowitz’sModern Portfolio Theory, which essentially encourage variety in investment over similar-ity, imply that individuals are rational and risk averse. However, experimental work inthe decades after the emergence of the classical theories of von Neumann and Morgenstern(1944) and Markowitz (1952) has shown that economic agents in reality systematicallyviolate the traditional rationality assumption when choosing among risky gambles. Inresponse to this, there has been an explosion of work on so-called non-expected-utilitytheories, all of which attempt to better match the experimental evidence. In particular,the last decade has witnessed the growing movement of behavioral economics, which ques-tions the assumptions of rational choice theoretic models and seeks to incorporate insightsfrom psychology, sociology and cognitive neuroscience into economic analysis. Behavioraleconomics has had some success in explaining how certain groups of investors behave, and,in particular, what kinds of portfolios they choose to hold and how they trade over time.As an example, there is evidence suggesting that investors diversify their portfolioholdings much less than is recommended by normative models of rational choice. In par-ticular, real-world diversification appears to be highly situational and context dependent;investors include a smaller number of assets in their portfolios than traditionally recom-mended; and some investors exhibit a pronounced home bias, which means that they holdonly a modest amount of foreign securities. Moreover, the portfolio construction method-ology recommended by Modern Portfolio Theory has some limitations. Consequently,alternative diversification paradigms have emerged in practice.Because of the empirical work suggesting that investors’ diversification behavior in real-ity deviates significantly from that implied by various rational models of choice, behavioralmodels assuming a specific form of irrationality have emerged. Behavioral economists turnto the extensive experimental evidence compiled by cognitive psychologists on the system-atic biases that arise when people form beliefs. By arguing that the violations of rationalchoice theory in practice are central to understanding a number of financial phenomena,new choice theoretic behavioral models have emerged.29 rospect Theory is arguably the most prominent such behavioral theory. It states thatagents make decisions based on the potential value of losses and gains rather than thefinal outcome, and that these losses and gains are evaluated using certain heuristics. Thetheory can be viewed as being descriptive, as it tries to model realistic observed and doc-umented choices rather than optimal rational decisions. The theory is due to Kahnemanand Tversky (1979) and is largely viewed as a psychologically more accurate descriptionof decision making compared to classical theories of rational choice. Since the originalversion of prospect theory gave rise to violations of first-order stochastic dominance, arevised version, called
Cumulative Prospect Theory (CPT), was introduced in Kahnemanand Tversky (1992). CPT overcomes this problem by using a probability weighting func-tion derived from rank-dependent expected utility theory.We refer the reader to De Giorgi and Mahmoud (2014) for a survey of the growingexperimental and empirical evidence that is in conflict with how diversification preferencesare traditionally viewed within classical models of choice, both under risk and underuncertainty, such as the one reviewed in this article. A particular focus is placed onCumulative Prospect Theory. Unlike economic agents whose preferences are consistentwith classical frameworks such as expected utility theory or Markowitz’s Modern PortfolioTheory, decision makers whose preferences conform to Cumulative Prospect Theory donot select portfolios that are well-diversified. This lack of diversification essentially stemsfrom inherent features of CPT, in particular those of framing and convex utility in thedomain of losses.
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