1 Ordinal utility of moments: foundations and financial behavior Riccardo Cesari Università di Bologna, Dip. MatemateS, viale Filopanti, 5 riccardo.cesari@unibo.it Carlo D’Adda Università di Bologna, Dip. Scienze economiche, strada Maggiore, 45 carlo.dadda@unibo.it Abstract: After Tobin (1958), a considerable effort has been devoted to connecting the expected utility approach to a utility function directly expressed in terms of moments. We follow the alternative route of providing, for the first time, the theoretical, autonomous foundation of an ordinal utility function of moments, representing rational choices under uncertainty, free of any ‘independence axiom’ and compatible with all the behavioral “paradoxes” documented in the economic literature. Keywords: mean-variance utility, expected utility, behavioral paradoxes 1. Introduction In this paper we develop for the first time - to our knowledge – the foundation of an ordinal utility of moments (section 2) as a rational and autonomous criterion of choice under uncertainty, showing that it explains all the best known behavioral paradoxes which are still embarrassing the expected utility theory (section 3). This ordinal approach is strongly reminiscent of standard microeconomic theory and it could be used to reset and generalize both assets demand and asset pricing models. 2. Foundations of an ordinal utility of moments for decisions under uncertainty It is well known that the theory of choice under uncertainty assumes that preferences are defined over the set of probability distribution functions (e.g. Savage, 1954 ch. 2 and DeGroot, 1970 ch. 7). In particular, let (Ω,ℑ,℘) be a standard probability space, Ω being the set of elementary events (states of the world), ℑ the set of subsets of Ω (events), ℘ a (subjective) probability measure of the events. Given the set A of all possible actions or decisions, all couples (ω,a) with ω∈Ω and a∈A, are mapped onto a real vector of monetary consequences c∈R n , the Euclidean space of n- dimensional real vectors, so that X(ω,a)=c or X a (ω)=c is a random variable and F a ∈ℱ is its probability distribution function. Clearly, the preferences over acts in A are, equivalently, preferences over the set of random variables X a and preferences over the set ℱ of distribution functions. Let us confine ourselves, for ease of exposition, to the case of univariate distributions (n=1) and assume that the essential information concerning any distribution F is contained in the m- dimensional vector of moments M≡(μ, μ (2) , μ (3) , ...., μ (m) ) where μ is the mean, and μ (s) is the s- order central moment in original units 1 : Definition of s-order modified central moment: ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ μ − = μ = ≡ μ ∫ 2 s dF ) x ( 1 s 0 s 1 s 1 s ) s ( (2.1) 1 Note that, instead of central moments, noncentral moments could, equivalently, be used. Moreover, scale, location and dispersion parameters can be considered in the case of distributions (e.g. stable) for which moments do not exist. MTISD 2008 – Methods, Models and Information Technologies for Decision Support Systems Università del Salento, Lecce, 1820 September 2008 ___________________________________________________________________________________________________________ © 2008 University of Salento - SIBA http://siba2.unile.it/ese 236