Skip to main content
SpringerLink
Log in

Worst case portfolio vectors and diversification effects

  • Published:
Finance and Stochastics Aims and scope Submit manuscript

Abstract

We consider the problem of identifying the worst case dependence structure of a portfolio X 1,…,X n of d-dimensional risks, which yields the largest risk of the joint portfolio. Based on a recent characterization result of law invariant convex risk measures, the worst case portfolio structure is identified as a μ-comonotone risk vector for some worst case scenario measure μ. It turns out that typically there will be a diversification effect even in worst case situations. The only exceptions arise when risks are measured by translated max correlation risk measures. We determine the worst case portfolio structure and the worst case diversification effect in several classes of examples as, e.g. in elliptical, Euclidean spherical, and Archimedean type distribution classes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aurenhammer, F., Hoffmann, F., Aronov, B.: Minkowski type theorems and least squares clustering. Algorithmica 20, 61–76 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insurance Math. Econom. 38, 289–297 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cheridito, P., Li, T.: Risk measures on Orlicz hearts. Math. Finance 19, 189–214 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cuesta-Albertos, J.A., Rüschendorf, L., Tuero-Diaz, A.: Optimal coupling of multivariate distributions and stochastic processes. J. Multivariate Anal. 46, 335–361 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics, pp. 1–37. Springer, Berlin (2002)

    Google Scholar 

  6. Ekeland, I., Galichon, A., Henry, M.: Comonotonic measures of multivariate risk, 2009. Preprint, http://ssrn.com/abstract=1115729

  7. Embrechts, P., Puccetti, G.: Bounds for functions of dependent risks. Finance Stoch. 10, 341–352 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Embrechts, P., Puccetti, G.: Bounds for functions of multivariate risks. J. Multivariate Anal. 97, 526–547 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Filipović, D., Svindland, G.: Convex risk measures on L p. Preprint, 2007. http://www.math.lmu.de/~filipo/PAPERS/crmlp.pdf

  10. Filipović, D., Svindland, G.: The canonical model space for law-invariant convex risk measures is L 1. Math. Financ. (to appear)

  11. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6, 429–447 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time, 2nd edn. de Gruyter, Berlin (2004)

    Book  MATH  Google Scholar 

  13. Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Banking Finance 26, 1473–1486 (2002)

    Article  Google Scholar 

  14. Frittelli, M., Rosazza Gianin, E.: Law invariant convex risk measures. Adv. Math. Econ. 26, 42–53 (2005)

    Google Scholar 

  15. Inoue, A.: On the worst case conditional expectation. J. Math. Anal. Appl. 286, 237–247 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jouini, E., Schachermayer, W., Touzi, W.: Law invariant risk measures have the Fatou property. In: Kusuoka, S., Yamazaki, Y. (eds.) Advances in Mathematical Economics. vol. 9, pp. 49–71 (2006)

    Chapter  Google Scholar 

  17. Kaina, M., Rüschendorf, L.: On convex risk measures on L p-spaces. Math. Methods Oper. Res. 69, 475–495 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    MathSciNet  Google Scholar 

  19. Meilijson, I., Nadas, A.: Convex majorization with an application to the length of critical paths. J. Appl. Probab. 16, 671–677 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. Puccetti, G., Scarsini, M.: Multivariate comonotonicity. J. Multivariate Anal. 101, 291–304 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rachev, S.T., Rüschendorf, L.: Mass Transportation Problems, vol. I: Theory. Springer, Berlin (1998)

    Google Scholar 

  22. Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433–452 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Rüschendorf, L.: Comparison of multivariate risks and positive dependence. J. Appl. Probab. 41, 391–406 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Rüschendorf, L.: Stochastic ordering of risks. In: Balakrishnan, N., Bairamov, I.G., Gebizlioglu, O.L. (eds.) Advances on Models, Characterization and Applications. Statistics: Textbooks and Monographs, vol. 180, pp. 19–56. Chapman & Hall/CRC, Boca Raton (2005)

    Chapter  Google Scholar 

  25. Rüschendorf, L.: Law invariant convex risk measures for portfolio vectors. Statist. Decisions 24, 97–108 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rüschendorf, L., Rachev, S.T.: A characterization of random variables with minimum L 2-distance. J. Multivariate Anal. 32, 48–54 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rüschendorf, L., Uckelmann, L.: On optimal multivariate couplings. In: Beneš, V., et al. (eds.) Distributions with Given Marginals and Moment Problems, Prague, Czech Republic. Proceedings of the 1996 Conference, pp. 261–273. Kluwer Academic, Dordrecht (1997)

    Chapter  Google Scholar 

  28. Rüschendorf, L., Uckelmann, L.: Numerical and analytical results for the transportation problem of Monge–Kantorovich. Metrika 51, 245–258 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Rüschendorf, L., Uckelmann, L.: On the n-coupling problem. J. Multivariate Anal. 81, 242–258 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Stochastics, University of Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    Ludger Rüschendorf

Authors
  1. Ludger Rüschendorf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ludger Rüschendorf.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rüschendorf, L. Worst case portfolio vectors and diversification effects. Finance Stoch 16, 155–175 (2012). https://doi.org/10.1007/s00780-010-0150-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00780-010-0150-8

Keywords

Mathematics Subject Classification (2000)

JEL Classification

Search

Navigation