Self‐Insurance, Self‐Protection, and Saving: On Consumption Smoothing and Risk Management

• Author: Richard Peter,Annette Hofmann
• DOI: http://doi.org/10.1111/jori.12060
• Date: 01 September 2016
• Published date: 01 September 2016

©2015 The Journal of Risk and Insurance. Vol.83, No. 3, 719–734 (2016).

DOI: 10.1111/jori.12060

Self-Insurance, Self-Protection, and Saving:

On Consumption Smoothing and Risk

Management

Annette Hofmann

Richard Peter

Abstract

This article studies the effect of risk preferences on self-insurance and self-

protection in a two-period expected utility framework. Here the investment

to reduce risk precedes its effect. In contrast to single-period models, self-

insurance and self-protection reactsimilarly when the agent’s utility function

becomes more concave. Effort is increasedif and only if current consumption

is sufficiently large. However, if we introduceendogenous saving, an agent

with more concave utility always selects more self-insurance, but will select

more self-protection if and only if the probability of loss is small enough.

These latter results concur with those in standard monoperiodic models with

no saving.

Introduction

In order to mitigate risks, individuals may invest in either reducing the size of a poten-

tial loss (loss reduction or self-insurance) or reducing the probability of a hazardous

event (loss prevention or self-protection).1Dionne and Eeckhoudt (1985) derive the

well-known comparative static result that more risk-averse agents invest more in

Annette Hofmann is at the Hamburg School of Business Administration. Hofmann can be

contacted via e-mail: annette.hofmann@hsba.de. Richard Peter is at the Institute for Risk Man-

agement and Insurance, Ludwig-Maximilians-Universit¨

at Munich. Peter can be contacted via

e-mail: peter@bwl.lmu.de. We thank Harris Schlesinger for very detailed and helpful com-

ments. We also want to thank Louis Eeckhoudt for stimulating discussions about an earlier

version of this paper. Furthermore, we acknowledge helpful comments by two anonymous

referees. Annette Hofmann thanks the University of Hamburg where part of the paper was

written while she worked there as a post-doc, and in particular Martin Nell and the Institute

for Risk and Insurance for their support. Richard Peter would like to thank the Munich Risk and

Insurance Center and Andreas Richter for their support. Also, we would like to thank partici-

pants in the 2013 meeting of the Deutscher Vereinfür Versicherungswissenschaft, particularly

Johannes Jaspersen, and participants in the 2013 EGRIE Seminar in Paris, particularly Peter

Zweifel, for helpful discussions. Finally,we acknowledge valuable feedback by Randy Dumm

and Ty Leverty that led to significant improvements of the exposition of the paper.

1See Ehrlich and Becker (1972). We use the term “risk reduction” as a generic term.

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self-insurance but not necessarily more in self-protection.2They conclude that self-

insurance and self-protection affect risks quite differently, and research has tried to

clarify the ambiguous link between risk aversion and self-protection. For instance,

Jullien et al. (1999) derive a utility-dependent threshold so that, for all loss probabili-

ties beneath it, higher risk aversion is associated with more self-protection.Chiu (2000)

analyzes how prudence affects this threshold. Eeckhoudt and Gollier (2005) further

document prudence as an important determinant of self-protection decisions. They

show that under certain assumptions a risk-neutral agent invests less in self-protection

than a prudent agent.

Interestingly, the timing of prevention decisions has not been much studied in the

economic literature. However, in many instances it makes sense to think of risk re-

duction as expenditures incurred today to mitigate future risks. Indeed, the term

“prevention” literally suggests that the investment precedes the risk exposure that is

to be mitigated.3Using a two-period framework, Menegatti (2009) shows that under

suitable technical conditions a prudent agent invests more in self-protection than the

risk-neutral benchmark agent. This result is opposite to the one obtained by Eeckhoudt

and Gollier (2005) in a single-period world where prudence and self-protection are

negatively related. This shows that the intertemporal relationship between risk-

reduction expenditures and benefits is decisive for the economic analysis of risk

reduction.

This article reinvestigates the impact of the concavity of an agent’s utility function on

self-insurance and self-protection decisions when they are modeled in a two-period

sense. In such an intertemporal context, the curvature of utility measures the resis-

tance to consumption fluctuations both across states of nature and across time.4In this

sense we use the concept of risk aversion to mean the aversion against consumption

fluctuations in general. Wefind that an agent with more concave utility increases risk-

reducing effort if and only if current consumption exceeds an endogenously deter-

mined threshold, that is, if more risk reduction is “affordable.” This resultis obtained

2Briys and Schlesinger (1990) rationalize this finding by showing that investing in actuarially

fair self-insurance is a mean-preserving contraction, whereas the investment in actuarially

fair self-protection is neither a mean-preserving contraction nor a mean-preserving spread;

rather, it consists of both at differentlevels of the final wealth distribution, which explains the

ambiguous effect of an increase in risk aversion.

3Two-period frameworks as proposed by Menegatti (2009) receive more and more attention

in the study of risk-reduction decisions: Courbage and Rey (2012) and Eeckhoudt, Huang,

and Tzeng (2012) use it to analyze the impact of background risk on self-protection deci-

sions, Menegatti (2014) investigates the interaction of self-protection and cure for health risks,

and Courbage, Louberg´

e, and Peter (2013) consider optimal self-protection in the context of

multiple correlated risks.

4To overcome this identification, Kreps and Porteus (1978) and Selden (1978) suggest using

a pair of utility functions so that the first one measures the attitude toward consumption

fluctuations across dates and the second one the attitude toward consumption fluctuations

across states of the world. To achieve comparability with the classical models that have been

used to study risk reduction, we will employ the standard expected utility model. Wethink it

is important to solve this benchmark case first before applying other intertemporal setups.