On optimal portfolio diversification with respect to extreme risks

Finance and Stochastics

Volumn 14, Issue 4, 2010, Pages 593-623

  Mainik, Georg ^a   Rüschendorf, Ludger ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Diversification effects; Multivariate extremes; Portfolio optimization; Risk management

Indexed keywords

EID: 78449260333     PISSN: 09492984     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00780-010-0122-z     Document Type: Article

References (40)

1. Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505-1518 (2002).
2. Barbe, P., Fougères, A.-L., Genest, C.: On the tail behaviour of sums of dependent risks. ASTIN Bull. 36, 361-373 (2006).
3. Basrak, B., Davis, R. A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908-920 (2002).
4. Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular Variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1987).
5. Böcker, K., Klüppelberg, C.: Modelling and measuring multivariate operational risk with Lévy copulas. J. Oper. Risk 3(2), 3-27 (2008).
6. Boman, J., Lindskog, F.: Support theorems for the Radon transform and Cramér-Wold theorems. J. Theor. Probab. 22, 683-710 (2009).
7. Daníelsson, J., Jorgensen, B. N., Sarma, M., de Vries, C. G.: Sub-additivity re-examined: the case for value-at-risk. EURANDOM Reports (2005). http://www. eurandom. nl/reports/2005/006-report. pdf.
8. Davis, R., Resnick, S.: Tail estimates motivated by extreme value theory. Ann. Stat. 12, 1467-1487 (1984).
9. de Haan, L., Ferreira, A.: Extreme Value Theory. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006).
10. de Haan, L., Resnick, S. I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 317-337 (1977).
11. de Haan, L., Resnick, S. I.: Estimating the limit distribution of multivariate extremes. Commun. Stat. Stoch. Mod. 9, 275-309 (1993).
12. de Haan, L., Sinha, A. K.: Estimating the probability of a rare event. Ann. Stat. 27, 732-759 (1999).
13. Dekkers, A. L. M., Einmahl, J. H. J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17, 1833-1855 (1989).
14. Drees, H.: Refined Pickands estimators of the extreme value index. Ann. Stat. 23, 2059-2080 (1995).
15. Drees, H., Ferreira, A., de Haan, L.: On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab. 14, 1179-1201 (2004).
16. Einmahl, J. H. J., de Haan, L., Huang, X.: Estimating a multidimensional extreme-value distribution. J. Multivar. Anal. 47, 35-47 (1993).
17. Einmahl, J. H. J., de Haan, L., Sinha, A. K.: Estimating the spectral measure of an extreme value distribution. Stoch. Process. Appl. 70, 143-171 (1997).
18. Einmahl, J. H. J., de Haan, L., Piterbarg, V. I.: Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Stat. 29, 1401-1423 (2001).
19. Embrechts, P., Lambrigger, D., Wüthrich, M. V.: Multivariate extremes and the aggregation of dependent risks: examples and counter-examples. Extremes 12, 107-127 (2009).
20. Embrechts, P., Nešlehová, J., Wüthrich, M. V.: Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness. Insur. Math. Econ. 44, 164-169 (2009).
21. Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events. DMV Seminar, vol. 23. Birkhäuser, Basel (1994).
22. Hajós, G., Rényi, A.: Elementary proofs of some basic facts concerning order statistics. Acta Math. Acad. Sci. Hungar. 5, 1-6 (1954).
23. Hauksson, H. A., Dacorogna, M. M., Domenig, T., Müller, U. A., Samorodnitsky, G.: Multivariate extremes, aggregation and risk estimation. SSRN eLibrary (2000). http://ssrn. com/paper=254392.
24. Hill, B. M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163-1174 (1975).
25. Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80(94), 121-140 (2006).
26. Joe, H.: Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997).
27. Klüppelberg, C., Resnick, S. I.: The Pareto copula, aggregation of risks, and the emperor's socks. J. Appl. Probab. 45, 67-84 (2008).
28. Malevergne, Y., Sornette, D.: Extreme Financial Risks. Springer, Berlin (2006).
29. McNeil, A. J., Frey, R., Embrechts, P.: Quantitative Risk Management. Princeton Series in Finance. Princeton University Press, Princeton (2005).
30. Moscadelli, M.: The modelling of operational risk: Experience with the analysis of the data collected by the Basel Committee. SSRN eLibrary (2004). http://ssrn. com/paper=557214.
31. Nešlehová, J., Embrechts, P., Chavez-Demoulin, V.: Infinite-mean models and the LDA for operational risk. J. Oper. Risk 1(1), 3-25 (2006).
32. Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119-131 (1975).
33. Pickands III, J.: Multivariate extreme value distributions. In: Proceedings of the 43rd session of the International Statistical Institute, Book 2, Buenos Aires, 1981, vol. 49, pp. 859-878, 894-902. ISI, 1981. With a discussion.
34. Resnick, S. I.: Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust, vol. 4. Springer, New York (1987).
35. Resnick, S.: The extremal dependence measure and asymptotic independence. Stoch. Mod. 20, 205-227 (2004).
36. Resnick, S. I.: Heavy-Tail Phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007).
37. Schmidt, R., Stadtmüller, U.: Non-parametric estimation of tail dependence. Scand. J. Stat. 33, 307-335 (2006).
38. Smith, R. L.: Estimating tails of probability distributions. Ann. Stat. 15, 1174-1207 (1987).
39. van der Vaart, A. W., Wellner, J. A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York (1996). Corrected 2nd printing 2000.
40. Wüthrich, M. V.: Asymptotic Value-at-Risk estimates for sums of dependent random variables. Astin Bull. 33, 75-92 (2003).