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Computing the Nondominated Surface in Tri-Criterion Portfolio Selection

Author

Listed:
  • Markus Hirschberger

    (Department of Mathematics, University of Eichstätt-Ingolstadt, 85072 Eichstätt, Germany)

  • Ralph E. Steuer

    (Department of Finance, University of Georgia, Athens, Georgia 30602)

  • Sebastian Utz

    (Department of Finance, University of Regensburg, 93040 Regensburg, Germany)

  • Maximilian Wimmer

    (Department of Finance, University of Regensburg, 93040 Regensburg, Germany)

  • Yue Qi

    (Department of Financial Management, Nankai University, Tianjin, China)

Abstract

Computing the nondominated set of a multiple objective mathematical program has long been a topic in multiple criteria decision making. In this paper, motivated by the desire to extend Markowitz portfolio selection to an additional linear criterion (dividends, liquidity, sustainability, etc.), we demonstrate an exact method for computing the nondominated set of a tri-criterion program that is all linear except for the fact that one of its objectives is to minimize a convex quadratic function. With the nondominated set of the resulting quad-lin-lin program being a surface composed of curved platelets, a multiparametric algorithm is devised for computing the platelets so that they can be graphed precisely. In this way, graphs of the tri-criterion nondominated surface can be displayed so that, as in traditional portfolio selection, a most preferred portfolio can be selected while in full view of all other contenders for optimality. Finally, by giving an example for socially responsible investors, we demonstrate that our algorithm can outperform standard portfolio strategies for multicriterial decision makers.

Suggested Citation

  • Markus Hirschberger & Ralph E. Steuer & Sebastian Utz & Maximilian Wimmer & Yue Qi, 2013. "Computing the Nondominated Surface in Tri-Criterion Portfolio Selection," Operations Research, INFORMS, vol. 61(1), pages 169-183, February.
    • Handle: RePEc:inm:oropre:v:61:y:2013:i:1:p:169-183
    DOI: 10.1287/opre.1120.1140
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    References listed on IDEAS

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    1. Ehrgott, Matthias & Klamroth, Kathrin & Schwehm, Christian, 2004. "An MCDM approach to portfolio optimization," European Journal of Operational Research, Elsevier, vol. 155(3), pages 752-770, June.
    2. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    3. Andras Niedermayer & Daniel Niedermayer, 2006. "Applying Markowitz's Critical Line Algorithm," Diskussionsschriften dp0602, Universitaet Bern, Departement Volkswirtschaft.
    4. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    5. Serpil Sayin, 2003. "A Procedure to Find Discrete Representations of the Efficient Set with Specified Coverage Errors," Operations Research, INFORMS, vol. 51(3), pages 427-436, June.
    6. Pekka Korhonen & Jyrki Wallenius, 1988. "A pareto race," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(6), pages 615-623, December.
    7. Renneboog, Luc & Ter Horst, Jenke & Zhang, Chendi, 2008. "Socially responsible investments: Institutional aspects, performance, and investor behavior," Journal of Banking & Finance, Elsevier, vol. 32(9), pages 1723-1742, September.
    8. Ledoit, Oliver & Wolf, Michael, 2008. "Robust performance hypothesis testing with the Sharpe ratio," Journal of Empirical Finance, Elsevier, vol. 15(5), pages 850-859, December.
    9. Frost, Peter A. & Savarino, James E., 1986. "An Empirical Bayes Approach to Efficient Portfolio Selection," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 293-305, September.
    10. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    11. Jobson, J D & Korkie, Bob M, 1981. "Performance Hypothesis Testing with the Sharpe and Treynor Measures," Journal of Finance, American Finance Association, vol. 36(4), pages 889-908, September.
    12. H. P. Benson & E. Sun, 2000. "Outcome Space Partition of the Weight Set in Multiobjective Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 105(1), pages 17-36, April.
    13. Steuer, Ralph E. & Na, Paul, 2003. "Multiple criteria decision making combined with finance: A categorized bibliographic study," European Journal of Operational Research, Elsevier, vol. 150(3), pages 496-515, November.
    14. Kirby, Chris & Ostdiek, Barbara, 2012. "It’s All in the Timing: Simple Active Portfolio Strategies that Outperform Naïve Diversification," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 47(2), pages 437-467, April.
    15. DeMiguel, Victor & Plyakha, Yuliya & Uppal, Raman & Vilkov, Grigory, 2013. "Improving Portfolio Selection Using Option-Implied Volatility and Skewness," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 48(6), pages 1813-1845, December.
    16. Dorfleitner, Gregor & Utz, Sebastian, 2012. "Safety first portfolio choice based on financial and sustainability returns," European Journal of Operational Research, Elsevier, vol. 221(1), pages 155-164.
    17. J. Tobin, 1958. "Liquidity Preference as Behavior Towards Risk," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(2), pages 65-86.
    18. David Disatnik & Saggi Katz, 2012. "Portfolio Optimization Using a Block Structure for the Covariance Matrix," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 39(5-6), pages 806-843, June.
    19. Swanson, Norman R. & White, Halbert, 1997. "Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models," International Journal of Forecasting, Elsevier, vol. 13(4), pages 439-461, December.
    20. Victor DeMiguel & Francisco J. Nogales, 2009. "Portfolio Selection with Robust Estimation," Operations Research, INFORMS, vol. 57(3), pages 560-577, June.
    21. Kempf, Alexander & Osthoff, Peer, 2007. "The effect of socially responsible investing on portfolio performance," CFR Working Papers 06-10, University of Cologne, Centre for Financial Research (CFR).
    22. Jorion, Philippe, 1986. "Bayes-Stein Estimation for Portfolio Analysis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 21(3), pages 279-292, September.
    23. Best, Michael J & Grauer, Robert R, 1991. "On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results," The Review of Financial Studies, Society for Financial Studies, vol. 4(2), pages 315-342.
    24. B. Curtis Eaves, 1971. "On Quadratic Programming," Management Science, INFORMS, vol. 17(11), pages 698-711, July.
    25. Alexander Kempf & Peer Osthoff, 2007. "The Effect of Socially Responsible Investing on Portfolio Performance," European Financial Management, European Financial Management Association, vol. 13(5), pages 908-922, November.
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