Dual Moments and Risk Attitudes
DDual Moments and Risk Attitudes ∗ Louis R. Eeckhoudt
I´ESEG School of ManagementCatholic University of Lilleand CORE
Roger J. A. Laeven
Amsterdam School of EconomicsUniversity of Amsterdam, EURANDOMand CentER
This Version: March 14, 2018
Abstract
In decision under risk, the primal moments of mean and variance play a central role todefine the local index of absolute risk aversion. In this paper, we show that in canonicalnon-EU models dual moments have to be used instead of, or on par with, their primalcounterparts to obtain an equivalent index of absolute risk aversion.
Keywords:
Risk Premium; Expected Utility; Dual Theory; Rank-Dependent Utility; LocalIndex; Absolute Risk Aversion.
JEL Classification:
D81.
OR/MS Classification:
Decision analysis: Risk. ∗ We are very grateful to Loic Berger (discussant), Sebastian Ebert, Glenn Harrison, Richard Peter (dis-cussant), Nicolas Treich, Ilia Tsetlin, Bob Winkler, and, in particular, to Christian Gollier for many detailedcomments and suggestions and to Harris Schlesinger ( † ) for discussions. We are also grateful to conference andseminar participants at the CEAR/MRIC Behavioral Insurance Workshop in Munich and City University ofLondon for their comments. This research was funded in part by the Netherlands Organization for ScientificResearch under grant NWO VIDI 2009 (Laeven). Research assistance of Andrei Lalu is gratefully acknowledged.Eeckhoudt: Catholic University of Lille, I´ESEG School of Management, 3 Rue de la Digue, Lille 59000, France.Laeven: University of Amsterdam, Amsterdam School of Economics, PO Box 15867, 1001 NJ Amsterdam, TheNetherlands. a r X i v : . [ q -f i n . R M ] M a r Introduction
In their important seminal work, Pratt [22] and Arrow [2, 3] (henceforth, PA) show that underexpected utility (EU) the risk premium π associated to a small risk ˜ ε with zero mean can beapproximated by the following expression: π (cid:39) m (cid:18) − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) (cid:19) . (1.1)Here, m is the second moment about the mean (i.e., the variance) of ˜ ε while U (cid:48) ( w ) and U (cid:48)(cid:48) ( w )are the first and second derivatives of the utility function of wealth U at the initial wealth level w . In the PA-approach, the designation “small” refers to a risk that has a probability massequal to unity but a small variance. The PA-approximation in (1.1) provides a very insightfuldissection of the EU risk premium, disentangling the complex interplay between the probabilitydistribution of the risk, the decision-maker’s risk attitude, and his initial wealth. This well-known result has led to many developments and applications within the EU model in manyfields.The aim of this paper is to show that a similar result can also be obtained outside EU,in the dual theory of choice under risk (DT; Yaari [36]) and, more generally and behaviorallymore relevant, under rank-dependent utility (RDU; Quiggin [25]). To achieve this, we substi-tute or complement the primal second moment m by its dual counterpart, and substitute orcomplement the derivatives of the utility function U by the respective derivatives of the prob-ability weighting function. , This modification enables us to develop for these two canonicalnon-EU models a simple and intuitive local index of risk attitude that resembles the one in(1.1) under EU. Our results allow for quite arbitrary utility and probability weighting functionsincluding inverse s -shaped functions such as the probability weighting functions in Prelec [24]and Wu and Gonzalez [33], which are descriptively relevant (Abdellaoui [1]). Thus, we allowfor violations of strong risk aversion (Chew, Karni and Safra [7] and Ro¨ell [26]) in the senseof aversion to mean-preserving spreads `a la Rothschild and Stiglitz [28] (see also Machina andPratt [18]). , For ease of exposition, we assume U to be twice continuously differentiable, with positive first derivative. Dual moments are sometimes referred to as mean order statistics in the statistics literature; see Section 2for further details. The RDU model encompasses both EU and DT as special cases and is at the basis of (cumulative) prospecttheory (Tversky and Kahneman [32]). According to experimental evidence collected by Harrison and Swarthout[16], RDU seems to emerge even as the most important non-EU preference model from a descriptive perspective. In a very stimulating strand of research, Chew, Karni and Safra [7] and Ro¨ell [26] have developed the “global”counterparts of the results presented here; see also the more recent Chateauneuf, Cohen and Meilijson [4, 5] andRyan [29]. Surprisingly, the “local” approach has received no attention under DT and RDU, except—to the bestof our knowledge—for a relatively little used paper by Yaari [35]. Specifically, Yaari exploits a uniformly orderedlocal quotient of derivatives (his Definition 4) with the aim to establish global results, restricting attention toDT. Yaari does not analyze the local behavior of the risk premium nor does he make a reference to dual moments.For global measures of risk aversion under prospect theory, we refer to Schmidt and Zank [30]. The insightful Nau [21] proposes a significant generalization of the PA-measure of local risk aversion in
The second dual moment about the mean of an arbitrary risk ˜ ε , denoted by ¯ m , is defined by¯ m := E (cid:104) max (cid:16) ˜ ε (1) , ˜ ε (2) (cid:17)(cid:105) − E [˜ ε ] , (2.1)where ˜ ε (1) and ˜ ε (2) are two independent copies of ˜ ε . The second dual moment can be interpretedas the expectation of the largest order statistic: it represents the expected best outcome amongtwo independent draws of the risk. Our analysis will reveal that for an RDU maximizer who evaluates a small zero-mean risk,the second dual moment stands on equal footing with the variance as a fundamental measureof risk. While the variance provides a measure of risk in the “payoff plane”, the second dualmoment can be thought of as a measure of risk in the “probability plane”. Indeed, for a risk ˜ ε with cumulative distribution function F , so m := E [˜ ε ] = (cid:90) x d F ( x ) , we have that m = (cid:90) ( x − m ) d F ( x ) , while ¯ m = (cid:90) ( x − m ) d( F ( x )) . another direction. He considers the case in which probabilities may be subjective, utilities may be state-dependent, and probabilities and utilities may be inseparable, by invoking Yaari’s [34] elementary definitionof risk aversion of “payoff convex” preferences, which agrees with the Rothschild and Stiglitz [28] concept ofaversion to mean-preserving spreads under EU. The definition and interpretation of the 2-nd dual moment readily generalize to the n -th order, n ∈ N > , byconsidering n copies. We refer to Meyer [19] and Eichner and Wagener [12] for insightful comparative statics results on themean-variance trade-off and its compatibility with EU. Formally, our integrals with respect to functions are Riemann-Stieltjes integrals. If the integrator is acumulative distribution function of a discrete (or non-absolutely continuous) risk, or a function thereof, thenthe Riemann-Stieltjes integral does not in general admit an equivalent expression in the form of an ordinaryRiemann integral. m , the maxiance by analogy to the variance . Our designation “small” in “smallzero-mean risk” will quite naturally refer to a risk with small maxiance under DT and to a riskwith both small variance and small maxiance under RDU.One readily verifies that for a zero-mean risk ˜ ε , E (cid:104) max (cid:16) ˜ ε (1) , ˜ ε (2) (cid:17)(cid:105) = − E (cid:104) min (cid:16) ˜ ε (1) , ˜ ε (2) (cid:17)(cid:105) . The miniance —the expected worst outcome among two independent draws—is perhaps a morenatural measure of “risk”, but agrees with the maxiance for zero-mean risks upon a sign change.Just like the first and second primal moments occur under EU when the utility function islinear and quadratic, the first and second dual moments correspond to a linear and quadraticprobability weighting function under DT. For further details on mean order statistics andtheir integral representations we refer to David [8]. In the stochastic dominance literature,these expectations of order statistics and their higher-order generalizations arise naturally inan interesting paper by Muliere and Scarsini [20], when defining a sequence of progressive n -thdegree “inverse” stochastic dominances by analogy to the conventional stochastic dominancesequence (see Ekern [13] and Fishburn [14]). , Consider a DT decision-maker. His evaluation of a risk A with cumulative distribution function F is given by (cid:90) x d h ( F ( x )) , (3.1) This is easily seen from the Riemann-Stieltjes representations of the miniance and maxiance. Indeed, − E (cid:104) min (cid:16) ˜ ε (1) , ˜ ε (2) (cid:17)(cid:105) = (cid:90) x d (1 − F ( x )) = − (cid:90) x d F ( x ) + (cid:90) x d ( F ( x )) = E (cid:104) max (cid:16) ˜ ε (1) , ˜ ε (2) (cid:17)(cid:105) , where the last equality follows because (cid:82) x d F ( x ) = 0 when ˜ ε is a zero-mean risk. In a related strand of the literature, Eeckhoudt and Schlesinger [9] (see also Eeckhoudt, Schlesinger andTsetlin [10]) and Eeckhoudt, Laeven and Schlesinger [11] derive simple nested classes of lottery pairs to sign the n -th derivative of the utility function and probability weighting function, respectively. Their approach can beseen to control the primal moments for EU and the dual moments for DT. Expressions similar (but not identical) to dual moments also occur naturally in decision analysis applications.For example, the expected value of information when the information will provide one of two signals is themaximum of the two posterior expected values (e.g., payoffs or utilities) minus the highest prior expected value.This generalizes to the case of n > h : [0 , → [0 ,
1] satisfies the usualproperties ( h (0) = 0, h (1) = 1, h (cid:48) ( p ) > , In order to develop the local index of absolute risk aversion under DT we start from alottery A given by the following representation: , Figure 1: Lottery A p − − p A We transform lottery A into a lottery B given by: Figure 2: Lottery B p − ε − ε x − p − ε B To obtain B from A we subtract a probability ε from the probabilities of both states of theworld in A without changing the outcomes and we assign these two probabilities jointly, i.e.,2 ε , to a new intermediate state to which we attach an outcome x with − < x <
1. If x ≡ E [ A ] = E [ B ] and B is a mean-preserving contraction of A .The value of x will be chosen such that the decision-maker is indifferent between A and B .Naturally the difference between 0 and x , denoted by ρ = 0 − x , represents the risk premiumassociated to the risk change from A to B . As we will show in Section 5.2 this definition ofthe risk premium can be viewed as a natural generalization of the PA risk premium to the For ease of exposition, we assume h to be twice continuously differentiable. Rather than distorting “decumulative” probabilities (as in Yaari [36]), we adopt the convention to distortcumulative probabilities. Our convention ensures that aversion to mean-preserving spreads corresponds to h (cid:48)(cid:48) < U (cid:48)(cid:48) < h ( p ) := 1 − h (1 − p ) would be convex when h is concave. In all figures, values along (at the end of) the arrows represent probabilities (outcomes). Of course, we assume 0 < p < We assume 0 < ε < min { p , − p } . h therisk premium ρ may be positive or negative. If (and only if) h (cid:48)(cid:48) ( p ) <
0, the corresponding DTmaximizer is averse to mean-preserving spreads, and would universally prefer B over A when x were 0. Thus, to establish indifference between A and B for such a decision-maker, x hasto be smaller than 0, in which case ρ is positive.In general, for x ≡ − ρ in B , indifference between A and B implies: h ( p ) ( w −
1) + (1 − h ( p )) ( w + 1) (3.2)= h ( p − ε ) ( w −
1) + ( h ( p + ε ) − h ( p − ε )) ( w − ρ ) + (1 − h ( p + ε )) ( w + 1) , where w is the decision-maker’s initial wealth level. From (3.2) we obtain the explicit solution ρ = ( h ( p ) − h ( p − ε )) − ( h ( p + ε ) − h ( p ))( h ( p + ε ) − h ( p − ε )) . (3.3)By approximating h ( p ± ε ) in (3.3) using second-order Taylor series expansions around h ( p ), we obtain the following approximation for the DT risk premium: ρ (cid:39) ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) . (3.4)Here, ¯ m is the unconditional maxiance of the risk ˜ ε that describes the mean-preserving spreadfrom B with x ≡ A . Unconditionally, ˜ ε takes the values ± ε .Furthermore, Pr is the total unconditional probability mass associated to ˜ ε ; see Figure 3.Figure 3: Mean-Preserving Spread from B with x ≡ A . p − ε − ε ε − p − ε Observe that lottery A is obtained from lottery B (with x ≡
0) by attaching the risk ˜ ε tothe intermediate branch of B . That is, the risk ˜ ε is effective conditionally upon realizationof the intermediate state of lottery B , which occurs with probability 2 ε . One readily verifiesthat, for this risk ˜ ε , we have that, unconditionally, ¯ m = 2 ε and Pr = 2 ε . We considerthe unconditional maxiance of the zero-mean risk ˜ ε to be “small” and compute the Taylor See the references in footnote 4 for global results on risk aversion under DT and RDU. ε . Henceforth, maxiances and variances are always understood tobe unconditional.It is important to compare the result in (3.4) to that obtained by PA presented in (1.1). InPA the local approximation of the risk premium is proportional to the variance, while underDT it is proportional to the maxiance.We note that the local approximation of the risk premium in (3.4) remains valid in general,for non-binary zero-mean risks ˜ ε with small maxiance, just like, as is well-known, (1.1) is validfor non-binary zero-mean risks with small variance. Under DT the local index arises from a risk change with small maxiance. To deal with the RDUmodel, which encompasses both EU and DT as special cases, we naturally have to considerchanges in risk that are small in both variance and maxiance. To achieve this, we start from alottery C given by: Figure 4: Lottery C p − ε − p ε C Similar to under DT, we transform lottery C into a lottery D by reducing the probabilities ofboth states in C by a probability ε and assigning the released probability 2 ε to an intermediatestate with outcome y , where − ε < y < ε . This yields a lottery D given by:Figure 5: Lottery D p − ε − ε ε y − p − ε ε D Detailed derivations are suppressed to save space. They are contained in an online appendix (available fromthe authors’ webpages; see ). We assume ε >
7f course, when y ≡ D is a mean-preserving contraction of C . All RDU decision-makersthat are averse to mean-preserving spreads therefore prefer D over C in that case. Formally,an RDU decision-maker evaluates a lottery A with cumulative distribution function F bycomputing (cid:90) U ( x ) d h ( F ( x )) , (4.1)and is averse to mean-preserving spreads if and only if U (cid:48)(cid:48) < h (cid:48)(cid:48) < In general, we can search for y such that indifference between C and D occurs. Thediscrepancy between the resulting y and 0 is the RDU risk premium associated to the riskchange from C to D and its value, denoted by λ = 0 − y , is the solution to h ( p ) U ( w − ε ) + (1 − h ( p )) U ( w + ε ) (4.2)= h ( p − ε ) U ( w − ε ) + ( h ( p + ε ) − h ( p − ε )) U ( w − λ ) + (1 − h ( p + ε )) U ( w + ε ) . It will be positive or negative depending on the shapes of U and h .Approximating the solution to (4.2) by Taylor series expansions, up to the first order in λ around U ( w ) and up to the second orders in ε and ε around U ( w ) and h ( p ), we obtainthe following approximation for the RDU risk premium: λ (cid:39) m Pr (cid:18) − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) (cid:19) + ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) . (4.3)Here, m and ¯ m are the unconditional variance and maxiance of the risk ˜ ε that dictates themean-preserving spread from D with y ≡ C . Unconditionally, ˜ ε takes the values ± ε eachwith probability ε . Furthermore, Pr is the total unconditional probability mass associated to˜ ε . Comparing (4.3) to (1.1) and (3.4) reveals that the local approximation of the RDU riskpremium aggregates the (suitably scaled) EU and DT counterparts, with the variance andmaxiance featuring equally prominently.As shown in online supplementary material, the local approximation of the RDU risk pre-mium in (4.3) also generalizes naturally to non-binary risks ˜ ε . Not only the local properties of the previous sections are valid under DT and RDU but alsothe corresponding global properties, just like in the PA-approach under the EU model (see, in See the references in footnote 4. It is straightforward to verify that for ˜ ε we have that, unconditionally, m = 2 ε ε , ¯ m = 2 ε ε , and Pr = 2 ε . ε and ε are “large”, as long as 0 < ε ≤ { p , − p } < ε > Proposition 5.1
Let U i , h i , λ i ( p , w , ε , ε ) be the utility function, the probability weightingfunction, and the risk premium solving (4.2) for RDU decision-maker i = 1 , . Then thefollowing conditions are equivalent:(i) − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) ≥ − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) and − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) ≥ − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) for all w and all p ∈ (0 , .(ii) λ ( p , w , ε , ε ) ≥ λ ( p , w , ε , ε ) for all < ε ≤ { p , − p } < , all w , and all ε > . Because the binary symmetric zero-mean risk ˜ ε in Section 4 induces a risk change that is aspecial case of a mean-preserving spread, the implication (i) ⇒ (ii) in Proposition 5.1 in principlefollows from the classical results on comparative risk aversion under RDU (Yaari [35], Chew,Karni and Safra [7], and Ro¨ell [26]). The reverse implication (ii) ⇒ (i) formalizes the connectionbetween our local risk aversion approach and global risk aversion under RDU.Due to the simultaneous involvement of both the utility function and the probability weight-ing function, the proof of the equivalences between (i) and (ii) under RDU is more complicatedthan that of the analogous properties under EU (and DT). Our proof of Proposition 5.1 (whichis contained in online supplementary material available from the authors’ webpages) is basedon the total differential of the RDU evaluation, and the sensitivity of the risk premium withrespect to changes in payoffs. Our definition of the risk premium under RDU in (4.2) can be viewed as a natural generalizationof the risk premium of Pratt [22] and Arrow [2, 3]. To see this, first note that the PA-definition, under which a risk is compared to a sure loss equal to the risk premium, occurswhen p = ε = . Then, lottery D becomes a sure loss of λ the value of which is such thatthe decision-maker is indifferent to the risk of lottery C .When ε < , our definition of the RDU risk premium provides a natural generalizationof the PA-definition. This becomes readily apparent if we omit the common components oflotteries D and C with the same incremental RDU evaluation and isolate the risk change, whichyields Recall that the probability ε and payoff ± ε in (4.2) can be “large” as long as 0 < ε ≤ { p , − p } < ε > D after Omitting the Components in Common with Lottery C . ε − λD \ C ∩ D and Figure 7: Lottery C after Omitting the Components in Common with Lottery D . ε − ε ε ε C \ C ∩ D The value of λ thus represents the risk premium for the risk change induced by a risk that,unconditionally, takes the values ± ε each with probability ε .When ε < , the original comparison between C and D is a comparison between two riskysituations as in Ross [27], Machina and Neilson [17], and Pratt [23]. The removal of commoncomponents, however, reveals that we essentially face a PA-comparison between a single lossand a risk with the same total probability mass, which is now allowed to be smaller than unity. Dual moments can be related to the Gini cofficient named after statistician Corrado Giniand used by economists to measure the dispersion of the income distribution of a population,summarizing its income inequality. In risk theory, the Gini coefficient G of a risk A is usuallydefined by G = E (cid:2) | A (1) − A (2) | (cid:3) E [ A ] , (5.4)which represents half the relative (i.e., normalized) expected absolute difference between twoindependent draws of the risk A . One can verify that, equivalently but less well-known, G = ¯ m m . (5.5)This alternative expression features the ratio of the maxiance and the first moment.Furthermore, n -th degree expectations of first order statistics also appear in Cherny andMadan [6] as performance measures in the context of portfolio evaluation. In this setting, theexpected maximal financial loss occurring in n independent draws of a risk is used as a measureto define an acceptability index linked to a tolerance level of stress.10 Examples
Owing to its local nature, our approximation is valid and can insightfully be applied when theprobability weighting function is not globally concave, as is suggested by ample experimentalevidence. Consider the canonical probability weighting function of Prelec [24] given by h ( p ) = 1 − exp {− ( − log (1 − p )) α } , < α < . (6.1)It captures the following properties which are observed empirically: it is regressive (first, h ( p ) > p , next h ( p ) < p ), is inverse s -shaped (first concave, next convex), and is asymmetric(intersecting the identity probability weighting function h ( p ) = p at p ∗ = 1 − / exp(1), theinflection point). The upper panel of Figure 8 plots this probability weighting function for α ∈ { . , . , . . . , . } . (Wu and Gonzalez [33] report estimated values of α between 0.03 and0.95.)Its local index − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) takes the form − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) = − − α (1 − ( − log(1 − p )) α ) + log(1 − p )(1 − p ) log(1 − p ) . (6.2)Figure 8, lower panel, plots this local index for α ∈ { . , . , . . . , . } . Recall our convention to distort cumulative probabilities rather than decumulative probabilities; see footnote12. Prelec’s original function is given by w ( p ) = 1 − h (1 − p ). Prelec’s function is characterized axiomatically as the probability weighting function of a sign- and rank-dependent preference representation that exhibits subproportionality, diagonal concavity, and so-called compoundinvariance . α ∈ { . , . , . . . , . } . - - The inverse s -shape of the probability weighting function (first concave, next convex) im-plies that its local index changes sign at the inflection point. More specifically, the local indexassociated with Prelec’s probability weighting function is decreasing (first positive, next neg-ative) in p for any 0 < α <
1. This property is naturally consistent with the inverse s -shapeproperty of the probability weighting function: the inverse s -shape property is meant to repre-sent a psychological phenomenon known as diminishing sensitivity in the probability domain(rather than the payoff domain), under which the decision-maker is less sensitive to changesin the objective probabilities when they move away from the reference points 0 and 1, andbecomes more sensitive when the objective probabilities move towards these reference points.A decreasing local index implies in particular that h (cid:48)(cid:48)(cid:48) >
0. (By Pratt [22], the sign of thederivative of the local index is the same as the sign of ( h (cid:48)(cid:48) ( p )) − h (cid:48) ( p ) h (cid:48)(cid:48)(cid:48) ( p ).) Inverse s -shapedprobability weighting functions, including Prelec’s canonical example, usually exhibit positivesigns for the odd derivatives and alternating signs (first negative, then positive) for the evenderivatives. For a probability weighting function that is inverse s -shaped (first concave, thenconvex) and has second derivative equal to zero at the inflection point, a positive sign of thethird derivative means that the function becomes increasingly concave when we move to the12eft of the inflection point and becomes increasingly convex when we move to the right of theinflection point.In Figure 12 in the online appendix we also plot the local index − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) of the probabilityweighting function proposed by Tversky and Kahneman [32] (see also Wu and Gonzalez [33])given by h ( p ) = 1 − (1 − p ) β (cid:16) (1 − p ) β + p β (cid:17) /β , < β < , (6.3)for values of the parameter β ∈ { . , . , . . . , . } as found in experiments (Wu and Gonzalez[33] report estimated values of β between 0.57 and 0.94). Observe the similarity between theshapes in Figure 8 and Figure 12.The analysis in this paper reveals that for a small risk the sign and size of the maxiance’scontribution to the RDU risk premium, given by the second term on the right-hand side of(4.3), i.e., ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) , varies with the probability level p , from strongly positive to strongly negative, in tandem withthe local index − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) to which it is proportional.We finally plot in Figure 9 our approximation to the RDU risk premium (4.3) of a risk withsmall variance and maxiance normalized to satisfy m Pr = ¯ m Pr = 1, as a function of both theinitial wealth level w and the probability level p . We suppose the utility function is given bythe conventional power utility (note that we consider a pure rank-dependent model) U ( x ) = x γ , with γ = 0 . α = 0 .
65. 13igure 9: Surface of the RDU Risk Premium Approximation. We consider a risk with smallvariance and maxiance normalized to satisfy m Pr = ¯ m Pr = 1 under power utility (with γ = 0 . α = 0 . (cid:16) − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) (cid:17) and (cid:16) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:17) evaluatedin the wealth and probability levels w and p , respectively. The orange surface represents ourapproximation (4.3) to the RDU risk premium λ , while the blue surface is the λ = 0-plane. Toillustrate the effect of a change in variance or maxiance, we also plot in Figure 13 in the onlineappendix the surface of the RDU risk premium approximation (4.3) for a small risk with ratiobetween the variance and maxiance equal to 3 (upper panel) and 1 / In order to illustrate how the concepts we have developed can be used we consider a simpleportfolio problem with a safe asset, the return of which is zero, and a binary risky asset withreturns expressed by the following representation: We assume 0 < R < R . p − R − p R Taking R R + R > p makes the expected return strictly positive.If an RDU investor has initial wealth w his portfolio optimization problem is given byarg max α { h ( p ) U ( w − αR ) + (1 − h ( p )) U ( w + αR ) } , (7.1)with first-order condition (FOC) given by − R h ( p ) U (cid:48) ( w − αR ) + R (1 − h ( p )) U (cid:48) ( w + αR ) ≡ . It is straightforward to show that the second-order condition for a maximum is satisfied pro-vided U (cid:48)(cid:48) < α ≡
0. Plugging α ≡ h ( p ) ≡ R R + R . (7.2)Without surprise, h ( p ) > p . This value of h ( p ) expresses the intensity of risk aversion thatinduces the choice of α ≡ p − ε − R ε ( R − R )2 p − ε R One may verify that such a mean-preserving contraction for a decision-maker who had decidednot to participate in the risky asset may induce him to select a strictly positive α .15ence, we raise the following question: By how much should we reduce the intermediatereturn R − R to induce the decision-maker to stick to the optimal α equal to zero? The answerto this question is denoted by ς .Because we are concentrating on the situation where α ≡ U thatappears in the FOC through different values of U (cid:48) becomes irrelevant at α ≡
0. The reasonto concentrate on α ≡ α (Gollier [15]).It turns out that, upon invoking Taylor series expansions and after several basic manipula-tions, the reduction ς that answers our question raised above is given by ς (cid:39) ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) , (7.3)where ¯ m is the maxiance of the risk that, unconditionally, takes the values ± R + R each withprobability ε , and where Pr is the total probability mass of this risk. Again the second dualmoment (instead of the primal one) appears, jointly with the intensity of risk aversion inducedby the probability weighting function. In particular, the mean-preserving contraction is animprovement and has made the risky asset attractive if and only of ς is positive. Under EU, the risk premium is approximated by an expression that multiplies half the varianceof the risk (i.e., its primal second central moment) by the local index of absolute risk aversion.This expression dissects the complex interplay between the risk’s probability distribution, thedecision-maker’s preferences and his initial wealth that the risk premium in general dependson. Surprisingly, a similar expression almost never appears in non-EU models.In this paper, we have shown that when one refers to the second dual moment—insteadof, or on par with, the primal one—one obtains quite naturally an approximation of the riskpremium in canonical non-EU models that mimics the one obtained within the EU model.The PA-approximation of the risk premium under EU has induced thousands of applicationsand results in many fields such as operations research, insurance, finance, and environmentaleconomics. So far, comparable developments have been witnessed to a much lesser extentoutside the EU model. Hopefully, the new and simple expression of the approximated riskpremium may contribute to a widespread analysis and use of risk premia for non-EU.
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UPPLEMENTARY MATERIAL(FOR ONLINE PUBLICATION) Generalization to Non-Binary Risks
In this supplementary material, we first show that the local approximation for the DT riskpremium in (3.4) remains valid for non-binary risks with small maxiance. Next, we prove thatthe RDU risk premium approximation in (4.3) also remains valid for non-binary risks withboth small variance and small maxiance. Throughout this supplement, we consider n -staterisks with probabilities p i associated to outcomes x i , i = 1 , . . . , n , with n ∈ N > . We orderstates from the lowest outcome state (designated by state number 1) to the highest outcomestate (designated by state number n ), which means that x ≤ · · · ≤ x n .We analyze the DT risk premium for a risk with n ≥ ε n , 0 < ε ≤ . The outcomes are, however,allowed to be the same among adjacent states; this would correspond to a risk with non-equalstate probabilities. Note the generality provided by this construction. We suppose that therisk has mean equal to zero, so (cid:80) ni =1 x i = 0. One may verify that the unconditional maxianceof this n -state risk is given by ¯ m = 4 ε n n (cid:88) i =1 (2 i − x i , (A.1)and that the total probability mass Pr = 2 ε . Observe that the maxiance is of the order ε ,i.e., ¯ m = O (cid:0) ε (cid:1) .Similar to the main text, this zero-mean risk is attached to the intermediate branch oflottery B (with x ≡
0) to induce a mean-preserving spread. (We normalize the outcomes of thezero-mean risk by restricting them to the interval [ − , B is not affected and can easily be generalized.) The DT risk premium ρ then occurs as the solution to( h ( p + ε ) − h ( p − ε )) ( w − ρ )= n (cid:88) i =1 (cid:18) h (cid:18) p − ε + i ε n (cid:19) − h (cid:18) p − ε + ( i −
1) 2 ε n (cid:19)(cid:19) ( w + x i ) . (A.2)From (A.2) we obtain the explicit solution ρ = − n (cid:88) i =1 (cid:0) h (cid:0) p − ε + i ε n (cid:1) − h (cid:0) p − ε + ( i − ε n (cid:1)(cid:1) h ( p + ε ) − h ( p − ε ) x i . (A.3)By invoking Taylor series expansions around h ( p ) up to the second order in ε we obtain20rom (A.3) the following approximation for the DT risk premium: ρ (cid:39) − n (cid:88) i =1 12 (2 i − ε n h (cid:48)(cid:48) ( p )2 ε h (cid:48) ( p ) x i = ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) , where the last equality follows directly from (A.1).Finally, turning to the risk premium under RDU, we consider, as under DT, an n -statezero-mean risk with unconditional state probabilities ε n , so (cid:80) ni =1 x i = 0 and Pr = 2 ε , nowassumed to satisfy additionally that m = n (cid:80) ni =1 x i = O (cid:0) ε (cid:1) for some ε >
0. Upon attachingthis zero-mean risk to the intermediate branch of lottery D (with y ≡ | x i | < ε ), the RDU risk premium λ occurs as the solution to( h ( p + ε ) − h ( p − ε )) U ( w − λ )= n (cid:88) i =1 (cid:18) h (cid:18) p − ε + i ε n (cid:19) − h (cid:18) p − ε + ( i −
1) 2 ε n (cid:19)(cid:19) U ( w + x i ) . (A.4)Invoking Taylor series expansions up to the first order in λ around U ( w ) and up to thesecond order in x i and ε around U ( w ) and h ( p ), we obtain from (A.4), at the leading orders,the desired approximation for the RDU risk premium: λ (cid:39) − n (cid:88) i =1 12 2 ε n U (cid:48)(cid:48) ( w )2 ε U (cid:48) ( w ) x i − n (cid:88) i =1 12 (2 i − ε n h (cid:48)(cid:48) ( p )2 ε h (cid:48) ( p ) x i = m Pr (cid:18) − U (cid:48)(cid:48) ( w ) U (cid:48) ( w ) (cid:19) + ¯ m Pr (cid:18) − h (cid:48)(cid:48) ( p ) h (cid:48) ( p ) (cid:19) . Proof of Proposition 5.1
First, note that (i) is equivalent to(iv) U ( U − ( t )) and h ( h − ( u )) are concave functions of t and u for all t and all u ∈ (0 , U ( y ) − U ( x ) U ( w ) − U ( v ) ≤ U ( y ) − U ( x ) U ( w ) − U ( v ) and h ( s ) − h ( r ) h ( q ) − h ( p ) ≤ h ( s ) − h ( r ) h ( q ) − h ( p ) for all v < w ≤ x < y and all0 < p < q ≤ r < s < ε > < ε ≤ { p , − p } < ε → λ i →
0. Define V i ( λ i , ε ) = ( h i ( p + ε ) − h i ( p − ε )) U i ( w − λ i ) − (( h i ( p ) − h i ( p − ε )) U i ( w − ε ) + ( h i ( p + ε ) − h i ( p )) U i ( w + ε )) . We compute the total differential d V i = ∂V i ∂λ i d λ i + ∂V i ∂ε d ε . It is given by − ( h i ( p + ε ) − h i ( p − ε )) U (cid:48) i ( w − λ i ) d λ i + (cid:0) ( h i ( p ) − h i ( p − ε )) U (cid:48) i ( w − ε ) − ( h i ( p + ε ) − h i ( p )) U (cid:48) i ( w + ε ) (cid:1) d ε . Equating the total differential to zero yieldsd λ i d ε = h i ( p ) − h i ( p − ε ) h i ( p + ε ) − h i ( p − ε ) U (cid:48) i ( w − ε ) U (cid:48) i ( w − λ i ) − h i ( p + ε ) − h i ( p ) h i ( p + ε ) − h i ( p − ε ) U (cid:48) i ( w + ε ) U (cid:48) i ( w − λ i ) . (B.5)From (i), as in Pratt [22], Eqn. (20), U (cid:48) ( x ) U (cid:48) ( w ) ≤ U (cid:48) ( x ) U (cid:48) ( w ) , for w < x, and U (cid:48) ( x ) U (cid:48) ( y ) ≥ U (cid:48) ( x ) U (cid:48) ( y ) , for x < y. Furthermore, from (v), U ( y ) − U ( x ) U ( w ) − U ( v ) + U ( w ) − U ( v ) U ( w ) − U ( v ) ≤ U ( y ) − U ( x ) U ( w ) − U ( v ) + U ( w ) − U ( v ) U ( w ) − U ( v ) , for v < w ≤ x < y. Taking w = x yields U ( y ) − U ( v ) U ( w ) − U ( v ) ≤ U ( y ) − U ( v ) U ( w ) − U ( v ) , for v < w < y, U ( w ) − U ( v ) U ( y ) − U ( v ) ≥ U ( w ) − U ( v ) U ( y ) − U ( v ) , and also U ( y ) − U ( w ) U ( y ) − U ( v ) ≤ U ( y ) − U ( w ) U ( y ) − U ( v ) , for v < w < y . In all inequalities in this paragraph, U i may be replaced by h i , with v, w, x and y restricted to (0 , λ d ε ≥ d λ d ε , (B.6)hence (ii).We have now proved that (ii) is implied by (the equivalent) (i), (iv) and (v). We finallyshow that (ii) implies (i), or rather that not (i) implies not (ii). This goes by realizing that, bythe arbitrariness of w , p , ε with 0 < ε ≤ { p , − p } <
1, and ε >
0, if (i) does not holdon some interval (of w or p ), one can always find feasible w , p , ε and ε , such that (B.6),hence (ii), hold on some interval but with the inequality signs strict and flipped. (cid:50) Figures
Figure 12: Tversky-Kahneman Probability Weighting Function (upper panel) and its LocalIndex (lower panel). We consider β ∈ { . , . , . . . , . } . - - ¯ m Pr = 1 with m / ¯ m = 3 (upper panel) and m Pr = 1with m / ¯ m = 1 / γ = 0 .
5) and Prelec’s probabilityweighting function (with α = 0 ..