The concept of comonotonicity in actuarial science and finance: theory

Dhaene, Jan; Denuit, Michel; Goovaerts, Marc; Kaas, Rob; Vyncke, David

doi:10.1016/s0167-6687(02)00134-8

Cited by 476 publications

(163 citation statements)

References 29 publications

Supporting

Mentioning

158

Contrasting

Order By: Relevance

“…Comonotone random variables have worst-case summation properties among all dependence structures, and, as such, have been used by Dhaene et al (2002) to compute upper bounds on sums of random variables. Coherent risk measures that are also comonotonic are linked to Choquet integrals, which we now define with some more terminology (originally introduced by Choquet 1954 himself).…”

Section: Comonotonicity Choquet Integrals and Law Invariancementioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

282

167

View full text Add to dashboard Cite

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.

show abstract

Section: Comonotonicity Choquet Integrals and Law Invariancementioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

282

167

View full text Add to dashboard Cite

show abstract

“…If ρ c (X ) = 1, then all Cov(X i , X j ) must be equal to Cov(X c i , X c j ) and thus (X i , X j ) are comonotonic for all i and j . From Theorem 4 in Dhaene et al (2002a) we know that comonotonicity of a random vector is equivalent with pairwise comonotonicity of its components, so X = d X c .…”

Section: Proof From (4) and The Fact That F Xmentioning

confidence: 99%

A multivariate dependence measure for aggregating risks

Dhaene

Linders

Schoutens

et al. 2014

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

To evaluate the aggregate risk in a financial or insurance portfolio, a risk analyst has to calculate the distribution function of a sum of random variables. As the individual risk factors are often positively dependent, the classical convolution technique will not be sufficient. On the other hand, assuming a comonotonic dependence structure will likely overrate the real aggregate risk. In order to choose between both approximations, or perhaps use a weighted average, we should have an indication on the accuracy. Clearly this accuracy will depend on the copula between the individual risk factors, but it is also influenced by the marginal distributions. In this paper we introduce a multivariate dependence measure that takes both aspects into account. This new measure differs from other multivariate dependence measures, as it focuses on the aggregate risk rather than on the copula or the joint distribution function itself. We prove several interesting properties of this new measure and discuss its relation to other dependence measures. We also give some comments on the estimation and conclude with examples and numerical results.

show abstract

“…, n}. For more details on comonotonicity in actuarial science and finance, see Dhaene et al (2000Dhaene et al ( , 2002.…”

Section: Preliminariesmentioning

confidence: 99%

Bounds on the value-at-risk for the sum of possibly dependent risks

Mesfioui

Quessy

2005

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

In this paper, explicit lower and upper bounds on the value-at-risk (VaR) for the sum of possibly dependent risks are derived when only partial information is available about the dependence structure and the individual behaviors. When the marginal distributions are known, a reformulation of a result of Embrechts et al. [Finan. Stoch. 7 (2003) 145-167] makes it possible, under some regularity conditions, to compute explicit bounds for the VaR under various dependence scenarios. In the case where only the means and the variances of the risks are available, explicit bounds are obtained from an optimization over all possible values of the correlation matrix associated with the vector of risks. Analytical and numerical investigations are presented in order to investigate the quality of these bounds.

show abstract

The concept of comonotonicity in actuarial science and finance: theory

Cited by 476 publications

References 29 publications

Constructing Uncertainty Sets for Robust Linear Optimization

Constructing Uncertainty Sets for Robust Linear Optimization

A multivariate dependence measure for aggregating risks

Bounds on the value-at-risk for the sum of possibly dependent risks

Contact Info

Product

Resources

About