The theory of orderings and risk probability functionals

2008

This paper studies and describes stochastic orderings of ri k/reward positions in order to define in a natural way risk/reward measures consistent/isotonic to investors’ preferences. We begin by discussing the connect ion between the theory of probability metrics, risk measures, distributional momen ts, and stochastic orderings. Then we examine several classes of orderings which are gener at d by risk probability functionals. Finally, we demonstrate how further ordering s could better specify the investor’s attitude toward risk. 2000 AMS Mathematics Subject Classification:Primary: 60E15, 91B16; Secondary: 91B28.

Applied Mathematical Finance, 2009

This paper unifies the classical theory of stochastic dominance and investor preferences with the recent literature on risk measures applied to the choice problem faced by investors. First we summarize the main stochastic dominance rules used in the finance literature. Then we discuss the connection with the theory of integral stochastic orders and we introduce orderings consistent with investors' preferences. Thus, we classify them, distinguishing several categories of orderings associated with different classes of investors. Finally we show how we can use risk measures and orderings consistent with some preferences to determine the investors' optimal choices.

This article presents various notions of risk generated by the intuitively appealing single-crossing operations between distribution functions. These stochastic orders, Bickel & Lehmann dispersion or (its equal-mean version) Quiggin's monotone mean-preserving increase in risk and Jewitt's location-independent risk, have proved to be useful in the study of Pareto allocations, ordering of insurance premia and other applications in the Expected Utility setup. These notions of risk are also relevant to the Quiggin-Yaari RDEU model of choice among lotteries. The Rankdependent Expected Utility model replaces expected utility by another functional, in which expectation is taken with respect to a distortion of the distribution of the lottery by a probability perception function. Risk aversion is modeled in the expected utility model by Rothschild & Stiglitz's meanpreserving increase in risk (MPIR). Realizing that in the broader rank-dependent setup this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose. This paper reviews four notions of mean-preserving increase in risk-MPIR, monotone MPIR and two versions of location-independent risk (renamed here left and right monotone MPIR)-and shows which choice questions are consistently modeled by each of these four orders. Keywords : Location-independent risk, monotone increase in risk, rank-dependent expected utility. JEL classi¯cation: D81 R ¶ esum ¶ e : Cet article pr ¶ esente di® ¶ erentes notions de risque, engendr ¶ ees par la notion intuitivement s ¶ eduisante de croisement unique entre fonctions de r ¶ epartition. Ces ordres stochastiques, la dispersion au sens de Bickel-Lehmann ou sa version µ a moyenne constante : l'accroissement de risque monotone de Quiggin, et le "risque ind ¶ ependant de la location" du µ a Jewiitt ont d ¶ ejµ a prouv ¶ e leur utilit ¶ e dans l' ¶ etude des allocations Pareto-optimales, dans la maniµ ere d'ordonner des primes d'assurance, et dans d'autres applications dans le cadre du modµ ele d'esp ¶ erance d'utilit ¶ e. Ces notions de risque ont aussi leur int ¶ erêt dans le modµ ele Quiggin-Yaari de d ¶ ecision dans le risque appel ¶ e Utilit ¶ e d ¶ ependant du rang Rank Dependent Utility model. Ce modµ ele remplace l'esp ¶ erance d'utilit ¶ e par une autre fonctionnelle dans laquelle l'esp ¶ erance est prise par rapport µ a une transformation (fonction de perception des probabilit ¶ es) de la distribution de probabilit ¶ e. Dans le modµ ele d'esp ¶ erance d'utilit ¶ e l'aversion pour le risque est mod ¶ elis ¶ ee par l'aversion pour "l'accroissement de risque µ a moyenne constante" (MPIR) due µ a Rothschild et Stiglitz. Quiggin r ¶ ealisant que cette notion d'accroissement de risque ¶ etait trop faible pour classer les choix dans le modµ ele RDEU a d ¶ evelopp ¶ e the concept pus fort d'accroissement monotone plus adapt ¶ e µ a son modµ ele. Cette article passe en revue quatre notions d'accroissement de risque µ a moyenne constantel'accroissement de risque µ a moyenne constante, MPIR, MPIR monotone, et deux versions de "risque ind ¶ ependant de la location" renomm ¶ ees ici monotone µ a gauche et µ a droite) et montre quels types de choix sont mod ¶ elis ¶ es au mieux par chacun de ces quatre ordres stochastiques. Mots cl ¶ es : Location-independent risk, accroissement monotone de risque, esp ¶ erance d'utilit ¶ e d ¶ ependant du rang.

Geneva Papers on Risk and Insurance Theory, 1998

This article examines the relationship between risk, return, skewness, and utility-based preferences. Examples are constructed showing that, for any commonly used utility function, it is possible to have two continuous unimodal random variables X and Y with positive and equal means, X having a larger variance and lower positive skewness than Y, and yet X has larger expected utility than Y, contrary to persistent folklore concerning U″′ > 0 implying skewness preference for risk averters. In additon, it is shown that ceteris paribus analysis of preferences and moments, as occasionally used in the literature, is impossible since equality of higher-order central moments implies the total equality of the distributions involved.

2015

The paper proposes a multivariate comparison among different financial markets, using risk/variability measures consistent with investors’ preferences. First of all, we recall a recent classification of multivariate stochastic orderings and properly define the selection problem among different financial markets. Then, we propose an empirical financial application, using multivariate stochastic orderings consistent with the non-satiable and risk averse investors’ preferences. For the empirical analysis we examine two different approaches; first, we assume that the return are normally distributed; second, we deal with the more generalassumption that the returns’ distribution follow a stable sub-Gaussian law.

Statistics: A Series of Textbooks and Monographs, 2005

In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fréchet-bounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the final part we discuss three important orderings of risks which arise from ∆-monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to "original sources" of these results.

SSRN Electronic Journal, 2021

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Valueat-Risk, a relevant issue for risk management practice.

SSRN Electronic Journal

We study various decision problems regarding short-term investments in risky assets whose returns evolve continuously in time. We show that in each problem, all risk-averse decision makers have the same (problem-dependent) ranking over shortterm risky assets. Moreover, in each of these problems, the ranking is represented by the same risk index as in the case of CARA utility agents and normally distributed risky assets.

SSRN Electronic Journal, 2007

This paper characterizes higher order risk e¤ects, such as prudence and temperance, via preferences that partially order a set of simple 50-50 lotteries. In particular, consider the random variables e X N ; e Y N ; e X M and e Y M , and assume that e X i dominates e Y i via i th-order stochastic dominance for i = M; N. We show that the 50-50 lottery [ e X N + e Y M ; e Y N + e X M ] dominates the lottery [ e X N + e X M ; e Y N + e Y M ] via (N +M) th-order stochastic dominance. A preference ranking over these lotteries is shown to generalize the concept of risk apportionment, as introduced by Eeckhoudt and Schlesinger (2006). We apply our results in several examples of decision making under risk.

2006

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables to the real line. Economically, a risk measure should capture the preferences of the decision-maker.