A multivariate tail covariance measure for elliptical distributions

Landsman, Zinoviy; Makov, Udi; Shushi, Tomer

doi:10.1016/j.insmatheco.2018.04.002

Cited by 35 publications

(25 citation statements)

References 5 publications

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“…In [50,51], a multivariate risk measure based on the first moment of X conditional on X > V@R θ (X) was proposed, where V@R θ (X) = (V@R θ 1 (X 1 ), . .…”

Section: Multivariate Excess Loss Tail Conditional Expectation and Ta...mentioning

confidence: 99%

“…From a practical point of view, it is also important to be able to fit the multivariate distributions to data using algorithms that can be easily implemented. Examples of multivariate distributions that have enjoyed success in some or all of these aspects include mixed Erlang [52,72] (which is also related to joining exponential, Erlang or mixed Erlang marginal distributions via the Farlie-Gumbel-Morgenstern copula or Sarmanov's family [20,22,62]), gamma [37,36], Pareto [6,64], elliptical [69,38,50,51], and phase-type [19].…”

Section: Introductionmentioning

confidence: 99%

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Multivariate matrix-exponential affine mixtures and their applications in risk theory

Cheung¹,

Peralta²,

Woo³

2022

Preprint

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In this paper, a class of multivariate matrix-exponential affine mixtures with matrixexponential marginals is proposed. The class is shown to possess various attractive properties such as closure under size-biased Esscher transform, order statistics, residual lifetime and higher order equilibrium distributions. This allows for explicit calculations of various actuarial quantities of interest. The results are applied in a wide range of actuarial problems including multivariate risk measures, aggregate loss, large claims reinsurance, weighted premium calculations and risk capital allocation. Furthermore, a multiplicative background risk model with dependent risks is considered and its capital allocation rules are provided as well. We finalize by discussing a calibration scheme based on complete data and potential avenues of research.

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“…In [50,51], a multivariate risk measure based on the first moment of X conditional on X > V@R θ (X) was proposed, where V@R θ (X) = (V@R θ 1 (X 1 ), . .…”

Section: Multivariate Excess Loss Tail Conditional Expectation and Ta...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multivariate matrix-exponential affine mixtures and their applications in risk theory

Cheung¹,

Peralta²,

Woo³

2022

Preprint

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“…The multivariate tail covariance (MTCov) measure provides a variation of the MTCE measure for the dispersion of the random vector of risks. This measure was introduced in Landsman-Makov-Shushi [9].…”

Section: Multivariate Tail Covariance For Log-elliptical Modelsmentioning

confidence: 99%

“…The first measure, E X|X > x q , has been introduced in Landsman-Makov-Shushi [8] and is called the multivariate tail conditional (MTCE) measure, where the second measure centralized around MTCE was introduced in Landsman-Makov-Shushi [9] in order to capture the dispersion of the random vector of risks when focusing on extreme losses. Analyzing the moments and the tail moments of random variables has been studied and well investigated in the literature, and there is a still active research in this area-in its applications in different fields, from data analysis to actuarial science (Loperfido [10], Loperfido-Mazur-Podgórski [11], Ogasawara [12]).…”

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confidence: 99%

Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

Landsman

Shushi

2021

Symmetry

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The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

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“…The set-valued risk measures were developed for quantifying and investigating systems of dependent risks having certain acceptance sets (Jouini et al 2004). These risk measures are defined as a map from subset V of possible outcomes of losses Ω to the Euclidean space R d , and are defined by Jouini et al 2004;Aubin and Frankowska 2009;Hamel et al 2011;Hamel et al 2013;Landsman et al 2016;Molchanov and Cascos 2016;Shushi 2018;Landsman et al 2018b). The acceptance set for set-valued risk measure R, is given by…”

Section: Introductionmentioning

confidence: 99%

Towards a Topological Representation of Risks and Their Measures

Shushi

2018

Risks

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In risk theory, risks are often modeled by risk measures which allow quantifying the risks and estimating their possible outcomes. Risk measures rely on measure theory, where the risks are assumed to be random variables with some distribution function. In this work, we derive a novel topological-based representation of risks. Using this representation, we show the differences between diversifiable and non-diversifiable. We show that topological risks should be modeled using two quantities, the risk measure that quantifies the predicted amount of risk, and a distance metric which quantifies the uncertainty of the risk.

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A multivariate tail covariance measure for elliptical distributions

Cited by 35 publications

References 5 publications

Multivariate matrix-exponential affine mixtures and their applications in risk theory

Multivariate matrix-exponential affine mixtures and their applications in risk theory

Multivariate Tail Moments for Log-Elliptical Dependence Structures as Measures of Risks

Towards a Topological Representation of Risks and Their Measures

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