Stochastic dominance relation and linear risk measures

Journal of Mathematical Economics, 2004

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 Rank-dependent Expected Utility (RDEU) model of choice among lotteries. Risk aversion is modeled in the vNM Expected Utility model by Rothschild & Stiglitz's Mean Preserving Increase in Risk (MPIR). Realizing that in the broader rank-dependent set-up this order is too weak to classify choice, Quiggin developed the stronger monotone MPIR for this purpose.

ZOR Zeitschrift f� Operations Research Methods and Models of Operations Research, 1992

This paper reviews recently proposed axiomatic models of choice under uncertainty and risk. The presentation focuses on the various models of transitive preferences which abandon the expected utility hypothesis by weakening the strong separability assumption known as the sure thing principle in the case of uncertainty and as the independence axiom in the case of risk. Special emphasis is on the remarkable similarities held in common by both the approaches to decisions under uncertainty and under risk.

Mathematical Programming, 2001

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically.

Social Choice and Welfare, 2007

In this paper, we propose the infimum of the Arrow–Pratt index of absolute risk aversion as a measure of global risk aversion of a utility function. We show that, for any given arbitrary pair of distributions, there exists a threshold level of global risk aversion such that all increasing concave utility functions with at least as much global risk aversion would rank the two distributions in the same way. Furthermore, this threshold level is sharp in the sense that, for any lower level of global risk aversion, we can find two utility functions in this class yielding opposite preference relations for the two distributions.

1998

We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The concept of α-consistency of these approaches is defined as the consistency within a bounded range of mean-risk trade-offs. We show that mean-risk models using central semideviations as risk measures are α-consistent with stochastic dominance relations of the corresponding degree if the trade-off coefficient for the semideviation is bounded by one.

Insurance: Mathematics and Economics, 1996

The non-expected-utility tbeories of decision under risk have fovond the appearance of new notions of incieasing risk like ntonotone increasing risk (based on the notion of comonotonic random vahablcs) or new notions of risk aversion like averaion to monotone increasing risk, in better agreement witb tbese new theories. After a survey of atl tbc possible notions of increasing risk and of risk aversion and tbeir intrinsic de^nitions, we show tbat cODtivy to expected-utility tbeory where all the notions of risk avenion bave the same characterization (u concave), in the ftunework of rank-dependent expected utility (one of the most well known of the non-expectedutility models), tbe characterizations of all these notions of risk aversion are different. Moreover, we sbow tbat, even in tbe expected-utility framcworic, the new notion of monotone increasing risk can give better answers to some pn^lems of comparative sutics sucb as in ponfoUo cboice or in partial insurance. This new notion also can suggest more intuitive approaches to inequalities measurement.

In this paper we compare overall as well as downside risk measures with respect to the criteria of first and second order stochastic dominance. While the downside risk measures, with the exception of tail conditional expectation, are consistent with first order stochastic dominance, overall risk measures are not, even if we restrict ourselves to two-parameter distributions. Most common risk measures preserve consistent preference orderings between prospects under the second order stochastic dominance rule, although for some of the downside risk measures such consistency holds deep enough in the tail only. In fact, the partial order induced by many risk measures is equivalent to sosd. Tail conditional expectation is not consistent with respect to second order stochastic dominance.

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.

Decision-making Process, 2009

Ce chapitre d'ouvrage collectif a pour but de présenter les bases de la modélisation de la prise de décision dans un univers risqué. Nous commençons par dé…nir, de manière générale, la notion de risque et d'accroissement du risque et rappelons des dé…nitions et catégorisations (valables en dehors de tout modèle de représentation) de comportements face au risque. Nous exposons ensuite le modèle classique d'espérance d'utilité de von Neumann et Morgenstern et ses principales propriétés. Les problèmes posés par ce modèle sont ensuite discutés et deux modèles généralisant l'espérance d'utilité brièvement présentés. Mots clé: risque, aversion pour le risque, espérance d'utilité, von Neumann et Morgenstern, Paradoxe d'Allais. JEL: D81

Report Eurandom, 2005

In this paper we compare two sets of risk measures with respect to the criteria of first and second order stochastic dominance. We observe that overall risk measures do not preserve consistent preference ordering between assets under the first order stochastic dominance rule, while the downside risk measures, with the exception of Expected Shortfall (es), do preserve a consistent preference ordering under first order stochastic dominance. Further, risk measures except es preserve consistent preference ordering between assets under the second order stochastic dominance rule, although for some of the downside risk measures such preference ordering is only partial.