Relative Risk Aversion as an Arc Elasticity

• Published date: 01 March 2010
• Date: 01 March 2010
• Author: Joseph G. Eisenhauer
• DOI: http://doi.org/10.1111/j.1540-6296.2010.01180.x

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Risk Management and Insurance Review, 2010, Vol.13, No. 1, 161-172

DOI: 10.1111/j.1540-6296.2010.01180.x

RELATIVE RISK AVERSION AS AN ARC ELASTICITY

Joseph G. Eisenhauer

ABSTRACT

Risk aversion is the central reason why individuals purchase insurance and

undertake other forms of risk management. But deriving the Pratt–Arrow coef-

ficient of relative risk aversion from a utility function requires familiarity with

differential calculus—a level of mathematics beyond the prerequisites for most

introductory risk management courses. Thus, students are not exposed to one

of the most important and fundamental concepts in the field unless and until

they take more advanced courses. The present article demonstrates that relative

risk aversion can be obtained as an arc elasticity using only elementary math-

ematics. This approach highlights the relationship between risk aversion and

the demand for insurance, and integrates concepts from the principles of eco-

nomics course, helping to unify the business curriculum. Numerical examples

are easily computed and graphed using electronic spreadsheets, providing stu-

dents with a hands-on learning experience. For sufficiently small risks, the arc

elasticity measure reduces to the Pratt–Arrow coefficient, providing a platform

for discussing the difference between large-scale and small-scale risk aversion

in upper-level courses.

INTRODUCTION

The concept of risk aversion is central to an understanding of behavior under uncer-

tainty. Decisions regarding gambling, investments, insurance, agricultural production,

crime, entrepreneurship, and other problems depend crucially on the extent to which

individuals are willing to bear risk. For more than four decades, the principal measure of

risk preference has been the Pratt–Arrow coefficient of relative risk aversion. This famil-

iar metric has been widely studied in the theoretical and empirical literatures, where it

has been invoked for both positive and normative purposes. Variations in risk aversion

across the population have been linked to differences in demographic characteristics,

and investigations continue into the magnitude of relative risk aversion and its response

to changes in wealth or income.

Although it is ubiquitous in the professional literature, relative risk aversion is largely

absent from most introductory textbooks on risk management and insurance (RMI).

The omission is easily justified: the Pratt–Arrow coefficient is the point elasticity of

Joseph G. Eisenhauer is Professor and Chair of Economics, 248 Rike Hall, Wright State

University, 3640 Colonel Glenn Highway, Dayton, OH 45435; phone: (937) 775-3070; e-mail:

Joseph.Eisenhauer@wright.edu. This article was subject to double-blind peer review.Two anony-

mous referees provided helpful comments on an earlier draft. Any errors are my own.

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