Risk Apportionment and Stochastic Dominance1

Louis Eeckhoudt2

Harris Schlesinger3

Ilia Tsetlin4

July 5, 2007

1 The authors thank Paul Kleindorfer, Claudio Mezzetti and Tim Van Zandt, as well

as seminar participants at the University of Paris - Dauphine, Paris School of Economics,
University of Verona and Université Louis Pasteur (Strasbourg) for helpful comments.
Ilia Tsetlin’s research was supported in part by the Center for Decision Making and Risk
Analysis at Insead.
2 Catholic Universities of Lille (France) and Mons (Belgium), and C.O.R.E.
3 University of Alabama (USA) and CESifo (Germany)
4 INSEAD (France and Singapore)Abstract

This paper characterizes higher order risk e¤ects, such as prudence and temperance,
via preferences that partially order a set of simple 50-50 lotteries.

In particular,eN ; YeN ; X
eM and YeM , and assume that X
ei dominates
consider the random variables X

Yei via ith -order stochastic dominance for i = M; N . We show that the 50-50 lotteryeN + YeM ; YeN + X
eM ] dominates the lottery [XeN + XeM ; YeN + YeM ] via (N +M )th -order[X

stochastic dominance. A preference ranking over these lotteries is shown to characterize higher orders of risk preference. We apply our results in several examples ofdecision making under risk.
Keywords: Downside Risk, Precautionary Savings, Prudence, Risk Apportionment,
Risk Aversion, Stochastic Dominance, Temperance
JEL classi…cation: D81

1

Introduction

Choice under uncertainty is often model speci…c. Much debate in the literature discusses the pros and cons of the various types of valuation methods used in modelingsuch decisions. The choice is made much easier when we have stochastic dominanceby one of the alternatives. In such a case, we will have agreement on the optimal
choice by a wide range of valuation methods. For example, suppose a corporation
believes its shareholders are all risk averse, de…ned as an aversion to mean-preserving
spreads.1 Its shareholders would then unanimously favor a decision yielding the rane over the alternative set of payo¤s Ye , whenever X
e dominates
dom payout variable X
Ye via second-order stochastic dominance (SSD). This holds true for any model ofpreferences that preserves a preference for SSD. For example, in an expected-utility
framework, this would hold whenever the utility function is increasing and concave.
The link between stochastic dominance and preferences within an expected-utility
1

See Rothschild and Stiglitz (1970).

1

framework is particularly well known.

If we restrict the utility function u to bee over Ye is equivalentdi¤erentiable, then N th -order stochastic dominance (NSD) of Xe over Ye by any individual whose utility exhibits certain
to unanimous preference of X
properties.
well.

This equivalence has implications for non-expected utility models as

For example, suppose that all individuals with expected-utility preferencese to Ye , then any non-expectedthat satisfy NSD preference unanimously prefer X
utility preference functional that satis…es NSD preference would also lead to thee over Ye .2
choice of X

This paper examines how concepts such as risk aversion, prudence and temper-

ance, as well as higher-order risk e¤ects, can be characterized by a simple lottery
preference based upon stochastic dominance rankings. In particular, consider theeN ; YeN ; X
eM and YeM , and assume that X
ei dominates
independent random variables X

Yei via ith -order stochastic dominance for i = M; N . We show that the 50-50 lotteryeN + YeM ; YeN + X
eM ] dominates the lottery [XeN + XeM ; YeN + YeM ] via (N + M )th [Xorder stochastic dominance. This ranking has implications for choice problems within
particular classes of valuation functionals over distribution functions. This characterization is shown to generalize the concept of risk apportionment, as introduced by
Eeckhoudt and Schlesinger (2006). We provide examples of such risk apportionment
2

A similar point was made by Zilcha and Chew (1990).

2

problems for distributions over wealth and over pro…ts.
The next two sections present our basic model and main results. The subsequent
section applies the main result to some choice problems involving risk apportionment.

Stochastic Dominance and N th Degree Risk

2

We start with a de…nition of stochastic dominance.3 Assume that all random variables have bounded supports contained within the interval [a; b]. Let F denote thecumulative distribution function for such a random variable. De…ne F (0) (x)and de…ne F (i) (x)Rxa

F (i 1) (t)dt for i

F (x)

1.

De…nition 1 We say that the distribution F weakly dominates the distribution G
in the sense of N th -order stochastic dominance if(i) F (N 1) (x)(ii) F (i) (b)G(N 1) (x) for all ax

G(i) (b) for i = 1; :::; N2:b

We write F NSD G to denote F dominates G via N th -order stochastic dominance.
e and Ye have cumulative distribution functions denoted by
If the random variables X
e NSD Ye .
F and G respectively, we will take the liberty to also say that X

For

N = 1; 2; 3, we will use the more common notations for …rst-, second-, and thirdorder stochastic dominance: F SD, SSD and T SD.
3

See, for example, Ingersoll (1987) and Jean (1980, 1984).

3

As a special case of stochastic dominance, Ekern (1980) considers the following:
De…nition 2 The distribution G has more N th -degree risk than the distribution F
if(i) F (N 1) (x)G(N 1) (x) for all ax(ii) F (i) (b) = G(i) (b) for i = 1; :::; N2:bNote that G has more N th -degree risk than F is equivalent to saying that F NSD
1 moments of F and G are identical.4

G and the …rst N

The following two theorems express the well-known links between stochastic dominance and expected utility, as well as Ekern’s extension of this result to increases
in N th degree risk.5 Here we let u(w) denote the individual’s utility function. Fornu(w)notational convenience, we use u(n) (w) to denote d dw
n .

Theorem 1 The following are equivalent:(i) F NSD G
(ii)

Rb

a

u(t)dFn = 1; :::; N .

Rbau(t)dG, for all functions u such that sgn u(n) (w) = ( 1)n+1 for

4

The second condition follows easily from part (ii) in the de…nition by integrating both F (i)and G(i) by parts. The case where N = 3 is labeled as an "increase in downside risk" and given
special attention in a paper by Menezes et al. (1980). The case where N = 4 is examined in partby Menezes and Wang (2005).
5
See Hadar and Russell (1969) and Hanoch and Levy (1969) who introduced this notion into theeconomics literature for SSD. See Jean (1980) and Whitmore (1989), as well as Ingersoll (1987) for
extensions to higher orders of stochastic dominance.

4

Theorem 2 The following are equivalent:(i) G has more N th degree risk than F
(ii)

3

Rb

u(t)dFaRbau(t)dG, for all functions u such that sgn u(N ) (w) = ( 1)N +1 .

Main Result

In this section, we examine a particular lottery preference over random wealth variables that can be ordered via stochastic dominance.

We show how this lottery

preference relates to higher order risk e¤ects, such as prudence and temperance. Ourcharacterization is a generalization of the concept "risk apportionment," as introduced by Eeckhoudt and Schlesinger (2006). In particular, their characterization isshown to be a special case of our results here.
Eeckhoudt and Schlesinger (2006) consider two basic "harms," namely a sureloss and a zero-mean risk. They start from an arbitrary (possibly random) initial
wealth level and consider two alternative 50-50 lotteries that will be added to theinitial wealth: the …rst consisting of a 50-50 chance of either one harm or the other
harm, and the second consisting of a 50-50 chance of adding either both harms
simultaneously or adding neither. An individual is de…ned as being "prudent" if heor she always would prefer the …rst of these two lotteries, which they label as a desire
to "disaggregate" these two harms. They then show how this de…nition, which they5

also label as "risk apportionment of order 3," is equivalent to the existing de…nitionof prudence as given by Kimball (1990).
Replacing the sure loss above with a second zero-mean risk that is statisticallyindependent from the …rst, they then show that a desire to disaggregate these twoharms (i.e., preference for a 50-50 lottery adding one risk or the other risk over a
50-50 lottery adding the sum of both risks or adding neither) is equivalent to thede…nition of "temperance" as described by Kimball (1992) and Gollier and Pratt(1996). This lottery preference is also labeled as "risk apportionment of order 4."
Proceeding recursively and allowing the 50-50 lotteries generated to be considered
the relative "harms," Eeckhoudt and Schlesinger characterize "risk apportionmentof order N:" Moreover, it is shown that preferences satisfy risk apportionment oforders 1; :::; N if and only if they also satisfy a preference for N th -order stochastic
dominance. In what follows, we essentially generalize the Eeckhoudt and Schlesinger
result to allow for the relative harms to be any random variables that can be ranked
via some order of stochastic dominance.
Let [A; B] denote a lottery that pays either A or B, each with probability one-half.
eN ; YeN ; X
eM and YeM , andConsider the mutually independent random variables X
ei dominates Yei via ith -order stochastic dominance for i = M; N . We
assume that X
eN + YeM ; YeN + X
eM ] and [XeN + XeM ; YeN + YeM ].
wish to compare the 50-50 lotteries [X6ei dominates Yei via ith -order stochastic dominance for i =Theorem 3 Suppose that X
eN + YeM ; YeN + X
eM ] dominates the lottery [XeN + XeM ; YeN + YeM ]M; N . The lottery [X

via (N + M )th -order stochastic dominance.

Proof. Let T be a positive integer and de…ne UT

fu j sgn u(n) (w) = ( 1)n+1 forn = 1; :::; T g. For an arbitrary function u 2 UN +M de…ne v(w)Eu(YeM + w)

eM +w), where E denotes the expectation operator. We …rst show that v 2 UN .
Eu(XTo see this, consider any integer k, 1

N . Observe that u(k) 2 UN +M kk

Now sgn v (k) (w) = sgn [Eu(k) (YeM + w)

UM .

eM + w)] = ( 1)k+1 . The second
Eu(k) (XeM MSD YeM . Thus, v 2 UN .
equality above follows since u(k) 2 UM and X

eN dominates YeN via N th -order stochastic dominance, toThe condition that XeN )gether with v 2 UN , implies that Ev(X

Ev(YeN ), which by the de…nition of v isequivalent to

eN + YeM )Eu(XeN + XeM )Eu(XEu(YeN + YeM )eM ):Eu(YeN + X(1)

Rearranging terms above, this inequality is equivalent to1
eN + YeM ) + Eu(YeN + X
eM )g
fEu(X
2

1
eN + XeM ) + Eu(YeN + YeM )g,fEu(X2which is precisely the lottery preference claimed in the theorem.
7(2)

The lottery preference expressed in Theorem 3 is analogous to the notion of"disaggregating the harms" discussed by Eeckhoudt and Schlesinger (2006), if wee random variables withinterpret the "harms" as sequentially replacing each of the X

eN + XeM .
a Ye random variable in the sum X

Or, said di¤erently, it expresses a

preference for lotteries that combine relatively good assets with relatively bad ones.
The preferences described here lead to a partial ordering of the four alternative
sums of random variables, based upon stochastic dominance criteria:

eN + XeM
X

ei + YejX

YeN + YeM for (i; j) 2 f(M; N ); (N; M )g.

(3)

eM + YeN and X
eN + YeM cannot be ordered via stochastic domiNote that the sums X

nance. In the spirit of Menezes and Wang (2005) we can refer to these two sums as
eN + XeM and YeN + YeM as the "outer risks." Theoremthe "inner risks" and the sums X

3 thus expresses a preference for a 50-50 lottery over the two inner risks as opposed
to a 50-50 lottery over the two outer risks.6
The following Corollary extends this result to Ekern’s ordering by N th -degreerisk. The proof follows easily from the proof of Theorem 3.

ei for i = M; N . Then
Corollary 4 Suppose that Yei has more ith -degree risk than X
6

A similar analogy is made by Eeckhoudt et al. (2007), who describe a lattice structure on the
ranking of the lottery components and subsequently de…ne a preference functional over the lattice
as being submodular.

8eN + XeM ; YeN + YeM ] has more (N + M )th -degree risk than the lottery
the lottery [XeN + YeM ; YeN + X
eM ].
[X

4

Applications

In this section we illustrate the applicability of our results. In particular, we …rst
demonstrate how Theorem 3 and its corollary can be used to gain insight into a few
extant concepts such as downside risk aversion and other higher-order e¤ects of risk
preferences. We then turn to two economic examples that directly apply our results
to decision making under risk.

4.1

Aversion to Downside Risk

We …rst use an illustration that is the result of an experiment by Mao (1970), andwas used by Menezes et al. (1980) to motivate the concept of aversion to downsiderisk (i.e. prudence). Consider the following two lotteries. Lottery A pays 1000 witha probability of 34 and pays 3000 with a probability of 14 . Lottery B pays zero witha probability of 41 and pays 2000 with a probability of 34 . Note that both lotteries
exhibit the same …rst two moments in their distributions.

9

Individuals who preferlottery A to lottery B exhibit "downside risk aversion."7This lottery preference follows from Corollary 4 by de…ning

eN
X

2000

YeN

1000

YeM

[ 1000; +1000], a 50-50 lottery.

eM
X

0

eN and YeM is a second-order increase
Clearly YeN is a …rst-order increase in risk over X

eM . It follows that A is the 50-50 lottery [XeN + YeM ; YeN + X
eM ] andin risk over X

eN + XeM ; YeN + YeM ]. Thus, from our Corollary, lottery B
B is the 50-50 lottery [Xdisplays more third-order risk, i.e. displays more downside risk, so that anyone who
is prudent (with u000 > 0 in an expected utility framework) would prefer lottery A.

4.2

Higher-order E¤ects of Risk

Within expected-utility models, the signs of the derivatives on the utility function
all have some economic meaning.

Temperance, u(4) < 0, is a fourth-order e¤ect.

7

A recent experimental paper by Baltussen et al. (2006) also supports this approach, thoughunknowingly. Although they interpret their experiments for Prospect Theory, they use mean
preserving spreads for "gains" and "losses," which we can reinterpret as "low wealth" and "highwealth" in our setting. We can interpret their results as showing a preference for attaching amean-preserving spread to the higher wealth level, which is the equivalent of aversion to downsiderisk in our setting.

10

Theorem 3 can be used to give two alternative equivalences to temperance.
Let Xa = Xb = 0 and let Yea and Yeb be independent zero-mean risks. Thus Xi

dominates Yei by SSD for i = a; b. Temperance implies that the 50-50 lottery [Xa +Yeb ; Xb +Yea ] is preferred to the lottery [Xa +Xb ; Yea +Yeb ]. Indeed, this lottery preference
de…ned "temperance" in Eeckhoudt and Schlesinger (2006). The individual prefers
a 50-50 gamble between Yea and Yeb over a 50-50 gamble between Yea + Yeb or neither.

By our Theorem 3, we know that this lottery preference is equivalent to temperance
ea ; Xeb ; Yeb ; Yea g with X
ei dominating Yei by SSD for i = a; b, not just thefor any fX
ea ; Xeb ; Yeb ; Yea g as described above.
particular set fX

We can obtain a second equivalence for temperance by letting Yea denote an in-ea (see Section 4.1 above), and letting X
eb FSD Yeb . The
crease in downside risk over X

eb helps to "temper" the e¤ects of the increased(stochastically) higher wealth in X

downside risk in Yea .

preference.

eb together with Yea in our lottery
Thus, we prefer to pair X

Lajeri (2004) studies the e¤ects of background risks on precautionary savings andin doing so, examines the condition of decreasing absolute temperance. A necessary
condition for this property is u(5) > 0, which she labels as "edginess." By choosing
M = 1 and N = 4, we can use Theorem 3 to interpret edginess as implying that a(stochastic) increase in wealth helps to temper the e¤ects of an increase in fourth-11

order risk. By choosing M = 2 and N = 3, we can alternatively interpret edginess as
implying that a decrease in risk (via SSD) helps to temper the e¤ects of an increasein downside risk.

4.3

Precautionary saving

Consider a simple two-period model of consumption and saving. An individual has
e at date t = 0 and income Ye at date t = 1. The individuala random income of X
decides to save some of her income at date t = 0 and to consume the rest.

Sheemust decide on how much to save before learning the realized value of X.

Thus,

e
her consumption at date t = 0 is Xs, where s is the amount saved. If s < 0, theconsumer is borrowing money (i.e. negative savings) and consuming more than thee at date t = 0. We assume that the interest rate for borrowing
realized value of X
or lending is zero and that there is no time-discounting for valuing consumption at
date t = 1. At this date the individual consumes her income plus any savings, Ye + s.

Let s denote the individual’s optimal choice for savings.

e dominates Ye via N th -order stochastic dominance.
Suppose that X

nonnegative scalar '0, since ' dominatesethat the 50-50 lottery [(XFor any' by FSD, it follows from Theorem 3

e + '); (Ye'); (Ye + ')] dominates [(X')] by (N+1)SD.

Reinterpreting the "50-50 lottery" [A; B] as sequential consumption of A at t = 0

12and B at t = 1, Theorem 3 implies that saving an arbitrary amount 'dominates saving the amount0 always', whenever preferences satisfy (N + 1)th -degree

stochastic dominance preference.

It follows that we must have s

0 for thisindividual.
For example, suppose that one’s income is risky in both periods, but it is riskier
in the sense of SSD next period.

Anyone with preferences satisfying third-orderstochastic dominance preference would prefer to save money rather than to borrowmoney.8

If instead of SSD we assume that next period’s income is stochasticallyhigher than this period’s via FSD, then any individual who is risk averse (i.e. satisfying SSD preference) will prefer to borrow rather than save.

4.4

After-tax pro…ts

e in country A and taxable
Consider a risk neutral corporation with taxable pro…t X
pro…t Ye in country B. We assume that the tax schedule is identical in both countries
e SSD Ye . The tax owed on realized pro…tand that X

is denoted by t( ). The tax

schedule is assumed to be increasing with a marginal tax rate that is also increasing,but at a decreasing rate.9 After tax pro…ts can thus be written as u( ) =t( ).

e is a constant equal to E Ye and that preferences are given by
For example, suppose that X
expected utility. Then this result coincides with that of Leland (1968) and Sandmo (1970), for thecase where preferences display prudence, u000 > 0.
9This assumption is realistic since the marginal rate is often bounded by some maximum, suchas …fty percent of additional pro…t.
8

13

If t( ) is di¤erentiable, our assumptions about t imply that u00 ( ) < 0 and u000 ( ) > 0.
Moreover, since we should also have t0 ( ) < 1 for any pro…t level , it follows thatu0 ( ) > 0 as well.
Suppose now that the corporation has a new project with a pre-tax distribution
e where Ze > 0 a.s. The corporation must decide whether to locate theof pro…t Z,

project in country A or in country B. The after-tax total pro…t of the corporation
is given by u( A ) + u( B ), where

i denotes the realized pre-tax pro…t in countryi, for i = A; B. Since Ze dominates zero by FSD, it follows from Theorem 3 thate + u(Ye + Z)]
e > E[u(Xe + Z)
e + u(Ye )], since the valuation function u (i.e. afterE[u(X)tax pro…ts) satis…es third-order stochastic dominance preference.

Thus, the …rmshould locate the new project in country B in order to maximize its global after-tax
pro…t.

5

Concluding Remarks

Hedging has long been a capstone of risk-management strategy.

Faced with an

initially risky wealth prospect, one can add a position in another asset whose payo¤
is negatively correlated to the original risky wealth. Such an asset will usually have apositive net payo¤ in states of the world in which the random wealth would otherwise
have been low. If we assume risk aversion, then marginal utility in low-wealth states14

is higher and therefore such a hedge is valuable to a risk averter. The cost of suchhedging is that the net payo¤ is usually negative in states of the world in which therandom wealth would otherwise have been high. In a certain sense, hedging pairs
up relatively good outcomes on one asset with relatively bad outcomes on anotherasset, and vice versa.
By contrast, risk apportionment pairs relatively good assets with relatively bad
ones. It assumes that the payo¤s on the assets are independent, which is antitheticalto the premise of hedging. Given the partial ordering in (3), we prefer a 50-50 gamble between the two "inner risks" as opposed to the two outer ones. Here relatively"good" and "bad" are determined via stochastic dominance rankings.

Under ex-pected utility, this ranking coincides with a utility function (or some other valuation
function) whose derivatives alternate in sign. If these rankings hold for all orders of
stochastic dominance, then they would coincide with preferences that are completely
monotone, as described by Brockett and Golden (1987) and Pratt and Zeckhauser(1987).
We provided several interpretations and applications of our main theoretical result, Theorem 3. Our results also can be used to add intuition to many other extant
concepts in the literature, such as skewness preference (see Chiu 2005) and transfer
principles in income redistribution (see Fishburn and Willig 1984 and Moyes 1999).

15

Moreover, our results can be useful in situations where preferences over the lotteries
are reversed. For example, a decision maker might prefer mean-preserving increases
in risk over the domain of losses, or might prefer increases in downside risk over thedomain of gains.10 The equivalences in our paper easily allow for such adaptation.
The lottery preference described in this paper generalizes the concept of "risk apportionment," as introduced by Eeckhoudt and Schlesinger (2006). One strength oftheir result is the simplicity of their characterization of lower levels of risk apportionment via a preference over simple 50-50 lotteries. However, their characterization isless simple for higher orders, since it requires that they construct larger and larger
nestings of simple lotteries. A strength of this paper is that we can characterize any
degree of risk apportionment by a preference between two simple 50-50 lotteries.

__________
10

For example, Wong (2007) essentially de…nes the …rst three orders of stochastic dominance overthe de-cumulative distribution function and refers to these respectively as "descending stochastic
dominance" of each order.

16

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