Tail Risk of Multivariate Regular Variation

Methodology and Computing in Applied Probability

Volumn 13, Issue 4, 2011, Pages 671-693

  Joe, Harry ^a   Li, Haijun ^b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b WASHINGTON STATE UNIVERSITY   (United States)

Author keywords

Coherent risk; Copula; Regularly varying; Tail conditional expectation; Tail dependence

Indexed keywords

EID: 80255132944     PISSN: 13875841     EISSN: 15737713     Source Type: Journal    
DOI: 10.1007/s11009-010-9183-x     Document Type: Article

References (33)

1. Albrecher H, Asmussen S, Kortschak D (2006) Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9: 107-130.
2. Alink S, Löwe M, Wüthrich MV (2004) Diversification of aggregate dependent risks. Insurance: Math Econom 35: 77-95.
3. Alink S, Löwe M, Wüthrich MV (2005) Analysis of the expected shortfall of aggregate dependent risks. ASTIN Bull 35(1): 25-43.
4. Alink S, Löwe M, Wüthrich MV (2007) Diversification for general copula dependence. Stat Neerl 61: 446-465.
5. Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risks. Math Financ 9: 203-228.
6. Bentahar I (2006) Tail conditional expectation for vector-valued risks. Discussion paper 2006-029, Technische Universität Berlin, Germany. http://sfb649. wiwi. hu-berlin. de.
7. Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge University Press, Cambridge, UK.
8. Cai J, Li H (2005) Conditional tail expectations for multivariate phase-type distributions. J Appl Probab 42: 810-825.
9. Cheridito P, Delbaen F, Klüppelberg C (2004) Coherent and convex monetary risk measures for bounded càdlàg processes. Stoch Process their Appl 112: 1-22.
10. Coles SG, Tawn JA (1991) Modelling extreme multivariate events. J R Stat Soc, B 53: 377-392.
11. Cook RD, Johnson ME (1981) A family of distributions for modelling non-elliptically symmetric multivariate data. J R Stat Soc, B 43: 210-218.
12. Delbaen F (2002) Coherent risk measure on general probability spaces. In: Sandmann K, Schönbucher PJ (eds) Advances in finance and stochastics-essays in honour of Dieter Sondermann. Springer-Verlag, Berlin, pp 1-37.
13. Embrechts P, Neslehová J, Wüthrich MV (2009) Additivity properties for value-at-risk under Archimedean dependence and heavy-tailedness. Insurance: Math Econom 44(2): 164-169.
14. Föllmer H, Schied A(2002) Convex measures of risk and trading constraints. Finance Stoch 6: 426-447.
15. Joe H (1993) Parametric family of multivariate distributions with given margins. J Multivar Anal 46: 262-282.
16. Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall, London.
17. Joe H, Hu T (1996) Multivariate distributions from mixtures of max-infinitely divisible distributions. J Multivar Anal 57: 240-265.
18. Joe H, Li H, Nikoloulopoulos AK (2010) Tail dependence functions and vine copulas. J Multivar Anal 101: 252-270.
19. Joe H, Smith RL, Weissman I (1992) Bivariate threshold methods for extremes. J R Stat Soc, B 54: 171-183.
20. Jouini E, Meddeb M, Touzi N (2004) Vector-valued coherent risk measures. Finance Stoch 8: 531-552.
21. Klüppelberg C, Kuhn G, Peng L (2008) Semi-parametric models for the multivariate tail dependence function - the asymptotically dependent. Scand J Statist 35(4): 701-718.
22. Kortschak D, Albrecher H (2009) Asymptotic results for the sum of dependent non-identically distributed random variables. Methodol Comput Appl Probab 11: 279-306.
23. Kousky C, Cooke RM (2009) Climate change and risk management: challenges for insurance, adaptation and loss estimation. Discussion paper RFF DP 09-03-Rev, Resources For the Future. http://www. rff. org/RFF/Documents/.
24. Landsman Z, Valdez E (2003) Tail conditional expectations for elliptical distributions. N Am Actuar J 7: 55-71.
25. Li H (2009) Orthant tail dependence of multivariate extreme value distributions. J Multivar Anal 100: 243-256.
26. Li H, Sun Y, (2009) Tail dependence for heavy-tailed scale mixtures of multivariate distributions. J Appl Probab 46(4): 925-937.
27. Mardia KV (1962) Multivariate Pareto distributions. Ann Math Stat 33: 1008-1015.
28. McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques, and tools. Princeton University Press, Princeton, New Jersey.
29. Nikoloulopoulos AK, Joe H, Li H (2009) Extreme value properties of multivariate t copulas. Extremes 12: 129-148.
30. Resnick S (1987) Extreme values, regular variation, and point processes. Springer, New York.
31. Resnick S (2007) Heavy-tail phenomena: probabilistic and statistical modeling. Springer, New York.
32. Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Stat Univ Paris 8: 229-231.
33. Takahasi K (1965) Note on the multivariate Burr's distribution. Ann Inst Stat Math 17: 257-260.