On law invariant coherent risk measures

Kusuoka, Shigeo

doi:10.1007/978-4-431-67891-5_4

Cited by 627 publications

(435 citation statements)

References 2 publications

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“…Note that the result is known (Kusuoka 2001) in the case of atomless distributions. Leitner (2005) proves a result for second-order stochastic dominance preserving (a stronger condition than law invariance) coherent risk measures on general probability spaces, but does not consider comonotonicity.…”

Section: Lemma 41 a Risk Measure Is A Distortion Risk Measure If Anmentioning

confidence: 93%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

290

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: Lemma 41 a Risk Measure Is A Distortion Risk Measure If Anmentioning

confidence: 93%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

290

167

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“…The second condition refines slightly the subadditivity property: subadditivity becomes additivity when X and Y are comonotone. We can now succinctly reformulate the characterization result of Kusuoka (2001) and Tasche (2002).…”

Section: Measuring Riskmentioning

confidence: 99%

Pessimistic Portfolio Allocation and Choquet Expected Utility

Bassett¹,

Koenker

Kordas

2004

Journal of Financial Econometrics

120

119

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Abstract. Recent developments in the theory of choice under uncertainty and risk yield a pessimistic decision theory that replaces the classical expected utility criterion with a Choquet expectation that accentuates the likelihood of the least favorable outcomes. A parallel theory has recently emerged in the literature on risk assessment. It is shown that pessimistic portfolio optimization based on the Choquet approach may be formulated as an exercise in quantile regression.

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“…Some of our results take a simpler form when further restricting attention to the class of distortion risk measures, which are all comonotonic risk measures additionally satisfying [P6]. Such measures have been examined in economics, actuarial science, and financial mathematics, and form a well-established class of risk metrics (see, e.g., [Schmeidler, 1986, Wang, 2000, Tsanakas, 2004, Cotter and Dowd, 2006, Kusuoka, 2001, Acerbi, 2002, Föllmer and Schied, 2004 for more references and details).…”

Section: Static Risk Measuresmentioning

confidence: 99%

“…The latter class has emerged as an axiomatically justified and computationally tractable alternative to several classical approaches, and has provided a strong bridge across a variety of parallel streams of research, including ambiguous representations of preferences in economics (e.g., Gilboa and Schmeidler [1989], Schmeidler [1989], Epstein and Schneider [2003], Maccheroni et al [2006]), axiomatic treatments of market risk in financial mathematics (Artzner et al [1999], Föllmer and Schied [2002]), actuarial science (Wirch and Hardy [1999], Wang [2000], Acerbi [2002], Kusuoka [2001], Tsanakas [2004]), operations research (Ben-Tal and Teboulle [2007]) and statistics (Huber [1981]). As such, our goal in the present paper is not to motivate the use of risk measures -rather, we take the framework as given, and investigate two distinct ways of using it to ascribe risk in dynamic decision settings.…”

Section: Introductionmentioning

confidence: 99%

Tight Approximations of Dynamic Risk Measures

Iancu

Petrik²,

Subramanian³

2015

Mathematics of OR

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This paper compares two different frameworks recently introduced in the literature for measuring risk in a multi-period setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, while the second involves applying a composition of one-step coherent risk mappings. We summarize the relative strengths of the two methods, characterize several necessary and sufficient conditions under which one of the measurements always dominates the other, and introduce a metric to quantify how close the two risk measures are.Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper-bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest possible upper-bound can be exactly characterized, and corresponds to a popular construction in the literature, while the tightest-possible lower bound is not readily available. We show that testing domination and computing the approximation factors is generally NP-hard, even when the risk measures in question are comonotonic and law-invariant. However, we characterize conditions and discuss several examples where polynomial-time algorithms are possible. One such case is the well-known Conditional Value-at-Risk measure, which is further explored in our companion paper Huang et al. [2012]. Our theoretical and algorithmic constructions exploit interesting connections between the study of risk measures and the theory of submodularity and combinatorial optimization, which may be of independent interest.

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On law invariant coherent risk measures

Cited by 627 publications

References 2 publications

Constructing Uncertainty Sets for Robust Linear Optimization

Constructing Uncertainty Sets for Robust Linear Optimization

Pessimistic Portfolio Allocation and Choquet Expected Utility

Tight Approximations of Dynamic Risk Measures

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