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Decreasing Risk Aversion and Mean-Variance AnalysisAuthor(s): Larry G. EpsteinSource: Econometrica, Vol. 53, No. 4 (Jul., 1985), pp. 945-961Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1912662

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Econometrica, Vol. 53, No. 4 (July, 1985)DECREASING RISK AVERSION AND MEAN-VARIANCEANALYSIS

BY LARRY G. EPSTEINIThis paper formulates a set of decreasing-absolute-risk-aversion postulates and showsthat only mean-variance utility functionals can satisfy them. These postulates are used toaxiomatize specific classes of mean-variance functionals. Finally, an equivalence is estab-lished between these postulates and corresponding comparative statics properties of assetdemands in two-asset portfolio problems.

1. INTRODUCTIONTHIS PAPERCONSIDERS wo common hypotheses in the literature on choice underuncertainty: (i) individuals exhibit declining absolute risk aversion (DARA), and(ii) the ranking of alternative distributions depends only on their means andvariances. The central result of the paper is that, under suitable regularityconditions, the latter hypothesis is implied by an appropriate formulation ofDARA. That is, mean-variance utility functions are the only ones which satisfythe notions of DARA described below.Mean-variance analysis is particularlyimportant in finance and largely becauseof the central role played by the capital asset pricing model in financial theory.A mean-variance framework is justified if the class of distributions is suitablyrestricted [2]; for example, if they are all normal. The capital asset pricing modelis also justified in a continuous time model where uncertainty is generated bydiffusion processes. (See [9].) But more generally the specification of mean-variance utility functions is an ad hoc functional form specification that is adoptedfor reasons of tractability. This paper provides a new justification for mean-variance analysis; namely, the basic DARA postulate.The most commonly used formulation of DARA is due to Arrow [1] and Pratt[10]. Their hypothesis is adopted in numerous papers in order to generatequalitative comparative statics results. Recently, Ross [11] and Machina [7] havestrengthened the Arrow-Pratt hypothesis in order to increase its predictive con-tent. Alternative formulations of DARA, some of which strengthen existinghypotheses, are investigated in this paper. They permit qualitative comparativestatics results to be derived in situations where earlier hypotheses have nopredictive power.

Suppose (x-,yj,J) is a trivariate distribution such that the conditional meansE[j/x] and E[e/y] vanish for all x and y.2 X-+ e and y + e define mean preservingl Mark Machina provided useful comments and an important proof. I have also benefited fromdiscussions with StuartTurnbull and from the comments of participants at the "Brown Bag" Theory

Workshop at the University of Toronto, a co-editor, and two referees.2 The notation adopted is consistent with that in [7]. Thus x denotes a random variable while x isa constant. The equation x = x indicates that the random variable x is constant at the level given byx. Fj,g, FX,Zdenote the cumulative distribution functions of i + e and (x, z) respectively; G, andGc,d denote the distributions which assign unit probability to the points c, and (c, d) respectively.All integrals are over [0, M]. For a cumulative density function F, yt(F) and u2(F) denote its meanand variance respectively. The terms increasing and decreasing when applied to a function, areintended in the weak sense.

945


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946 LARRY G. EPSTEINspreads [12] of x and y respectively. Think of e as defining the "same" additivegamble for either base wealth x or -.Let vr be a constant such that the decisionmaker is indifferent between x- and x-+ e + r and define v; similarly. Thenexisting hypotheses may be described as follows:3

DARA1 (Arrow-Pratt):vr - v.yifZx= x, -y= y, y > x.DARA2 (Ross): ir ; if; =x+w, w-O.DARA3 (Machina): vr ; ifjy= x +w, wIn the Arrow-Pratt hypothesis the two base wealths are nonstochastic. Thustheir hypothesis is too weak to have relevance to a real world setting where not

all risks are insurable. With this motivation Ross permits the base wealth xZobe random, but he maintains that the increment w is nonstochastic. Machinaargues that this restriction is unrealistic and deletes it in his formulation. Herequires only that the two base wealth distributions are comparable accordingto the criterion of first degree stochastic dominance.Restrict distributions to lie in D[O, M], the set of all cumulative distributionfunctions over [0, M], where M , vr if (Fx, Fy) E R.

In DARAI-3 particular specifications for R are adopted. In this paper alternativerestrictions for R are considered. In all cases, it is shown that R-DARA impliesmean-variance utility. Three specifications in particular, it is argued, lead tointuitively plausible-hypotheses. These specifications provide complete charac-terizations of appropriate subclasses of mean-variance utility functions.The paper proceeds as follows: Section 2 describes the weakest form of DARAconsidered in this paper. Its relationship with mean variance analysis is establishedin Section 3. Three stronger hypotheses are considered in Section 4. Section 5establishes the equivalence, under suitable regularity conditions, of some specialcases of R-DARA and corresponding comparative statics properties of assetdemands in two asset portfolio problems.Proofs are gathered in the Appendix. Some are adaptations of arguments from[7] to which the reader is referred for more detail. The extended discussion in[7] of motivation and related research and much of the intuition supplied thereare also relevant to the present analysis.

3Usually the risk premium v,j is defined by indifference between x + e and x - v,j. The differencebetween v,j and 7rj is analogous to the difference between equivalent and compensating variationsof a price change in demand theory. For each of DARA1-DARA3 replacing 7rj by v,j leads to anequivalent hypothesis. But the distinction between the two premia is significant for the hypothesesinvestigated in this paper as shown below.


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DECREASINGRISK AVERSION 9472. DECREASINGRISK AVERSION

An individual is assumed to rank elements of D[O, M] by means of a preferencefunctional V. Most frequently in the literature on choice under uncertainty anexpected utility specification is adopted for V But Machina [7] has shown thatthe expected utility hypothesis is inconsistent with DARA3, unless there is riskneutrality. It is similarly inconsistent with the R-DARA hypotheses consideredhere. Thus follow [6,7] and maintain only that V is differentiable. To be precise,view D[O, M] as a topological subspace of L'[0, M], the space of absolutelyintegrable functions on [0, M] with the L' norm; and assume that V is onceFrechet differentiable. In [6] it was shown that there exists at each Fo in D[O, M]a "local utility function" U(; FO),defined on [0, M], such that for any F inD[O, M],(1) V(F)-V(Fo)=j| U(w; Fo)[dF(w)-dFo(w)] + o(IIF - FoII)-Here o(*) denotes a function of higher order than its argument and 1 I s theL' norm.The following preliminary assumption is adopted.

ASSUMPTION 1: V is Frechet differentiableand the local utility function U issuch that U1(x; F) and U11(x; F) exist and are continuous in x for each F inD[O, M]. Moreover, U1 x; F) > 0 and U1 x; F) - 0 everywhere.

U1> 0 guarantees that V is monotonic in the sense of first degree stochasticdominance. The nonpositive second order derivative is equivalent to risk aversionin the sense that V is averse to all mean preserving increases in risk. (See [6].)The R-DARA postulates considered in this paper are those for which the setR satisfies both of the following conditions:R.1: If (Fe, Fj-) (=R, y, y*E [0, M], then there exists p, 0< j < 1, such that

((1-p)F +pG,, (1-p)Fj*+pGy*)G R for all O


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948 LARRY G. EPSTEINRepresent random variables by 4-dimensional vectors whose components rep-resent realizations in the various states of the world. Let z = (z, z, z, z) andz* = (z*, z*, z*, z*) be two nonstochastic distributions with z < z*. Suppose that(Fe, F;*) E R. (That is the case for each of the sets R corresponding to DARA1-3.)Let x=(z, z, y, y) and x*=(z*, z*, y*, y*). Then Fz=(1-p)F +pGy, F,z*=(1 -p)F1*+pGy*, and R.1 implies that (Fl, F,z*)G R for small p, and thereforethat ITi ? ITZ*. Moreover, this inequality is valid for all additive gambles e andfor all values of y and y*. In particular, it must be true if y > y* and if e = (0,0, 1, -1). Thus vz - vz* even for a gamble e that is nontrivial only in states ofthe world (3 and 4) where x- is larger than x*! In spite of the fact that thosestates occur with low probability, this implication appears counterintuitive, atleast given the intuition provided by the Arrow-Pratt DARA postulate and theindependence axiom. In contrast, vz - vz* is not implied by Machina's DARA3,for example, because Fz* first order stochastically dominates FZ only if y S y*.The example shows clearly that R.1 is potentially acceptable only if theindependence axiom of expected utility theory is dropped; in particular, only ifit is accepted that an individual's aversion to gambles that are confined to aspecific subset of states of the world can be influenced by the entire basedistribution of wealth. Though the independence axiom is intuitively appealingand has been dominant for several decades in the theory of choice underuncertainty, it is no longer universally adopted. Machina [4] surveys the empiricaland theoretical arguments against the independence axiom and surveys severalnonexpected utility models of choice, including his own corresponding to (1)above.4 Moreover, as noted earlier, the independence axiom must be abandonedeven if the more straightforward DARA3 postulate is adopted. Thus interest inR.1 cannot be ruled out on the basis of the above intuition.Further justification for the general requirement R.1 is not attempted directly.Rather, Section 4 presents three examples of R-DARA postulates which arereadily interpreted and which have some intuitive plausibility. These examplesserve to establish that the analysis below is mathematically nonvacuous andeconomically relevant.Since Theorem 1 below may be more easily appreciated by the reader if hehas a concrete example in mind, an example is presented here. It may be motivatedby firstconsidering DARA1-DARA3. The latter suggest numerous other formula-tions. For example, DARA3 could be strengthened by requiring that Z?-whenever - dominates x by second order stochastic dominance. A commonfeature of all these formulations is that y is preferable or indifferent to x. Thusthe following hypothesis would appear to merit particular attention:5

DARA4: vz - v; if V(Fx ) - V(F, ).Clearly DARA4 is stronger than DARA3. It is also a special case of the generalR-DARA hypothesis where R.1 and R.2 are satisfied. To see this let R =

4The theoretical arguments include temporal risks [5] and group risk sharing [8].S Willig [13] considers a weaker hypothesis, that is similar in spirit, in an expected utility framework.


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DECREASING RISK AVERSION 949{(F, G) ED2[0, M]: V(F) < V(G)}, and note that the Frechet differentiability ofV (Assumption 1) implies that V is continuous and hence, in particular, that Ris open.

3. MEAN VARIANCE UTILITYIn this section it is shown that the R-DARA hypothesis, where R satisfiesconditions R.1and R.2, implies that utility can depend only on mean and variance.But another preliminary assumption is required. It concerns the following func-tional:

(2) A(y; F)--U11I(Y; F)/ U,(w; F) dF(w),y e [0, M], Fe D[O, M].

ASSUMPTION 2: A(y; ) is continuous on D[O, M] for each ye [0, M].A(y; F) resembles an Arrow-Pratt risk aversion measure which admits thewell-known interpretation as twice the risk premium per unit of variance for asmall gamble. Note that A(y; Gy)= - Ul(y; GY)/Ul(y; Gy), which reduces tothe Arrow-Pratt index if the local utility function U is independent of the base

distribution. A comparable interpretation may be provided for A(y; F) in thegeneral case of a nonexpected utility functional. To see this, proceed informally.For small p > 0 and e > 0, and fixed z and y, let ff = 7r(e,p) be the unique solutionto(3) V[(1-p)Fr+ +2GY+??+g+2 Gy-1+j, ==V[(l-p)Fj+pGy].Differentiate6 totally with respect to p and evaluate at p = 0 to derive

-(e, 0) =[ U(y +,l; FF)+ U( y-1e; FF)-U(y; Fj)] U(w; F) dF(w).

For small e, the numerator is approximately e - U11 y; F2)/2. Thus ir(e, p) canbe approximated by ir(e, O)+p* aw(e, O)/1p, or by ep* A(y; F5)/2. In short,IT-ep *A(y; Fj)/2.

Finally, note that (3) has the form(3') V(F+j+v) =V(FR),

6To differentiate, make use of (1), i.e., that locally V is approximately an expected utility functional.See [6].


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950 LARRY G. EPSTEINwhere E[e/x] = 0 identically, and e(g)= ep. Thus A(y; F,) is approximated by2 iri/r-2()e which provides the desired interpretation.Theorem 1 is the central result of this section.

THEOREM 1: Let V satisfy Assumptions 1 and 2. Suppose that V satisfiesR-DARA where R satisfies R.1 and R.2. Then the functional A satisfies(4) A(y; F),A(y*; G) Vy, y*E[O, M] and V(F, G)e R.In addition, V can be expressed in the form7(5) V(F) = v(,u(F), o"2(F)),for some differentiable unction v with domain S --{(,(F), o2(F)): F E D[O, M]},wherev,>O, V2_f0; and(6) A(y; F) = -2v2(A (F), &2(F))/ v1 ,u (F), -2(F)).

Consider first the conclusions of the Theorem. The central conclusion is (5)that V is a mean-variance utility function. This is proven by first establishing theimplication (4) of R-DARA. From (4) it follows that the risk aversion measureA is in fact independent of its first argument. But that implies (because of (2))that the local utility function U( ; F) is quadratic for each F.The mean-variancespecification then follows.8 Of course, (6) relates the risk aversion measureA(y; F) to the slope of the , _- - indifference curve, which is the standardmeasure in mean-variance analysis.The key to the proof of the Theorem is the proof of (4) and an informal andintuitive sketch of the latter may be provided: Let (Fe, Fj*) ER and y, y* G 0, M].Define iT by (3). It was pointed out above that ir= 1Ti, where Fz is definedimplicitly by (3) and (3'). Define iT*= iTR* and FR*by a similar set of equationswhere - and y are replaced by z* and y* respectively. Then for small p, (Fi, Fx*)ER because of R. 1. Thus w7R 1T* because of R-DARA. Finally, (4) follows fromthe risk premium interpretation provided above for the functional A(*, ). Given(4), R.2 and the continuity of A (Assumption 2) imply thatA(y;F)>A(y*;F) VFeD[0,M].Since y and y* are arbitraryA( ; F) must be constant, i.e., U1(, ; F) is constant,which implies the mean variance specification.The fact that the local utility function U is quadratic in its first argument hastwo interesting implications. First, U1 *; F) and U1 *; F) S 0 can be satisfiedfor all y in [0, oo] only if U1 vanishes identically, in which case V is risk neutral.Thus the DARA hypothesis in the Theorem, monotonicity and strict risk aversionon D[O, M] for all M > 0, are inconsistent. But these properties are consistent ifdistributions are restricted to have supports lying in a given compact set [0, M],which restriction is imposed in this paper.

7Throughout the paper, equality between utility functionals is to be interpreted modulo ordinalequivalence.8 I am indebted to Mark Machina for providing me with a proof of this step.


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DECREASING RISK AVERSION 951Secondly, recall that in the expected utility framework quadratic utility indicesare often criticized because they imply that the Arrow-Pratt measure is increasingin violation of DARAI. In terms of U above, this observation takes the formthat - U I(y; F)! U1 y; F) is increasing in y if U1 does not vanish. But in the

more general Machina framework where local utility functions may depend onF, the above noted property of the Arrow-Pratt measure is not relevant torisk-taking behavior. (See also [6; p. 300].) Thus not only does quadratic localutility not violate DARAI, it is necessary and, if suitably restricted, also sufficientfor the much stronger hypotheses described in the next section.In the introduction it was noted that except for the trivial case of risk neutrality,expected utility theory is inconsistent with the new DARA postulates consideredin this paper. In fact, to prove this inconsistency it is necessary to add a mildrequirement for the set R. (Otherwise, a counterexample is provided by theexpected utility functional V corresponding to a quadratic utility index. It exhibitsR-DARA, where R = {(F, G) E D2[0, M]: ,u(F)> ,u(G)} satisfies R.1 and R.2;and on a suitable domain, V is monotonic and risk averse.)

R.3: There exists (Fo, F,) E R such that Fo(x) > F, (x) for all x E (0, M).The condition imposed on Fo and F, is stronger than requiring that they becomparable by strict first degree stochastic dominance. (The latter would require

Fo(x) - F, (x) for all x E[0, M] with strict inequality for at least one x.) But R.3is satisfied by the specializations of R-DARA considered in the paper. Now theinconsistency may be proven as follows: If V is an expected utility functional,its local utility function U is independent of the base distribution. Thus writeU(y; F) = u(y). Condition (4) implies that

u'(w)[dG(w) - dF(w)] >-O V(F, G) E R.But take (F, G) = (FogF1). Since u' is decreasing, the above integral is nonpositiveand equals zero only if u' is constant. Thus R-DARA, with R.1, R.2 and R.3,implies risk neutrality if V is an expected utility functional.It is evident that condition R.1 does not necessarily, at this level of generality,reflect any form of decreasing risk aversion. Indeed, it is shown in the next sectionthat preferences which everyone would agree to call increasingly risk averse alsosatisfy R-DARA with R.1 and R.2. The quadratic expected utility functionaldescribed prior to the statement of R.3 provides another such example. Therestrictions imposed by the R-DARA hypotheses are twofold: (a) for all (F, G)in the set R the relative magnitudes of the risk premia for the base distributionsF and G are independent of the gamble e, and (b) R is "large" because of R.1and the closure condition R.2.These are the properties which imply mean-varianceutility. Thus Theorem 1 could be loosely interpreted as proving that "systemati-cally changing risk aversion" implies mean-variance utility. But since hypothesesabout the systematic variation in (absolute) risk aversion in practice are usuallyDARA postulates, the interpretation described in the introduction and title tothis paper has been adopted.


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952 LARRY G. EPSTEIN4. THREE PARTICULAR HYPOTHESES

4.1. Constant Risk AversionFor a first example consider a strong form of constant (absolute) risk aversion.Suppose that risk premia are independent of base wealth; then for a given gamblee, rj= iy for all distributions Fi and Fy. Then the set R is all of D[O, M] xD[O, M] and the hypotheses R. 1 and R.2 are trivially satisfied. Theorem 1 showsthat if V exhibits constant risk aversion in this sense then V is a mean-varianceutility function having straight line and parallel indifference curves, i.e., for all F

(7) V(F) = v(,uL(F), 2(F)), where VQ(1,o-2)= - ao-2, a -O.In fact, Theorem 2 provides a complete characterizationof this common functionalform specification.

THEOREM 2: Let V satisfy Assumptions 1 and 2. Then thefollowing statementsare equivalent: (a) V satisfies R-DARA whereR = D[O, M] x D[O, M]. (b) V hasthe functional form in (7).The expression "constant absolute risk aversion" immediately evokes theexpected utility functional with an exponential von Neumann-Morgenstern utilityindex. For this specification risk premia are invariant to the base distribution F,

but only for gambles J that are distributed independently of i. For the functionalforms in (7) the constancy of risk premia applies for the much larger class ofgambles with zero conditional means, which class corresponds to the set of meanpreserving increases in risk [12]. (Machina [7; footnote 4] has emphasized therestrictiveness of the assumption of an independently distributed gamble e.)4.2. Risk Aversion and the Level of Utility

A natural weakening of the hypothesis of constant absolute risk aversion isprovided by DARA4, defined in Section 3. Since Theorem 1 is applicable, onlymean-variance utility functions are consistent with DARA4. In fact the class ofutility functionals that satisfy DARA4 can be completely characterized.THEOREM 3: Let Vsatisfy Assumptions 1 and 2. Then thefollowing statementsare equivalent: (a) V satisfies DARA4. (b) V has the mean-variancefunctionalform (5) where V2(0, u2)/VI(b, u2) = k(v(,V 2)) for all (g, 0.2) in the domainof v and for some continuous, decreasing, and non-negative function q defined onthe range of V.The indifference map of the typical mean-variance utility function correspond-ing to DARA4 is shown in Figure 1. ,u- (r2 indifference curves are straight linesbut not necessarily parallel. Higher indifference curves have smaller slopes. Theintersection point of these curves lies outside the domain where the utilityfunctional V is well-behaved.


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DECREASING RISK AVERSION 9531-=

domain fv

~~~~~~~~~~,J2FIGURE 1-Mean-varianceindifference map for DARA4 utility functional.

Utility functions consistent with Figure 1 may be constructed analytically asfollows: Let 4)be a differentiable decreasing function with domain (- 00, oo)andrange lying in (8, M-1) for some 8 > 0. Let K > 0 be a constant and suppose thatd(l/4(z))/dz < KM-" for zE ( -oo, c). Let2(8) D(z, ,u,o ) =- -fZK,4)(z) 2for z E(- oo, oo) and (,u,o.2) E {(A(F), u2(F)): F E D[O, M]}.Then D is continuous and strictlydecreasing in z and limz,(0 D(z; g, o-2) = 00)limz _.o D(z; A, a.2) = 00. Thus the equation D(z; ,u, o-2) = 0 has a unique solutionz. Define v(,u, .2) to be that unique solution; that is, v(,t, 0.2) is defined implicitlyby(9) D(v(, 2,0) ) = 0.Total differentiation of (9) yields immediately that - v2/v1= +(v) and thus Figure1 and DARA4 are valid. In the special case where 4 is a constant function, (8)and (9) reduce to the constant absolute risk aversion specification (7).Next consider two hypotheses which are variants of DARA4. First, refer tothe alternative risk premium definition mentioned in footnote 3. That is, supposeVX is the constant such that x-+ and x - Pz are indifferent. The followingalternative to DARA4 seems worthy of consideration:

DARA4': v;- Py whenever V(Fj) S V(F;).DARA4' also implies mean-variance utility, but it is not equivalent to DARA4.In fact, it is equivalent to the much stronger constant absolute risk aversionhypothesis. This is proven in the following theorem:


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954 LARRY G. EPSTEINTHEOREM 4: Let V satisfy Assumptions 1 and 2. Then thefollowing statementsare equivalent: (a) V satisfies DARA4'. (b) V has thefunctional form (7).Finally, consider the following antithesis of DARA4:DARA4": vrj> ir, if V(Fj) ? V(F;).Here risk premia are larger for preferred base distributions. This hypothesisis a special case of R-DARA where R = {(Fi, F;) E D2[O, M]: V(Fj) > V(F;)}.Theorem 1applies and so V must be a mean-variance utility function even thoughDARA4"requires increasing ratherthan decreasing risk aversion. This hypothesisis included here to illustrate the point made in the last section regarding the

proper interpretation of Theorem 1.

4.3. A Final HypothesisThe final specialization of R-DARA considered is DARA5:DARA5: vrr if either (a) ,u(Fj)> tt(F;) and V(Fj)< V(F,), or (b).LFj) < ,L(F;) and V(Fj,4) < V(F;), A (F;) - g(Fj)DARA5 requires that 17j 7 in either of two cases. In the first (a), y ispreferable even though x has a larger mean. Thus intuitively speaking x involvesgreater variability at least as the latter is measured by V. The risk premium issmaller for the preferred base distribution -, given that it has smaller variability.In the second case (b), - is preferredtox + A even though they have equal means.Speaking loosely again, - is less variable than x + A and hence also than x. Againthe risk premium is smaller for the preferred base distribution if it has smallervariability.This interpretation of DARA5 shows clearly the intuitively plausible way in

which it weakens DARA4, where the risk premium was always required to besmaller for the preferred base distribution. DARA5 is also stronger than boththe Arrow-Pratt and Ross hypotheses, but it is not comparable to Machina'sDARA3. It is, however, a special case of the R-DARA postulate of Theorem 1,since R can be defined to be the union {(Fi, F4) E D2[O, M]: , (F) > ,u(F;) andV(F;) > V(F, )} u {(Fi, F;) E D2[, M]: a (F) - (F) > 0 and V(F;) >V(Fj+A)}. R defined in this way satisfies R.1 and R.2 and R-DARA is equivalentto DARA5 (as long as Assumptions 1 and 2 are maintained). Thus only mean-variance utility functions are consistent with DARA5. The next theorem describesprecisely the class of utility functions that satisfy DARA5.

THEOREM 5: Let Vsatisfy Assumptions 1 and 2. Then thefollowing statementsare equivalent: (a) V satisfies DARA5. (b) V has the mean-variancefunctionalform (5) where v is quasiconcave and - v2/ vI is increasing in cr2 and decreasingin ,.


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DECREASING RISK AVERSION 955The mean-variance utility functions described in (b) are those having convexindifference curves and for which both ,t and U2 are normal goods. DARA5provides an axiomatization for this common specification.

5. ASSET DEMAND CONDITIONSPrevious formulations of DARA have been shown to be equivalent to appropri-ate forms of decreasingly risk averse behavior in the context of simple portfolioproblems. This section establishes such an equivalence for each of the specialR-DARA postulates of the last section.The typical portfolio problem considered involves two assets, represented byx and -, where z = x + r + e, r> 0, and E[e/x] = 0 for all x. The asset correspondingto z has a higher mean and greatervariability (in the sense of [12]). The portfolioproblem is to choose the value of a which yields the most preferred distributionof the form F(1a)j?aj = Fz+a(r+J).If a is unrestricted, then FR+a,(r+9)will beoutside D[O, M], the domain of V,for some values. Thus there must be an implicitconstraint on a such that FR+a(r+0) lie in D[O, M]. The constraint set alwaystakes the form of a closed interval. To rule out cases in which the intervaldegenerates to a point assume that Fj has compact support. If also Fz is restrictedto lie in D[M, M] for some 0 < M < M < M, then the interval will include zeroin its interior. Given risk aversion the optimal value of a is nonnegative so thatonly nonnegative values of a in the constraint set need be considered. Thus itmay be assumed that a varies over an interval [0, bo], where bo-max {a: FX+a,(r+j) ED[O,M]}. An optimal value ao of a will be said to be aninteriorsolution if ao < bo.Assets demands in the above portfolio problem are compared to demands ina new problem where the same assets are available but where a stochastic expost increment to wealth, Ax, is expected. Similar assumptions, notation, andterminology are adopted for this problem; for example, b,max {a: FR,+,+a(r+e)E D[O, M]}, and a solution al is interior if it is less than bl.Fix a set R as in Theorem 1. The following property of asset demands(decreasingly risk averse behavior, or DRAB) is the focus of this section:R-DRAB: Suppose F* and Fz+,dzare in D[M, M] for some 0 < M < M < M,r 0, Fe has compact support, and E[e/x] = E[e/Ax]= 0 for all x and Ax. Letz = x r+e and suppose that ao and al yield the most preferred distributionsof the form F(1-a)j+, and F(l-a)x+zlx+a! respectively. Denote by Fo and F1 thecorresponding optimal distributions. Then (FO, F1) eR implies that ao0 a1 or

ao> a, = bl.The interpretation of this condition is straightforward: the expectation of thestochastic increment Ax will cause a revision of the portfolio. But if (FO,F1)ER,and if it is not the case that a0> b1= a1, then ao< a1 necessarily. In particular,if al is an interiorsolution, then (FO,F1) E R implies that the new optimal portfoliocannot involve less risk taking, which is a form of decreasingly risk averse


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956 LARRY G. EPSTEINbehavior. The need to include the alternative possibility, where a0o>a1, is easilyunderstood. The relative sizes of ao and a1Idepend not only on whether preferen-ces are decreasingly risk averse but also on the relative severity of the constraintsimposed on a in the two problems. In the event that ao> a1 = bl, the reverseinequality ao> a1 reflects the tighter (b, < bo) and binding (a1 = b,) constrainton a in the problem with the ex post increment, rather than increasingly riskaverse behavior.Interpret R-DRAB again for each of the three specifications for R adopted inthe last section. (Throughout restrict attention to interior solutions.) If R =D2[0, M], the constant absolute risk aversion case, then R-DRAB asserts thatthe demand for the riskier asset is unaffected by the ex post increment.For the R that corresponds to DARA4, R-DRAB asserts that preferred optimalportfolios cannot involve less risk taking, i.e., V(FO) V(F1) implies a0o a1. Therankingindicated by V(FO) V(F1) is valid if ax 0, which is the case consideredin [7]. But the restriction Ax - 0 is severe, e.g., it excludes all cases in which theex post increment assumes small negative values only on a set of small measure.Thus DARA3 (and a fortiori DARAI and DARA2) are not strong enough toimply "decreasingly risk averse" behavior in a large variety of realistic situations,where the hypotheses formulated in this paper have predictive power.Finally, let R correspond to DARA5. Then R-DRAB requires that the demandfor the riskier asset increases as a result of the ex post increment if F1is preferredto Fo and if F1 is also less variable than Fo in the sense of the discussion inSection 3.3.To prove the equivalence between R-DARA and R-DRAB, the followingadditional assumption, adapted from [7], is required:

AssuMPTION 3: For all r, x, and e as in the statement of R-DRAB, preferencesover {FE+a(r+e) e D[O, M]}af are strictly quasiconcave in a.

Machina calls this property diversification. It ensures that there is a uniqueoptimal value for a and that preferences are strictly decreasing in a below thisvalue and, if the solution is interior, strictly decreasing above it.The final result of this paper is that, under suitable assumptions, R-DARAand R-DRAB are equivalent hypotheses.THEOREM 6: Let V satisfy Assumptions 1 and 2 and let R satisfy R. 1 and R.2of Theorem 1. Then R-DARA implies R-DRAB. Moreover, if V also satisfiesAssumption 3, and if R is the set for the constant absolute risk aversionpostulate,DARA4 or DARAS, then R-DARA and R-DRAB are equivalent hypothesesabout V.A primejustification for considering the constrained portfolio problems aboveis that they permit the characterization, in terms of asset demand conditions, ofdecreasing risk aversion on a domain D[O, M], M


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DECREASING RISK AVERSION 957the domain of the utility functional is taken to be suitably unrestricted. Thus itmay be worthwhile to provide further justification for the analysis in this section.Firstly, note simply that the constraints are irrelevant given interior solutions,in which case Theorem 6 shows that R-DARA implies decreasingly risk aversebehavior, and in environments in which DARA1-DARA3 have no predictivepower. Indeed, since mean-variance utility implies the separation property, R-DARA has predictive power even in models with more than two risky assets.(See [3].) The complexity introduced by boundary conditions can be avoided ifit is desired only to establish asset demand conditions that are necessary, but notsufficient, for R-DARA.Secondly, it is possible to modify the preceding analysis slightly so that itconforms more closely to standard formulations of portfolio problems. Forconcreteness, consider the special cases of R-DARA and R-DRAB that corre-spond to the R of DARA4. Adopt the usual assumption that the domain of theutility functional is sufficiently large that constraints on a are unnecessary. Indeedutility functionals often admit natural extensions from some D[O, M] to a largersubset of D[O, ac] which does not require that the supports of all distributionslie in a given compact set. (For example, V(F) from (9) is well-defined for anyF that has finite mean and variance.) Denote by V the extension of V. Theorem1 and the subsequent discussion show that V cannot satisfy DARA4 and beincreasing and risk averse, over its entire domain. Suppose therefore, thatmonotonicity is violated on D[O, M'] for any M'> M But the interpretation ofDARA4 as a description of decreasing risk aversion depends on utility beingincreasing. In fact, if preferences are decreasing then DARA4 describes a formof increasing risk aversion. It follows that V exhibits decreasing risk aversiononly on D[O, M] where it coincides with V.Turn to the appropriate unconstrained optimization problemsmaxa V(FR+a(r+,))and maxa V(FR+=?*+a(r+,)).Denote optimal values of a by aoand aI respectively and consider what statement about these demands correspondsto the decreasing risk aversion of V on D[O, M]. Let bo and b, be defined asabove. Then bo, for example, is the maximum value of a in the initial portfolioproblem for which the corresponding wealth distribution F*+a(r+0) remains inthe region where V is increasing. It is quite conceivable that a> bo or a > bl,or both, in which case the corresponding wealth distributions do not lie in theregion where V exhibits declining risk aversion. Thus one would not expect tobe able to prove that a 6 a in all cases. But it can be shown that, given theappropriate diversification assumption, the following modification of R-DRABis both necessary and sufficient for V to exhibit DARA4 on D[O, M]:V(F+.+61 (r+-) ) > V(FR+&0(r+9)) implies either (i) a &a, or (ii) a&> &, >b1.

University of Toronto

Manuscripteceivedanuary, 984;revision eceived uly,1984.


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958 LARRY G. EPSTEIN

APPENDIXProofs of all theorems are collected here.PROOF OF THEOREM 1: Adapt the arguments in the proof that (ii)=*(i) in Theorem I of [7;

pp. 1076-7]. Fix 0 < M < M< M. Suppose hatA(y; F2) < 2/ < A(y*; FF.) wherey $ y*, (F2, FzF) ER, y and y* lie in [M, M] and FF, FF lie in D[M, M]. Thend {V[(l -p)Fi+pei3+ Gy+fe++pef3+?Gy0

for sufficiently small e > 0. (Since F2, F2F, Gq,and Gy lie in D[M, M], the firstargument of V willlie in D[O, M] if p and e are sufficiently small.) Repeat with (y, FF) replaced by (y*, FF-) to deducethat for all sufficiently small p, e > 0,(10) V[( l-p)Fj+pe3 +- Gy+fe+pe + 2 GV.f+pe j> V[(I -p)F;F+pG], and(I 1) V[(I I-P) F+pej3 2 )K*+/+pefl 2 y-,Ie+pe< V[(I-p)FZ-+pG0.V].There exists a distribution F;, for which F; = FF and Fp+g= FF*. (Machina required that i 2 O, butthat is not necessary here.) Define Fz ax and Fj, ,x( ./ x, Ax) precisely as in [7]. In particular,(12) Fx=(1-p)Fj+pGq, and Fj+,AX=(1-p)F2*+pG,*.Both Fj and FX+ j are in D[O, M]. Moreover, F,+j+pe,0 and F;+A;X+j+Pe,3are given by the argumentsof V on the left sides of (10) and (11) respectively. Thus, as noted above, they also lie in D[O, M].Finally, it follows from (10) and ( 11) that vrz < pe,83


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DECREASING RISK AVERSION 959(a)?(b): From (4),

(14) A(y; F):A(y; G) if V(F)< V(G), and(15) A(y; F)=A(y; G) if V(F)= V(G).Equations (5), (6), and (15)=X - V2 V =(v).i is decreasing because of (6) and (14). The remainingproperties of q follow from Assumption 1. Q.E.D.

PROOF OF THEOREM 4: (b)?i(a) is trivial.(a)?(b): A slight adaptation of the proofs of Theorems I and 3 proves (14), (15) and hence thatU(. ; F) is quadratic, V is a mean-variance functional and that -V2VI = + (v) as in Theorem 3.Indeed this adaptation is closer to Machina's original arguments [7; pp. 1076-7] since he uses therisk premium vF rather than ire. It remains only to prove that k is constant.Fix A, a-2and 6-v(,u, a2). Define H(a, A) v(f(a), - , wheref is definedimplicitly by v(f(a), o.2+ a) = 6. Clearly, f(O) = A and'(0) =-v2( , -2)/vI(A,' c2). Differentiate Hwith respect to a and evaluate at a = 0 to obtain(16) Ha(0, a) = vl(,c, 2+A)[+(6) - f(V(., a2+a))].Suppose that 0 is not constant in a neighborhood of v. Since + is decreasing 3a and a > 0 such that(17) +(v) #5 (v( (,A'-2+ a)) $0(v(f(a), a-2+ a+A)) VO< a


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960 LARRY G. EPSTEINSuppose 3y, y*, F7, Fw*such that (Fw, Fw*)E R, y, y* E [M, M], F, Fw* D[A, M] for some0 < M < M < M, and such that A(y; F -) < A(y*; F -*). Adapt Machina's [7; pp. 1078-9] proof that(iii) =(i) in his Theorem 1, in the same way that the proof of Theorem 1of this paper was constructedfrom his arguments. In particular, find a > 0 and 3> 0 such that

2-8[U,y + C; Fw,) - U,(y - ; Fw,)] --2[U,(y + d; Fw*-)- U,(y*; *)| Uj(e; Fw-)dFW,(6) |U,(e; Fw*-)FT,.(6)Fix 8>0 such that 8


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DECREASING RISK AVERSION 961[3] HART, 0. D.: "Some Negative Results on the Existence of Comparative Statics Results inPortfolio Theory," Review of EconomicStudies, 42(1975), 615-621.[4] MACHINA, M. J.: "The Economic Theory of Individual Behaviour Toward Risk: Theory,Evidence and New Directions," Report 433, Stanford University, IMSS, 1983.[5] "Temporal Risk and the Nature of Induced Preferences," Journal of Economic Theory,

33(1983), 199-231.[6] : "'Expected Utility' Analysis Without theIndependence Axiom," Econometrica,50(1982),277-323.[7] "A Stronger Characterization of Declining Risk Aversion," Econometrica, 50(1982),1069- 1079.[8] : "The Behaviour of Risk Sharers," mimeo, University of California at San Diego, 1984.[9] MERTON, R.: "Optimum Consumption and Portfolio Rules in a Continuous-Time Model,"Journal of Economic 7heory, 3(1971), 373-413.[10] PRATT, J. W.: "Risk Aversion in the Small and the Large," Econometrica, 32(1964), 122-136.[11] Ross, S. A.: "Some Stronger Measures of Risk Aversion in the Small and the Large withApplications," Econometrica, 49(1981), 621-638.[12] ROTHSCHILD,M., AND J. STIGLITZ:"Increasing Risk: I. A Definition," Journal of EconomicTheory, 2(1970), 225-243.[13] WILLIG,R.: "Risk Invariance and Ordinally Additive Utility Functions," Econometrica,45(1977),621-640.

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