The goal of this paper is the examination of the conditions on preferences to guarantee the monotonicity of asset prices, when their returns change in the sense of first- and second-order stochastic dominances.Asset Price; Comparative Statics; First-order Stochastic Dominance; Second-order Stochastic Dominance.

Masamitsu Ohnishi

Yusuke Osaki

English

Research Papers in Economics

 
 
 
Discussion Papers In Economics 
And Business 
 













Graduate School of Economics and 
Osaka School of International Public Policy (OSIPP) 
Osaka University, Toyonaka, Osaka 560-0043, JAPAN

The Monotonicity of Asset Prices 
with Changes in Risk 
 
 
Masamitsu Ohnishi Yusuke Osaki 
 
 
Discussion Paper 05-14  
 
 
 
 
 
 
 
 
 
 
 
 
 

 
May 2005 

͜ͷݚڀ͸ʮେֶӃܦࡁֶݚڀՊɾܦࡁֶ෦ه೦ࣄۀʯ
جۚΑΓԉॿΛड͚ͨɺهͯ͠ײँ͢Δɻ
 
Graduate School of Economics and 
Osaka School of International Public Policy (OSIPP) 
Osaka University, Toyonaka, Osaka 560-0043, JAPAN 

The Monotonicity of Asset Prices 
with Changes in Risk 
 
 
Masamitsu Ohnishi Yusuke Osaki 
 
 
Discussion Paper 05-14 The Monotonicity of Asset Prices
with Changes in Risk¤
Masamitsu Ohnishiyz
Graduate School of Economics, Osaka University
Daiwa Securities Chair, Graduate School of Economics, Kyoto University
Yusuke Osakix
Graduate School of Economics, Osaka University
Abstract
The goal of this paper is the examination of the conditions on preferences
to guarantee the monotonicity of asset prices, when their returns change in
the sense of ﬁrst– and second–order stochastic dominances.
JEL Classiﬁcation: D81, G12.
Keywords: Asset Price; Comparative Statics; First–order Stochastic Domi-
nance; Second–order Stochastic Dominance.
¤The authous thank Marc Bremer and Katsushige Sawaki for their comments and suggestions.
Of course, all remaining errors in the paper are mine.
yCorresponding Author.
zAddreess: Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toy-
onaka, Osaka 560–0043, Japan. E–mail: ohnishi@econ.osaka-u.ac.jp. Phone Number: +81–6–
6850–5234. Fax Number: +81–6–6850–5277
xE–mail: osakiyusuke@srv.econ.osaka-u.ac.jp1 Introduction
Many studies have examined the eﬀects of changes in risk on optimal risk–taking
behavior. Rothschild and Stiglitz (1971) and Fishburn and Porter (1976) obtained
two counterintuitive results in their seminal papers. Although risk averters improve
their utilities for ﬁrst– and second–order changes in risk, these stochastic domi-
nances only lead to ambiguous comparative static results of optimal decisions for
risk averters. In order to resolve these results, the development of this topic has
followed two directions: restrictions on changes in risk and preferences.1 An end
of the research for the former direction is made by Gollier (1995) and that for the
latter direction is made by Hadar and Seo (1990). It is natural to question how
changes in risk aﬀect asset prices. Gollier and Schlesinger (2002) gave the result for
asset prices corresponding to the former direction. They determined the stochastic
dominance that is an equivalent condition for asset prices to be monotone. This
stochastic dominance cannot be compared with ﬁrst– and second–order stochastic
dominances. This means that asset prices do not necessarily have monotonicity when
their returns change in ﬁrst– and second–order stochastic dominance shifts. Hence
the following question araises: What conditions on preferences must be imposed
to guarantee monotone changes in asset prices for ﬁrst– and secon–order changes
in risk? This paper answers to this question. To obtain the answer, we only add
a natural condition from empirical and theoretical viewpoints to the conditions on
preferences for optimal risk–taking behavior: No additional condition is imposed for
ﬁrst–order stochastic changes in risk and only prudence is required for second–order
stochastic changes in risk.2
2 Stochastic Dominance
As an introduction, we give the deﬁnition and properties of ﬁrst– and second–order
stochastic dominances, and denote a Cumulative Distribution Function (CDF) over
1Gollier and Eeckhoudt (2000) provided a survey of this topic along these directions.
2Kimball (1990) showed that prudence households have positive precautionary savings. From
empirical observations, households save money for future uncertainty. This means that prudence
is justiﬁed from the descriptive viewpoints. See Kimball (1990) for further discussions.
1a bounded support [a;b] as F(x) := P(˜ x · x).
Deﬁnition 2.1.
² F(2) dominates F(1) in the sense of First–order Stochastic Dominance (FSD)
if F(x;2) · F(x;1) holds for all x 2 [a;b]. We denote this as F(2) ¸FSD F(1);
² F(2) dominates F(1) in the sense of Second–order Stochastic Dominance
(SSD) if
R x
a F(t;2)dt ·
R x
a F(t;1)dt holds for all x 2 [a;b] and
R b
a F(t;2)dt =
R b
a F(t;1)dt. We denote this as F(2) ¸SSD F(1).
The following properties are well known in the theory of stochastic dominance.
Hence we give the following theorem without proofs. Readers may refer to Ch. 3 in
Gollier (2001) for detailed discussions.
Theorem 2.1.
² F(2) ¸FSD F(1), if and only if E[g(˜ x(2))] ¸ E[g(˜ x(1))] for every increasing
function g.
² F(2) ¸SSD F(1), if and only if E[g(˜ x(2))] ¸ E[g(˜ x(1))] for every concave
function g.
3 Comparative Statics
3.1 Equilibrium Asset Price
Let us consider a static version of a Lucas (1978) economy. The economy is a
two–date competitive and pure exchange economy with a representative investor.
The representative investor has an expected utility representation with a strictly in-
creasing, strictly concave and suﬃciently smooth von Neumann–Morgenstern utility
function (utility function) u, which means that all of required higher order deriva-
tives are assumed to exist. The endowment of the investor is w units of a risk–free
asset and one unit of a risky asset. The risk–free asset is the numeraire and the gross
risk–free rate is normalized to one. The return on the risky asset is represented by
a random variable ˜ x with a CDF F deﬁned over a bounded support [a;b]. The price
of the risky asset is denoted as q.
2Following Gollier and Schlesinger (2002), the equilibrium asset price is given as
q =
E[˜ xu0(z(˜ x))]
E[u0(z(˜ x))]
; (1)
where z(x) is the ﬁnal wealth in equilibrium deﬁned by z(x) := w + x
3.2 First–order Stochastic Dominance
We consider an economy i (= 1;2) with a returen on a risky asset ˜ xi distributed
according to a CDF F(i), and suppose that F(2) dominates F(1) in the sense of
FSD: F(1) ·FSD F(2). In this subsection, we examine what conditions on utility
functions guarantee q1 · q2. First, we show the following lemma.
Lemma 3.1. Consider a random variable ˜ xi (i = 1;2) with a CDF F(i), and suppose
that F(2) dominates F(1) in the sense of FSD: F(1) ·FSD F(2). If relative risk
aversion deﬁned by R(x) := ¡xu00(x)=u0(x) is less than unity, then E[˜ x1u0(z(˜ x1))] ·
E[˜ x2u0(z(˜ x2))]
Proof. Since the proof is similar to those of Lemma 1 in Hadar and Seo (1990) and
Proposition 9 in Gollier (2001), we provide only an intuition of the proof. We ﬁnd a
condition on preference to guarantee that the function xu0(x) is increasing in x.
We obtain the next proposition.
Proposition 3.1. Consider an economy i (= 1;2) with a risky asset return ˜ xi
distributed according to a CDF F(i), and suppose that F(2) dominates F(1) in the
sense of FSD. If relative risk aversion is less than unity, then q1 · q2.
Proof. By the above lemma, we have that
E[˜ x1u
0(z(˜ x1))] · E[˜ x2u
0(z(˜ x2))]: (2)
Since u0(z(x)) is a decreasing function of x, we have that
E[u
0(z(˜ x1))] ¸ E[u
0(z(˜ x2))]: (3)
Combining Eqs. (2) and (3), we obtain
q1 =
E[˜ x1u0(z(˜ x1))]
E[u0(z(˜ x1))]
·
E[˜ x2u0(z(˜ x2))]
E[u0(z(˜ x2))]
= q2: (4)
33.3 Second–order Stochastic Dominance
We consider an economy i (= 1;2) with a return on a risky asset ˜ xi distributed
according to a CDF F(i), and suppose that F(2) dominates F(1) in the sense of
SSD: F(1) ·SSD F(2). In this subsection, we examine what conditions on utility
functions guarantee q1 · q2. The analysis is parallel to the previous subsection.
Lemma 3.2. Consider a random variable ˜ xi (i = 1;2) with a CDF F(i) and suppose
that F(2) dominates F(1) in the sense of SSD: F(1) ·SSD F(2). If
² absolute risk aversion deﬁned by A(x) := ¡u00(x)=u0(x) is a decreasing function
of x, and relative risk aversion is less then unity and increasing function of x;
and/or
² relative prudence deﬁned by xP(x) := ¡xu000(x)=u00(x) is positive and less than
2,
then E[˜ x1u0(z(˜ x1))] · E[˜ x2u0(z(˜ x2))].
Proof. Since the proof is similar to those of Lemma 1 in Hadar and Seo (1990) and
Proposition 9 in Gollier (2001), we provide only an intuition of the proof. We ﬁnd a
condition on preferences to guarantee that the function xu0(x) is a concave function
of x.
We obtain the next proposition.
Proposition 3.2. Consider an economy i (= 1;2) with a risky asset return ˜ xi
distributed according to a CDF F(i). Suppose that F(2) dominates F(1) in the
sense of SSD and that the representative investor is prudent, i.e. u000(x) ¸ 0. If
² absolute risk aversion is a decreasing function, and relative risk aversion is less
then unity and an increasing function; and/or
² relative prudence is positive and less than 2,
then q1 · q2.
4Proof. By the above lemma, we have that
E[˜ x1u
0(z(˜ x1))] · E[˜ x2u
0(z(˜ x2))]: (5)
Since u0(z(x)) is a convex function of x,
E[u
0(z(˜ x1))] ¸ E[u
0(z(˜ x2))]: (6)
Combining Eqs. (5) and (6), we obtain
q1 =
E[˜ x1u0(z(˜ x1))]
E[u0(z(˜ x1))]
·
E[˜ x2u0(z(˜ x2))]
E[u0(z(˜ x2))]
= q2: (7)
4 Concluding Remarks
We examine the conditions on preferences to guarantee the monotonicity of asset
prices, when their returns change in the sense of FSD and SSD. Our motivation
stems from the counterintuitive results obtained by Gollier and Schlesinger (2002):
the FSD and SSD changes in risk only yield ambiguous comparative static results
of asset prices. Whereas their approach to this result is the restrictions on changes
in risk, our approach introduces the restrictions on preferences. Compared with the
conditions on preferences to guarantee the unambiguous comparative static results
of optimal risk–taking behavior, an additional condition on preferences for asset
prices is prevalent from both empirical and theoretical viewpoints: risk aversion for
the FSD changes in risk and prudence for the SSD changes in risk.3
3Since risk aversion is necessary for the existence of optimal portfolio and equilibrium, it is not
explicitly appeared in conditions on preferences.
5References
Fishburn, P. and B. Porter, 1976, Optimal portfolios with one safe and
one risky asset: Eﬀects of changes in rate of return and risk, Man-
agement Science 22, 1064–1073.
Gollier, C., 1995, The comparative statics of changes in risk revisited,
Journal of Economic Theory 66, 522–536.
Gollier, C., 2001, The economics of risk and time (MIT Press, Cam-
bridge).
Gollier, C. and L. Eeckhoudt, 2000, The eﬀects of changes in risk on
risk taking: A survey, in: G. Dionne, eds., Handbook of Insurance
(Kluwer Academic Publishers, Boston) Ch. 4, 117–131.
Gollier, C. and H. Schlesinger, 2002, Changes in risk and asset prices,
Journal of Monetary Economics 49, 747–760.
Hardar, J. and T. K. Seo, 1990, The eﬀects of shifts in a return dis-
tribution on optimal portfolios, International Economic Review 31,
721–736.
Kimball, M. S., 1990, Precautionary savings in the small and in the large,
Econometrica 61, 53–73.
Lucas, R. E., 1978, Asset prices in an exchange economy, Econometrica
46, 1429–1446.
Rothschild, M. and J. Stiglitz, 1971, Increasing risk: II Its economic
consequences, Journal of Economic Theory 3, 66–84,
6

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