Advertisement

Statistical estimation of composite risk functionals and risk optimization problems

  • Darinka Dentcheva
  • Spiridon PenevEmail author
  • Andrzej Ruszczyński
Article

Abstract

We address the statistical estimation of composite functionals which may be nonlinear in the probability measure. Our study is motivated by the need to estimate coherent measures of risk, which become increasingly popular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit theorems for composite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal values when an estimator of the risk functional is used. While the mathematical structures accommodate commonly used coherent measures of risk, they have more general character, which may be of independent interest.

Keywords

Risk measures Composite functionals Central limit theorem 

Notes

Acknowledgments

The first author was partially supported by the NSF Grant DMS-1311978. The second author was partially supported by a research Grant PS27205 of The University of New South Wales and by Australian Research Council’s Discovery Project funding scheme (Project DP160103489). The third author was partially supported by the NSF Grant DMS-1312016.

References

  1. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.Google Scholar
  2. Belomestny, D., Krätschmer, V. (2012). Central limit theorems for law-invariant coherent risk measures. Journal of Applied Probabability, 49(1), 1–21.Google Scholar
  3. Ben-Tal, A., Teboulle, M. (2007). An old-new concept of risk measures: The optimized certainty equivalent. Mathematical Finance, 17(3), 449–476.Google Scholar
  4. Beutner, E., Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. Journal of Multivariate Analysis, 101(10), 2452–2463.Google Scholar
  5. Bonnans, J. F., Shapiro, A. (2000). Perturbation analysis of optimization problems. New York: Springer.Google Scholar
  6. Brazauskas, V., Jones, B. L., Puri, M. L., Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. Journal of Statistical Planning and Inference, 138(11), 3590–3604.Google Scholar
  7. Cheridito, P., Li, T. H. (2009). Risk measures on Orlicz hearts. Mathematical Finance, 19, 189–214.Google Scholar
  8. Dentcheva, D., Penev, S. (2010). Shape-restricted inference for Lorenz curves using duality theory. Statistics & Probability Letters, 80, 403–412.Google Scholar
  9. Dentcheva, D., Penev, S., Ruszczyński, A. (2010). Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181, 325–335.Google Scholar
  10. Dentcheva, D., Ruszczyński, A. (2014). Risk preferences on the space of quantile. Mathematical Programming, 148(1–2), 181–200.Google Scholar
  11. Dentcheva, D., Stock, G. J., Rekeda, L. (2011). Mean-risk tests of stochastic dominance. Statistics & Decisions, 28, 97–118.Google Scholar
  12. Föllmer, H., Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447.Google Scholar
  13. Föllmer, H., Schied, A. (2011). Stochastic finance. An introduction in discrete time (3rd ed.). Berlin: de Gruyter.Google Scholar
  14. Frittelli, M., Rosazza Gianin, E. (2005). Law invariant convex risk measures. Advances in mathematical economics (Vol. 7, pp. 33–46). Tokyo: Springer.Google Scholar
  15. Gülten, S., Ruszczyński, A. (2015). Two-stage portfolio optimization with higher-order conditional measures of risk. Annals of Operations Research, 229(1), 409–427.Google Scholar
  16. Jahn, J. (2011). Vector optimization. Theory, applications and extensions (2nd ed.). Berlin: Springer-Verlag.Google Scholar
  17. Jones, B. L., Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 44–54.Google Scholar
  18. Jones, B. L., Zitikis, R. (2007). Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics, 41(2), 279–297.Google Scholar
  19. Kijima, M., Ohnishi, M. (1993). Mean-risk analysis of risk aversion and wealth effects on optimal portfolios with multiple investment possibilities. Annals of Operations Research, 45, 147–163.Google Scholar
  20. Krokhmal, P. (2007). Higher moment coherent risk measures. Quantitative Finance, 7, 373–387.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77–91.Google Scholar
  23. Markowitz, H. M. (1987). Mean-variance analysis in portfolio choice and capital markets. Oxford: Blackwell.zbMATHGoogle Scholar
  24. Matmoura, Y., Penev, S. (2013). Multistage optimization of option portfolio using higher order coherent risk measures. European Journal of Operational Research, 227, 190–198.Google Scholar
  25. Ogryczak, W., Ruszczyński, A. (1999). From stochastic dominance to mean-risk models: Semideviations and risk measures. European Journal of Operational Research, 116, 33–50.Google Scholar
  26. Ogryczak, W., Ruszczyński, A. (2001). On consistency of stochastic dominance and mean-semideviation models. Mathematical Programming, 89, 217–232.Google Scholar
  27. Ogryczak, W., Ruszczyński, A. (2002). Dual stochastic dominance and related mean-risk models. Society for Industrial and Applied Mathematics Journal of Optimization, 13(1), 60–78.Google Scholar
  28. Pflug, G., Römisch, W. (2007). Modeling, measuring and managing risk. Hackensack, New Jersey: World Scientific.Google Scholar
  29. Pflug, G., Wozabal, N. (2010). Asymptotic distribution of law-invariant risk functionals. Finance and Stochastics, 14, 397–418.Google Scholar
  30. Rockafellar, R. T. (1974). Conjugate duality and optimization. Volume 16 of CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
  31. Rockafellar, R. T., Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26, 1443–1471.Google Scholar
  32. Rockafellar, R. T., Uryasev, S., Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance and Stochastics, 10, 51–74.Google Scholar
  33. Römisch, W. (2003). Stability of stochastic programming problems. In A. Ruszczynski, A. Shapiro (Eds.), Stochastic Programming. Handbooks in Operations Research and Management Science, Vol. 10 (pp. 483–554). Amsterdam: Elsevier.Google Scholar
  34. Römisch, W. (2006) Delta method, infinite dimensional. In S. Kotz, C. B. Read, N. Balakrishnan, B. Vidakovic (Eds.), Encyclopedia of statistical sciences (2nd ed.). Wiley Online Library. Published online 15 Aug 2006.Google Scholar
  35. Ruszczyński, A., Shapiro, A. (2006). Optimization of convex risk functions. Mathematics of Operations Research, 31, 433–452.Google Scholar
  36. Shapiro, A. (2008). Asymptotics of minimax stochastic programs. Statistics & Probability Letters, 78(2), 150–157.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Shapiro, A., Dentcheva, D., Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory. Philadelphia: Society for Industrial and Applied Mathematics Publications.Google Scholar
  38. Stoyanov, S., Racheva-Iotova, B., Rachev, S., Fabozzi, F. (2010). Stochastic models for risk estimation in volatile markets: A survey. Annals of Operations Research, 176, 293–309.Google Scholar
  39. Tsukahara, H. (2013). Estimation of distortion risk measures. Journal of Financial Econometrics, 12(1), 213–235.CrossRefGoogle Scholar
  40. Van der Vaart, A. W. (1998). Asymptotic statistics. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2016

Authors and Affiliations

  • Darinka Dentcheva
    • 1
  • Spiridon Penev
    • 2
    Email author
  • Andrzej Ruszczyński
    • 3
  1. 1.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA
  2. 2.Department of Statistics, School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia
  3. 3.Department of Management Science and Information SystemsRutgers UniversityNew BrunswickUSA

Personalised recommendations