Finance and Stochastics

, Volume 14, Issue 4, pp 593–623 | Cite as

On optimal portfolio diversification with respect to extreme risks

  • Georg Mainik
  • Ludger Rüschendorf


Extreme losses of portfolios with heavy-tailed components are studied in the framework of multivariate regular variation. Asymptotic distributions of extreme portfolio losses are characterized by a functional γ ξ =γ ξ (α,Ψ) of the tail index α, the spectral measure Ψ, and the vector ξ of portfolio weights. Existence, uniqueness, and location of the optimal portfolio are analysed and applied to the minimization of risk measures. It is shown that diversification effects are positive for α>1 and negative for α<1. Strong consistency and asymptotic normality are established for a semiparametric estimator of the mapping ξ γ ξ . Strong consistency is also established for the estimated optimal portfolio.


Portfolio optimization Risk management Diversification effects Multivariate extremes 

Mathematics Subject Classification (2000)

62G32 62G05 62G20 62P05 

JEL Classification

C13 C14 G11 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002) CrossRefGoogle Scholar
  2. 2.
    Barbe, P., Fougères, A.-L., Genest, C.: On the tail behaviour of sums of dependent risks. ASTIN Bull. 36, 361–373 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 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) zbMATHGoogle Scholar
  5. 5.
    Böcker, K., Klüppelberg, C.: Modelling and measuring multivariate operational risk with Lévy copulas. J. Oper. Risk 3(2), 3–27 (2008) Google Scholar
  6. 6.
    Boman, J., Lindskog, F.: Support theorems for the Radon transform and Cramér–Wold theorems. J. Theor. Probab. 22, 683–710 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 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).
  8. 8.
    Davis, R., Resnick, S.: Tail estimates motivated by extreme value theory. Ann. Stat. 12, 1467–1487 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    de Haan, L., Ferreira, A.: Extreme Value Theory. Springer Series in Operations Research and Financial Engineering. Springer, New York (2006) zbMATHGoogle Scholar
  10. 10.
    de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 317–337 (1977) zbMATHCrossRefGoogle Scholar
  11. 11.
    de Haan, L., Resnick, S.I.: Estimating the limit distribution of multivariate extremes. Commun. Stat. Stoch. Mod. 9, 275–309 (1993) zbMATHCrossRefGoogle Scholar
  12. 12.
    de Haan, L., Sinha, A.K.: Estimating the probability of a rare event. Ann. Stat. 27, 732–759 (1999) zbMATHCrossRefGoogle Scholar
  13. 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) zbMATHCrossRefGoogle Scholar
  14. 14.
    Drees, H.: Refined Pickands estimators of the extreme value index. Ann. Stat. 23, 2059–2080 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Drees, H., Ferreira, A., de Haan, L.: On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab. 14, 1179–1201 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Einmahl, J.H.J., de Haan, L., Huang, X.: Estimating a multidimensional extreme-value distribution. J. Multivar. Anal. 47, 35–47 (1993) zbMATHCrossRefGoogle Scholar
  17. 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) zbMATHCrossRefGoogle Scholar
  18. 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) zbMATHCrossRefGoogle Scholar
  19. 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) CrossRefMathSciNetGoogle Scholar
  20. 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) zbMATHCrossRefGoogle Scholar
  21. 21.
    Falk, M., Hüsler, J., Reiss, R.-D.: Laws of Small Numbers: Extremes and Rare Events. DMV Seminar, vol. 23. Birkhäuser, Basel (1994) zbMATHGoogle Scholar
  22. 22.
    Hajós, G., Rényi, A.: Elementary proofs of some basic facts concerning order statistics. Acta Math. Acad. Sci. Hungar. 5, 1–6 (1954) zbMATHCrossRefGoogle Scholar
  23. 23.
    Hauksson, H.A., Dacorogna, M.M., Domenig, T., Müller, U.A., Samorodnitsky, G.: Multivariate extremes, aggregation and risk estimation. SSRN eLibrary (2000).
  24. 24.
    Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975) zbMATHCrossRefGoogle Scholar
  25. 25.
    Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80(94), 121–140 (2006) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Joe, H.: Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997) zbMATHGoogle Scholar
  27. 27.
    Klüppelberg, C., Resnick, S.I.: The Pareto copula, aggregation of risks, and the emperor’s socks. J. Appl. Probab. 45, 67–84 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Malevergne, Y., Sornette, D.: Extreme Financial Risks. Springer, Berlin (2006) zbMATHGoogle Scholar
  29. 29.
    McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management. Princeton Series in Finance. Princeton University Press, Princeton (2005) zbMATHGoogle Scholar
  30. 30.
    Moscadelli, M.: The modelling of operational risk: Experience with the analysis of the data collected by the Basel Committee. SSRN eLibrary (2004).
  31. 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) Google Scholar
  32. 32.
    Pickands III, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 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 Google Scholar
  34. 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) zbMATHGoogle Scholar
  35. 35.
    Resnick, S.: The extremal dependence measure and asymptotic independence. Stoch. Mod. 20, 205–227 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Resnick, S.I.: Heavy-Tail Phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007) zbMATHGoogle Scholar
  37. 37.
    Schmidt, R., Stadtmüller, U.: Non-parametric estimation of tail dependence. Scand. J. Stat. 33, 307–335 (2006) zbMATHCrossRefGoogle Scholar
  38. 38.
    Smith, R.L.: Estimating tails of probability distributions. Ann. Stat. 15, 1174–1207 (1987) zbMATHCrossRefGoogle Scholar
  39. 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 zbMATHGoogle Scholar
  40. 40.
    Wüthrich, M.V.: Asymptotic Value-at-Risk estimates for sums of dependent random variables. Astin Bull. 33, 75–92 (2003) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematical StochasticsUniversity of FreiburgFreiburgGermany

Personalised recommendations