Extremes for coherent risk measures |
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Institution: | 1. Cass Business School, City University London, London EC1Y 8TZ, United Kingdom;2. School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, PR China;1. Risk Methodologies, UniCredit Spa, Milan, Italy;2. Univ Lyon, ISFA, LSAF EA2429, F-69007 Lyon, France;1. Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada;2. Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada;1. The Key Lab of Financial Engineering of Jiangsu Province, Nanjing Audit University, Nanjing, 211815, China;2. School of Economics and Management, Southeast University, Nanjing, 210096, China;3. Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong;1. School of Economics and Management, Southeast University, Nanjing, 210029, PR China;2. Department of Statistics, Nanjing Audit University, Nanjing, 211815, PR China;3. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania;1. School of Statistics and Mathematics, Zhejiang Gongshang University, 310018, PR China;2. School of Mathematical Sciences, Soochow University, Suzhou, 215006, PR China;3. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, PR China;1. Department of Mathematics, University of the Aegean, Karlovassi, Greece;2. School of Mathematical Science and LPMC, Nankai University, Tianjin 300071, PR China |
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Abstract: | Various concepts appeared in the existing literature to evaluate the risk exposure of a financial or insurance firm/subsidiary/line of business due to the occurrence of some extreme scenarios. Many of those concepts, such as Marginal Expected Shortfall or Tail Conditional Expectation, are simply some conditional expectations that evaluate the risk in adverse scenarios and are useful for signaling to a decision-maker the poor performance of its risk portfolio or to identify which sub-portfolio is likely to exhibit a massive downside risk. We investigate the latter risk under the assumption that it is measured via a coherent risk measure, which obviously generalizes the idea of only taking the expectation of the downside risk. Multiple examples are given and our numerical illustrations show how the asymptotic approximations can be used in the capital allocation exercise. We have concluded that the expectation of the downside risk does not fairly take into account the individual risk contribution when allocating the VaR-based regulatory capital, and thus, more conservative risk measurements are recommended. Finally, we have found that more conservative risk measurements do not improve the fairness of the cost of capital allocation when the uncertainty with parameter estimation is present, even at a very high level. |
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Keywords: | Capital allocation Coherent/Distortion risk measure Conditional tail expectation Extreme value theory Marginal expected shortfall Rapid variation Regular variation |
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