Risk‐Taking‐Neutral Background Risks

• Date: 01 June 2018
• Author: Richard C. Stapleton,Guenter Franke,Harris Schlesinger
• Published date: 01 June 2018
• DOI: http://doi.org/10.1111/jori.12235

©2018 The Journal of Risk and Insurance. Vol.85, No. 2, 335–353 (2018).

DOI: 10.1111/jori.12235

Risk-Taking-Neutral Background Risks

Guenter Franke

Harris Schlesinger

Richard C. Stapleton

Abstract

This article examines how decision making under uncertainty is affected by

the presence of a linearly dependent background risk, for individuals with

HARA utility. A linearly dependent background risk is a background risk

that increases linearly in the chosen tradable outcome. In order to do this, we

construct a parametric class of background risks that we label as risk-taking-

neutral (RTN). These background risks have the property that they will not

alter the decision made with respect to the market risk. As such, these RTN

background risks provide a benchmark. In many situations, a background

risk that is faced by an investor can be compared to one from the RTN class

in order to predict qualitative changes in the investor’s choice decision. As

this benchmark is easily available, it is convenient to use to predict these

changes.

Introduction

Consider an individual who must make an economic decision in the face of risk. For

example, the individual might be an investor deciding on how to allocate wealth

between a risky and a risk-free asset. Alternatively, the individual might be deciding

on how to insure or how to otherwise hedge a risky asset. Often, such a decision is

modeled in isolation, where there is only the one source of risk. For the sake of clarity,

we will refer to this risk as the “primary risk.” However, more realistically, there are

other risks that are also faced by the individual. One type of such risk is often referred

to as a “background risk,” meaning that there is no market for trading directly on this

second risk. One question that has been given considerable attention in the literature

is “how does the presence of this background risk affect behavior toward the primary

risk?” Obviously, many types of dependence between the risks might be indirectly

treated via trading on the primary risk. For example, contracts on the primary risk

Guenter Franke is at the University of Konstanz, Germany. Franke can be contacted via

e-mail: guenter.franke@uni-konstanz.de. Harris Schlesinger was at the University of Alabama

(deceased). Richard C. Stapleton is at the University of Manchester,United Kingdom. Stapleton

can be contacted via e-mail: richard.stapleton1@btinternet.com. Wethank seminar participants

at the University of Pennsylvania, Imperial College, and the University of Konstanz, as well as

Jens Jackwerth for helpful comments. We are also grateful to three referees for their valuable

comments.

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might partially mitigate effects of the backgroundrisk via “cross-hedging” techniques.

However, even in cases wheresuch techniques are not possible, such as when the two

risks are independent, it is now well known that a background risk can still affect

decisions about the treatment of the primary risk.1

For example, one might hypothesize that an independent background risk with a non-

positive mean will lead to less risk taking with respect to the primary risk. However,

as shown by Gollier and Pratt (1996), risk aversion alone is not sufficient to guarantee

such behavior. Preferences that do guarantee such behavior are labeled “risk vul-

nerable” by Gollier and Pratt, who also derive the complex necessary and sufficient

conditions that lead to such behavior. Luckily, relevant sufficient conditions aremuch

easier to satisfy. Our analysis provides a novel intuition for risk vulnerability.

A number of articles look at more general settings of background risk. Franke,

Schlesinger, and Stapleton (2006) consider a multiplicative background risk such that

the tradable outcome is multiplied by an independent risk factor, and they derive

conditions for multiplicative risk vulnerability. Tsetlin and Winkler (2005) analyze

a dependent additive and a dependent multiplicative background risk and derive

sufficient conditions under which a risky project is desirable or undesirable in the

presence of such correlated background risks. Dana and Scarsini (2007) analyze the

effects of stochastic dependence between an insurable and an uninsurable risk on effi-

cient insurance contracts. Malevergne and Rey (2009) generalize risk vulnerability to

cross-risk vulnerability that captures the impact of nonfinancial background risks on

financial risk taking. They derive the conditions for additive resp. multiplicative risk

vulnerability as special cases. Franke, Schlesinger, and Stapleton (2011) also consider

an additive and a multiplicative background risk for a constant relative risk averse

agent and ask under what conditions the multiplicative background risk reinforces

or weakens the effect of an additive background risk on risk taking.

In this article, we compare risk taking in the presence of background risk to risk

taking in the absence of background risk. To simplify this task, we first construct a

parametric class of background risks for an individual with the property that they

will not affect the individual’s decision making on the primary risk. In particular, we

construct this class of background risks, which we label “risk-taking-neutral (RTN)

background risk,” for expected-utility preferences with so-called hyperbolic absolute

risk aversion (HARA). Second, we compare the actual background risk to an RTN

background risk to infer the impact of the actual background risk on the individual’s

risk taking. The closure of the HARA class includes most of the commonly used utility

functions, including constant absolute risk aversion (CARA) and constant relative risk

aversion (CRRA). The HARA class is important in finance as it is precisely the class

1This line of research began with Kihlstrom, Romer, and Williams (1981), Ross (1981), and

Nachman (1982). These articles all consider an individual who is more risk averse than another

one with respect to the primary risk and examine whether the introduction of background risk

preserves comparative risk aversion. Doherty and Schlesinger (1983) show how such a back-

ground risk might affect an individual’s decision toward the primary risk. Good summaries

of how such background risks embed into economic and financial decisions can be found in

Gollier (2001) and Campbell and Viceira (2002).