On multivariate extensions of Conditional-Tail-Expectation |
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| Institution: | 1. Université de Lyon, Université Lyon 1, ISFA, Laboratoire SAF, 50 avenue Tony Garnier, 69366 Lyon, France;2. CNAM, Paris, Département IMATH, Laboratoire Cédric EA4629, 292 rue Saint-Martin, Paris Cedex 03, France;1. Institut Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg and CNRS, 7, rue René-Descartes, 67084 Strasbourg cedex, France;2. Dipartimento di Scienze delle Decisioni, Università Bocconi, via Roentgen 1, 20136 Milano, Italy;1. Department of Statistics, Purdue University, West Lafayette, IN 47907, United States;2. Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada;1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China;2. Department of Finance, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong;1. Department of Statistics, Nanjing Audit University, Nanjing, 211815, China;2. School of Economics and Management, Southeast University, Nanjing, 210096, China;3. Department of Statistics and Actuarial Science, Chongqing University, 401331, China;4. School of Statistics and Mathematics, Zhejiang Gongshang University, 310018, China;5. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China;1. École d’Actuariat, Université Laval, Québec, Canada;2. Department of Mathematics, Université Libre de Bruxelles (ULB), Bruxelles, Belgium |
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| Abstract: | In this paper, we introduce two alternative extensions of the classical univariate Conditional-Tail-Expectation (CTE) in a multivariate setting. The two proposed multivariate CTEs are vector-valued measures with the same dimension as the underlying risk portfolio. As for the multivariate Value-at-Risk measures introduced by Cousin and Di Bernardino (2013), the lower-orthant CTE (resp. the upper-orthant CTE) is constructed from level sets of multivariate distribution functions (resp. of multivariate survival distribution functions). Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogeneous in nature and cannot be aggregated together. Several properties have been derived. In particular, we show that the proposed multivariate CTE-s satisfy natural extensions of the positive homogeneity property, the translation invariance property and the comonotonic additivity property. Comparison between univariate risk measures and components of multivariate CTE is provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Sub-additivity of the proposed multivariate CTE-s is provided under the assumption that all components of the random vectors are independent. Illustrations are given in the class of Archimedean copulas. |
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| Keywords: | Multivariate risk measures Level sets of distribution functions Multivariate probability integral transformation Stochastic orders Copulas and dependence |
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