Skip to main content
Log in

On Extending the LP Computable Risk Measures to Account Downside Risk

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

A mathematical model of portfolio optimization is usually quantified with mean-risk models offering a lucid form of two criteria with possible trade-off analysis. In the classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. Following Sharpe’s work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving linear programming (LP) problems. Typical LP computable risk measures, like the mean absolute deviation (MAD) or the Gini’s mean absolute difference (GMD) are symmetric with respect to the below-mean and over-mean performances. The paper shows how the measures can be further combined to extend their modeling capabilities with respect to enhancement of the below-mean downside risk aversion. The relations of the below-mean downside stochastic dominance are formally introduced and the corresponding techniques to enhance risk measures are derived.

The resulting mean-risk models generate efficient solutions with respect to second degree stochastic dominance, while at the same time preserving simplicity and LP computability of the original models. The models are tested on real-life historical data.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. W.J. Baumol, “An expected gain-confidence limit criterion for portfolio selection,” Management Science, vol. 10, pp. 174–182, 1964.

    Google Scholar 

  2. D.R. Carino, D.H. Myers, and W.T. Ziemba, “Concepts,technical issues and uses of the Russel-Yasuda Kasai financial planning model,” Operations Research, vol. 46, pp. 450–463, 1998.

    Google Scholar 

  3. E.J. Elton and M.J. Gruber, Modern Portfolio Theory and Investment Analysis (Third Edition), John Wiley & Sons, New York, 1987.

    Google Scholar 

  4. C.D. Feinstein and M.N. Thapa, “A Reformulation of a mean-absolute deviation portfolio optimization model,” Management Science, vol. 39, pp. 1552–1553, 1993.

    Google Scholar 

  5. P.C. Fishburn, “Mean-risk analysis with risk associated with below target returns,” American Economic Review, vol. 67, pp. 116–126, 1977.

    Google Scholar 

  6. P.C. Fishburn, “Stochastic dominance and moments of distributions,” Mathematics of Operations Research, vol. 5, pp. 94–100, 1980.

    Google Scholar 

  7. F. Glover and D. Klingman, “The simplex SON method for LP/embedded network problems,” Mathematical Programming Study, vol. 15, pp. 148–176, 1981.

    Google Scholar 

  8. H. Grootveld and W. Hallerbach, “Variance vs Downside Risk: Is There Really That Much Difference,” European Journal of Operational Research, vol. 114, pp. 304–319, 1999.

    Article  Google Scholar 

  9. ILOG Inc., Using the CPLEX Callable Library, ILOG Inc., CPLEX Division, Incline Village, 1997.

  10. H. Konno, “Piecewise linear risk function and portfolio optimization,” Journal of the Operations Research Society of Japan, vol. 33, pp. 139–156, 1990.

    Google Scholar 

  11. H. Konno and H. Yamazaki, “Mean-absolute deviation portfolio optimization model and its application to Tokyo stock market,” Management Science, vol. 37, pp. 519–531, 1991.

    Google Scholar 

  12. H. Konno and A. Wijayanayake, “Mean-absolute deviation portfolio optimization model under transaction costs,” Journal of the Operations Research Society of Japan, vol. 42, pp. 422–435, 1999.

    Article  Google Scholar 

  13. H. Levy, “Stochastic dominance and expected utility: Survey and analysis,” Management Science, vol. 38, pp. 555–593, 1992.

    Google Scholar 

  14. R. Mansini and M.G. Speranza, “Heuristic algorithms for the portfolio selection problem with minimum transaction lots,” European Journal of Operational Research, vol. 114, pp. 219–233, 1999.

    Article  Google Scholar 

  15. H.M. Markowitz, “Portfolio selection,” Journal of Finance, vol. 7, pp. 77–91, 1952.

    Google Scholar 

  16. H.M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, New York, 1959.

    Google Scholar 

  17. W. Michalowski and W. Ogryczak, “Extending the MAD portfolio optimization model to incorporate downside risk aversion,” Naval Research Logistics, vol. 48, pp. 185–200, 2001.

    Article  Google Scholar 

  18. W. Ogryczak and A. Ruszczyński, “From stochastic dominance to mean-risk models: Semideviations as risk measures,” European Journal of Operational Research, vol. 116, pp. 33–50, 1999.

    Article  Google Scholar 

  19. W. Ogryczak and A. Ruszczyński, On Consistency of Stochastic Dominance and Mean–Semideviation Models. Mathematical Programming 89 (2001) 217–232.

    Google Scholar 

  20. W. Ogryczak and A. Ruszczyński, “Dual stochastic dominance and related mean-risk models,” SIAM J. Optimization, vol. 13, pp. 60–78, 2002.

    Article  Google Scholar 

  21. W. Ogryczak and A. Ruszczyński, “Dual Stochastic Dominance and Quantile Risk Measures,” International Transactions in Operational Research, vol. 9, pp. 661–680, 2002.

    Article  Google Scholar 

  22. M. Rothschild and J.E. Stiglitz, “Increasing risk: I. A definition,” Journal of Economic Theory, vol. 2, pp. 225–243, 1969.

    Article  Google Scholar 

  23. A.D. Roy, “Safety-first and the holding of assets,” Econometrica, vol. 20, pp. 431–449, 1952.

    Google Scholar 

  24. H. Shalit and S. Yitzhaki, “Mean-Gini, Portfolio Theory, and the Pricing of Risky Assets,” Journal of Finance, vol. 39, pp. 1449–1468, 1984.

    Google Scholar 

  25. W.F. Sharpe, “A linear programming approximation for the general portfolio analysis problem,” Journal of Financial and Quantitative Analysis, vol. 6, pp. 1263–1275, 1971.

    Google Scholar 

  26. W.F. Sharpe, “Mean-absolute deviation characteristic lines for securities and portfolios,” Management Science, vol. 18, pp. B1–B13, 1971.

    Google Scholar 

  27. F.A. Sortino and H.J. Forsey, “On the use and misuse of downside risk,” Journal of Portfolio Management, pp. 35–42, Winter 1996.

  28. M.G. Speranza, “Linear programming models for portfolio optimization,” Finance, vol. 14, pp. 107–123, 1993.

    Google Scholar 

  29. B.K. Stone, “A linear programming formulation of the general portfolio selection problem,” Journal of Financial and Quantitative Analysis, vol. 8, pp. 621–636, 1973.

    Google Scholar 

  30. G.A Whitmore and M.C. Findlay (Eds.), Stochastic Dominance: An Approach to Decision-Making Under Risk, D.C. Heath, Lexington MA, 1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The research was supported by the grant PBZ-KBN-016/P03/99 from The State Committee for Scientific Research.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krzemienowski, A., Ogryczak, W. On Extending the LP Computable Risk Measures to Account Downside Risk. Comput Optim Applic 32, 133–160 (2005). https://doi.org/10.1007/s10589-005-2057-4

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-005-2057-4

Keywords

Navigation