Academic journal article Journal of Risk and Insurance

Direction and Intensity of Risk Preference at the Third Order

Academic journal article Journal of Risk and Insurance

Direction and Intensity of Risk Preference at the Third Order

Article excerpt


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-Desgagne, 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 (Bramoulle 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 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 use Taylor series for small risks to show that 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.


It is well known that a negative second derivative for a utility function u(y) implies 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 [R.sub.u](y) [equivalent to] -u"(y)/u'(y) introduced 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. …

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