Greater downside risk aversion in the large

Journal of Economic Theory 144 (2009) 1092–1101

www.elsevier.com/locate/jet

Greater downside risk aversion in the large

Donald C. Keenan ∗, Arthur Snow

Department of Economics, University of Georgia, Fifth Floor Brooks Hall, Athens, GA 30602, United States

Received 5 June 2008; accepted 9 August 2008

Available online 12 November 2008

Abstract

In this paper, we advance a definition of greater downside risk aversion that applies to both large and smallchanges in risk preference, and thereby complements the results for small changes reported previously. Weshow that a downside risk-averse transformation of a utility function results in a function that is moredownside risk averse in the same manner that a risk-averse transformation increases risk aversion. Ourdemonstration is conducted first by using the compensated approach introduced by Diamond and Stiglitz[P. Diamond, J. Stiglitz, Increases in risk and in risk aversion, J. Econ. Theory 8 (1974) 337–360] and thenby using an adaptation of the risk premium approach taken by Pratt [J. Pratt, Risk aversion in the small andin the large, Econometrica 32 (1964) 122–136].© 2008 Elsevier Inc. All rights reserved.

JEL classification: D81

Keywords: Downside risk; Risk aversion; Arrow–Pratt; Diamond–Stiglitz

1. Introduction

In the expected utility model of saving with time-separable preferences, Leland [8] shows thata convex von Neumann–Morgenstern marginal utility function (u′′′ � 0) is indicative of a pre-cautionary saving motive, whereby uncertainty about future income increases current saving.1

* Corresponding author. Fax: +1 706 542 3376.E-mail address: [email protected] (D.C. Keenan).

1 Throughout, we use primes to denote derivatives with respect to income y.

0022-0531/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2008.08.007

D.C. Keenan, A. Snow / Journal of Economic Theory 144 (2009) 1092–1101 1093

Menezes et al. [9] identify this characteristic of risk preferences as aversion to downside risk,or a dislike of any mean-and-variance preserving third-order spread of the income distribution.Kimball [6] shows that the index of prudence (−u′′′/u′′) measures the strength of the precaution-ary saving motive, and Jindapon and Neilson [4] relate greater prudence to greater willingness toforego expected utility in exchange for a reduction in downside risk. While the results of Kim-ball and of Jindapon and Neilson link greater prudence with behaviors motivated by downsiderisk aversion, neither shows that the index of prudence measures the strength of downside riskaversion.

We provide an analysis of downside risk aversion that mirrors the standard results for riskaversion. Specifically, a function u is risk averse if u′′ � 0, and v = ϕ(u) is more risk aversethan u if the mapping ϕ is, itself, risk averse. In like manner, u is downside risk averse if u′′′ � 0,and we say that v is more downside risk averse than u if ϕ′′′ � 0. Given arbitrary functions u

and v, verifying these properties of the transformation function ϕ may be difficult. However, forthe case of risk aversion, the concavity of ϕ is equivalent to the condition that the Arrow–Prattindex of risk aversion r is greater for v than for u. For downside risk aversion, we provide anindex that is also defined directly on u and v such that v is more downside risk averse than u ifthis index is greater for v than for u.

Thus, our results concerning greater downside risk aversion contrast with those of Kimball [6]concerning greater prudence. Kimball shows that v is more prudent than u if marginal utility v′is a transformation of u′ that has a positive second derivative. By contrast, we establish that v

is more downside risk averse than u if utility v is a transformation of u that has a positive thirdderivative.

Keenan and Snow [5] and Modica and Scarsini [10] have previously addressed the charac-terization of the strength of downside risk aversion. By confining attention to small increasesin downside risk aversion, Keenan and Snow develop a measure characterizing greater aversionto compensated increases in downside risk. Modica and Scarsini characterize greater aversion touncompensated increases in downside risk in the same manner as Ross [12] characterized greateraversion to Rothschild and Stiglitz [13] increases in risk; however, the resulting Ross-like restric-tions on preferences are quite strong and cannot be expressed in terms of any local risk measure.In this paper we offer a characterization of greater downside risk aversion in the large, that is,for large increases in downside risk aversion, which complements the local results reported byKeenan and Snow. We also identify the limits to relating this characterization to a measure thatindicates greater downside risk aversion.

In the next section, we reexamine the approach taken by Diamond and Stiglitz [3], whereincompensated increases in risk for one utility function reduce the expected value of another if andonly if the second is a concave transformation of the first, and extend this result to the next orderto establish that a compensated increase in downside risk for one utility function reduces theexpected value of another if and only if the second is a downside risk-averse transformation ofthe first. In Section 3, we show that the measure of downside risk aversion introduced by Keenanand Snow [5] is a sufficient, although not a necessary, indicator of greater downside risk aversionin the large. That it is both necessary and sufficient for small increases in downside risk aversionsuggests that no measure fully characterizes the degree of downside risk aversion. In Section 4,we adapt the risk premium approach to characterizing greater risk aversion, and show that anappropriately specified downside risk premium is greater for one utility function than anotherif and only if the second is a downside risk-averse transformation of the first. Conclusions arepresented in Section 5.

1094 D.C. Keenan, A. Snow / Journal of Economic Theory 144 (2009) 1092–1101

2. Large increases in downside risk aversion using the compensated approach

While the global condition for greater risk aversion, that v is more concave that u, is attrac-tive in its own right, it is more compelling that it is confirmed by thought experiments involvingintroductions or increases in risk. For introductions of risk, Pratt [11] shows that the risk pre-mium, being the amount one is willing to pay to avoid assuming a risk, is always greater for v

than it is for u if and only if v is a concave transformation of u. For increases in risk, Diamondand Stiglitz [3] (D&S) characterize the role of greater risk aversion by considering the class ofdistributional changes in risk that induce mean preserving spreads in the utility distribution. Itis straightforward to show, using this approach, that v dislikes all mean preserving spreads inthe utility distribution for u if and only if v is a concave transformation of u. This result is notactually proven in D&S, who restrict attention to small changes in risk preference, so we recordthe result here.2

We assume, throughout, that any utility function u or v is positively monotonic in y and thatthe supports for all cumulative distribution functions F(y) are contained in the compact interval[a, b], with F(a) ≡ 0. Recall that, given the distribution function F(y, r) and a shift parameter r ,a D&S compensated increase in risk Fr(y) for u is characterized by the condition:

y∫a

u′(x)Fr(x) dx � 0, for all y ∈ [a, b], with equality at y = b. (2.1)

Throughout, we shall assume that φ is the transformation such that v = φ(u).

Lemma 1. All D&S compensated increases in risk for u result in lower expected utility for v ifand only if v is a risk-averse transformation of u, i.e. φ′′ � 0 everywhere.

Proof. Using integration by parts, we have

Er [v] =b∫

a

v(y) dFr(y) =b∫

a

φ(u(y)

)dFr(y) (2.2)

=b∫

a

−φ′(u(y))u′(y)Fr(y) dy (2.3)

=b∫

a

φ′′(u(y))u′(y)

y∫a

u′(x)Fr(x) dx dy. (2.4)

Sufficiency of φ′′ � 0 for Er [v] � 0 is immediate, given that Fr(y) satisfies (2.1). For necessity,suppose φ′′(u(y)) > 0 were true, for some y. Then there would exist a surrounding interval on

2 While the arguments in D&S do show that φ′′ � 0 is equivalent to rv � ru in the large, this is not enough, givenonly their results, to relate either inequality condition to compensated spreads under large changes in risk preferences.It might appear that one could chain together their infinitesimal preference changes (or those of Keenan & Snow [5])to get global results, but in the case of compensated utility spreads, or their analysis of risk premia, this is not possible,since the parametric shifts in preferences studied in those papers also affect the reference utility on which their resultsare based. The global results proven here do not suffer from this difficulty.


which φ′′ > 0 would continue to hold, and on choosing Fr to be concentrated on that interval,one would then obtain Er [v] > 0, contradicting the hypothesis Er [v] � 0. �2.1. The definition of an increase in downside risk aversion

Menezes et al. [9] introduce downside risk aversion by altering the notion of Rothschild andStiglitz [13] (R&S) mean-preserving spreads in y to ones that are also variance-preserving third-order spreads. They then show that a dislike of all such spreads is characterized by u′′′ � 0.By analogy to the second-order case, this strongly suggests that the correct notion of greaterdownside risk aversion in the large should be that φ′′′ � 0 throughout. Indeed, henceforth, thiswill be our definition of v being more downside risk averse than u, and the rest of the paper willsupport this chosen formulation through various experiments involving changes in risk. We willalso investigate whether there exists any local risk measure indicating that one utility function ismore downside risk averse than another.

2.2. The characterization of large increases in downside risk aversion under the compensatedapproach

Keenan and Snow [5] (K&S) note that, just as with R&S spreads in terms of the underlyingvariable y, it is straightforward to adapt the notion of D&S mean-preserving spreads in utility u

to those which are third-order mean-and-variance-preserving spreads in the utility distribution.In analogy with (2.1), a compensated increase in downside risk for u is characterized by theconditions

b∫a

u′(y)Fr(y) dy = 0 (2.5)

andy∫

a

u′(x)

x∫a

u′(z)Fr(z) dz dx � 0 for all y ∈ [a, b],with equality at y = b. (2.6)

The following result then shows that, as with the second-order case, this compensated ap-proach leads to the global notion of greater downside risk aversion proposed above.

Theorem 1. All compensated increases in downside risk for u result in lower expected utility forv if and only if v is a downside risk-averse transformation of u, i.e. φ′′′ � 0 everywhere.

Proof. Using integration by parts, and taking account of (2.5), we have

Er [v] =b∫

a

v(y) dFr(y) =b∫

a

φ(u(y)

)dFr(y) (2.7)

=b∫−φ′(u(y)

)u′(y)Fr(y) dy =

b∫φ′′(u(y)

)u′(y)

y∫u′(x)Fr(x) dx dy (2.8)

a a a


=b∫

a

−φ′′′(u(y))u′(y)

y∫a

u′(x)

x∫a

u′(z)Fr(z) dz dx dy. (2.9)

Sufficiency of φ′′′ � 0 for Er [v] � 0 is immediate, given that Fr(y) satisfies (2.6). Necessity isassured by the same type of argument used in the proof Lemma 1. �

For small increases in downside risk aversion, K&S show in their Theorem 1 that φ′′′ � 0 isnecessary and sufficient for v to dislike all compensated increases in downside risk for u. Thepresent result shows that this conclusion applies in the large as well as in the small.

3. Measures indicating large increases in downside risk aversion

While the characterization of greater downside risk aversion in terms of φ′′′ � 0 would, thus,seem to be the correct one, this condition, like the one in terms of spreads, is not always so easilyverified. Fortunately, in the classical case of risk aversion, there is a local measure, in terms ofjust the utility function, such that an increase in this measure, for each income, in going from u

to v, completely characterizes a global increase in risk aversion, in the sense that φ is concave.This role, of course, is fulfilled by the Arrow–Pratt measure of risk aversion (Pratt [11], Arrow[1]), ru(y) = −u′′(y)/u′(y), and thus, the required condition is that rv(y) � ru(y) for all y.

For the case of downside risk, K&S show that the measure su(y) = du(y) − (3/2)r2u(y), indi-

cates greater aversion to downside risk for small increases in downside risk aversion, where, inthe notation of Modica and Scarsini [10] (M&S), du(y) = u′′′(y)/u′(y).3 The issue then ariseswhether anything can be said of the measure in the case of large changes in risk preference. Now,it is easily seen that, as with d , or any other such measure, a uniform increase in s does not charac-terize greater global downside risk aversion, in the sense that φ′′′ � 0. One does, however, obtainan alternative characterization of the ranking of preferences by s, expressible in terms of φ. Forcomparison, we begin with the corresponding result in the classical case of risk aversion.

Lemma 2. rφ(u) ≡ −φ′′φ′ (u) � 0 for u = u(y) if and only if rv(y) � ru(y).

Proof. With v = φ(u), we have v′ = φ′u′ and v′′ = φ′′(u′)2 + φ′u′′ and so,(−v′′

v′

)−

(−u′′

u′

)= −φ′′

φ′ u′. � (3.1)

We now extend this result to the case of downside risk aversion.

Lemma 3. sφ(u) ≡ φ′′′φ′ (u) − 3

2 (rφ(u))2 � 0 for u = u(y) if and only if sv(y) � su(y).

Proof. With v = φ(u), in addition to v′ = φ′u′ and v′′ = φ′′(u′)2 + φ′u′′, we have v′′′ =φ′′′(u′)3 + 3φ′′u′u′′ + φ′u′′′, and so,(

v′′′

v′ − 3

2

(v′′

v′

)2)−

(u′′′

u′ − 3

2

(u′′

u′

)2)=

[φ′′′

φ′ − 3

2

(φ′′

φ′

)2](u′)2. � (3.2)

3 M&S show that the Arrow–Pratt risk premium for bearing a small risk increases more for v than for u with a smallincrease in downside risk if and only if dv(y0) � du(y0) at initial sure income y0.


Note that we have thus proved that the ranking yielded by the measure s is an intrinsic concept,in that it does not depend on the reference utility, and concerns only the relation between twopreferences. That is, if we take another utility function u and consider v = φ(u), then sv(y) �su(y) exactly to the extent that sv(y) � su(y). This in itself seems a desirable property, andindeed, is one satisfied by the classical notion of risk aversion r . On the other hand, the rankingby the measure d(y) = u′′′(y)/u′(y) cannot be similarly characterized in terms of φ alone, andindeed, s is the only apparent candidate for a measure of downside risk aversion whose inducedranking can be so expressed.4

With this characterization of the ranking by s in terms of φ, we are able to prove the followingresult concerning global changes in downside risk aversion in going from u to v.

Theorem 2. The condition sv(y) � su(y) for all y is sufficient for v to be more downside riskaverse than u, in the sense that we always have φ′′′ � 0. This condition on s is also necessary inthose cases where the change in risk preferences is small.

Proof. Sufficiency: This follows directly from Lemma 3, since, with that result, sv(y) � su(y)

for all y implies sφ(u) � 0, and so φ′′′/φ′ � (3/2)(rφ)2 � 0.Necessity: Being an argument confined to the case of small changes in preferences, the result

is demonstrated by part 2 in the proof of Theorem 1 in K&S. Here we present an argument usingLemma 3.

Let utility perturbations v = φ(u, δ) from u be parameterized by δ, with these perturba-tions vanishing when δ = 0, so that φ(u,0) = u, the identity transformation. We then haveφ′(u,0) ≡ 1, with φ′′(u,0) ≡ 0 and φ′′′(u,0) ≡ 0. Letting sv(y, δ) signify sv(y), for v = φ(u, δ),we have from Lemma 3,

∂sv(y,0)

∂δ= ∂(

φ′′′(u(y,0))φ′(u(y,0))

− 32 (rφ(u(y,0)))2)

∂δ= ∂φ′′′(u(y),0)

∂δ. � (3.3)

While its sufficiency is heartening, the condition sv(y) � su(y) is not a necessary one forall global changes from u to v to be increases in downside risk aversion since, as shown inTheorem 1, only the weaker condition φ′′′ � 0 is required.5 Thus, it does not produce a globalcharacterization of greater downside risk aversion, as was obtained in the second-order case,where the Arrow–Pratt risk measure r fully characterizes the degree of risk aversion. However,the necessity portion of Theorem 2 indicates that this sufficient condition in terms of s is indeedtight, in that the necessity of the s condition for small increases in downside risk aversion showsthat one could never find a weaker condition still yielding greater downside risk aversion. Weconclude that there is no measure that fully characterizes greater downside risk aversion. Observethat, in comparison to s, the rival condition dv(y) � du(y) is neither necessary nor sufficient forgreater downside risk aversion, whether for small or large changes in risk preference.6

4 As may be shown of r , not only does s have this intrinsic property, but when its intrinsic nature is expressed in termsof φ, the resulting measure sφ takes the same form as does su in terms of u.

5 Similarly, s � 0 is sufficient for downside risk aversion, but not necessary, except in the case of small departures fromrisk neutrality. Rather than indicating downside risk aversion, the proper role of s is to indicate greater downside riskaversion by its increase.

6 Both Modica and Scarsini [10] and Crainich and Eeckhoudt [2] offer d as a candidate for the appropriate indicator ofthe degree of downside risk aversion.


Another interesting sufficient condition for globally greater downside risk aversion, assumingr to always be nonnegative, is that −r ′

v(y) � −r ′u(y) for all y, where −r ′ is the natural measure

of decreasing absolute risk aversion. Since a uniform increase in −r ′ implies a uniform decreasein r , and s = −r ′ − (1/2)r2, the condition −r

′v(y) � −r

′u(y) implies sv(y) � su(y).7 Clearly,

then, greater decreasing absolute risk aversion is a stronger, not a weaker condition than sv(y) �su(y).

4. Large increases in downside risk aversion using the risk premium approach

Calling the method of analysis initiated by D&S the compensated approach correctly sug-gests analogies with the familiar Hicksian approach to the law of demand. While it is unlikely,in practice, that a price change is going to be accompanied by compensation, nonetheless, byrestricting the analysis of consumer behavior to income-compensated changes, one isolates di-minishing marginal rate of substitution as the key property of preferences that supports the lawof demand. Even without actual compensation, the crucial role of diminishing marginal rate ofsubstitution remains, though other properties of preference then enter as well, confounding itseffect.

Similarly, it is well recognized that admitting all possible uncompensated increases in riskdoes not permit one to identify the underlying property of risk preferences characterizing greaterrisk aversion, since, for arbitrary increases in risk, other aspects of risk preference intervene.However, as D&S recognize, by restricting attention to compensated increases in risk, one doesisolate the key global property indicating that v is more risk averse than u, namely that v be arisk-averse transformation of u. The present analysis, by again using the compensated approach,identifies the key global property for greater downside risk aversion, namely that v be a downsiderisk-averse transformation of u.

The main alternative to restricting changes in risk to compensated utility spreads, when iden-tifying greater risk aversion, is to restrict them, instead, to introductions of risk, where thecompensation then takes the form of a risk premium. Since this technique indeed isolates thecorrect notion of increased risk aversion in the classical case, one might surmise that this al-ternative approach could be used to identify increased downside risk aversion. However, sucha strategy immediately runs into difficulties. The first problem is that, while one can introducerisk, one cannot introduce downside risk; one can only change the downside risk of a risk that isalready present.

4.1. Adapting the risk premium approach to the case of downside risk aversion

A solution to this dilemma, as suggested by K&S, is to introduce the notion of a differentialrisk premium, compensating for an increase in risk.8 This then subsumes the notion of the full

7 Corollary 1 of K&S is the same conclusion expressed for small changes in risk preference. The essential argument,however, is seen to be unaffected by the size of the change.

8 An alternative, less orthodox, strategy is to keep the full risk premium, and not consider a change in downside risk,but to, instead, compare the risk premia of two introductions of risk, one with more downside risk than the other. M&Sdo this in the case of two infinitesimal risks introduced to a known income, one an uncompensated downside spread ofthe other. As always, this uncompensated approach becomes unwieldy, even in the second-order case, if the introductionsof risk are large. However, if one pursues a compensated approach, it is easy to show that, even using full risk premia,the characterizing condition is still φ′′′ � 0.


risk premium for an introduction of risk, since the latter is nothing but a differential risk premiumwhen there happens to be no initial risk. The next issue is to identify the class of risk increasespermitted; as mentioned, it is well known that, unlike general risk introductions, the whole classof increases in risk is inappropriate, even in the second-order case. K&S propose using the classof patent increases in risk, introduced by Kimball [7], and indeed, demonstrate that, in the caseof small changes in risk preference, a uniform increase in r characterizes required increases inrisk premia for all patent increases in risk. Note that, since an introduction of risk is itself a patentincrease in risk, this characterization then encompasses the classical result concerning full riskpremia in the case of risk introductions.

It is then a straightforward matter to extend the notion of a differential risk premium to coverincreases in downside risk, by simply providing the corresponding notion of a patent increase indownside risk. As would be expected, K&S are able to show that a uniform marginal increase ins identifies those persons who require greater differential risk premia for all patent increases indownside risk. As is the case throughout that paper, though, this is entirely an analysis under theassumption that the changes in risk preferences are small. The obvious question, then, is: In thecase of large changes in risk preference, does the differential risk premium approach once againidentify φ′′′ � 0 as the condition characterizing greater downside risk aversion?

4.2. The characterization of large increases in downside risk aversion using the risk premiumapproach

For completeness, we begin with the second-order case. Recall, following Kimball [7], thata distribution F(y) has “patently more risk for u” than F 0(y) if, when fully compensated toneutralize any effect on expected utility, F(y) is a compensated increase in risk for u, i.e. Fr(y) ≡F(y) − F 0(y + πu2) satisfies (2.1), where the differential risk premium πu2 for u bearing riskF(y) over F 0(y) is defined implicitly by

b∫a

u(y − πu2) dF 0(y) =b∫

a

u(y) dF (y). (4.1)

A differential risk premium πv2 for v is defined in the analogous manner.

Lemma 4. For distribution F(y) patently more risky for u than F 0(y), one has πv2 � πu2 if andonly if v is more concave than u, i.e. φ′′ � 0 everywhere.

Proof. Introduce the distribution F 0(y) = F 0(y + πu2), and extend both F(y) and F 0(y) to in-clude the range [a −πu2, b]. Then observe that πv2 � πu2 if and only if

∫ b

av(y −πu2) dF 0(y) �∫ b

av(y − πv2) dF 0(y). Using (4.1), but stated for v, one can reexpress this inequality as

0 �b∫

a

v(y) dF (y) −b∫

a

v(y − πu2) dF 0(y) (4.2)

=b∫

v(y) dF (y) −b∫

v(y) dF 0(y) (4.3)

a−πu2 a−πu2


=b∫

a−πu2

φ(u(y)

)d[F(y) − F 0(y + πu2)

](4.4)

=b∫

a−πu2

−φ′(u(y))u′(y)

[F(y) − F 0(y + πu2)

]dy (4.5)

=b∫

a−πu2

φ′′(u(y))u′(y)

y∫a−πu2

u′(x)[F(x) − F 0(x + πu2)

]dx dy. (4.6)

Sufficiency follows from condition (2.1) since F(y) − F 0(y + πu2) is a compensated in-crease in risk for u, and necessity follows by the same type of argument used in the proof ofLemma 1. �

We now show that the differential risk premium approach identifies φ′′′ � 0 as the conditioncharacterizing greater downside risk aversion. In the manner of Kimball, a distribution F(y) isdefined in K&S to have “patently more downside risk for u” than distribution F 0(y) if, whenfully compensated to neutralized any effect on expected utility, F(y) is a compensated increasein downside risk for u in the sense of (2.5) and (2.6). A differential downside risk premium πu3for u, is then defined in the same manner as (4.1), but where the change from F 0(y) to F(y) istaken to be a patent increase in downside risk for u.

Theorem 3. For distribution F(y) having patently more downside risk for u than F 0(y), one hasπv3 � πu3 if and only if φ′′′ � 0 everywhere.

Proof. All but the last step of the argument is the same as in Lemma 4, except, of course, thepremia are now called πu3 and πv3. Thus, skipping from the beginning of the chain of equalitiesto the end, we have

0 �b∫

a

v(y) dF (y) −b∫

a

v(y − πu3) dF 0(y) (4.7)

=b∫

a−πu3

φ′′(u(y))u′(y)

y∫a−πu3

u′(x)[F(x) − F 0(x + πu3)

]dx dy (4.8)

=b∫

a−πu3

−φ′′′(u(y))u′(y)

y∫a−πu3

u′(x) (4.9)

×x∫

a−πu3

u′(z)[F(z) − F 0(z + πu3)

]dzdx dy. (4.10)

Sufficiency and necessity now follow by the same type of arguments used in the proof ofLemma 4. �


5. Conclusions

Just as a risk-averse transformation of one utility function yields another that is more riskaverse, we show that a downside risk-averse transformation results in a utility function thatis more downside risk averse. Specifically, when the compensated approach to characterizinggreater risk aversion developed by Diamond and Stiglitz is extended from mean preservingspreads of the distribution of utility to mean-and-variance preserving spreads, we find that greaterdownside risk aversion is characterized by downside risk-averse transformations of utility. Whenthe alternative risk premium approach is used, we find that the differential risk premium for apatent increase in risk is greater after utility undergoes a risk-averse transformation, and in likemanner the differential downside risk premium for a patent increase in downside risk is greaterafter utility undergoes a downside risk-averse transformation. Thus, not only is the character-istic of being a downside risk-averse transformation an intuitive extension of the classical caseof a risk-averse transformation, but the two approaches used to identify greater risk aversion,when applied to downside risk, confirm that this characteristic is the correct indicator of greaterdownside risk aversion.

We also find that, while Keenan and Snow establish that an increase in the measure su =du − (3/2)r2

u is necessary and sufficient for a small increase in downside risk aversion, a highervalue for sv over su is sufficient, although not necessary, for v to exhibit greater downside riskaversion than u in the large. Thus, whereas the Arrow–Pratt measure of risk aversion r providesa global characterization of greater risk aversion, the same is not true of the measure s. Thefact that a higher value for s is necessary when the change in risk preference is small, however,indicates that no other measure fully characterizes greater downside risk aversion.

Acknowledgments

We are grateful for the helpful suggestions of an anonymous referee and of the editor.

References

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approach, J. Econ. Theory 136 (2007) 719–728.[5] D. Keenan, A. Snow, Greater downside risk aversion, J. Risk Uncertainty 24 (2002) 267–277.[6] M. Kimball, Precautionary saving in the small and in the large, Econometrica 58 (1990) 53–73.[7] M. Kimball, Standard risk aversion, Econometrica 61 (1993) 589–611.[8] H. Leland, Saving and uncertainty: The precautionary demand for saving, Quart. J. Econ. 82 (1968) 465–473.[9] C. Menezes, C. Geiss, J. Tressler, Increasing downside risk, Amer. Econ. Rev. 70 (1980) 921–932.

[10] S. Modica, M. Scarsini, A note on comparative downside risk aversion, J. Econ. Theory 122 (2005) 267–271.[11] J. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964) 122–136.[12] S. Ross, Some stronger measures of risk aversion in the small and the large with applications, Econometrica 49

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Greater downside risk aversion in the large