DIRECTION AND INTENSITY OF RISK PREFERENCE AT THE THIRD ORDER

• DOI: http://doi.org/10.1111/jori.12232
• Date: 01 June 2018
• Published date: 01 June 2018
• Author: Arthur Snow,Donald C. Keenan

DIRECTION AND INTENSITY OF RISK PREFERENCE AT THE

THIRD ORDER

Donald C. Keenan

Arthur Snow

ABSTRACT

In expected utility theory, aversion to risk, greater aversion, and the desire to

substitute away from risk are each characterized by properties of the Arrow–

Pratt index of absolute risk aversion, with comparative statics implications

for such decisions as saving. At the third order, however, no single index

suffices. We contrast alternative indices of third-order risk preference and

show that the substitution effect of downside risk is governed by the

Schwarzian, and that where the degree of prudence governs the magnitude

of precautionary saving, the Schwarzian governs the effect of background

risk on the marginal rate of time preference.

INTRODUCTION

In expected utility theory, higher-order risk preferences, such as prudence at the third

order and temperance at the fourth, are becoming increasingly important in

establishing comparative statics predictions for behavioral responses to risk and in

framing empirical studies of the links between risk and risk preferences. Considerable

interest focuses on the third order, where the early results of Leland (1968) and

Sandmo (1970), tying prudence, or equivalently downside risk aversion, to the

precautionary motive for saving, are now augmented by more recent studies showing

that prudence implies precautionary self-protection in a temporal context (Wang and

Li, 2015) and revealing the importance of prudence for predicting the effect of risk on

the ranking of monitoring systems when randomized monitoring is practicable

(Fagart and Sinclair-Desgagn

e, 2007), on patience in bargaining (White, 2008), on

precautionary bidding in auctions (Kocher, Pahlke, and Trautmann, 2015), and on the

private supply of public goods (Bramoull

e and Treich, 2009; Eichner and Pethig,

2015), among many other applications.

In the terminology of Eeckhoudt (2012), the linking of prudence to precautionary

saving and precautionary self-protection is governed by the “direction” of third-order

Donald C. Keenan, Department of Economics and Management, Universit

edeCergy-

Pontoise & THEMA, Cergy-Pontoise, France 95011. Keenan can be contacted via e-mail:

dkeenan@uga.edu. Arthur Snow, Department of Economics, University o f Georgia,

Athens, GA 30602. Snow can be contacted via e-mail: snow@uga.edu.

© 2017 The Journal of Risk and Insurance. Vol. 85, No. 2, 355–378 (2018).

DOI: 10.1111/jori.12232

355

risk preference, as the precautionary motive hinges on a positive sign for the third

derivative of the von Neumann–Morgenstern utility function. In contrast, the

remaining predictions cited concern the “intensity” of third-order risk preference, as

they turn on the magnitude of the index of prudence introduced by Kimball (1990). In

this article we reexamine direction and intensity of risk preference at the third order,

and establish the Schwarzian index, introduced by Keenan and Snow (2002, 2012), as

a complementary measure of third-order preference intensity identified with

substitution effects of downside risk.

We introduce the various indicators of direction and intensity in the second

section, and in the third section, as a heuristic in the manner of Pratt (1964), we

useTaylorseriesforsmallriskstoshowthat each of these indices measures the

willingness to trade off a distinct pair of orders of risk. In the fourth section, we

summarize known results and apply them to establish that the substitution effect

of an increase in downside risk on the choice of an optimal control reduces the

degree of absolute downside risk aversion as measured by the Schwarzian. In the

fifth section, applications in the context of saving relate the Schwarzian to the

strength of the precautionary response to risk through the substitution effect of

downside risk and the response of the marginal rate of time preference to

background risk. Conclusions are offered in the sixth section.

INDICATORS OF DIRECTION AND INTENSITY

It is well known that a negative second derivative for a utility function uðyÞimplies a

positive willingness to pay for the elimination of risk about income y, while a positive

third derivative implies positive precautionary saving in response to a background

risk to future income. Eeckhoudt and Schlesinger (2006) equate these directional

attitudes with consistent preference for combining good with bad in simple 50:50

lotteries, providing a choice-theoretic foundation for identifying nth-degree risk

aversion with a negative (positive) sign for the nth even (odd) derivative of utility, as

suggested by Ekern (1980).

A measure of intensity in risk preference is, in contrast, described by Eeckhoudt

(2012) as one that indicates when, and the extent to which, one decision maker is more

averse to bearing risk than another. At the second order, the index of absolute risk

aversion RuðyÞu00ðyÞ=u0ðyÞintroduced by Arrow (1965) and Pratt (1964) serves

this purpose, providing a measure that yields transitive rankings of utility functions

and intuitive comparative statics predictions for decisions involving portfolio choice,

demand for insurance, and in many other contexts.

1

At the third order, Menezes, Giess, and Tressler (1980) show that dislike of any

increase in downside risk, defined as a mean-and-variance preserving spread (MVPS)

in the distribution for income, is unique to decision makers with u000 >0, thereby

1

Throughout, we assume that utility functions are strictly increasing, real-valued functions of

income y>0 and that transformations of utility are strictly increasing, real-valued functions

defined on a subset of the real numbers. We use primes to denote the first three derivatives of

uðyÞand denote the fourth derivative by uivðyÞ.

356 THE JOURNAL OF RISK AND INSURANCE