Introduction to the Theory of Probabilistic Functions and Percentiles (Value-at-Risk)

  • S. Uryasev
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 49)


Probabilistic and quantile (percentile) functions are commonly used for the analysis of models with uncertainties or variabilities in parameters. In financial applications, the percentile of the losses is called Value-at-Risk (VaR). VaR, a widely used performance measure, answers the question: what is the maximum loss with a specified confidence level? Percentiles are also used for defining other relevant performance measures, such as Conditional Value-at-Risk (CVaR). CVaR (also called Mean Excess Loss, Mean Shortfall, or Tail VaR) is the average loss for the worst x% scenarios (e.g., 5%). CVaR risk measure has more attractive properties compared to VaR. This introductory paper gives basic definitions and reviews several topics:
  • sensitivities of probabilistic functions;

  • sensitivities of percentiles (VaR);

  • optimization approaches for CVaR.

The emphasis of this paper is on issues which have been relatively recently developed.


Probability Function Credit Risk Constraint Function Quantile Function Stochastic Programming Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Andersson, F., Mausser, H., Rosen, D., and S. Uryasev (2000) Credit Risk Optimization With Conditional Value-At-Risk Criterion. Mathematical Programming. To appear.Google Scholar
  2. [2]
    Artzner, P., Delbaen F., Eber, J.M., and D. Heath (1999) Coherent Measures of Risk. Mathematical Finance, 9, 203–228.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Birge J.R. and F. Louveaux (1997) Introduction to Stochastic Programming. Springer, New York.MATHGoogle Scholar
  4. [4]
    Bucay, N. and D. Rosen (1999) Credit Risk of an International Bond Portfolio: a Case Study. ALGO Research Quarterly. Vol.2, No. 1, 9–29.Google Scholar
  5. [5]
    Dembo, R.S. and A.J. King (1992) Tracking Models and the Optimal Regret Distribution in Asset Allocation. Applied Stochastic Models and Data Analysis. Vol. 8, 151–157.CrossRefGoogle Scholar
  6. [6]
    Duffie, D. and J. Pan (1997) An Overview of Value-at-Risk. Journal of Derivatives. 4, 7–49.CrossRefGoogle Scholar
  7. [7]
    Ermoliev, Yu. (1976) Stochastic Programming Methods. Nauka, Moscow, (in Russian).Google Scholar
  8. [8]
    Ermoliev, Yu. (1983) Stochastic Quasi-Gradient Methods and Their Applications to System Optimization. Stochastics, 4, 1–36.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Ermoliev, Yu. and R. J-B Wets (Eds.) (1988): Numerical Techniques for Stochastic Optimization, Springer Series in Computational Mathematics, 10.MATHGoogle Scholar
  10. [10]
    Glasserman P. (1991) Gradient Estimation Via Perturbation Analysis. Kluwer, Boston.MATHGoogle Scholar
  11. [11]
    Ho, Y.C. and X.R. Cao (1991) Perturbation Analysis of Discrete Event Dynamic Systems. Kluwer, Boston.MATHCrossRefGoogle Scholar
  12. [12]
    Ioffe, A.D. and V.L. Levin (1972) Sub differentials of Convex Functions. Papers of Moscow Mathematical Society(Trudy MMO), V. 26, 3–73 (in Russian).MathSciNetMATHGoogle Scholar
  13. [13]
    Kail, P. and S.W. Wallace (1994) Stochastic Programming. Willey, Chichester.Google Scholar
  14. [14]
    Kan, Y.S., and Kibzun, A.I. (1996). Stochastic Programming Problems with Probability and Quantile Functions, John Wiley & Sons, 316.MATHGoogle Scholar
  15. [15]
    Kast, R., Luciano, E., and L. Peccati (1998) VaR and Optimization. 2nd International Workshop on Preferences and Decisions, Trento, July 1–3 1998.Google Scholar
  16. [16]
    Kibzun, A.I., Malyshev, V.V., and D.E. Chernov (1988) Two Approaches to Solutions of Probabilistic Optimization Problems. Soviet Journal of Automaton and Information Silences, V.20, No.3, 20–25.MathSciNetGoogle Scholar
  17. [17]
    Kibzun, A.I. and G.L. Tretyakov (1996) Probability Function Differentiability. Doklady RAN, (in Russian).Google Scholar
  18. [18]
    Kibzun A.I. and S. Uryasev. (1998) Differentiability of Probability Functions. Stochastic Analysis and Applications. 16(6), 1101–1128.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Konno, H. and H. Yamazaki (1991): Mean Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market. Management Science. 37, 519–531.CrossRefGoogle Scholar
  20. [20]
    Litterman, R. (1997) Hot Spots and Hedges (I). Risk. 10 (3), 42–45.Google Scholar
  21. [21]
    Litterman, R. (1997) Hot Spots and Hedges (II). Risk, 10 (5), 38–42.Google Scholar
  22. [22]
    Lucas, A., and Klaassen, P. (1998) Extreme Returns, Downside Risk, and Optimal Asset Allocation. Journal of Portfolio Management, Vol. 25, No. 1, 71–79.CrossRefGoogle Scholar
  23. [23]
    Marti, K. (1994) Approximations and Derivatives of Probability Functions. In Approximation, Probability and Related Fields, edited by G. Anastassiou and S.T. Rachev, Plenum Press, New York., 367–377.CrossRefGoogle Scholar
  24. [24]
    Marti K. (1995) Differentiation Formulas for Probability Functions: The Transformation Method. Mathematical Programming Journal, Series B, Vol. 75, No. 2.Google Scholar
  25. [25]
    Markowitz, H.M. (1952) Portfolio Selection. Journal of finance. Vol.7, 1, 77–91.Google Scholar
  26. [26]
    Mausser, H. and D. Rosen (1998) Beyond VaR: From Measuring Risk to Managing Risk, ALGO Research Quarterly, Vol. 1, No. 2, 5–20.Google Scholar
  27. [27]
    Mausser, H. and D. Rosen (1999) Applying Scenario Optimization to Portfolio Credit Risk, ALGO Research Quarterly, Vol. 2, No. 2, 19–33.Google Scholar
  28. [28]
    Palmquist, J., Uryasev, S., and P. Krokhmal (2000) Portfolio Optimization with Conditional Value-At-Risk Objective and Constraints. The Journal of Risk. To appear.Google Scholar
  29. [29]
    Pflug, G.Ch. (1996) Optimization of Stochastic Models: The Interface Between Simulation and Optimization. Kluwer Academic Publishers, Dordrecht, Boston.MATHCrossRefGoogle Scholar
  30. [30]
    Pflug, G.Ch. (2000) Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. In.”Probabilistic Constrained Optimization: Methodology and Applications”, Ed. S. Uryasev, Kluwer Academic Publishers, 2000Google Scholar
  31. [31]
    Prékopa, A. (1970) On Probabilistic Constrained Programming, in: Proceedings of the Princeton Symposium on Mathematical Programming, (Princeton University Press, Princeton, N.J.), 113–138.Google Scholar
  32. [32]
    Prekopa, A. (1995) Stochastic Programming, Kluwer Academic Publishers.Google Scholar
  33. [33]
    Pulkkinen, A. and S. Uryasev (1991) Optimal Operational Strategies for an Inspected ComponentSolution Techniques. Collaborative Paper CP-91–13, International Institute for Applied Systems Analysis, Laxenburg, Austria.Google Scholar
  34. [34]
    Prékopa, A. and T. Szântai (1978) A New Multivariate Gamma Distribution and Its Fitting to Empirical Streamflow Data. Water Resources Research, 14, 19–24.CrossRefGoogle Scholar
  35. [35]
    Pritsker, M. (1997) Evaluating Value at Risk Methodologies, Journal of Financial Services Research, 12:2/3, 201–242.CrossRefGoogle Scholar
  36. [36]
    Raik, E. (1975) The Differentiability in the Parameter of the Probability Function and Optimization of the Probability Function via the Stochastic Pseudogradient Method. Eesti NSV Teaduste Akadeemia Toimetised. Füüsika-Matemaatika, 24, 1, 3–6 (in Russian).MathSciNetMATHGoogle Scholar
  37. [37]
    RiskMetrics™ (1996) Technical Document, 4-th Edition, New York, NY, J.P.Morgan Inc., December.Google Scholar
  38. [38]
    Rockafellar, R.T. (1970): Convex Analysis. Princeton Mathematics, Vol. 28, Princeton Univ. Press.MATHGoogle Scholar
  39. [39]
    Rockafellar, R.T. and S. Uryasev (2000) Optimization of Conditional Value-At-Risk. The Journal of Risk, Vol. 2, No. 3.Google Scholar
  40. [40]
    Rockafellar, R.T. and R. J.-B. Wets (1982) On the Interchange of Subdifferenti-ation and Conditional Expectation for Convex Functionals. Stochastics, 7, 173–182.MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Roenko, N. (1983) Stochastic Programming Problems with Integral Functionals over Multivalued Mappings, (Ph.D. Thesis, Kiev, Ukraine) (in Russian).Google Scholar
  42. [42]
    Rubinstein, R. (1992) Sensitivity Analysis of Discrete Event Systems by the “Push Out” Method. Annals of Operations Research, 39.Google Scholar
  43. [43]
    Rubinstein, R. and A. Shapiro (1993) Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization via the Score Function Method. Willey, Chichester.Google Scholar
  44. [44]
    Simon, J. (1989) Second Variation in Domain Optimization Problems. In International Series of Numerical Mathematics, F. Kappel, K. Kunish and W. Schappacher, Birkhauser Verlag, 91, 361–378.Google Scholar
  45. [45]
    Simons, K. (1996) Value-at-Risk New Approaches to Risk Management. New England Economic Review, Sept/Oct, 3–13.Google Scholar
  46. [46]
    Stambaugh, F. (1996) Risk and Value-at-Risk. European Management Journal, Vol. 14, No. 6, 612–621.CrossRefGoogle Scholar
  47. [47]
    Thompson, K.M., D.E. Burmasater and E.A.C. Crouch (1992) Monte Carlo Techniques for Quantitative Uncertainty Analysis in Public Risk Assessments. Risk Analysis, 12.Google Scholar
  48. [48]
    Uryasev, S. (1987) Adaptive Algorithms for Stochastic Optimization and Game Theory. Nauka, Moscow (in Russian).Google Scholar
  49. [49]
    Uryasev, S. (1989) A Differentiation Formula for Integrals over Sets Given by Inclusion. Numerical Functional Analysis and Optimization, 10 (7 & 8), 827–841.MathSciNetCrossRefGoogle Scholar
  50. [50]
    Uryasev, S. (1994) Derivatives of Probability Functions and Integrals over Sets Given by Inequalities. J. Computational and Applied Mathematics, 56, 197–223.MathSciNetCrossRefGoogle Scholar
  51. [51]
    Uryasev, S. (1995) Derivatives of Probability Functions and some Applications. Annals of Operations Research, 56, 287–311.MathSciNetMATHCrossRefGoogle Scholar
  52. [52]
    Uryasev, S. (2000) Conditional Value-at-Risk: Optimization Algorithms and Applications. Financial Engineering News, No. 14, February.Google Scholar
  53. [53]
    Uryasev, S. and A. Shlyakhter (1994) A Procedure for Simultaneous Calculation of Sensitivities in Probabilistic Risk Analysis. In: Abstracts Society for Risk Analysis Annual Conference and Exposition, Baltimore, Maryland, December 1994.Google Scholar
  54. [54]
    Young, M.R. (1998): A Minimax Portfolio Selection Rule with Linear Programming Solution. Management Science. Vol.44, No. 5, 673–683.MATHCrossRefGoogle Scholar
  55. [55]
    Ziemba, W.T. and J.M. Mulvey (Eds.) (1998): Worldwide Asset and Liability Modeling, Cambridge Univ. Pr.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • S. Uryasev
    • 1
  1. 1.University of FloridaGainesvilleUSA

Personalised recommendations