Finance and Stochastics

, Volume 11, Issue 3, pp 373–397 | Cite as

Multivariate risks and depth-trimmed regions



We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this axiomatic framework.

It is shown that the concept of depth-trimmed (or central) regions from multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.


Acceptance set Cone Depth-trimmed region Multivariate risk Risk measure 

Mathematics Subject Classification (2000)

91B30 91B82 60D05 62H99 


C60 C61 


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.
    Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bäuerle, N., Müller, A.: Stochastic orders and risk measures: consistency and bounds. Insur. Math. Econ. 38, 132–148 (2006) MATHCrossRefGoogle Scholar
  4. 4.
    Burgert, C., Rüschendorf, L.: Consistent risk measures for portfolio vectors. Insur. Math. Econ. 38, 289–297 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Cascos, I.: Depth functions based on a number of observations of a random vector. Working paper 07-29. Statistics and Econometrics Series, Universidad Carlos III de Madrid (2007). Available from
  6. 6.
    Cascos, I., López-Díaz, M.: Integral trimmed regions. J. Multivar. Anal. 96, 404–424 (2005) MATHCrossRefGoogle Scholar
  7. 7.
    Cherny, A.S., Madan, D.B.: CAPM, rewards, and empirical asset pricing with coherent risk. Arxiv:math.PR/0605065 (2006) Google Scholar
  8. 8.
    Davydov, Y., Molchanov, I., Zuyev, S.: Strictly stable distributions on convex cones. ArXiv math.PR/0512196 (2005) Google Scholar
  9. 9.
    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
  10. 10.
    Embrechts, P., Puccetti, G.: Bounds for functions of multivariate risks. J. Multivar. Anal. 97, 526–547 (2006) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stochastics 6, 429–447 (2002) MATHCrossRefGoogle Scholar
  12. 12.
    Hamel, A.H.: Translative sets and functions and their applications to risk measure theory and nonlinear separation. Available from (2006)
  13. 13.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic, Boston (1994) MATHGoogle Scholar
  14. 14.
    Jarrow, R.: Put option premiums and coherent risk measures. Math. Finance 12, 135–142 (2002) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jaschke, S., Küchler, U.: Coherent risk measures and good-deal bounds. Finance Stochastics 5, 181–200 (2001) MATHCrossRefGoogle Scholar
  16. 16.
    Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stochastics 8, 531–552 (2004) MATHMathSciNetGoogle Scholar
  17. 17.
    Kabanov, Y.M.: Hedging and liquidation under transaction costs in currency markets. Finance Stochastics 3, 237–248 (1999) MATHCrossRefGoogle Scholar
  18. 18.
    Koshevoy, G.A., Mosler, K.: Zonoid trimming for multivariate distributions. Ann. Stat. 25, 1998–2017 (1997) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. 1. North-Holland, Amsterdam (1971) Google Scholar
  20. 20.
    Massé, J.C., Theodorescu, R.: Halfplane trimming for bivariate distribution. J. Multivar. Anal. 48, 188–202 (1994) MATHCrossRefGoogle Scholar
  21. 21.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005) MATHGoogle Scholar
  22. 22.
    Mosler, K.: Multivariate Dispersion, Central Regions and Depth. The Lift Zonoid Approach. Lecture Notes in Statistics, vol. 165. Springer, Berlin (2002) MATHGoogle Scholar
  23. 23.
    Rousseeuw, P.J., Ruts, I.: The depth function of a population distribution. Metrika 49, 213–244 (1999) MATHMathSciNetGoogle Scholar
  24. 24.
    Zuo, Y., Serfling, R.: General notions of statistical depth function. Ann. Stat. 28, 461–482 (2000) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Zuo, Y., Serfling, R.: Structural properties and convergence results for contours of sample statistical depth functions. Ann. Stat. 28, 483–499 (2000) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversidad Carlos III de MadridLeganés, MadridSpain
  2. 2.Department of Mathematical Statistics and Actuarial ScienceUniversity of BerneBerneSwitzerland

Personalised recommendations