Discussion Papers In Economics
And BusinessComparative Risk Aversion under Background
Risks Revisited

Masamitsu Ohnishi and Yusuke Osaki
Discussion Paper 06-16

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANComparative Risk Aversion under Background
Risks Revisited

Masamitsu Ohnishi and Yusuke Osaki
Discussion Paper 06-16
June 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANComparative Risk Aversion under Background
Risks Revisited∗
Masamitsu Ohnishi1† and Yusuke Osaki1,2ohnishi@econ.osaka-u.ac.jp/osakiyusuke@srv.econ.osaka-u.ac.jp
1

Graduate School of Economics, Osaka University
2
JSPS Research Fellow

Abstract
This note determines a sufficient condition on (von Neumann–Morgenstern)utility functions to preserve (reserve) comparative risk aversion under generalbackground risks. Our condition is weaker than the one determined by Nachman (1982, Journal of Economic Theory). Nachman’s condition requires themonotonicity in the global sense, in other hand our condition only requires itin the local sense. And this generalization may make the condition on utility
functions to hold the desirable property consisitent with the recent empiricalobservation.
JEL Classification: D81.
Keywords: Background risk, comparative risk aversion, single crossing condition.

∗

The authors greatly acknowledge the financial support from the MEXT of the Japanese government (Ohnishi) and the JSPS Fellowships for Young Scientists (Osaki).
†
The corresponding author.
Addreess: Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toyonaka, Osaka 560–0043, Japan. Phone Number: +81–6–6850–5234. Fax Number: +81–6–6850–5277

1Introduction

It is doubtless that seminal papers by Pratt (1964) and Arrow (1971) are mostimportant contributions to decision analysis under expected utility theory. They
introduced a notion of risk aversion and an associated order of (von NeumannMorgenstern) utility functions. Comparative risk aversion, which is the order of risk
aversion, has unambiguous qualitative properties for many decision making problems such as asset demand, insurance demand and other many problems. While
comparative risk aversion has many intuitive properties for decision making undersingle source of a risk, it is well known that it has many counterintuitive ones for
decision making under multiple sources of risks. Since most decision making under
uncertainty involves multiple sources of risks, these properties have been taken special interests. Over the last two decades, a considerable number of studies have been
devoted to the investigations of conditions on utility functions to obtain intuitive
qualitative properties for decision making problems with multiple sources of risks.
Most decision making problems with multiple sources of risks deal with the situations in the presence of double sources of risks: One risks are endogenous and otherrisks are exogenous. Exogenous risks are usually called background risks. Two counterintuitive properties of risk aversion under background risks are well known: The
first is that comparative risk aversion under background risks may be reversed, andthe second is that risk aversion under background risks may decrease. We are concerned with the first property. Interseted readers can see Part IV in Gollier (2001)
for an excellent survey about researches concerning the second property. Nachman(1982) derived a condition on utility functions for the preservation of comparativerisk aversion under background risks with general payoff functions. Independently,
Kihlstrom, Romer, and Williams (1981) derived the same condition for backgroundrisks with additive payoff functions. Hence their result is a special case of Nachman(1982). And then, Pratt (1988) obtained a necessary and sufficient condition on
utility functions in the additive background risk case. Decreasing Absolute Risk
Aversion (DARA) is a sufficient condition for background risks with additive payoff
functions. DARA has been viewed as a relevant property for utility functions for along time. However, Jackwerth (2000) observed U-shaped risk aversion implied by
option prices written on index portfolios. DARA is not consistent with this empiricalobservation. On the other hand, we cannot imagine shapes of utility functions via
the condition derived by Pratt (1988) because his condition is a technical one. The
above things suggest that we need the further theoretical investigation of the conditions on utility functions to preserve comparative risk aversion under backgroundrisks, which have the possibility of the consistency with Jackwerth’s empirical observations (2000). We consider the background risks with general payoff functions,and obtain a weaker sufficient condition than Nachman’s one (1988). Our condition
for background risk with additive payoff functions requires only the monotonicity ofrisk aversion in a local sense, hence it may be consistent with Jackwerth’s empiricalobservation (2000).
The organization of the paper is as follows. In section 2, we give some preliminaries for the analysis. In section 3, we determine the condition on utility functions
to preserve (reserve) comparative risk aversion under background risks and compare1

our result to Nachman’s one. In section 4, we give the condition in the case that
payoff functions have the additive form, and discuss the relation between our theoretical result and the recent empirical observation. In section 5, our results are also
applied in the stochastic dominance changes in background risks. In last section,
we give concluding remarks.

2

Preliminaries

Since our setting is identical with Nachman (1982), we borrow his notation. Let
us consider a utility function u : R → R of a Decision Maker (DM). The utility
function u is a strictly increasing function, and higher order derivatives required in
the analysis assumed to exist. We note that the DM is not necessarily risk averter,that is concavity of the utility function is not required for the analysis. Let us
consider a payoff function g : X × Y → R. The payoff function g is a strictlyincreasing function of x. x ∈ X is a realization of a decision variable and y ∈ Y
is that of an exogenous variable. The exogenous risk ỹ called background risk, isa random variable followed by a probability measure m defined over support Y .
As considering financial markets with one risk–free asset and one risky asset as
an example, endogenous risks are market portfolios, and exogenous risks stand for
nontraded labor income risks (Weil; 1992).
Let us define the derived utility function asZv(x) :=u(g(x, s))m(ds).
(1)
Y

The derivatives of the derived utility function are written as follows:Z0
v (x) =
gx (x, y)u0 (g(x, y))m(dy),

(2)

Y

Z
00

{gxx (x, y)u0 (g(x, y)) + g(x, y)2 u0 (g(x, y))}m(dy),v (x) =

(3)

Y

where prime denotes derivatives, and gx and gxx denote the first and second partial
derivatives of g with respect to x. The derived utility function v is an strictlyincreasing function of x by the above assumptions, gx > 0 and u0 > 0.
Let us define the function
h(x, y) := gx (x, y)A(g(x, y); u) + A((x, y); g),(4)where A(u) := −u00 /u0 , and A(g) := −g 00 /g 0 . Recall that A(u) is the Arrow–Prattabsolute risk aversion of the utility function u (Pratt; 1964, and Arrow; 1971). Using
the function
gx (x, y)u(g(x, y))m(y),g (x, s)u(g(x, s))m(ds)Y xm̂(y) := R2(5)the Arrow–Pratt absolute risk aversion of the utility function u under the backgroundrisk, or equivalently that of the utility function v, can be rewritten asZv 00 (x)=A(x; v) = − 0h(x, s)m̂(ds).
(6)
v (x)
Y
Ry
We note that the function M̂ (y) :=m̂(s)ds can be viewed as the cumulative
distributionfunctiondefinedoversupportY , since m̂(y) > 0 for all y ∈ Y andRm̂(s)ds = 1. In the case the function h(x, y) = h(x) is a constant function forYy, it is clear that the Arrow–Pratt absolute risk aversion under background risk isconstant for all ỹ, that is A(x, v) = h(x) for all ỹ. This is a slight generalization of
Proposition 1 in Gollier and Schlesinger (2003).

3

Main Result

Pratt (1964) and Arrow (1971) introduced the notion of comparative risk aversion
defined as follows: u1 is more risk–averse than u2 , if A(x; u1 ) ≥ A(x; u2 ). We denote
this as u1 ≥A u2 . The goal of the paper is to give a sufficient condition to guarantee
comparative risk aversion under background risks, v1 ≥A v2 . Our sufficient condition
is weaker than Nachman’s one (1982), hence our result is a generalization of hisanalysis.

3.1

Theorem

Before giving a theorem, we define the notion of the single crossing condition: afunction f : X → R satisfies the single crossing condition at x0 from the above(below), if there exists x0 ∈ X such that f (x) ≥ (≤) 0 for all x ≤ x0 and f (x) ≤(≥) 0 for all
R x0 ≤ x. And, we define y0 ∈ Y as follows: There exists y0 such thath(x, y0 ) = Y h(x, s)m̂(ds)(= A(x; v)). The following theorem is our main result.
Theorem 3.1. Assume that g(x, y) is an increasing function of y for all x ∈ X. If
u1 ≥A (≤A ) u2 and either of functions hi (x, y) − h(x, y0 ) satisfies the single crossing
condition at y0 from the above (below), then v1 ≥A v2 .

3.2

Two Lemmas

For the preparation of the proof, we give the following two lemmas. A similar result
to the first lemma was obtained such as Osaki (2005), Ohnishi and Osaki (2006), andothers in various contexts. We note that Pratt (1988) obtained a similar result, andgave a different proof for the result of Kihlstrom, et al. (1981) using the relationof stochastic dominance. Before providing the first lemma, we define a notion ofstochastic dominance, the Monotone Likelihood Ratio Dominance (MLRD). For thesake of simplicity, we consider that two random variables have a same support X.
Definition 3.1. M (2) dominates M (1) in the sense of MLRD, if m(x; 2)/m(x; 1) isincreasing in x ∈ X. We denote this as M (1) ≤MLRD M (2).
Lemma 3.1. u1 ≥A (≤A ) u2 , if and only if M̂ (1) ≤MLRD (≥MLRD )M̂ (2).
3

Proof. Since the proof is similar, we give only the first case: u1 ≥A u2 , if and only
if M̂ (1) ≤MLRD M̂ (1).
It follows a straightforward calculation that¶µ∂ u02 (g(x, y))gy (x, y)(u0 (g(x, y))u002 (g(x, y)) − u001 (g(x, y))u02 (g(x, y))). (7)= 00∂y u1 (g(x, y)){u1 (g(x, y)}2 1
Since gy (x, y)/{u01 (g(x, y)}2 ≥ 0,
½ µ 0¶¾
∂ u2 (g(x, y))sgn= sgn{A(g(x, y); u1 ) − A(g(x, y); u2 )}.
∂y u01 (g(x, y))(8)

Therefore, A(g(x, y); u1 ) ≥ A(g(x, y); u2 ), if and only if u02 (g(x, y))/u01 (g(x, y)) is anincreasing function of y, or equivalently
u02 (g(x, y))u02 (g(x, z))≤u01 (g(x, y))u01 (g(x, z))(9)for all y, z ∈ Y with y ≤ z because of g(x, y) being an increasing function of y. On
the other hand, we have thatm̂(y; 2)m̂(z; 2)gx (x, y)u02 (g(x, y))m(y)gx (x, z)u02 (g(x, z))m(z)
≤
⇔
≤
(10)
m̂(y; 1)m̂(z; 1)gx (x, y)u01 (g(x, y))m(y)gx (x, z)u01 (g(x, z))m(z)u0 (g(x, z))u0 (g(x, y))
≤ 20
.
(11)
⇔ 02
u1 (g(x, y))u1 (g(x, z))Combining the above two discussions, we obtain the following:
A(g(x, y); u1 ) ≥ A(g(x, y); u2 ) ⇔m̂(y; 2)m̂(z; 2)≤∀ y, z ∈ Y with y ≤ z. (12)m̂(y; 1)m̂(z; 1)Hence we complete the proof.
The following lemma is known as the variation diminishing property. Hence
we give the following lemma without a proof. It was given by Karlin and Novikoff(1963), and Karlin (1968). Jewitt (1987) and Athey (2002) discussed some economic
applications. Before giving the lemma, we give the definition of TP2 function as
follows: h : X × Y → R is a TP2 function of x and y, if for x1 ≤ x2 and y1 ≤ y2 ,h(x1 , y2 )h(x2 , y1 ) ≤ h(x1 , y1 )h(x2 , y2 ).1)
Lemma 3.2. Let us consider that a function g : R → R satisfies the single crossing
condition at x0 from above (below), that is, there exists x0 ∈ X such that g(x) ≥(≤) 0 for x ≤ x0 and g(x) ≤ R(≥) 0 for x0 ≤ x. If a function
R h(x; i) : X → R is a
TP2 fuction for x and i, then X g(s)h(s; 1)dt = 0 implies X g(s)h(s; 2)dt ≤ (≥) 0.

1)

TP2 is also called log–supermodularity.

4

3.3

Proof

In this subsection, we give a proof of the Theorem 3.1. The proof is a similar way,hence we give only the proof for the first statement: If u1 ≥A u2 and either offunctions hi (x, y) − h(x, y0 ) satisfies the single crossing condition at x0 from theabove, then v1 ≥A v2 .
First, let us consider the case: h1 (x, y) − h(x, y0 ) satisfies the single crossing
condition at x0 from the above. By Lemma 3.1,
u1 ≥A u2 ⇔ M̂ (1) ≤MLRD M̂ (2).

(13)

Or equivalently, the probability measure m̂(y, i) is a TP2 function for y and i. We
have the following inequality:Z
0 ={h1 (x, y) − h1 (x, y0 )}m̂(dy; 1)
(14)
ZY
≥
{h1 (x, y) − h1 (x, y0 )}m̂(dy; 2)
(15)
Y
Z
≥
{h2 (x, y) − h1 (x, y0 )}m̂(dy; 2).
(16)
Y

The first inequality comes from the variation diminishing property (Lemma 3.2). It
follows that by a direct calculation h1 (x, y) − h2 (x, y) = gy (x, y){A(g(x, y); u1 ) −A(g(x, y); u2 ), hence A(g(x, y); u1 ) ≥ A(g(x, y); u2 ) is equivalent to h1 (x, y) ≥h2 (x, y). Therefore, we have the second inequality. The above inequality can be
rewritten asZ
0 ≥ {h2 (x, y) − h1 (x, y0 )}m̂(dy; 2) ⇔ A(x; v1 ) ≥ A(x, v2 ).
(17)
YSince the proof for the case of h2 is similar, we omit the proof. Hence we complete
the proof. ¤3.4

RemarkGiving a comment on the relation between our and Nachman’s analysis (1982),
we close this section. He determined another sufficient condition to guarantee thecomparative risk aversion under background risks. First, we review his analysis.
Nachman (1979) proved that if u1 ≥A u2 , then M̂ (2) dominates M̂ (1) in the senseof First–order Stochastic Dominance (FSD), M̂ (y; 2) ≤ M̂ (y; 1) for all y ∈ Y . It is
well known that the following two conditions are equivalent:• F (2) dominates F (1) in the sense of FSD;
• E[g(x̃1 )] ≤ E[g(x̃2 )] for every increasing function g(See e.g. Gollier; 2001, and Müller and Stoyan; 2002.). Applying the above condition, we have that if u1 ≥A u2 and either of hi (x, y) is a decreasing function of y,then v1 ≥A v2 .
5

Since functions satisfying the monotonicity condition imply ones satisfying thesingle crossing condition, the condition determined in this paper is weaker thanthat detrmined by Nachman (1982). The reason why our condition is weaker thanNachman’s one, is that our analysis is based on the stronger stochastic dominancethan his analysis, that is the MLRD is a stronger stochastic dominance than the
FSD. We summarize the above discussion into the table.
Table 1: A summary of the relation our and Nachman’s analysis

Nachman’s paper
relationour paper

4

stochastic dominance
FSD
⇑
MLRD

condition of hmonotonicity⇓
single crossing

Additive form

Over the past two decades, many studies have been concerned with the effectsof background risks with additive payoff functions, g(x, y) := x + y. Hence weinvestigate that which conditions on utility functions guarantee v1 ≥A v2 underbackground risks which have the additive form. When payoff functions have theadditive form, h(x, y) = A(x + y; u). Hence the Arrow–Pratt absolute risk aversionof the utility function u under additive background risks, that is, that of the utility
function v is given byZ
A(x; v) =A(x + s; u)m̂(ds).
(18)
Y

R
We define y0 ∈ Y as a similar way in the previous section: There exists y0 = Y A(x+s; u)m̂(ds)(= A(x; v)). We obtain the following corollary applied by Theorem 3.1.
Corollary 4.1. Assume that g(x, y) is an increasing function of y for all x ∈ X.
If u1 ≥A (≤A ) u2 and either of functions A(x + y; ui ) − A(x + y0 , ui ) satisfies the
single crossing condition at y0 from the above (below), then v1 ≥A v2 .
Any functions satisfying the monotonicity condition imply also ones satisfying
the single crossing condition. Hence Corollary 4.1 holds under the monotonicity
condition, and this result was obtained by Kihlstrom, et al. (1981). Corollary 4.1
can be viewed as a generalization of Kihlstrom, et al. (1981). And this generalization
is not only important from a technical viewpoint with respect to previous studies,but also from a empirical viewpoint. Jackwerth (2000) observed U-shaped risk
aversion. This empirical observation means that DARA has not any predictions
about conditions of utility functions to guarantee the desirable property, u1 ≤A
u2 ⇒ v1 ≤A v2 . On the other hand, when absolute risk aversion satisfies the single
crossing condition from the above, we do not require that absolute risk aversion isdecreasing in the global sense, in other words absolute risk aversion is only decreasing
6

in the neighborhood of the single crossing point. Corollary 4.1 may imply theexistence of utility functions such that• they preserve comparative risk aversion under additive background risks;• they are consistent with recent empirical observations, e.g. DARA for low
wealth and IARA for high wealth.

5

Stochastic Dominance

As another direction of studies concerning comparative risk aversion under background risks, Eeckhoudt, Gollier, and Schlesinger (1996, hereafter EGS) determined
conditions on utility functions to guarantee that first– and second–order deteriorations in background risks increase risk aversion. As we note that MLRD andcomparative risk aversion have the same property, that is both can be represented
by TP2 function, we can obtain the parallel result in comparative risk aversion with
MLRD changes in risk. Hence we only offer the result without a proof.
Theorem 5.1. Assume that g(x, y) is an increasing function of y for all x ∈ X. If
M (1) ≤MLRD (≥MLRD ) M (2) and h(x, y) − h(x, y0 ) satisfies the single crossing
condition at y0 from the above (below), then v1 ≥A v2 .
In the payoff functions being the additive form g(x, y) = x + y, EGS (1996)
claimed that DMs with DARA utility functions in the sense of Ross (1981) behave
more risk averse manner when background risks change in the sense of FSD. Sincecomparative risk aversion in the sense of Ross is stronger than that of Arrow–Pratt,
DARA in the Ross means DARA in the Arrow–Pratt. When payoff functions are
additive, a sufficient condition to preserve comparative risk aversion is that Arrow–
Pratt absolute risk aversion satisfies the single crossing condition from the above by
Theorem 5.1. And it is clear that this condition is a weaker condition than DARA
in the sense of Ross determined in EGS (1996). This observation is consistent withthe fact that the MLRD is a stronger notion of the FSD. We summarize the above
discussion into the table.
Table 2: A summary of the relation our and EGS analysis

EGS paper
relationour paper

6

stochastic dominance
FSD
⇑
MLRD

condition of hRoss DARA⇓
single crossing

Concluding Remarks

We determine a sufficient condition on utility functions to preserve (reserve) comparative risk aversion under background risks in this note. Nachman’s condition
7

(1982) requires the monotonicity in the global sense, on the other hand our condition only requires it in the local sense. This generalization does not only comes fromtheoretical importance, but also empirical one, because Jackwerth (2000) observed
U–shaped risk aversion in the asset market. Our condition has some predictions
concernign comparative risk aversion under additive background risks, in contrast
Nachman’s condition does not have them. We also give some comments on the effectof risk aversion on changes in background risks.

8

References
Arrow, K. J., 1971, Essays in the theory of risk bearing, Markham Publishing, Chicago.
Athey, S., 2002, Monotone comparative statics under uncertainty, Quarterly Journal of Economics, 117, 187–223.
Eeckhoudt, L., C. Gollier, and H. Schlesinger, 1996, Changes in background risk, and risk taking behaviour, Econometrica, 64, 683–690.
Gollier, C., 2001, The economics of risk and time. MIT Press, Cambridge.
Gollier, C. and H. Schlesinger, 2003, Preserving preference rankings under background risk, Economics Letter, 80, 337–341.
Jackwerth, J. C., 2000, Recovering risk aversion from option prices andrealized returns, Review of Financial Studies, 13, 433–451.
Jewitt, I., 1987, Risk aversion and the choice between risky prospects:
The preservation of comparative statics results, Reivew of Economic
Studies, 54, 73–85.
Karlin, S., 1968, Total positivity, Stanford Univercity Press, Stanford.
Karlin, S., and A. Novikoff, 1963 Generalized convex inequalities, Pacific
Journal of Mathematics, 13, 1251–1279.
Kihlstrom, R., D. Romer, and S. Williams, 1981, Risk aversion withrandom initial wealth. Econometrica, 49, 747–760.
Müller, A., and D. Stoyan, 2002, Comparison methods for stochastic
models and risks. John Wiley & Sons, New York.
Nachman, D. C., 1979, On the theory of risk aversion and the theory of
risk, Journal of Economic Theory, 21, 317–335.
Nachman, D. C., 1982, Preservation of “more risk averse” under expectations, Journal of Economic Theory, 28, 361–368.
Ohnishi, M., and Y. Osaki, 2006, The comparative statics of asset prices
based bull and bear market measure, European Journal of Operational Research, 168, 291–300.
Osaki, Y., 2005, Dependent background risks and asset prices, Economics Bulletin 4.8, 1–8.
Pratt, J. W., 1964, Risk aversion in the small and in the large, Econometrica, 32, 122–136.
Pratt, J. W., 1988, Aversion to one risk in the presence of others, Journalof Risk and Uncertainty, 1, 395–413.
Ross, S. A., 1981, Some stronger measures of risk aversion in the small
and large with applications, Econometrica, 49, 621–638.
Weil, P., 1992, Equilibrium asset prices with undivecifiable labor income
risk. Journal of Economic Dynamics and Control, 16, 769–790.
9