Discussion Papers In Economics
And Business

Dependent Background Risks
and Asset Prices
Yusuke Osaki
Discussion Paper 05-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDependent Background Risks
and Asset Prices
Yusuke Osaki
Discussion Paper 05-13

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Dependent Background Risks
and Asset Prices∗
Yusuke Osaki†
Graduate School of Economics, Osaka University

Abstract
Dependent background risks which have functional forms are introduced
into Lucas economies. This paper determines the conditions on preferences toguarantee the monotonicity of asset prices, when dependent background risks
satisfy the monotonicity and the single crossing conditions.
JEL Classification: D51, D81, G12.
Keywords: Asset Price, Dependent Background Risk, Monotonicity, Single
Crossing Condition.

∗

I am grateful to Marc Bremer, Masamitsu Ohnishi, and Katushige Sawaki for their advice andsuggestion. Needless to say, all remaining errors in this paper are mine.
†
Address: Graduate School of Economics, Osaka University, 1–7 Machikaneyama–Machi, Toyonaka, Osaka 560–0043, Japan. E–mail: OsakiYusuke@srv.econ.osaka-u.ac.jp. Phone Number: +81–6–6850–5234. Fax Number: +81–6–6850–5277

1Introduction

In recent years, many studies concerning the economic theory of decision making
under uncertainty have been analyzing situations with multiple sources of risk. In
typical settings, decision makers face two sources of risk, controllable risks and uncontrollable risks, which are mutually independent. Uncontrollable risks are usuallycalled background risks. However, independent background risks are controversial
matters from the descriptive viewpoint. Surely, we do not face independent, but dependent background risks in most situations with uncertainty. Actually, the equity
premium puzzle (Mehra and Prescott, 1985) cannot be explained by independent
background risks.1) Therefore, it is necessary to analyze situations with dependent
background risks. Yet, very few papers have investigated this topic. Motivated by
the state of the art, we examine the effect of dependent background risks on asset
prices in this paper. Of course, our analysis can be applied to other situations withdependent background risks. Since negatively dependent background risks have an
insurance device for risk averse investors, we have the following intuition: positively(negatively) dependent background risks make asset prices lower (higher). We confirm that our intuition is correct at least for a class of preference justified from a
descriptive viewpoint.
Since analyses of decision making under uncertainty with background risks are
based on independence assumption, dealing with dependent background risks encounters some difficulties. To overcome these difficulties, we must restrict dependent
background risks to have some specific forms. In this paper, dependent backgroundrisks have functional forms, that is, dependent background risks are represented asfunctions of marketable risks. An advantage of these forms is that situations withdependent background risks are changed to be ones with only one source of risks.
The monotonicity of asset prices is examined when the difference between dependent background risks satisfy the motonocity and the single crossing conditions.
The reasons for using the single crossing condition are the following: First, it iswell known that the single crossing condition plays an important role in analyses ofdecision making under uncertainty, for example, the monotone comparative statics
of decision making under uncertainty (Jewitt, 1987; Athey, 2002). Our paper canalso seem to be an application of monotone comparative statics under uncertainty.
Second, the analyses under other conditions satisfying the monotonicity condition
are given by a direct application of that under the single crossing condition. Finally,
the single crossing condition includes the case when dependent background risks
have the same means. The same mean is a natural criterion for the comparison. By
the covariance inequality (Wagener, 2003), the monotonicity condition can be ordered by its covariance with marketable risks. The purpose is achieved by using themethod of comparative statics based on the property of the risk–neutral probability.
Although this method is similar to work by Milgrom (1981), and Ohnishi and Osaki(2005), it is clear that our analysis is different from previous articles because these
are not concerned with background risks.
The rest of this paper is organized as follows. In Section 2, we derive the equilibrium asset price in a static version of a Lucas (1978) economy with a dependent1)

See section 9.3 in Gollier (2001) and references therein for details.

1background risk. In Section 3, we show the conditions on preferences to guarantee
the monotonicity of asset prices, when the difference between dependent backgroundrisks satisfy the monotonicity and the single crossing conditions. Finally, we provide
discussions on the results.

2

Asset Market

Let us consider a static version of a Lucas (1978) economy, that is, a two–datepure exchange economy with homogeneous investors. Every investor has an identical expected utility representation with a strictly increasing, strictly concave andsufficiently smooth von Neumann–Morgenstern utility function (utility function) u,
which means that all of the required higher order derivatives are assumed to exist.2)
One risk–free asset and one risky asset are traded in the asset market. The investor
is endowed with w units of the risk–free asset, and one unit of the risky asset. Let
us consider that the risk–free asset is the numeraire, and the gross risk–free rate
is normalized to one without loss of any generality. The return on the risky asset
at the final date is a random variable x̃ with a Cumulative Distribution Function(CDF) F (x) defined over a support [a, b]. By limited liability, the risky asset returnis typically defined over a positive reigion, a ≥ 0. The CDF F (x) is assumed to bedifferentiable, that is, the Probability Density Function (PDF) F 0 (x) := f (x) exists, where primes denote derivatives. The investor also faces a non–marketable risk,
which is dependent on the risky asset return, and is called the dependent backgroundrisk. The dependent relation between the marketable and dependent backgroundrisks is represented by a function ²(x), therefore the dependent background risk isgiven by the random variable ²̃ := ²(x̃). The function ²(x) is assumed so that the final wealth in equilibrium w+²(x)+x is an increasing function of x. This assumption
is sufficient for ²(x) to be an increasing function of x. An economic interpretation
is given as: By following the covariance inequality,3) this assumption means thatthe dependent background risk has positive covariance with the marketable risk.
Since the economy satisfies the one–fund separation theorem, the risky asset can beconsidered as the market portfolio. The dependent background risk can be considered as the variation of gross domestic product (GDP). Therefore the assumption
implies that the variation of GDP has positive covariance with the market portfolio.
Considering an actual economy, we can consider that the assumption is a naturalone.
The investor buys the portfolio α = (α0 , α1 ) to maximize her expected utility
from final wealth, where α0 and α1 stands for investments in the risk–free asset andrisky asset respectively. The asset prices are given by q = (1, q), where the risk–
free asset price is equal to one because of normalization and the risky asset price is

2)

We use the term “increasing” and “decreasing” in the weak sense.
The covariance inequality claims the following: if both g and h are increasing functions, thenE[g(x̃)h(x̃)] ≥ E[g(x̃)]E[h(x̃)] holds for all random variable x̃. See Wagener (2003) for the details
and applications of the covariance inequality.
3)2

denoted as q. The investor’s problem is given as follows:P

max E[u(α0 + ²(x̃) + α1 x̃)]{ }s.t. α0 + α1 q ≤ w + q(1)

Let us define the Lagrangean L(α, λ) := E[u(α0 + ²(x̃) + α1 x̃)] − λ(α0 + α1 q − w − q),
where λ is the Lagrange multiplier. Since the objective function is concave and theconstraint is linear, the first order conditions meet the necessary and sufficient conditions for the optimality. By the homogeneity of investors, a no–trade equilibrium
occurs. Then the demands for assets in equilibrium are qual to the endowments:
α0 = w, α1 = 1. The solutions of investor’s problem in equilibrium are given as
follows:∂L= E[u0 (z(x̃))] − λ = 0∂α0∂L= E[x̃u0 (z(x̃))] − λq = 0,
∂α1

(2)
(3)

where z(x) is the final wealth in equilibrium defined by z(x) := w + ²(x) + x, and isan increasing function of x. By Eqs. (2) and (3), the equilibrium asset price is given
as follows:q=E[x̃u0 (z(x̃))].
E[u0 (z(x̃))](4)

Let us define the function0u (z(x))f (x)fˆ(x; ²) :=,E[u0 (z(x̃))]x ∈ [a, b],(5)where ² represents the parameter of fˆ. As fˆ(x; ²) ≥ 0 for all x ∈ [a, b] andRbfˆ(t; ²)dt = 1, fˆ(x; ²) can be regarded as a PDF defined over the support [a, b]. Byataking the expectation with respect to the PDF fˆ, we can rewrite the equilibrium
asset price as
q = Ê[x̃],(6)where Ê denotes the expectation operator with respect to the PDF fˆ. Since equilibrium asset prices become equal to the expected values
R xof their returns under the
expectation with respect to the probability F̂ (x; ²) = a fˆ(t; ²)dt, the probabilityF̂ (²) is called the risk–neutral probability.

3

Main Results

In this section, we examine the effects of dependent background risks on asset prices.
Let us consider two different economies with dependent background risk ²1 (x) and²2 (x), say economy 1 and 2 respectively. First, we assume the monotonicity condition:²1 (x) − ²2 (x) is an increasing function of x.
(7)3

Applying the covariance inequality, the monotonicity condition (7) means thatcov(²1 (x̃) − ²2 (x̃), x̃) = cov(²1 (x̃), x̃) − cov(²2 (x̃), x̃) ≥ 0.

(8)

A condition satisfying the monotonicty condition (7) is the following:²1 (x) − ²2 (x) ≤ 0 ∀x ∈ [a, x0 );²1 (x) − ²2 (x) = 0 ∀x ∈ [x0 , x0 ];²1 (x) − ²2 (x) ≥ 0 ∀x ∈ (x0 , b].

(9)

The above condition is called the single crossing condition. We determine the condition on prferences to guarantee the monotonicity of asset prices under the monotonicity condition (7) and the single crossing condition (9). There are two reasons
why we are interested in the case of the single crossing condition. The first is thatanalyses under two other conditions satisfying the monotonicty condition are justdirict applications for that under the single crossing condition. The other conditions are: ²1 (x) − ²2 (x) ≤ 0, ∀x ∈ [a, b], and ²1 (x) − ²2 (x) ≥ 0, ∀x ∈ [a, b]. The
second reason is that the single crossing condition includes the interesting case,E[²1 (x̃)] = E[²2 (x̃)]. The same mean can be considered to be a natural criterion forthe comparison. Preparing for the analysis, we give some definitions and properties
for the risk aversion and stochastic dominance.
Definition 3.1. Absolute risk aversion of u at wealth x is defined by A(x) :=−u00 (x)/u0 (x). We call absolute risk aversion as simply risk aversion.
We call that utility functions u exhibit Increasing, Constant, and Decreasing
Absolute Risk Aversion (IARA, CARA, and DARA) respectively, if risk aversion isincreasing, constant, and decreasing functions of x. Note that IARA and DARA
include CARA as a special case.
Definition 3.2.
• F (2) dominates F (1) in the sense of Monotone Likelihood Ratio Dominance(MLRD) if f (y; 2)/f (y; 1) ≥ f (x; 2)/f (x; 1) holds for all x, y ∈ [a, b] such thatx ≤ y; We denote this as F (2) ≥MLRD F (1).
• F (2) dominates F (1) in the sense of First–order Stochastic Dominance (FSD)
if F (x; 1) ≥ F (x; 2) holds for all x ∈ [a, b]. We denote this as F (2) ≥FSD F (1).
The following properties are well known in the theory of stochastic dominance.
Hence we give the following theorem without proofs. The reader may refer to Gollier(2001) and Shaked and Shanthikumar (1994) for more details.
Theorem 3.1.
• If F (2) ≥MLRD F (1), then F (2) ≥FSD F (1);
• F (2) ≥FSD F (1), if and only if E[g(x̃(2))] ≥ E[g(x̃(1))] for every increasing
function g.
For the first step, we show the following lemma.
4Lemma 3.1. If utility function u displays DARA for x ≤ x0 and IARA for x0 ≤ x,then F̂ (²1 ) ≤MLRD F̂ (²2 ).
Proof. We have thatfˆ(xt ; ²2 )fˆ(xt ; ²1 )≥fˆ(xs ; ²2 )u0 (z2 (xt ))u0 (z2 (xs ))
⇔ 0
≥ 0
.
u (z1 (xt ))u (z1 (xs ))fˆ(xs ; ²1 )(10)Hence, we can replace the condition F̂ (²1 ) ≤MLRD F̂ (²2 ) with the right–hand sideof Eq. (10) for all xs , xt ∈ [a.b] such that xs ≤ xt . To shorten the proof, we consider
the case x0 = x0 = x0 .
(i) xs < xt < x0 :It follows by the straightforward calculation that
½µ 0
¶0 ¾
u (z2 (x))
sgn
= sgn{A(z1 (x))z10 (x) − A(z2 (x))z20 (x)}.
0u (z1 (x))(11)

Since z1 (x) ≤ z2 (x) and u displays DARA for all x ≤ x0 , we have thatA(z1 (x)) ≥ A(z2 (x)).

(12)

Since ²1 (x) − ²2 (x) is an increasing function of x, we have that
²01 (x) ≥ ²02 (x) ⇔ z10 (x) ≥ z20 (x).

(13)

Combining Eqs. (12) and (13), we obtain that A(z1 (x))z10 (x) − A(z2 (x))z20 (x) ≥ 0,that is u0 (z2 (x))/u0 (z1 (x)) is an increasing function. Hence,u0 (z2 (xt ))u0 (z2 (xs ))≥, for all xs ≤ xt ≤ x0 .
u0 (z1 (xt ))u0 (z1 (xs ))(14)(ii) xs < x0 < xt :Since u0 (x) is a decreasing function of x, z1 (xs ) ≤ z2 (xs ) means thatu0 (z2 (xt )),u0 (z1 (xt ))(15)

u0 (z2 (xt )).
u0 (z1 (xt ))(16)

u0 (z2 (xs ))u0 (z2 (xt ))≤, for all xs ≤ x0 ≤ xt .
u0 (z1 (xs ))u0 (z1 (xt ))(17)

u0 (z1 (xs )) ≥ u0 (z2 (xt )) ⇔ 1 ≥and z1 (xt ) ≥ z2 (xt ) means thatu0 (z1 (xs )) ≤ u0 (z2 (xt )) ⇔ 1 ≤Combining Eqs. (15) with (16), we obtain that(iii) x0 < xs < xt :We can obtain a similar proof in the manner as case (i) except for changing thesign, hence we omit the proof.
Through the above discussion, we obtain the right–hand side of Eq. (10) for all
xs , xt ∈ [a.b] such that xs ≤ xt . Hence, we complete the proof.
5

Recalling that MLRD is a sufficient condition for FSD, we can obtainq2 = Ê2 [x̃] ≥ Ê1 [x̃] = q1 ,(18)

if utility function u displays DARA for x ≤ x0 and IARA for x ≤ x. We summarize
the discussion with the following proposition:
Proposition 3.1. Let us consider economy 1 and 2 with dependent background risk
²1 and ²2 , and denote asset price by q1 and q2 . Suppose that ²1 (x) − ²2 (x) satisfies
the monotonicty condition (7) and the single crossing condition (9). If preferences
display DARA for x ≤ x0 and IARA for x0 ≤ x, then q1 ≤ q2 .
Let us consider the case where there does not exist a single crossing point. By
the above analysis, we have the following corollary.
Corollary 3.1. Let us consider economy 1 and 2 with dependent background risk
²1 and ²2 , and denote asset price by q1 and q2 . Suppose that ²1 (x) − ²2 (x) satisfies
the monotonicty condition (7) and the condition, ²1 (x) − ²2 (x) ≤ (≥) 0, ∀x ∈ [a, b].
If preferences display DARA (IARA), then q1 ≤ q2 .

4

Discussion

We provide two discussions: the first concerns the validity of the conditions on preferences, and the second is connected to the equity premium puzzle. The conditions
on preferences in Prop. 3.1 are that risk aversion is a decreasing function for low
wealth and an increasing function for high wealth. This condition has the validityfrom a descriptive viewpoint as suggested by Pratt (1964).4) Moreover, our results
are consistent with the following intuition for asset prices: ‘Risk averse investors havedesirability for consumption–smoothing. They decrease (increase) demands for assets with higher (lower) positively dependent background risks. As a result, higher(lower) positively dependent background risks decrease (increase) asset prices.’ On
the other hand, DARA is the least demanding condition for most problems withindependent background risks. Clearly, our conditions on preferences do not satisfy
DARA in the global sense. Through the above two opposite discussions, the validityof the conditions on preferences remains an open question. However, this is a difficult question because we have no consensus for the conditions on preferences fromempirical and/or experimental observations.
Second, an important implication of the result concernes the equity premium
puzzle (Mehra and Prescott, 1985). The equity premium puzzle means that theequity premium in a Lucas economy (1978) is too small compared to the observed
one. As a theoretical resolution, Weil (1992) modified the economy by adding an
independent background risk to a standard Lucas economy (1978) and concluded
that this tends to solve the puzzle under standard risk aversion which means thatboth risk aversion and prudence are decreasing functions. Here, (absolute) prudence
4)

Pratt (1964, pp.123) “And consideration of the yield and riskiness per investment dollar ofinvestor’s portfolios may suggest, at least in some context, description by utility functions forwhich r∗ (x) is first decreasing and then increasing.”6

is defined by P := −u000 /u00 . Through empirical studies, however, Lucas (1994) and
Telmer (1993) observed that this effect is too small to explain the puzzle. Our results
imply that the equity premium in an economy with both positively dependent andindependent background risks is higher than the premium in the economy consideredby Weil (1992) with an additional restriction on preferences that is CARA for highwealth.5) Here, the positively dependent background risk means an increasing function ²(x) with single crossing points at the x–axis. In other words, the puzzle can be
explained by the introduction of the positively dependent background risk. Recall
that background risk and risky asset can be considered as the variation of GDP
and the market portfolio respectively in an actual economy. Hence the positively
dependent background risk is a natural requirement from an empirical viewpoint.

5)

Recall that CARA is the negative exponential preference that means constant absolute prudence (CAP), and the DARA and decreasing absolute prudence include both CARA and CAP.

7

References
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