Skewness Preference, Risk Taking and Expected Utility Maximisation

The Geneva Risk and Insurance Review, 2010, 35, (108–129) r 2010 The International Association for the Study of Insurance Economics 1554-964X/10 www.palgrave-journals.com/grir/ Skewness Preference, Risk Taking and Expected Utility Maximisation W. Henry Chiu Economics, School of Social Sciences, University of Manchester, Manchester, M13 9PL, U.K. Available empirical evidence suggests that skewness preference plays an important role in understanding asset pricing and gambling. This paper establishes a skewness-comparability condition on probability distributions that is necessary and sufficient for any decision-maker’s preferences over the distributions to depend on their means, variances, and third moments only. Under the condition, an Expected Utility maximizer’s preferences for a larger mean, a smaller variance, and a larger third moment are shown to parallel, respectively, his preferences for a first-degree stochastic dominant improvement, a mean-preserving contraction, and a downside risk decrease and are characterized in terms of the von Neumann-Morgenstern utility function in exactly the same way. By showing that all Bernoulli distributions are mutually skewness comparable, we further show that in the wide range of economic models where these distributions are used individuals’ decisions under risk can be understood as trade-offs between mean, variance, and skewness. Our results on skewness-inducing transformations of random variables can also be applied to analyze the effects of progressive tax reforms on the incentive to make risky investments. The Geneva Risk and Insurance Review (2010) 35, 108–129. doi:10.1057/grir.2009.9; published online 23 March 2010 Keywords: skewness preference; risk aversion; downside risk; moment; gambling Introduction Do individual decision-makers, other things being equal, prefer a more positively skewed distribution? There is a substantial and growing body of empirical evidence suggesting that they do. Building on the earlier seminal contributions of Arditti (1967) and Kraus and Litzenberger (1976), Harvey and Siddique (2000),1 for example, show in an asset pricing model that systematic skewness is economically important and commands a substantial premium. Studying the data from horse race betting and from state lotteries (in the U.S.), respectively, Golec and Tamarkin (1998) and Garrett and Sobel (1999) find 1 See also the references therein for a sample of other related empirical work. W. Henry Chiu Skewness Preference, Risk Taking and Expected Utility Maximisation 109 evidence supporting the contention that gamblers are not necessarily risk lovers but skewness lovers. So far, however, skewness preference has no firm choice theoretic foundation. Skewness has been treated as synonymous with the (unstandardized) third central moment but it is well-known that preference for a larger third moment is in general not consistent with Expected Utility (EU) maximisation unless the utility function is cubic. As a result, in studies of skewness preference to date, either a cubic utility function is assumed2 or a cubic Taylor approximation of the EU is taken (i.e., the utility function is approximated by a Taylor series truncated to three terms before taking expectations). The limitations of these approaches are obvious. A truncated Taylor series, for instance, can be a reasonable approximation only for small risks.3 Menezes et al. (1980) come closest to establish a formal linkage between skewness preference and EU maximisation by showing that a distribution having more ‘‘downside risk’’ implies, but is not implied by, its (unstandardized) third moment being smaller, and that downside risk aversion is characterized by a von Neumann-Morgenstern (VNM) utility function with a positive third derivative. In the statistics literature, Van Zwet (1964) defines a distribution F to be more positively skewed than G if R(x)F 1(G(x)) is convex and it has become widely accepted that a good skewness measure should preserve the skewness ordering so defined (see, for example, Oja (1981) and Arnold and Groeneveld (1995)). Oja (1981) proposes a condition in terms of the number of crossings of two standardized distribution functions that relaxes Van Zwet’s (1964) skewness-comparability condition. The preferences of EU maximizing decision-makers over skewness-comparable distributions as defined by these authors, on the other hand, have not been explored and characterized. This paper establishes a skewness-comparability condition on probability distributions that is necessary and sufficient for any decision-maker’s preferences over the distributions to depend on their means, variances, and third moments only. Under the condition, a EU maximizer’s preferences for a larger mean, smaller variance, and a larger third moment are shown to parallel, respectively, his preferences for a first-degree stochastic dominant (FSD) improvement, a mean-preserving contraction (MPC), and a downside risk decrease and are characterized in terms of the VNM utility function in exactly the same way. The condition generalizes not just the skewness-comparability conditions proposed by Van Zwet (1964) and Oja (1981) but also the condition 2 3 Hanoch and Levy (1970) is an early example of using the cubic utility function in portfolio choice theory. Other pitfalls of these approaches are discussed in the text. The Geneva Risk and Insurance Review 110 for two distributions to be comparable in terms of downside risk defined by Menezes et al. (1980). Furthermore, distributions satisfying the ‘‘location-scale’’ or ‘‘linear class’’ condition of Meyer (1987) and Sinn (1983), which they show to be sufficient for the consistency between the mean-variance analysis and EU maximisation, are shown to be skewness-comparable distributions with identical standardized third moments. By showing that all Bernoulli distributions are mutually skewness comparable, we further show that in the wide range of economic models where these distributions are used individuals’ decisions under risk can be understood as trade-offs between mean, variance, and skewness. Our basic characterizations also immediately imply that a concave transformation of a random variable reduces the skewness of the distribution and hence, other things being equal, the attractiveness of the distribution to a skewness-preferring decision-maker. An application of this general regularity addresses the issue of whether a progressive tax reform reduces the incentive to take risks. The rest of the paper is organized as follows. Skewness comparability and expected utility maximisation section sets out the basic definitions and main results on skewness comparability. Skewness of the Bernoulli distributions section establishes the skewness comparability of the widely used Bernoulli distributions and examines its implications. Comparison with the existing approach and implications for gambling and tax reforms section concludes with discu (...truncated)