Risk apportionment via bivariate stochastic dominance

Social Choice and Welfare, 2010

Incomplete preferences over lotteries on a finite set of alternatives satisfying, besides independence and continuity, a property called bad outcome aversion are considered. These preferences are characterized in terms of their specific multi-expected utility representations (cf. Dubra et al., 2004), and can be seen as generalized stochastic dominance preferences.

2017

This paper extends the theory between Kappa ratio and stochastic dominance (SD) and risk-seeking SD (RSD) by establishing several relationships between first- and higher-order risk measures and (higher-order) SD and RSD. We first show the sufficient relationship between the (n+1)-order SD and the n-order Kappa ratio. We then find that, in general, the necessary relationship between SD/RSD and the Kappa ratio cannot be established. Thereafter, we find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, we can get the necessary relationships between the (n+1)-order SD with the n-order Kappa ratio when we impose some conditions on the means. Our findings enable academics and practitioners to draw better decision in their analysis.

2011

There is a duality theory connecting certain stochastic orderings between cumulative distribution functions F 1 , F 2 and stochastic orderings between their inverses F −1 1 , F −1 2 . This underlies some theories of utility in the case of the cdf and deprivation indices in the case of the inverse. Under certain conditions there is an equivalence between the two theories. An example is the equivalence between second order stochastic dominance and the Lorenz ordering. This duality is generalised to include the case where there is "distortion" of the cdf of the form v(F ) and also of the inverse. A comprehensive duality theorem is presented in a form which includes the distortions and links the duality to the parallel theories of risk and deprivation indices. It is shown that some wellknown examples are special cases of the results, including some from the Yaari social welfare theory and the theory of majorization.

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.

Economic Theory, 2009

In this paper, we consider a décision-maker facing a financial risk flanked by a background risk, possibly non-financial, such as health or environmental risk. A decision has to be made about the amount of an investment (in the financial dimension) resulting in a future benefit either in the same dimension (savings) or in the order dimension (environmental quality or health improvement). In the first case, we show that the optimal amount of savings decreases as the pair of risks increases in the bivariate increasing concave dominance rules of higher degrees which express the common preferences of all the decision-makers whose twoargument utility function possesses direct and cross derivatives fulfilling some specific requirements. Roughly speaking, the optimal amount of savings decreases as the two risks become "less positively correlated" or marginally improve in univariate stochastic dominance. In the second case, a similar conclusion on optimal investment is reached under alternative conditions on the derivatives of the utility function.

1998

Two methods are frequently used for modeling the choice among uncertain prospects: stochastic dominance relation and mean-risk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. The seminal Markowitz model uses the variance as the risk measure in the mean-risk analysis which results in a formulation of a quadratic programming model. Following the pioneering work of Sharpe, many attempts have been made to linearize the mean-risk approach. There were introduced risk measures which lead to linear programming mean-risk models. This paper focuses on two such risk measures: the Gini's mean (absolute) difference and the mean absolute deviation. Consistency of the corresponding mean-risk models with the second degree stochastic dominance (SSD) is reexamined. Both the models are in some manner consistent with the SSD rules, provided that the trade-off coefficient is bounded by a certain constant. However, for the Gini's mean difference the consistency turns out to be much stronger than that for the mean absolute deviation. The analysis is graphically illustrated within the framework of the absolute Lorenz curves.

RAIRO - Operations Research, 1999

In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

Comparing uncertain prospects is one of fundamental interests of the economic decision theory. Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on the axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. It is, in fact, a multiple criteria model with a continuum of criteria. The mean-risk approach quantifies the problem in a lucid form of only two criteria: the mean, representing the expected outcome, and the risk: a scalar measure of the variability of outcomes. The mean-risk model is appealing to decision makers and allows a simple trade-off analysis, analytical or geometrical. On the other hand, for typical dispersion statistics used as risk measures, the mean-risk approach may lead to inferior conclusions. Several risk measures, however, can be combined with the mean itself into the robust optimization criteria thus generating SSD consistent performances (safety) measures. In this paper we introduce general conditions for risk measures sufficient to provide the SSD consistency of the corresponding safety measures.

2012

This paper studies some properties of stochastic dominance (SD) for risk-averse and risk-seeking investors, especially for the third order SD (TSD). We call the former ascending stochastic dominance (ASD) and the latter descending stochastic dominance(DSD). We first discuss the basic property of ASD and DSD linking the ASD and DSD of the first three orders to expected-utility maximization for risk-averse and risk-seeking investors. Thereafter, we prove that a hierarchy exists in both ASD and DSD relationships and that the higher orders of ASD and DSD cannot be replaced by the lower orders of ASD and DSD. Furthermore, we study conditions in which third order ASD preferences will be 'the opposite of' or 'the same as' their counterpart third order DSD preferences. In addition, we construct examples to illustrate all the properties developed in this paper. The theory developed in this paper provides investors with tools to identify first, second, and third order ASD and ...

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.

Wiley-Blackwell eBooks, 2011

The goals of this chapter are the following: • To explore the relationship between preference relations and quasi-semidistances. • To introduce a universal description of probability quasisemidistances in terms of a Hausdorff structure. • To provide examples with first-, second-, and higher-order stochastic dominance and to introduce primary, simple, and compound stochastic orders. • To explore new stochastic dominance rules based on a popular risk measure. • To provide a utility-type representation of probability quasisemidistances and to describe the degree of violation utilized in almost stochastic orders in terms of quasi-semidistances.

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.

CRC Press eBooks, 2014

European Journal of Operational Research, 1999

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible tradeoff analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean-variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome-Risk diagram, which appears to be particularly useful for comparing uncertain outcomes.