Allocative downside risk aversion

doi: 10.1111/j.1742-7363.2013.12019.x

Allocative downside risk aversion

Richard Watt∗ and Francisco J. Vazquez†

The literature on the intensity of downside risk aversion has been clear on the point that greaterprudence is not equivalent to greater downside risk aversion, although the two concepts arelinked. In the present paper, we present a new concept of the downside risk aversion of a decisionmaker, namely the fraction of a zero-mean risk that the decision maker would optimally place onthe upside. We then consider how this measure can be used to identify the intensity of downsiderisk aversion. Specifically, we show that greater downside risk aversion in our model can beaccurately measured by a relationship that is very similar to, although somewhat stronger than,greater prudence.

Key words risk aversion, prudence, downside risk

JEL classification D8

Accepted 8 May 2013

1 Introduction

Downside risk aversion (DSRA) has now become a standard inclusion in the literature of the studyof choice under risk. It is, therefore, natural that a measure of the intensity of DSRA be known. Thishas proved to be a rather more difficult task than would be expected, and more than one measurehas been suggested. Concretely, two measures of local DSRA can be found in the literature—theratio of third to first derivatives of utility (e.g., Modica and Scarsini 2005; Crainich and Eeckhoudt2007), and the “Schwartzian derivative" (e.g., Keenan and Snow 2002, 2009a,b).1 Globally, Keenanand Snow (2009b) show that the intensity of DSRA is found by a positive third derivative of thefunction that transforms one utility function (the less downside risk averse one) into another (themore downside risk averse one).

In the present paper, we reconsider the issue of a measure of downside risk aversion, and weprovide a new concept of how we can understand what DSRA is. While this new definition certainlydeparts from the traditional methodology of looking only for strict relationships between utility func-tions to characterize greater DSRA, given that the traditional methodology has been unable to cometo a consensus as to a unique measure, we argue that our alternative methodology is thereby justified.Then we consider some comparative statics of this new definition of DSRA, and above all we considerhow utility functions can be ranked according to the new definition of intensity of DSRA. Concretely,we find that intensity can be reliably measured by a concept that is very similar to prudence.

∗ Department of Economics and Finance, University of Canterbury, Christchurch, New Zealand. Email: [email protected]

† Department of Economic Analysis, Autonomous University of Madrid, Madrid, Spain.1 Although Menezes et al. (1980) identified prudence as aversion to downside risk, prudence cannot be shown to measure

the intensity of DSRA (e.g., Keenan and Snow 2010). In fact, since the ratio of third to first derivatives of utility is theproduct of prudence and risk aversion, the Crainich and Eeckhoudt measure uses prudence implicitly.

International Journal of Economic Theory 9 (2013) 267–277 © IAET 267

Allocative downside risk aversion Richard Watt and Francisco J. Vazquez

2 Traditional downside risk aversion

Traditionally, the literature on DSRA starts with a comparison of two lotteries, each of which containstwo independent risks. In both lotteries, the decision maker faces a first risk which is a one-half chanceof the loss of x2 and a one-half chance of a gain of x1, where xi ≥ 0, i = 1, 2, and xi > 0 for at leastone i = 1, 2.2 We denote this as the “x risk,” or the “primary risk.” Then, a second lottery, definedby a random variable y, which is characterized by having zero mean Ey = 0 and positive variance(we denote this second risk as the “y risk,” or the “secondary risk”), is placed either on the “upside”or the “downside” of the first risk. That is, the decision maker is asked to rank the two followingexpected utility measures:3

1. y is on the upside, giving expected utility of 12Eu(w + x1 + y) + 1

2u(w − x2);2. y is on the downside, giving expected utility of 1

2u(w + x1) + 12Eu(w − x2 + y).

It is not difficult to show that, conditional upon marginal utility being convex (i.e., u′′′ > 0), theexpected utility of having y on the upside is always greater than the expected utility of having it onthe downside (see Appendix I). That is,

u′′′ > 0 ⇒ 1

2Eu(w + x1 + y) + 1

2u(w − x2) >

1

2u(w + x1) + 1

2Eu(w − x2 + y). (1)

A decision maker displaying such preferences is then characterized as suffering DSRA.In order to measure the local intensity of DSRA in terms of the shape of the utility function u,

the traditional strategy has been to define a concept that is akin to a risk premium, such that theinequality in (1) is replaced by an equality. This is the methodology adopted by, for example, Crainichand Eeckhoudt (2007). Imagine that the secondary risk was initially placed on the downside, andthat the decision maker is compensated by a sure-thing payment of, say, m, in the good state suchthat the decision maker is indifferent to the lottery in which the zero-mean risk is placed on theupside. That is, m must satisfy

1

2Eu(w + x1 + y) + 1

2u(w − x2) = 1

2u(w + x1 + m) + 1

2Eu(w − x2 + y). (2)

Using a second-order Taylor expansion, Crainich and Eeckhoudt (2007) show that m can beexpressed as a function of u′′′

u′ , and therefore u′′′u′ can be taken as being a measure of the intensity of

DSRA, since the greater is u′′′u′ , the greater would have to be the upside compensation, m, for having

the risk located on the downside.4

2 In several studies, x1 is set at 0. This restriction is not really needed—all that is required is that the lottery has an upsideand a downside.

3 Here, u represents a strictly increasing and concave von Neumann–Morgenstern indirect utility function for wealth, andw is an amount of non-random initial wealth, which is assumed to be greater than or equal to 0.

4 There is, actually, a difficulty with the Crainich and Eeckhoudt approach. Concretely, although the left-hand side of (1)is greater than its right-hand side, and by introducing the compensation m we can increase the value of the right-handside, it is not clear that the equality that we seek can ever be achieved. That is, there is no guarantee that there actuallyexists an m that satisfies (2). In fact, it can be shown that it is generally true that there is no universal solution. This issue,however, will not be addressed in the current paper.

268 International Journal of Economic Theory 9 (2013) 267–277 © IAET

Richard Watt and Francisco J. Vazquez Allocative downside risk aversion

The standard methodology for measuring the intensity of DSRA has several shortcomings. First,it can be shown that the equation that is used as the foundation for the standard methodology, namelythe equation that defines the compensation to the decision maker for allocating the secondary riskon the upside, does not actually have a solution for all risks and all utility functions. Second, thestandard methodology begins with the assumption that the secondary risk can only be allocatedeither entirely on the upside or entirely on the downside of the primary risk. This is in stark contrastwith the general theory of risk bearing, in which risks are thought of as being spread over severalstates of nature. Third, since the standard methodology relies on Taylor’s approximations, it is onlyvalid for small enough risks. We would like to find a methodology that is valid for all risks. Fourth,the literature has yet to come to a general consensus that the standard methodology is able to give avalid measure of intensity of DSRA, with the measure of prudence, the Schwartzian derivative, andthe ratio of third to first derivatives of the utility function all having been championed as measuresof the intensity of DSRA in the literature. Given this, we would like to investigate a methodologythat is not based upon a Taylor’s expansion, or on a compensation equation. As we will see, thisdeparture from the standard methodology is guaranteed to supply a measurable concept of DSRAthat is independent of the size of the downside risk, and that certainly exists and that is uniquewithin that methodology, and that fits better within the general theory of risk bearing in termsof the assumption on how the secondary risk is spread over the states of nature of the primaryrisk.

3 A new DSRA methodology: allocative downside risk aversion

Imagine that, instead of looking for a simple preference of where to locate the secondary lottery asabove, we ask the following hypothetical question of our decision maker. Given the choice, whatfraction of the zero-mean risk would you like to locate on the upside of the primary risk? That is,we allow our decision maker to locate a fraction λ of y on the upside, and thus a fraction (1 − λ)on the downside, and then ask what the optimal value of λ is. Throughout, we make the traditionalassumptions that all derivatives of utility are monotone, and of course that utility is strictly increasingand concave.

To study this choice, we define the indirect utility function

U(λ) = 1

2Eu(w1 + λy) + 1

2Eu(w2 + (1 − λ)y)

= 1

2[Eu(w1 + λy) + Eu(w2 + (1 − λ)y)] ,

where w1 = w + x1 and w2 = w − x2, so that w1 > w2.The first-order condition for an optimal choice of λ is

U ′(λ∗) = 0 =⇒ 1

2

[Eu′(w1 + λ∗y)y − Eu′(w2 + (1 − λ∗)y)y

] = 0

The second-order condition, U ′′(λ∗) < 0, is satisfied by concavity of u. Note that the first-ordercondition can be expressed as

λ∗ ← Eu′(w1 + λ∗y)y − Eu′(w2 + (1 − λ∗)y)y = 0.



Define G(λ) ≡ Eu′(w1 + λy)y − Eu′(w2 + (1 − λ)y)y. By definition, we now have G(λ∗) = 0.Consider G(0), given by

G(0) = Eu′(w1)y − Eu′(w2 + y)y

= u′(w1)Ey − Eu′(w2 + y)y

= −Eu′(w2 + y)y,

where we have used the fact that Ey = 0.Now, from elementary statistics, we know that, for any two random variables a and b it is true

that cov(a, b) = Eab − EaEb. That is, Eab = cov(a, b) + EaEb. Given that, we know that

Eu′(w2 + y)y = cov(u′(w2 + y), y) + Eu′(w2 + y)Ey

= cov(u′(w2 + y), y),

where again we have used the fact that Ey = 0.Under concavity of u, it must be true that the greater is y, the smaller is u′(w2 + y), that

is, cov(u′(w2 + y), y) < 0. Finally, then, we note that G(0) = −Eu′(w2 + y)y = −cov(u′(w2 +y), y) > 0. So the slope of the first derivative of the indirect utility function at λ = 0 is strictlypositive.

Using an identical analysis, it is straight forward to show that G(1) < 0, that is, the slope of thefirst derivative of the indirect utility function at λ = 1 is strictly negative. Together with the strictconcavity of the indirect utility function, we now know that the following is true:

Lemma 1 0 < λ∗ < 1, that is, neither extreme is a solution to the problem.

Note that this result is due only to concavity of the utility function, not to convexity of marginalutility.

In and of itself, Lemma 1 is interesting. It says that moving all of the zero-mean risk to theupside of the primary risk is not actually an optimal allocation for the decision maker. Or in otherwords, a comparison between having the zero-mean risk on either the upside or the downside, as istraditionally done in the DSRA literature, is a comparison of two sub-optimal risk allocations.

In what follows, the following lemma will be useful on several occasions:

Lemma 2 Let h(w + λy) be an increasing (decreasing) function, with w > 0 and λ > 0, and let y bea zero-mean random variable. Then Eh(w + λy)y > 0 (< 0).

PROOF: We prove here the case of h increasing. For any y < 0 it is true that h(w + λy) < h(w), andfor any y > 0 it is true that h(w + λy) > h(w). But then

∀y < 0, h(w + λy)y > h(w)y,

∀y > 0, h(w + λy)y > h(w)y.

Thus, for all y, we have h(w + λy)y ≥ h(w)y, with equality only when y = 0. Given that, takeexpectations over y to get Eh(w + λy)y > h(w)Ey = 0. The case of h decreasing is proved in ananalogous manner. �



We now state and prove the following theorem:

Theorem 1 Convex marginal utility is both implied by and implies λ∗ > 12 ; that is, u′′′ > 0 ⇔ λ∗ > 1

2 .

PROOF: First, we prove that convex marginal utility implies λ∗ > 12 . We only need to prove that

G(

12

)> 0. That is, we need to show that Eu′(w1 + 1

2 y)y − Eu′(w2 + 12 y)y > 0, or that Eu′(w1 +

12 y)y > Eu′(w2 + 1

2 y)y. Consider how the function Eu′(w + 12 y)y changes with w. The derivative

with respect to w is Eu′′(w + 12 y)y. Since we are assuming that marginal utility is convex, we must

have u′′′ > 0, that is, u′′ is an increasing function. Thus, from Lemma 2, Eu′′(w + 12 y)y > 0, or

in short, Eu′(w + 12 y)y is an increasing function of w. Thus Eu′(w1 + 1

2 y)y > Eu′(w2 + 12 y)y, as

required.Second, we prove that λ∗ > 1

2 implies convex marginal utility. Since λ∗ is an optimal risk allo-cation, it must happen that λ∗ > 1

2 is strictly preferred to λ = 12 . Since expected utility is strictly

concave in λ, this directly implies that G(

12

)> 0, that is, Eu′(w1 + 1

2 y)y > Eu′(w2 + 12 y)y. But

then from Lemma 2, it is impossible for Eu′(w1 + 12 y)y > Eu′(w2 + 1

2 y)y to hold under eitherconstant or decreasing u′′.5 Thus it must be that marginal utility is indeed convex. �

This theorem provides us with our new definition of DSRA; if the optimal choice of risk allocationis to place more than half of the zero-mean risk on the upside of the other risk, then the decisionmaker displays DSRA in the sense that he has convex marginal utility. In order to differentiate fromthe traditional measure of DSRA, and to highlight the fact that our measure of DSRA is foundedupon an optimal risk allocation, we define “allocative” DSRA:

Definition 1 If λ∗ > 12 , the decision maker displays allocative DSRA.

It also seems natural to say that the greater is λ∗, that is, the more of the zero-mean risk thedecision maker would like to place on the upside, the greater is the intensity of allocative DSRA thatthis decision maker displays:

Definition 2 If λ∗A > λ∗

B > 12 , then decision maker A displays greater allocative DSRA than does

decision maker B.

A comment is in order. The main departure from the traditional methodology has been tocharacterize a specific utility feature (DSRA) by an optimal decision, rather than directly by lookingat curvatures of utility. This is akin to the statement that a positive demand for insurance under afair premium defines risk aversion. However, while for the case of risk aversion there is no need torely upon hypothetical decisions in order to classify risk aversion, since the Arrow–Pratt measure oflocal risk aversion has proved to be universally accepted, the same cannot be said for DSRA, whichhas resisted the adoption of a universally accepted measure based only on measures of curvature ofutility. Thus, we feel that for the case of DSRA it is justified to use the methodology above.

5 Recall that we are assuming that u′′ is monotone.



4 Comparative statics

In this section, we will discuss some comparative statics issues. First, we will show how λ∗ changeswith the parameters of the other risk, that is, x1 and x2. Second, we will consider the relationshipbetween λ∗ and the characteristics of the utility function. Throughout we retain the assumption thatmarginal utility is convex (i.e., u′′′ > 0) in order that the decision maker is indeed downside riskaverse.

Theorem 2 An increase in either x1 or x2 leads to a greater optimal value of λ.

PROOF: From the implicit function theorem, and the fact that the second derivative of the objectivefunction is negative from the second-order condition, the sign of ∂λ∗

∂x1is the same as the sign of

∂G∂x1

, where G(λ) ≡ Eu′(w + x1 + λ∗y)y − Eu′(w − x2 + (1 − λ∗)y)y. Carrying out the impliedderivative yields

∂G

∂x1= Eu′′(w + x1 + λ∗y)y

However, under the condition that marginal utility is convex, u′′ is an increasing function, and so,from Lemma 2, Eu′′(w + x1 + λ∗y)y > 0. Thus, we have ∂λ∗

∂x1> 0. The proof of ∂λ∗

∂x2> 0 proceeds

in an analogous manner. �

This theorem points to an interesting aspect of DSRA. When the primary risk (the x risk)increases,6 then the optimal allocative response is to place a greater part of the secondary risk (thatdefined by y) on the upside. That is, a riskier x lottery leads to greater allocative DSRA. This resultcan be related to the concept of temperance under a background risk (see Gollier and Pratt 1996),and it shows that allocative DSRA is aggravated by an increase in the primary risk.

Second, what happens to the optimal risk allocation as the size of risk-free wealth, w, increases? Asit happens, we can sign this effect only under a dubious assumption on the higher-order derivativesof utility. Concretely, we have the following result:

Theorem 3 If u′′′′(z) > 0, then λ∗ increases with w.

PROOF: From the implicit function theorem, and the first-order condition for λ∗, and recalling thatw1 = w + x1 > w2 = w − x2, we have

sign∂λ∗

∂w= sign

[Eu′′(w1 + λ∗y)y − Eu′′(w2 + (1 − λ∗)y)y

].

Define G(λ) ≡ Eu′′(w1 + λy)y − Eu′′(w2 + (1 − λ)y)y, so that the expression on the right-hand side is just G(λ∗). At λ = 1, we get G(1) = Eu′′(w1 + y)y > 0, where the sign is givenby Lemma 2 and the fact that u′′ is an increasing function. Second, at λ = 1

2 , we get G( 12 ) =

Eu′′(w1 + 12 y)y − Eu′′(w2 + 1

2 y)y. However, consider H(z) = Eu′′(z + 12 y)y. We have H ′(z) =

Eu′′′(z + 12 y)y. In order to sign this, we need to know if u′′′ is increasing or decreasing, and then we

can use Lemma 2. Concretely, if u′′′ is increasing (i.e., u′′′′ > 0), then H ′(z) = Eu′′′(z + 12 y)y > 0,

6 The x risk increases when x1 − (−x2) = x1 + x2 increases, which occurs if either x1 or x2 increases. Since the probabilityof each xi is set at one-half, it is not possible to alter the riskiness of the x lottery by changing its probabilities.



and G(z) would be an increasing function of z. In this case (since w1 > w2), we would haveEu′′(w1 + 1

2 y)y − Eu′′(w2 + 12 y)y > 0. Given that, in the case of u′′′′ > 0, it would be true that

G(λ) > 0 for all λ between 12 and 1, if G(λ) were a monotone function. But since G′(λ) =

Eu′′′(w1 + λy)y2 + Eu′′′(w2 + (1 − λ)y)y2 > 0, it turns out that G is indeed monotone (increas-ing, in fact), and so in this case we would have G(λ∗) > 0, and correspondingly ∂λ∗

∂w> 0. �

A similar result applies if we consider the effect of an increase in the size of the risk y. To do so,set y = ks, with k > 0. Then an increase in k corresponds to an increase in the size of the risk y.

Theorem 4 If u′′′′ > 0, then λ∗ increases with k.

PROOF: From the implicit function theorem, and the first-order condition for λ∗, we have

sign∂λ∗

∂k= sign

[Eu′′(w1 + λ∗ks)λ∗ks − Eu′′(w2 + (1 − λ∗)ks)(1 − λ∗)ks

].

Now define G(λ) ≡ Eu′′(w1 + λks)λks − Eu′′(w2 + (1 − λ)ks)(1 − λ)ks, so that the expressionon the right-hand side is just G(λ∗). At λ = 1, we get G(1) = Eu′′(w1 + ks)ks > 0, where thesign is given by Lemma 2 and the fact that u′′ is an increasing function. Second, at λ = 1

2 , weget G( 1

2 ) = Eu′′(w1 + 12ks) 1

2ks − Eu′′(w2 + 12ks) 1

2ks. Now consider H(z) = Eu′′(z + 12ks) 1

2ks.We have H ′(z) = Eu′′′(z + 1

2ks) 12ks. If u′′′ is increasing (i.e., u′′′′ > 0), then H ′(z) > 0, and G(z)

would be an increasing function of z. In this case (since w1 > w2), we would have G( 12 ) > 0. Given

that, in the case of u′′′′ > 0, it would be true that G(λ) > 0 for all λ between 12 and 1, if G(λ)

were a monotone function. But since G′(λ) = Eu′′′(w1 + λks)λ(ks)2 + Eu′′′(w2 + (1 − λ)y)(1 −λ)(ks)2 + Eu′′(w1 + λks)ks + Eu′′(w2 + (1 − λ)ks)ks > 0 (where again we have used the fact thatu′′ is an increasing function and Lemma 2) it turns out that G is indeed monotone (increasing, infact), and so in this case we would have G(λ∗) > 0, and correspondingly ∂λ∗

∂k> 0. �

The assumption needed to derive a crisp comparative static result for risk-free wealth and the sizeof the allocatable risk, that the fourth derivative of utility is positive, can be thought to be somewhatdubious. It is more common to see the assumption that the derivatives of utility alternate in sign,with the odd-numbered derivatives (first, third, etc.) being positive in sign, and the even-numberedderivatives (second, fourth, etc.) being negative in sign. Concretely, a positive fourth derivativeimplies that absolute prudence is not necessarily decreasing in wealth, but does not alter the morecommon assumptions of positive and decreasing absolute risk aversion.

Indeed, it is well known that in order to get an unambiguous wealth effect in ordinary problemsin which a single risk is to be undertaken, one needs to make a non-standard assumption on thethird derivative of utility—concretely that it is negative, which would in turn imply that the decisionmaker would not suffer DSRA. A positive fourth derivative, while certainly not the most comfortableof assumptions, is certainly far less obtrusive than non-standard assumptions on the third derivative.

Finally, we now consider the relationship between allocative DSRA and the shape of the utilityfunction u, in order to discuss the intensity of allocative DSRA. This is of importance, since in theliterature to date two measures of the intensity of DSRA have been proposed, so here we have theprincipal result of our paper. The first measure of local DSRA in the literature is the product of

risk aversion and prudence,(−u′′

u′) (

−u′′′u′′

)= u′′′

u′ , and the second is the Schwartzian derivative,

u′′′/u′ − (3/2)(u′′/u′)2 = (−u′′/u′)[−(u′′′/u′′) − (3/2)(−u′′/u′)]. Positive prudence, which sharesthe same sufficient condition as does positive DSRA, namely that marginal utility is convex, has also



been closely linked by some authors to behavior related to DSRA (e.g., Menezes et al. 1980; Kimball1990; Jindapon and Neilson 2007), but it is not true that greater prudence is equivalent to greaterDSRA in the traditional sense. Indeed, Keenan and Snow (2010) show that “a uniform increase inprudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicategreater DSRA, provided prudence is greater (less) than three times the degree of risk aversion.” Onthe other hand, an increase in u′′′

u′ alone (which, for example, happens if there is a uniform increasein both prudence and risk aversion, regardless of the relative sizes of these two measures) does leadto a greater willingness to accept the secondary risk to the downside rather than having it on theupside, as was shown by Crainich and Eeckhoudt (2007). In short, both of the accepted measuresindicate that we cannot say anything about traditional DSRA without implying the measures of bothabsolute risk aversion and absolute prudence.

For what follows, we need to define a couple of terms:

Definition 3 u2 is “strongly” more prudent than u1 if, for all w, we have that u′′′2 (w) ≥ u′′′

1 (w) and−u′′

2(w) ≤ −u′′1(w), with strict inequality for at least one of the two relationships. If both inequalities

are strict, then u2 is “strictly strongly” more prudent than u1.

We have the following result:

Theorem 5 If we have two utility functions, u1 and u2 such that u2 is strongly more prudent than u1,

then 12 < λ∗

1 < λ∗2 < 1.

PROOF: See Appendix II. �

Note that this theorem says nothing about how the two utility functions rank in terms of theEeckhoudt and Crainich measure of intensity of DSRA, or the Schwartzian derivative.7

5 Conclusions

In this paper, we have reconsidered the definition and the comparative statics of downside riskaversion. There has been considerable debate concerning the meaning and characterization of greaterDSRA. Here, rather than looking at willingness to pay or to accept (for which we find certaintheoretical difficulties regarding existence), we have modeled DSRA as an optimal allocation of asecondary, zero-mean, risk over the states of nature of a primary risk. Using this methodology, wefind a natural definition of DSRA as the choice of holding more than one-half of the secondary riskon the upside of the primary risk. We then interpret greater “allocative” DSRA as a greater share ofthe secondary risk that is held on the upside of the primary lottery.

Is this interpretation of DSRA superior to the more traditional methodology in which oneconsiders willingness to pay, or to accept, to locate all of a zero-mean risk on one side or the otherof a primary binomial risk? We believe so, since it is a general fact that risks can be allocated—overparties, over states of nature, etc. Indeed, the general theory of optimal risk bearing and insurance isfounded on such an assumption. Since the traditional DSRA methodology considers a sub-optimal

7 It is also interesting to conjecture that the Keenan and Snow (2009a) measure of the intensity of global DSRA (i.e., thatu2 is a downside risk averse transform of u1) implies that u2 is more allocative downside risk averse than u1. However,we have been unable to prove this result generally.



allocation of the allocatable risk, we find that our approach (which corrects for this sub-optimalallocation) does indeed have a good deal of theoretical logic attached.

Besides introducing and defining allocative DSRA, we have looked at some of the comparativestatics of allocative DSRA. We find that it is unambiguously aggravated by an increase in the size ofthe primary risk in a way that appears to be related to the concept of temperance. We also find thatallocative DSRA is increasing in wealth and in the size of the allocatable risk if the fourth derivative ofutility is positive. Finally, we define the concept of strongly greater prudence and prove that a strongincrease in prudence aggravates allocative DSRA.

Our research agenda on allocative DSRA is still rather populated. The present study has suggestedthat the results presented are only the starting point of what may turn out to be a promising researchpath. We wonder what sort of comparative statics effects can be proved under an assumption ofnegative fourth derivative of utility. We also remain interested in discovering the type of utilitytransformation that captures a strong increase in prudence.

Appendix I

Define

U(λ) = 1

2Eu(w + x1 + λy) + 1

2Eu(w − x2 + (1 − λ)y)

= 1

2

[Eu(w + x1 + λy) + Eu(w − x2 + (1 − λ)y)

].

We want to show that if marginal utility, u′, is convex, then U(0) < U(1).We have

2U(0) = Eu(w + x1) + Eu(w − x2 + y) = u(w + x1) + Eu(w − x2 + y),

2U(1) = Eu(w + x1 + y) + Eu(w − x2) = Eu(w + x1 + y) + u(w − x2).

Thus we need to show that

u(w + x1) + Eu(w − x2 + y) < Eu(w + x1 + y) + u(w − x2).

This rearranges as

Eu(w − x2 + y) − u(w − x2) < Eu(w + x1 + y) − u(w + x1)

or

Eu(w2 + y) − u(w2) < Eu(w1 + y) − u(w1),

where w1 = w + x1 and w2 = w − x2, so that w1 > w2.Now, define J(w) ≡ Eu(w + y) − u(w). What we need to show is that J(w1) > J(w2). The slope of J is J ′(w) =

Eu′(w + y) − u′(w). Now, if u′ is convex, then we know that Eu′(w + y) > u′(w + Ey). Since, for the case at hand, we haveEy = 0, this now reads Eu′(w + y) > u′(w). However, this says that the function J(w) is increasing, and so J(w1) > J(w2)as required.

Appendix II

In order to prove the result it is useful to rewrite the problem in terms of the density and distribution of the secondary risk.Concretely, the marginal utility of λ was given above as

U ′(λ) = 1

2Eu′(w + x1 + λy)y − 1

2Eu′(w − x2 + (1 − λ)y)y.



However, assuming that the density corresponding to y is f (y), and that this variable is defined on support (a, b), we get

U ′(λ) = 1

2

∫ b

a

u′(w + x1 + λy)yf (y)dy − 1

2

∫ b

a

u′(w − x2 + (1 − λ)y)yf (y)dy.

Integrating by parts, this becomes

U ′(λ) = − 1

2

∫ b

a

λu′′(w + x1 + λy)H(y)dy + 1

2

∫ b

a

(1 − λ)u′′(w − x2 + (1 − λ)y)H(y)dy, (3)

where H(y) =∫ y

asf (s)ds. Note that: (1) H(a) = H(b) = 0; (2) H ′(y) = yf (y), and so H ′(y) < 0 if y < 0, and H ′(y) > 0

if y > 0; (3) therefore, it follows that H(y) < 0 for y ∈ (a, b).Now, in order to consider how a change in the utility function u(x) affects the value of λ∗, we assume that utility depends

upon a shift variable α. That is, we now assume that, when wealth is z, rather than u(z) utility is u(α, z). In this way, a valueof α defines a family, or class, of utility functions (see Diamond and Stiglitz 1974). Given this, we now get λ = λ∗(α), and wecan consider how a change in α affects the optimal solution.

Using standard notation for derivatives, the optimal solution λ∗(α) is the solution to∫ b

a

ux(α, w + x1 + λy)yf (y)dy −∫ b

a

ux(α, w − x2 + (1 − λ)y)yf (y)dy = 0.

Taking the derivative with respect to α gives∫ b

a

uαx(α, w + x1 + λy)yf (y)dy +∫ b

a

uxx(α, w + x1 + λy)λαy2f (y)dy −∫ b

a

uαx(α, w − x2 + (1 − λ)y)yf (y)dy

+∫ b

a

uxx(α, w − x2 + (1 − λ)y)λαy2f (y)dy = 0.

Solving for λα = ∂λ∗∂α

, we find that

∂λ∗

∂α= −

∫ b

auαx(α, w + x1 + λy)yf (y)dy −

∫ b

auαx(α, w − x2 + (1 − λ)y)yf (y)dy∫ b

auxx(α, w + x1 + λy)y2f (y)dy +

∫ b

auxx(α, w − x2 + (1 − λ)y)y2f (y)dy

,

the denominator of which satisfies

D2(α, λ) =∫ b

a

[uxx(α, w + x1 + λy) + uxx(α, w − x2 + (1 − λ)y)]y2f (y)dy < 0.

Thus, the sign of ∂λ∗∂α

is the same as the sign of:

D1(α, λ) =∫ b

a

[uαx(α, w + x1 + λy) − uαx(α, w − x2 + (1 − λ)y)]yf (y)dy

= −∫ b

a

[λuαxx(α, w + x1 + λy)

−(1 − λ)uαxx(α, w − x2 + (1 − λ)y)]H(y)dy,

where the second line is found by deriving by parts. Now, using the second expression for D1(α, λ), we get (recall thatH(y) < 0, and of course w + x1 + 1

2 y > w − x2 + 12 y for all y)

D1

(α,

1

2

)= − 1

2

∫ b

a

[uαxx

(α, w + x1 + 1

2y

)− uαxx

(α, w − x2 + 1

2y

)]H(y)dy ≥ 0

if uαxxx ≥ 0. And using the first expression for D1(α, λ), we get

∂

∂λD1(α, λ) =

∫ b

a

[uαxx(α, w + x1 + λy) + uαxx(α, w − x2 + (1 − λ)y)]y2f (y)dy ≥ 0



if uαxx ≥ 0. Therefore, if uαxx ≥ 0 (a family of utility with uxx non-decreasing in α) and uαxxx ≥ 0 (a family of utilitywith uxxx non-decreasing in α) it turns out that D1(α, λ) is non-decreasing in λ. And since D1(α, 1

2 ) ≥ 0, we must have thatD1(α, λ) ≥ 0 for all λ > 1

2 . In particular, if at least one of the inequalities uαxx ≥ 0 and uαxxx ≥ 0 holds, then it follows thatD1(α, λ) > 0 for all λ > 1

2 . In such a case, we can conclude that

∂λ∗

∂α= −D1(α, λ∗)

D2(α, λ∗)> 0.

This proves that, under the assumptions of u′′ < 0 and u′′′ > 0, given changes in the utility function such that u′′′2 ≥ u′′′

1and u′′

1 ≤ u′′2 , with strict inequality for at least one of these relationships, then 1

2 < λ∗1 < λ∗

2 < 1. But this is what we havedefined as a strong increase in prudence.

References

Crainich, D., and L. Eeckhoudt (2007), “On the intensity of DSRA,” Journal of Risk and Uncertainty 36, 267–76.

Diamond, P., and J. Stiglitz (1974), “Increases in risk and in risk aversion,” Journal of Economic Theory 8, 337–60.

Gollier, C., and J. Pratt (1996), “Risk vulnerability and the tempering effect of background risk,” Econometrica 64, 1109–23.

Jindapon, P., and W. Neilson (2007), “Higher order generalizations of Arrow-Pratt and Ross risk aversion: A comparative

statics approach,” Journal of Economic Theory 136, 719–28.

Keenan, D., and A. Snow (2002), “Greater downside risk aversion,” Journal of Risk and Uncertainty 24, 267–77.

Keenan, D., and A. Snow (2009a), “Greater downside risk aversion in the large,” Journal of Economic Theory 144, 1092–101.

Keenan, D., and A. Snow (2009b), “The Schwartzian derivative as a ranking of downside risk aversion,” Working Paper,

University of Georgia.

Keenan, D., and A. Snow (2010), “Greater prudence and greater downside risk aversion,” Journal of Economic Theory 145,

2018–26.

Kimball, M. (1990), “Precautionary saving in the small and in the large,” Econometrica 58, 53–73.

Menezes, C., C. Geiss, and J. Tressler (1980), “Increasing downside risk,” American Economic Review 70, 921–32.

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Allocative downside risk aversion