A Class of Distortion Operators for Pricing Financial and Insurance Risks

Wang, Shaun S.

doi:10.2307/253675

Cited by 544 publications

(366 citation statements)

References 29 publications

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“…Choquet integrals have been used in the application of pricing insurance premia (e.g., Denneberg 1990, Wang 2000. One direction of the following result is originally due to Dellacherie (1970);Schmeidler (1986) later completed it.…”

Section: Comonotonicity Choquet Integrals and Law Invariancementioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

291

167

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In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203-228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

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Section: Comonotonicity Choquet Integrals and Law Invariancementioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

291

167

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“…It is called distortion risk measure with distortion function g and has been studied several times in the literature, cf. [1,13,14,15,16,24,25,26] and references cited therein. We emphasize that most of the popular risk measures in practice can be represented as in (2), cf.…”

Section: Introductionmentioning

confidence: 99%

Rates of almost sure convergence of plug-in estimates for distortion risk measures

Zähle

2010

Metrika

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In this article, we consider plug-in estimates for distortion risk measures as for instance the Value-at-Risk, the Expected Shortfall or the Wang transform. We allow for fairly general estimates of the underlying unknown distribution function (beyond the classical empirical distribution function) to be plugged in the risk measure. We establish strong consistency of the estimates, we investigate the rate of almost sure convergence, and we study the small sample behavior by means of simulations.

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“…10 The class of spectral risk measures includes the ES as a special case, obtained by giving tail losses the same weight and other observations a zero weight. It is also very closely related to the class of distortion risk measures introduced by Shaun Wang that have become widely used in the actuarial risk literature (see, e.g., Wang, 1996Wang, , 2000. The relationship between these measures is discussed further in Dowd (2005, Chapter 3).…”

Section: Measures Of Financial Riskmentioning

confidence: 96%