Annals of Operations Research

, Volume 229, Issue 1, pp 591–605 | Cite as

Kusuoka representations of coherent risk measures in general probability spaces

  • Nilay NoyanEmail author
  • Gábor Rudolf


Kusuoka representations provide an important and useful characterization of law invariant coherent risk measures in atomless probability spaces. However, the applicability of these results is limited by the fact that such representations do not always exist in probability spaces with atoms, such as finite probability spaces. We introduce the class of functionally coherent risk measures, which allow us to use Kusuoka representations in any probability space. We show that this class contains every law invariant risk measure that can be coherently extended to a family containing all finite discrete distributions. Thus, it is possible to preserve the desirable properties of law invariant coherent risk measures on atomless spaces without sacrificing generality. We also specialize our results to risk measures on finite probability spaces, and prove a denseness result about the family of risk measures with finite Kusuoka representations.


Kusuoka representation Coherent risk measures Spectral risk measures Acceptability functional Law invariance Comonotonicity 



The second author has been funded by TUBITAK-2216 Research Fellowship Programme. The authors thank the Associate Editor and the anonymous referees for their valuable comments and suggestions.


  1. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
  2. Bacharach, M. (1966). Matrix rounding problems. Management Science, 12(9), 732–742.CrossRefGoogle Scholar
  3. Bertsimas, D., & Brown, D. B. (2009). Constructing uncertainty sets for robust linear optimization. Operations Research, 57(6), 1483–1495.CrossRefGoogle Scholar
  4. Cheridito, P., & Li, T. (2008). Dual characterization of properties of risk measures on Orlicz hearts. Mathematics and Financial Economics, 2(1), 29–55.CrossRefGoogle Scholar
  5. Dana, R.-A. (2005). A representation result for concave schur concave functions. Mathematical Finance, 15(4), 613–634.CrossRefGoogle Scholar
  6. Dentcheva, D., Penev, S., & Ruszczyński, A. (2010). Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181, 325–335.CrossRefGoogle Scholar
  7. Dentcheva, D., & Ruszczyński, A. (2013a). Common mathematical foundations of expected utility and dual utility theories. SIAM Journal on Optimization, 23(1), 381–405.CrossRefGoogle Scholar
  8. Dentcheva, D., & Ruszczyński, A. (2013b). Risk preferences on the space of quantile functions. Mathematical Programming, Ser: B (online first). doi: 10.1007/s10107-013-0724-2.
  9. Föllmer, H., & Schied, A. (2004). Stochastic finance. Number 27 in De Gruyter studies in mathematics. Berlin: de Gruyter, 2, rev. and extended edition.Google Scholar
  10. Frittelli, M., & Rosazza Gianin, E. (2005). Law invariant convex risk measures. In Kusuoka, S., Maruyama, T. (Eds.), Advances in mathematical economics (Vo. 7, pp. 33–46).Google Scholar
  11. Grechuk, B., & Zabarankin, M. (2012). Schur convex functionals: Fatou property and representation. Mathematical Finance, 22(2), 411–418.CrossRefGoogle Scholar
  12. Jouini, E., Schachermayer, W., & Touzi, N. (2006). Law-invariant risk measures have the Fatou property. Advances in Mathematical Economics, 9(1), 49–71.CrossRefGoogle Scholar
  13. Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.CrossRefGoogle Scholar
  14. Leitner, J. (2005). A short note on second-order stochastic dominance preserving coherent risk measures. Mathematical Finance, 15(4), 649–651.CrossRefGoogle Scholar
  15. McNeil, A., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton series in finance. Princeton: Princeton University Press.Google Scholar
  16. Nelsen, R. B. (1999). An introduction to copulas. New York: Springer.CrossRefGoogle Scholar
  17. Noyan, N., & Rudolf, G. (2012a). Kusuoka representations of coherent risk measures in finite probability spaces. Technical report, RUTCOR-Rutgers Center for Operations Research, RRR 33-2012.
  18. Noyan, N., & Rudolf, G. (2012b). Optimization with multivariate conditional value-at-risk-constraints. Technical report, Optimization online.
  19. Noyan, N., & Rudolf, G. (2013). Optimization with multivariate conditional value-at-risk-constraints. Operations Research, 61(4), 990–1013.Google Scholar
  20. Pflug, G., & Wozabal, D. (2009). Ambiguity in portfolio selection. In A. H. Dempster, G. Pflug, & G. Mitra (Eds.), Quantitative fund management, Chapman & Hall/CRC financial mathematics series (pp. 377–391). Boca Raton, FL: CRC Press.Google Scholar
  21. Pflug, G. C. (2000). Some remarks on the value-at-risk and the conditional value-at-risk. In S. Uryasev (Ed.), Probabilistic constrained optimization: Methodology and applications. Dordrecht: Kluwer.Google Scholar
  22. Pflug, G. C., & Römisch, W. (2007). Modeling, managing and measuring risk. Singapore: World Scientific publishing.CrossRefGoogle Scholar
  23. Pichler, A., & Shapiro, A. (2012). Uniqueness of Kusuoka representations.
  24. Rockafellar, R. T. (2007). Coherent approaches to risk in optimization under uncertainty. Tutorials in operations research, 3, 38–61.Google Scholar
  25. Ruszczyński, A., & Shapiro, A. (2006). Optimization of convex risk functions. Mathematics of Operations Research, 31(3), 433–452.CrossRefGoogle Scholar
  26. Shapiro, A. (2013). On Kusuoka representation of law invariant risk measures. Mathematics of Operations Research, 38, 142–152.CrossRefGoogle Scholar
  27. Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory. Philadelphia, USA: The society for industrial and applied mathematics and the mathematical programming society.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Manufacturing Systems/Industrial Engineering ProgramSabancı UniversityIstanbulTurkey
  2. 2.Department of Industrial EngineeringKoc UniversityIstanbulTurkey

Personalised recommendations