Discussion Papers In Economics
And Business

The Monotonicity of Asset Prices
with Changes in Risk
Masamitsu Ohnishi Yusuke Osaki

Discussion Paper 05-14

Graduate School of Economics andOsaka 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
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The Monotonicity of Asset Prices
with Changes in Risk∗
Masamitsu Ohnishi†‡
Graduate School of Economics, Osaka University
Daiwa Securities Chair, Graduate School of Economics, Kyoto UniversityYusuke Osaki§
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 inthe sense of first– and second–order stochastic dominances.
JEL Classification: D81, G12.
Keywords: Asset Price; Comparative Statics; First–order Stochastic Dominance; 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.
†Corresponding Author.
‡Addreess: Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toyonaka, Osaka 560–0043, Japan. E–mail: ohnishi@econ.osaka-u.ac.jp. Phone Number: +81–6–6850–5234. Fax Number: +81–6–6850–5277§E–mail: osakiyusuke@srv.econ.osaka-u.ac.jp

1Introduction

Many studies have examined the effects of changes in risk on optimal risk–takingbehavior. Rothschild and Stiglitz (1971) and Fishburn and Porter (1976) obtained
two counterintuitive results in their seminal papers. Although risk averters improvetheir utilities for first– and second–order changes in risk, these stochastic dominances only lead to ambiguous comparative static results of optimal decisions forrisk 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 endof the research for the former direction is made by Gollier (1995) and that for thelatter direction is made by Hadar and Seo (1990). It is natural to question howchanges 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. Hencethe 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 forfirst–order stochastic changes in risk and only prudence is required for second–orderstochastic changes in risk.2

2

Stochastic Dominance

As an introduction, we give the definition and properties of first– and second–orderstochastic dominances, and denote a Cumulative Distribution Function (CDF) over1

Gollier and Eeckhoudt (2000) provided a survey of this topic along these directions.
Kimball (1990) showed that prudence households have positive precautionary savings. Fromempirical observations, households save money for future uncertainty. This means that prudence
is justified from the descriptive viewpoints. See Kimball (1990) for further discussions.
21

a bounded support [a, b] as F (x) := P(x̃ ≤ x).
Definition 2.1.
• 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 ∈ [a, b]. We denote this as F (2) ≥FSD F (1);
• F (2) dominates F (1) in the sense of Second–order Stochastic Dominance
Rx
Rx
Rb(SSD) if a F (t; 2)dt ≤ a F (t; 1)dt holds for all x ∈ [a, b] and a F (t; 2)dt =RbF (t; 1)dt. We denote this as F (2) ≥SSD F (1).
aThe 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 inGollier (2001) for detailed discussions.
Theorem 2.1.
• F (2) ≥FSD F (1), if and only if E[g(x̃(2))] ≥ E[g(x̃(1))] for every increasing
function g.
• F (2) ≥SSD F (1), if and only if E[g(x̃(2))] ≥ E[g(x̃(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 atwo–date competitive and pure exchange economy with a representative investor.
The representative investor has an expected utility representation with a strictly increasing, strictly concave and sufficiently smooth von Neumann–Morgenstern utility
function (utility function) u, which means that all of required higher order derivatives are assumed to exist. The endowment of the investor is w units of a risk–freeasset and one unit of a risky asset. The risk–free asset is the numeraire and the grossrisk–free rate is normalized to one. The return on the risky asset is represented bya random variable x̃ with a CDF F defined over a bounded support [a, b]. The priceof the risky asset is denoted as q.
2

Following Gollier and Schlesinger (2002), the equilibrium asset price is given as
E[x̃u0 (z(x̃))]q=,E[u0 (z(x̃))](1)

where z(x) is the final wealth in equilibrium defined by z(x) := w + x

3.2

First–order Stochastic DominanceWe consider an economy i (= 1, 2) with a returen on a risky asset x̃i distributed
according to a CDF F (i), and suppose that F (2) dominates F (1) in the sense ofFSD: 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 x̃i (i = 1, 2) with a CDF F (i), and supposethat 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[x̃1 u0 (z(x̃1 ))] ≤E[x̃2 u0 (z(x̃2 ))]Proof. Since the proof is similar to those of Lemma 1 in Hadar and Seo (1990) andProposition 9 in Gollier (2001), we provide only an intuition of the proof. We find acondition 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 x̃i
distributed according to a CDF F (i), and suppose that F (2) dominates F (1) in thesense of FSD. If relative risk aversion is less than unity, then q1 ≤ q2 .
Proof. By the above lemma, we have thatE[x̃1 u0 (z(x̃1 ))] ≤ E[x̃2 u0 (z(x̃2 ))].

(2)

Since u0 (z(x)) is a decreasing function of x, we have thatE[u0 (z(x̃1 ))] ≥ E[u0 (z(x̃2 ))].

(3)

Combining Eqs. (2) and (3), we obtainq1 =E[x̃2 u0 (z(x̃2 ))]E[x̃1 u0 (z(x̃1 ))]≤= q2 .
E[u0 (z(x̃1 ))]E[u0 (z(x̃2 ))]

3(4)

3.3

Second–order Stochastic DominanceWe consider an economy i (= 1, 2) with a return on a risky asset x̃i 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 x̃i (i = 1, 2) with a CDF F (i) and supposethat F (2) dominates F (1) in the sense of SSD: F (1) ≤SSD F (2). If• absolute risk aversion defined by A(x) := −u00 (x)/u0 (x) is a decreasing functionof x, and relative risk aversion is less then unity and increasing function of x;
and/or• relative prudence defined by xP(x) := −xu000 (x)/u00 (x) is positive and less than2,then E[x̃1 u0 (z(x̃1 ))] ≤ E[x̃2 u0 (z(x̃2 ))].
Proof. Since the proof is similar to those of Lemma 1 in Hadar and Seo (1990) andProposition 9 in Gollier (2001), we provide only an intuition of the proof. We find acondition on preferences to guarantee that the function xu0 (x) is a concave functionof x.
We obtain the next proposition.
Proposition 3.2. Consider an economy i (= 1, 2) with a risky asset return x̃i
distributed according to a CDF F (i). Suppose that F (2) dominates F (1) in thesense of SSD and that the representative investor is prudent, i.e. u000 (x) ≥ 0. If• absolute risk aversion is a decreasing function, and relative risk aversion is lessthen unity and an increasing function; and/or• relative prudence is positive and less than 2,
then q1 ≤ q2 .

4

Proof. By the above lemma, we have thatE[x̃1 u0 (z(x̃1 ))] ≤ E[x̃2 u0 (z(x̃2 ))].

(5)

Since u0 (z(x)) is a convex function of x,E[u0 (z(x̃1 ))] ≥ E[u0 (z(x̃2 ))].

(6)

Combining Eqs. (5) and (6), we obtainq1 =4

E[x̃2 u0 (z(x̃2 ))]E[x̃1 u0 (z(x̃1 ))]≤= q2 .
E[u0 (z(x̃1 ))]E[u0 (z(x̃2 ))](7)

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 resultsof asset prices. Whereas their approach to this result is the restrictions on changes
in risk, our approach introduces the restrictions on preferences. Compared with theconditions on preferences to guarantee the unambiguous comparative static resultsof 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.

5References
Fishburn, P. and B. Porter, 1976, Optimal portfolios with one safe andone risky asset: Effects of changes in rate of return and risk, Management 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, Cambridge).
Gollier, C. and L. Eeckhoudt, 2000, The effects of changes in risk onrisk 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 distribution 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,

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