Beyond Risk Aversion: Why, How and What's Next?

The Geneva Risk and Insurance Review, 2012, 37, (141–155) r 2012 The International Association for the Study of Insurance Economics 1554-964X/12 www.palgrave-journals.com/grir/ EGRIE Keynote Address Beyond Risk Aversion: Why, How and What’s Next?n Louis Eeckhoudta,b a Iéseg School of Management, 3 rue de la Digue, Lille 59000, France. CORE, 34 Voie du Roman Pays, Louvain-la-Neuve 1348, Belgique. E-mail: b Risk attitudes other than risk aversion (e.g. prudence and temperance) are becoming important both in theoretical and empirical work. While the literature has mainly focused its attention on the intensity of such risk attitudes (e.g. the concepts of absolute prudence and absolute temperance), I consider here an alternative approach related to the direction of these attitudes (i.e. the sign of the successive derivatives of the utility function). The Geneva Risk and Insurance Review (2012) 37, 141–155. doi:10.1057/grir.2012.1; published online 10 July 2012 Keywords: risk aversion; prudence; temperance; moments of a distribution Introduction In the 18th century, without using terms such as risk aversion, marginal utility or concavity, which are so familiar today, Bernoulli and Cramer1 had already anticipated the notion of risk aversion. While the link between risk aversion and the concavity of the utility function (u(x), i.e. utility of wealth) was regularly re-examined since then,2 it is not before the mid-sixties that the notions of absolute ((u00 /u0 )) and relative (x(u00 /u0 )) risk aversion were firmly established.3 As is well known, Arrow and Pratt4 made independently a central contribution about these definitions. Indeed they not only analysed the properties of these notions but they also made assumptions about their behaviour that are still widely used today. For our purpose, it is important to distinguish Bernoulli’s approach (and subsequent ones) from that of Arrow and Pratt. Bernoulli defined an attitude * This paper was prepared for the 23rd Geneva Risk Economics lecture delivered at the 38th Seminar of the European Group of Risk and Insurance Economists (EGRIE) in Vienna in 2011. 1 Bernoulli (1738) and Cramer (1728). 2 For a lively account of such developments, see Borch (1990). 3 As usual u0 and u00 stand for the first and second derivatives of u with respect to wealth (x). 4 Arrow (1965) and Pratt (1964). The Geneva Risk and Insurance Review 142 “risk aversion” that could be contrasted later on with other ones such as risk neutrality and risk loving. In a terminology that I sometimes use Bernoulli discusses a direction. Building upon Bernoulli’s results Arrow and Pratt consider the intensity of such an attitude. They not only want to know if a decision maker (D.M.) is risk averse: their purpose is mainly to determine when and to which extent a D.M. is more risk averse than his neighbour. If one looks at the papers posterior to Arrow’s and Pratt’s contributions, it seems pretty obvious that they mainly focused on intensities of attitudes beyond risk aversion such as absolute (or relative) prudence and absolute temperance. For instance, through his well-known contribution on precautionary savings, Kimball5 defined the coefficient of absolute prudence (u000 /u00 where u000 stands for the third derivative of u) and made the assumption that it is decreasing in wealth (D.A.P).6 Combining this assumption with that of decreasing absolute risk aversion (D.A.R.A.) the way was opened for the analysis of the very rich concepts of properness, standardness and risk vulnerability of the utility function.7 Relatively to this important literature the current paper partly steps back a pace. Essentially, I’ll apply to Bernoulli’s “directions” what Kimball and followers did to Arrow–Pratt’s intensities. To pursue this goal this paper is organised as follows. I’ll first indicate why it is interesting to pay attention to the sign of higher order derivatives of the utility function (beyond the second order). Then in the next section I’ll contrast two ways of interpreting these signs and I’ll explain why one dominates the other. Finally in the third section I’ll discuss some potentially interesting extensions of the existing body of literature. Why? There are at least three main reasons that justify the interest for the interpretation of higher order derivatives of u. The first one is immediate and related to the discussion in the introduction. The absolute intensities of risk aversion, prudence, temperance are all ratios of higher order derivatives of u (of different order).8 When exogenous shocks occur—such as for example an increase in wealth—the behaviour of the ratio is 5 Kimball (1990). On this see also Eeckhoudt-Kimball (1992). 7 Well-known contributions in this approach are Kimball (1992), Gollier and Pratt (1996) and Pratt and Zeckhauser (1987). See also Gollier (2001) for a synthesis. 8 For instance, the ratio for temperance is defined by u0000 /u000 where u0000 is the fourth derivative of u. 6 Louis Eeckhoudt Beyond Risk Aversion: Why, How and What’s Next? 143 more difficult to interpret than that of each of its constituents, the numerator and the denominator. Hence by having an interpretation of the constituents and of their behaviour one can obtain a better understanding and a better evaluation of assumptions made about the behaviour of absolute intensities. An illustration of this fact can be found in Eeckhoudt and Schlesinger.9 Consider for example the exponential utility that is characterised by a constant absolute risk aversion (CARA). When wealth increases, its numerator (u00 )— which is related to the pain induced by the presence of a zero-mean risk— decreases iff u000 >0 (a third order effect). At the same time at higher wealth u0 is also smaller under u00 o0 (a second order effect). It just happens for the exponential utility that the numerator and the denominator of absolute risk aversion decrease at the same rate yielding the CARA assumption. Using similar arguments it can be shown that the quadratic utility, which exhibits increasing absolute risk aversion (IARA) (a property usually considered as undesirable), is more attractive than one could expect. Indeed for this function u000 ¼0 so that the numerator of the absolute risk aversion coefficient is constant when wealth increases. This means that under the quadratic utility the pain induced by a zero-mean risk is constant at all wealth levels. The property of IARA then simply results from the fall in u0 under increasing wealth. It is also worth stressing that the attention paid to higher order risk attitudes surprisingly leads to a much better interpretation of the coefficients of relative risk aversion, relative prudence and relative temperance. The reader is referred to Danthine and Donaldson, Eeckhoudt et al., Chiu et al., and Eeckhoudt and Schlesinger10 for details about this relationship. These simple examples illustrate a more general fact: the analysis of higher order risk a (...truncated)