Tail Risk of Multivariate Regular Variation

Tail risk refers to the risk associated with extreme values and is often affected by extremal dependence among multivariate extremes. Multivariate tail risk, as measured by a coherent risk measure of tail conditional expectation, is analyzed for multivariate regularly varying distributions. Asymptotic expressions for tail risk are established in terms of the intensity measure that characterizes multivariate regular variation. Tractable bounds for tail risk are derived in terms of the tail dependence function that describes extremal dependence. Various examples involving Archimedean copulas are presented to illustrate the results and quality of the bounds.

[1]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[2]  Richard A. Davis,et al.  Regular variation of GARCH processes , 2002 .

[3]  Richard A. Davis,et al.  A characterization of multivariate regular variation , 2002 .

[4]  Haijun Li,et al.  Orthant tail dependence of multivariate extreme value distributions , 2009, J. Multivar. Anal..

[5]  Paul Embrechts,et al.  Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness , 2009 .

[6]  Diversification for general copula dependence , 2007 .

[7]  S. Resnick.  Heavy-Tail Phenomena: Probabilistic and Statistical Modeling , 2006 .

[8]  Koiti Takahasi,et al.  Note on the multivariate burr’s distribution , 1965 .

[9]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[10]  Haijun Li,et al.  Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions , 2009, Journal of Applied Probability.

[11]  Aristidis K. Nikoloulopoulos,et al.  Extreme value properties of multivariate t copulas , 2009 .

[12]  F. Delbaen,et al.  Coherent and convex monetary risk measures for bounded càdlàg processes , 2004 .

[13]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[14]  Matthias Löwe,et al.  Diversification of aggregate dependent risks , 2004 .

[15]  Christine M. Anderson-Cook,et al.  Book review: quantitative risk management: concepts, techniques and tools, revised edition, by A.F. McNeil, R. Frey and P. Embrechts. Princeton University Press, 2015, ISBN 978-0-691-16627-8, xix + 700 pp. , 2017, Extremes.

[16]  S. Rachev.  Handbook of heavy tailed distributions in finance , 2003 .

[17]  Aristidis K. Nikoloulopoulos,et al.  Tail dependence functions and vine copulas , 2010, J. Multivar. Anal..

[18]  Hansjörg Albrecher,et al.  Tail asymptotics for the sum of two heavy-tailed dependent risks , 2006 .

[19]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[20]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[21]  Lei Si Ni Ke Resnick.S.I..  Extreme values. regular variation. and point processes , 2011 .

[22]  Samuel Kotz,et al.  Multivariate Pareto Distributions , 2005 .

[23]  Jun Cai.  CONDITIONAL TAIL EXPECTATIONS FOR MULTIVARIATE PHASE-TYPE DISTRIBUTIONS , 2005 .

[24]  Harry Joe,et al.  Bivariate Threshold Methods for Extremes , 1992 .

[25]  Nizar Touzi,et al.  Vector-valued coherent risk measures , 2002, Finance Stochastics.

[26]  S. Coles,et al.  Modelling Extreme Multivariate Events , 1991 .

[27]  Emiliano A. Valdez,et al.  Tail Conditional Expectations for Elliptical Distributions , 2003 .

[28]  Harry Joe,et al.  Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions , 1996 .

[29]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[30]  Harry Joe,et al.  Parametric families of multivariate distributions with given margins , 1993 .

[31]  R. Cooke,et al.  Climate Change and Risk Management: Challenges for Insurance, Adaptation, and Loss Estimation , 2009 .

[32]  C. Klüppelberg,et al.  Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case , 2008 .

[33]  J. Geluk.  Π-regular variation , 1981 .

[34]  Matthias Löwe,et al.  Analysis of the Expected Shortfall of Aggregate Dependent Risks , 2005, ASTIN Bulletin.

[35]  Imen Bentahar,et al.  Tail Conditional Expectation for vector-valued Risks , 2006 .

[36]  R. Nelsen.  An Introduction to Copulas , 1998 .

[37]  H. Albrecher,et al.  Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables , 2009 .

[38]  M. E. Johnson,et al.  A Family of Distributions for Modelling Non‐Elliptically Symmetric Multivariate Data , 1981 .

[39]  R. Bhar,et al.  Advances in finance and stochastics : essays in honour of Dieter Sondermann , 2002 .