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Annals of Operations Research

, Volume 152, Issue 1, pp 227–256 | Cite as

Conditional value at risk and related linear programming models for portfolio optimization

  • Renata Mansini
  • Włodzimierz OgryczakEmail author
  • M. Grazia Speranza
Article

Abstract

Many risk measures have been recently introduced which (for discrete random variables) result in Linear Programs (LP). While some LP computable risk measures may be viewed as approximations to the variance (e.g., the mean absolute deviation or the Gini’s mean absolute difference), shortfall or quantile risk measures are recently gaining more popularity in various financial applications. In this paper we study LP solvable portfolio optimization models based on extensions of the Conditional Value at Risk (CVaR) measure. The models use multiple CVaR measures thus allowing for more detailed risk aversion modeling. We study both the theoretical properties of the models and their performance on real-life data.

Keywords

Portfolio optimization Mean-risk models Linear programming Stochastic dominance Conditional Value at Risk Gini’s mean difference 

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References

  1. Acerbi, C. (2002).“Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion.” Journal of Banking & Finance, 26, 1505–1518.CrossRefGoogle Scholar
  2. Acerbi, C. and P. Simonetti. (2002). “Portfolio Optimization with Spectral Measures of Risk.” Working Paper (http://gloriamundi.org).
  3. Andersson, F., H. Mausser, D. Rosen, and S. Uryasev. (2001). “Credit Risk Optimization with Conditional Value-at-Risk Criterion.” Mathematical Programming, 89, 273–291.CrossRefGoogle Scholar
  4. Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). “Coherent Measures of Risk.” Mathematical Finance, 9, 203–228.CrossRefGoogle Scholar
  5. Mansini, R. and M.G. Speranza. (2005). “An Exact Approach for the Portfolio Selection Problem with Transaction Costs and Rounds.” IIE Transactions, 37, 919–929.Google Scholar
  6. Chiodi, L., R. Mansini, and M.G. Speranza. (2003). “Semi-Absolute Deviation Rule for Mutual Funds Portfolio Selection.” Annals of Operations Research, 124, 245–265.CrossRefGoogle Scholar
  7. Embrechts, P., C. Klüppelberg, and T. Mikosch. (1997). Modelling Extremal Events for Insurance and Finance. New York: Springer-Verlag.Google Scholar
  8. Haimes, Y.Y. (1993). “Risk of Extreme Events and the Fallacy of the Expected Value.” Control and Cybernetics, 22, 7–31.Google Scholar
  9. Jorion, P. (2001). Value-at-Risk: The New Benchmark for Managing Financial Risk. NY: McGraw-Hill.Google Scholar
  10. Kellerer, H., R. Mansini, and M.G. Speranza. (2000). “Selecting Portfolios with Fixed Costs and Minimum Transaction Lots.” Annals of Operations Research, 99, 287–304.CrossRefGoogle Scholar
  11. Konno, H., and H. Yamazaki. (1991). “Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market.” Management Science, 37, 519–531.Google Scholar
  12. Konno, H., and A. Wijayanayake. (2001). “Portfolio Optimization Problem under Concave Transaction Costs and Minimal Transaction Unit Constraints.” Mathematical Programming, 89, 233–250.CrossRefGoogle Scholar
  13. Levy, H., and Y. Kroll. (1978). “Ordering Uncertain Options with Borrowing and Lending.” Journal of Finance, 33, 553–573.CrossRefGoogle Scholar
  14. Mansini, R., W. Ogryczak, and M.G. Speranza. (2003a). “On LP Solvable Models for Portfolio Selection.” Informatica, 14, 37–62.Google Scholar
  15. Mansini, R., W. Ogryczak, and M.G. Speranza. (2003b). “LP Solvable Models for Portfolio Optimization: A Classification and Computational Comparison.” IMA J. of Management Mathematics, 14, 187–220.CrossRefGoogle Scholar
  16. Mansini, R., W. Ogryczak, and M.G. Speranza. (2003c). “Conditional Value at Risk and Related Linear Programming Models for Portfolio Optimization.” Tech. Report 03–02, Warsaw Univ. of Technology.Google Scholar
  17. Mansini, R., and M.G. Speranza. (1999). “Heuristic Algorithms for the Portfolio Selection Problem with Minimum Transaction Lots.” European J. of Operational Research, 114, 219–233.CrossRefGoogle Scholar
  18. Markowitz, H.M. (1952). “Portfolio Selection.” Journal of Finance, 7, 77–91.CrossRefGoogle Scholar
  19. Ogryczak, W. (1999). “Stochastic Dominance Relation and Linear Risk Measures.” In A.M.J. Skulimowski (ed.), Financial Modelling—Proceedings of the 23rd Meeting of the EURO Working Group on Financial Modelling. Cracow: Progress & Business Publ., 191–212.Google Scholar
  20. Ogryczak, W. (2000). “Multiple Criteria Linear Programming Model for Portfolio Selection.” Annals of Operations Research, 97, 143–162.CrossRefGoogle Scholar
  21. Ogryczak, W. (2002). “Multiple Criteria Optimization and Decisions under Risk.” Control and Cybernetics, 31, 975–1003.Google Scholar
  22. Ogryczak, W. and A. Ruszczyński. (1999). “From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures.” European J. of Operational Research, 116, 33–50.CrossRefGoogle Scholar
  23. Ogryczak, W. and A. Ruszczyński. (2001). “On Stochastic Dominance and Mean-Semideviation Models.” Mathematical Programming, 89, 217–232.CrossRefGoogle Scholar
  24. Ogryczak, W. and A. Ruszczyński. (2002a). “Dual Stochastic Dominance and Related Mean-Risk Models.” SIAM J. on Optimization, 13, 60–78.CrossRefGoogle Scholar
  25. Ogryczak, W. and A. Ruszczyński. (2002b). “Dual Stochastic Dominance and Quantile Risk Measures.” International Transactions in Operational Research, 9, 661–680.CrossRefGoogle Scholar
  26. Ogryczak, W. and A. Tamir. (2003). “Minimizing the Sum of the k-Largest Functions in Linear Time.” Information Processing Letters, 85, 117–122.CrossRefGoogle Scholar
  27. Pflug, G.Ch. (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 A.P.Google Scholar
  28. Rockafellar, R.T. and S. Uryasev. (2000). “Optimization of Conditional Value-at-Risk.” Journal of Risk, 2, 21–41.Google Scholar
  29. Rockafellar, R.T. and S. Uryasev. (2002). “Conditional Value-at-Risk for General Distributions.” Journal of Banking & Finance, 26, 1443–1471.CrossRefGoogle Scholar
  30. Rockafellar, R.T., S. Uryasev, and M. Zabarankin. (2002). “Deviation Measures in Generalized Linear Regression.” Research Report 2002-9, Univ. of Florida, ISE.Google Scholar
  31. Rothschild, M. and J.E. Stiglitz. (1969). “Increasing Risk: I. A Definition.” Journal of Economic Theory, 2, 225–243.CrossRefGoogle Scholar
  32. Shalit, H. and S. Yitzhaki. (1994). “Marginal Conditional Stochastic Dominance.” Management Science, 40, 670–684.Google Scholar
  33. Sharpe, W.F. (1971a). “A Linear Programming Approximation for the General Portfolio Analysis Problem.” Journal of Financial and Quantitative Analysis, 6, 1263–1275.CrossRefGoogle Scholar
  34. Sharpe, W.F. (1971b). “Mean-Absolute Deviation Characteristic Lines for Securities and Portfolios.” Management Science, 8, B1–B13.Google Scholar
  35. Shorrocks, A.F. (1983). “Ranking Income Distributions.” Economica, 50, 3–17.CrossRefGoogle Scholar
  36. Simaan, Y. (1997). “Estimation Risk in Portfolio Selection: The Mean Variance Model and the Mean-Absolute Deviation Model.” Management Science, 43, 1437–1446.Google Scholar
  37. Speranza, M.G. (1993). “Linear Programming Models for Portfolio Optimization.” Finance, 14, 107–123.Google Scholar
  38. Topaloglou, N., H. Vladimirou, and S.A. Zenios. (2002). “CVaR Models with Selective Hedging for International Asset Allocation.” Journal of Banking & Finance, 26, 1535–1561.CrossRefGoogle Scholar
  39. Whitmore, G.A., and M.C. Findlay (eds.). (1978). Stochastic Dominance: An Approach to Decision–Making Under Risk. Lexington MA: D.C.Heath.Google Scholar
  40. Yaari, M.E. (1987). “The Dual Theory of Choice under Risk.” Econometrica, 55, 95–115.CrossRefGoogle Scholar
  41. Yitzhaki, S. (1982). “Stochastic Dominance, Mean Variance, and Gini’s Mean Difference.” American Economic Revue, 72, 178–185.Google Scholar
  42. Young, M.R. (1998). “A Minimax Portfolio Selection Rule with Linear Programming Solution.” Management Science, 44, 673–683.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Renata Mansini
    • 1
  • Włodzimierz Ogryczak
    • 2
    Email author
  • M. Grazia Speranza
    • 1
  1. 1.Department of Electronics for AutomationUniversity of BresciaBresciaItaly
  2. 2.Institute of Control and Computation EngineeringWarsaw University of TechnologyWarsawPoland

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