Skip to main content
SpringerLink
Log in

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

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Acerbi, C. (2002).“Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion.” Journal of Banking & Finance, 26, 1505–1518.

    Article  Google Scholar 

  • Acerbi, C. and P. Simonetti. (2002). “Portfolio Optimization with Spectral Measures of Risk.” Working Paper (http://gloriamundi.org).

  • Andersson, F., H. Mausser, D. Rosen, and S. Uryasev. (2001). “Credit Risk Optimization with Conditional Value-at-Risk Criterion.” Mathematical Programming, 89, 273–291.

    Article  Google Scholar 

  • Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). “Coherent Measures of Risk.” Mathematical Finance, 9, 203–228.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Embrechts, P., C. Klüppelberg, and T. Mikosch. (1997). Modelling Extremal Events for Insurance and Finance. New York: Springer-Verlag.

    Google Scholar 

  • Haimes, Y.Y. (1993). “Risk of Extreme Events and the Fallacy of the Expected Value.” Control and Cybernetics, 22, 7–31.

    Google Scholar 

  • Jorion, P. (2001). Value-at-Risk: The New Benchmark for Managing Financial Risk. NY: McGraw-Hill.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Konno, H., and A. Wijayanayake. (2001). “Portfolio Optimization Problem under Concave Transaction Costs and Minimal Transaction Unit Constraints.” Mathematical Programming, 89, 233–250.

    Article  Google Scholar 

  • Levy, H., and Y. Kroll. (1978). “Ordering Uncertain Options with Borrowing and Lending.” Journal of Finance, 33, 553–573.

    Article  Google Scholar 

  • Mansini, R., W. Ogryczak, and M.G. Speranza. (2003a). “On LP Solvable Models for Portfolio Selection.” Informatica, 14, 37–62.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Markowitz, H.M. (1952). “Portfolio Selection.” Journal of Finance, 7, 77–91.

    Article  Google Scholar 

  • 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.

  • Ogryczak, W. (2000). “Multiple Criteria Linear Programming Model for Portfolio Selection.” Annals of Operations Research, 97, 143–162.

    Article  Google Scholar 

  • Ogryczak, W. (2002). “Multiple Criteria Optimization and Decisions under Risk.” Control and Cybernetics, 31, 975–1003.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ogryczak, W. and A. Ruszczyński. (2001). “On Stochastic Dominance and Mean-Semideviation Models.” Mathematical Programming, 89, 217–232.

    Article  Google Scholar 

  • Ogryczak, W. and A. Ruszczyński. (2002a). “Dual Stochastic Dominance and Related Mean-Risk Models.” SIAM J. on Optimization, 13, 60–78.

    Article  Google Scholar 

  • Ogryczak, W. and A. Ruszczyński. (2002b). “Dual Stochastic Dominance and Quantile Risk Measures.” International Transactions in Operational Research, 9, 661–680.

    Article  Google Scholar 

  • Ogryczak, W. and A. Tamir. (2003). “Minimizing the Sum of the k-Largest Functions in Linear Time.” Information Processing Letters, 85, 117–122.

    Article  Google Scholar 

  • 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 

  • Rockafellar, R.T. and S. Uryasev. (2000). “Optimization of Conditional Value-at-Risk.” Journal of Risk, 2, 21–41.

    Google Scholar 

  • Rockafellar, R.T. and S. Uryasev. (2002). “Conditional Value-at-Risk for General Distributions.” Journal of Banking & Finance, 26, 1443–1471.

    Article  Google Scholar 

  • Rockafellar, R.T., S. Uryasev, and M. Zabarankin. (2002). “Deviation Measures in Generalized Linear Regression.” Research Report 2002-9, Univ. of Florida, ISE.

  • Rothschild, M. and J.E. Stiglitz. (1969). “Increasing Risk: I. A Definition.” Journal of Economic Theory, 2, 225–243.

    Article  Google Scholar 

  • Shalit, H. and S. Yitzhaki. (1994). “Marginal Conditional Stochastic Dominance.” Management Science, 40, 670–684.

    Google Scholar 

  • Sharpe, W.F. (1971a). “A Linear Programming Approximation for the General Portfolio Analysis Problem.” Journal of Financial and Quantitative Analysis, 6, 1263–1275.

    Article  Google Scholar 

  • Sharpe, W.F. (1971b). “Mean-Absolute Deviation Characteristic Lines for Securities and Portfolios.” Management Science, 8, B1–B13.

    Google Scholar 

  • Shorrocks, A.F. (1983). “Ranking Income Distributions.” Economica, 50, 3–17.

    Article  Google Scholar 

  • 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 

  • Speranza, M.G. (1993). “Linear Programming Models for Portfolio Optimization.” Finance, 14, 107–123.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Yaari, M.E. (1987). “The Dual Theory of Choice under Risk.” Econometrica, 55, 95–115.

    Article  Google Scholar 

  • Yitzhaki, S. (1982). “Stochastic Dominance, Mean Variance, and Gini’s Mean Difference.” American Economic Revue, 72, 178–185.

    Google Scholar 

  • Young, M.R. (1998). “A Minimax Portfolio Selection Rule with Linear Programming Solution.” Management Science, 44, 673–683.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electronics for Automation, University of Brescia, via Branze 38, 25123, Brescia, Italy

    Renata Mansini & M. Grazia Speranza

  2. Institute of Control and Computation Engineering, Warsaw University of Technology, Nowowiejska 15/19, 00-665, Warsaw, Poland

    Włodzimierz Ogryczak

Authors
  1. Renata Mansini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Włodzimierz Ogryczak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Grazia Speranza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Włodzimierz Ogryczak.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mansini, R., Ogryczak, W. & Speranza, M.G. Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res 152, 227–256 (2007). https://doi.org/10.1007/s10479-006-0142-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-006-0142-4

Keywords

Search

Navigation