Comparative risk aversion in two periods: An application to self‐insurance and self‐protection

• Published date: 01 March 2022
• Author: Tobias Huber
• Date: 01 March 2022
• DOI: http://doi.org/10.1111/jori.12353

Journal of Risk and Insurance. 2022;89:97–130. wileyonlinelibrary.com/journal/JORI

|

97

Received: 15 September 2019

|

Revised: 14 February 2021

|

Accepted: 29 May 2021

DOI: 10.1111/jori.12353

ORIGINAL ARTICLE

Comparative risk aversion in two periods:

An application to self‐insurance and

self‐protection

Tobias Huber

Institute for Risk Management and

Insurance, Munich Risk and Insurance

Center, Ludwig‐Maximilians‐Universität

München, Munich, Germany

Correspondence

Tobias Huber, Institute for Risk

Management and Insurance, Munich

Risk and Insurance Center, Ludwig‐

Maximilians‐Universität München,

Geschwister‐Scholl‐Platz 1, 80539

Munich, Germany.

Email: tobias.huber@lmu.de

Abstract

Risk management decisions provide a means to elicit

individuals' risk preferences empirically. In such a

context, the literature often presumes that the decision

to invest in risk management and the benefit of this

investment occur contemporaneously. There is, how-

ever, no consensus in the theoretical literature that

one‐period results can be transferred to intertemporal

settings. To address this gap, we study the effect of an

increase in risk aversion on the demand for risk

management in a two‐period context. Our findings

reproduce the one‐period results and, thus, support the

focus of previous empirical literature on the structure

of the risk rather than on the timing of investments

and benefits. We also contrast our results with those

obtained by employing widely used but limited pre-

ferences to examine risk aversion in intertemporal

settings (standard additive expected utility setting,

Selden, Epstein and Zin).

KEYWORDS

comparative risk aversion, saving, self‐insurance, self‐protection

JEL CLASSIFICATION

D61, D81, D91

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and

reproduction in any medium, provided the original work is properly cited.

© 2021 The Authors. Journal of Risk and Insurance published by Wiley Periodicals LLC on behalf of American Risk and Insurance

Association.

1|INTRODUCTION

The decision to invest in risk management involves the consideration of risk and uncertainty.

Individuals can prevent car accidents by driving carefully or by applying what is learned in a

defensive driving course, reduce the risk of a fire by installing sprinkler systems or lightning

conductors, and eat well and exercise regularly to reduce health risks. Companies invest in

cybersecurity to prevent successful cyberattacks, install burglar alarms to reduce the risk of

theft, and, in the case of farming, store extra water in the event of a drought. Moreover, both

individuals and companies obtain insurance to mitigate financial losses. Given the numerous

examples of risk management activities, how can we use observed behavior to draw inferences

about the underlying risk preferences of decision makers?

Mitigation decisions usually involve activities to reduce either the size or the probability of loss.

We follow Ehrlich and Becker (1972) and call the former self‐insurance and the latter self‐

protection.

1

While previous research supports to employ self‐insurance decisions to infer risk

preferences, it identifies intricacies to draw inferences from self‐protection activities. Theoretical

findings show that greater risk aversion raises optimal self‐insurance but has an ambiguous effect

on optimal self‐protection in one‐period models (Briys & Schlesinger, 1990; Dionne &

Eeckhoudt, 1985). In this setting, the investment in risk management and the induced risk re-

duction are assumed to occur contemporaneously, but, often, for example, when buying insurance

or when making safety investments, they occur temporally separated.

This paper employs a two‐period model to incorporate the time structure of investments

in risk management and their benefits. The objective is to study how optimal risk man-

agement changes with increased risk aversion. This comes with the conceptual challenge of

how to compare risk preferences in a two‐period model. Kihlstrom and Mirman (1974)

demonstrate that comparing agents in terms of their risk aversion requires keeping ordinal

preferences unchanged. Preferences also need to fulfill a monotonicity property to ensure

that agents do not prefer first‐order stochastically dominated lotteries (Bommier

et al., 2012,2017). Theoretical research, however, widely employs risk preferences that are

either not consistent with ordinal preferences (standard additive expected utility setting) or

not necessarily monotone (Epstein & Zin, 1989;Selden,1978). In contrast, our approach to

comparative risk aversion in intertemporal settings fulfills both requirements.

First, we find that increased risk aversion unambiguously raises optimal self‐insurance,

using a framework provided by Bommier et al. (2012), which works only with ordinal pre-

ferences and does not assume a particular representation of risk preferences. To study the

interaction between saving and self‐insurance, we further employ the preferences proposed by

Kihlstrom and Mirman (1974). In this setting, we demonstrate that a more risk‐averse agent

invests more in self‐insurance with and without endogenous saving. Second, we study self‐

protection decisions. While Bommier et al.'s approach remains silent about the effect of greater

risk aversion on self‐protection, Kihlstrom and Mirman's preferences enable us to draw con-

clusions about this effect. Both with and without endogenous saving, we show that, if the loss

probability is sufficiently small, a more risk‐averse agent invests more in self‐protection. We

further contrast our results with those obtained by employing widely used but limited

1

Insurance is, for example, a special case of self‐insurance and has similar comparative statics (Courbage et al., 2013).

Additionally, note that decision makers in Ehrlich and Becker's (1972) setting are assumed to have precise information

about the benefits of risk mitigation. In contrast, Li and Peter (2021) analyze self‐insurance and self‐protection activities

where the effectiveness of risk mitigation is subject to exogenous environmental factors.

98

|

HUBER

preferences to examine risk aversion in intertemporal settings (standard additive expected

utility setting, Selden, Epstein and Zin).

Our results contribute to both theoretical and empirical literature. Bommier et al. (2012)show

that standard additive expected utility settings as well as Selden's (1978) and Epstein and Zin's

(1989) preferences are not well ordered in terms of risk aversion. We explore how the lack thereof

affects optimal risk management under an increase in risk aversion. Without the consistency of

ordinal preferences (standard additive expected utility setting) or the lack of monotonicity (Selden,

Epstein and Zin), we find that two‐period results for risk management hinge on the consideration

of endogenous saving. Agents who optimize risk management and saving separately (employing

two mental accounts) would then behave differently than would agents, who optimize risk man-

agement and saving jointly (employing only one mental account). Using preferences that are well

ordered in terms of risk aversion, however, we reproduce the one‐period results in our two‐period

models independent of the consideration of endogenous saving. Our findings are thus good news

for empiricists. Empirical studies often ignore the intricacies of the time structure when analyzing

mitigation decisions and simply assume all costs and benefits to appear contemporaneously. They

thus make the implicit assumption that results of one‐period models can be transferred to inter-

temporal settings. Our analysis, therefore, supports the focus of the empirical literature on the

structure of the risk rather than on the timing of investments and benefits.

Risk management in two‐period settings has received only scant attention in the literature.

Menegatti (2009) was the first to study self‐protection in a two‐period setting, finding that

prudence is positively related to investments in self‐protection. By introducing endogenous

saving, Peter (2017) reproduces the one‐period result and explains his finding through the

substitution effect between self‐protection and saving (Menegatti & Rebessi, 2011). The paper

most closely related to ours is Hofmann and Peter (2016), who study self‐insurance and self‐

protection in a two‐period expected utility framework with and without saving. In the absence

of saving, the authors find that, if first‐period consumption is sufficiently high, a more concave

utility function raises optimal self‐insurance and optimal self‐protection. They show that, in the

presence of saving, if the loss probability is sufficiently low, higher concavity unambiguously

increases optimal self‐insurance and optimal self‐protection. Increasing the concavity of the

utility function, however, changes ordinal preferences in a standard additive expected utility

setting, which makes it impossible to study the effect of risk aversion on optimal self‐insurance

and optimal self‐protection. We address this gap and contribute to the literature by focusing on

the role of greater risk aversion on optimal risk management in isolation.

In Section 2, we first summarize Bommier et al.'s (2012) framework of comparative risk

aversion. We then examine the effect of greater risk aversion on self‐insurance and self‐

protection decisions, using monotone preferences in Section 3and employing nonmonotone

preferences in Section 4. We discuss our findings in view of the extant literature in Section 5.

The paper ends with concluding remarks and directions for future research in Section 6. All

proofs appear in Appendix A.

2|COMPARATIVE RISK AVERSION IN TWO PERIODS

2.1 |Overview

According to comparative Arrow‐Pratt risk aversion, an agent A is more risk‐averse than agent

B if A's utility function is more concave than B's (Arrow, 1963; Pratt, 1964). While greater

HUBER

|

99