Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation

  • Matthew NortonEmail author
  • Valentyn Khokhlov
  • Stan Uryasev
S.I.: Recent Developments in Financial Modeling and Risk Management


Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered probability of exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distributions. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distributions. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distributions simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantiles (MOS), a simple variation of the method of moments where moments are replaced by superquantiles at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.


Conditional value-at-risk Buffered probability of exceedance Superquantile Density estimation Portfolio optimization 



This research was supported by the Naval Postgraduate School’s Research Initiation Program.


  1. Andreev, A., Kanto, A., & Malo, P. (2005). On closed-form calculation of CVaR. Helsinki School of Economics working paper W-389.Google Scholar
  2. Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
  3. Davis, J. R., & Uryasev, S. (2016). Analysis of tropical storm damage using buffered probability of exceedance. Natural Hazards, 83(1), 465–483.CrossRefGoogle Scholar
  4. Everitt, B. S. (2006). The Cambridge dictionary of statistics. Cambridge: Cambridge University Press.Google Scholar
  5. Karian, Z. A., & Dudewicz, E. J. (1999). Fitting the generalized lambda distribution to data: A method based on percentiles. Communications in Statistics: Simulation and Computation, 28(3), 793–819.CrossRefGoogle Scholar
  6. Landsman, Z. M., & Valdez, E. A. (2003). Tail conditional expectation for elliptical distributions. North American Actuarial Journal, 7(4), 55–71.CrossRefGoogle Scholar
  7. Mafusalov, A., Shapiro, A., & Uryasev, S. (2018). Estimation and asymptotics for buffered probability of exceedance. European Journal of Operational Research, 270(3), 826–836.CrossRefGoogle Scholar
  8. Mafusalov, A., & Uryasev, S. (2018). Buffered probability of exceedance: Mathematical properties and optimization. SIAM Journal on Optimization, 28(2), 1077–1103.CrossRefGoogle Scholar
  9. Norton, M., Mafusalov, A., & Uryasev, S. (2017). Soft margin support vector classification as buffered probability minimization. The Journal of Machine Learning Research, 18(1), 2285–2327.Google Scholar
  10. Norton, M., & Uryasev, S. (2016). Maximization of AUC and buffered AUC in binary classification. Mathematical Programming, 174(1–2), 575–612.Google Scholar
  11. Rockafellar, R., & Royset, J. (2010). On buffered failure probability in design and optimization of structures. Reliability Engineering & System Safety, 95, 499–510.CrossRefGoogle Scholar
  12. Rockafellar, R., & Uryasev, S. (2000). Optimization of conditional value-at-risk. The Journal of Risk, 2(3), 21–41.CrossRefGoogle Scholar
  13. Rockafellar, R. T., & Royset, J. O. (2014). Random variables, monotone relations, and convex analysis. Mathematical Programming, 148(1–2), 297–331.CrossRefGoogle Scholar
  14. Rockafellar, R. T., & Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443–1471.CrossRefGoogle Scholar
  15. Sgouropoulos, N., Yao, Q., & Yastremiz, C. (2015). Matching a distribution by matching quantiles estimation. Journal of the American Statistical Association, 110(510), 742–759.CrossRefGoogle Scholar
  16. Shang, D., Kuzmenko, V., & Uryasev, S. (2018). Cash flow matching with risks controlled by buffered probability of exceedance and conditional value-at-risk. Annals of Operations Research, 260(1–2), 501–514.CrossRefGoogle Scholar
  17. Uryasev, S. (2014). Buffered probability of exceedance and buffered service level: Definitions and properties. Department of Industrial and Systems Engineering, University of Florida, research report 3.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Matthew Norton
    • 1
    Email author
  • Valentyn Khokhlov
    • 2
  • Stan Uryasev
    • 3
  1. 1.Operations Research DepartmentNaval Postgraduate SchoolMontereyUSA
  2. 2.CFA SocietyKyivUkraine
  3. 3.Department of Industrial and Systems Engineering, Risk Management and Financial Engineering LaboratoryUniversity of FloridaGainesvilleUSA

Personalised recommendations