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An analytical derivation of the efficient surface in portfolio selection with three criteria

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  • Yue Qi

    (Nankai University)

  • Ralph E. Steuer

    (University of Georgia)

  • Maximilian Wimmer

    (University of Regensburg)

Abstract

In standard mean-variance bi-criterion portfolio selection, the efficient set is a frontier. While it is not yet standard for there to be additional criteria in portfolio selection, there has been a growing amount of discussion in the literature on the topic. However, should there be even one additional criterion, the efficient frontier becomes an efficient surface. Striving to parallel Merton’s seminal analytical derivation of the efficient frontier, in this paper we provide an analytical derivation of the efficient surface when an additional linear criterion (on top of expected return and variance) is included in the model addressed by Merton. Among the results of the paper there is, as a higher dimensional counterpart to the 2-mutual-fund theorem of traditional portfolio selection, a 3-mutual-fund theorem in tri-criterion portfolio selection. 3D graphs are employed to stress the paraboloidic/hyperboloidic structures present in tri-criterion portfolio selection.

Suggested Citation

  • Yue Qi & Ralph E. Steuer & Maximilian Wimmer, 2017. "An analytical derivation of the efficient surface in portfolio selection with three criteria," Annals of Operations Research, Springer, vol. 251(1), pages 161-177, April.
    • Handle: RePEc:spr:annopr:v:251:y:2017:i:1:d:10.1007_s10479-015-1900-y
    DOI: 10.1007/s10479-015-1900-y
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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. Cabello, J.M. & Ruiz, F. & Pérez-Gladish, B. & Méndez-Rodríguez, P., 2014. "Synthetic indicators of mutual funds’ environmental responsibility: An application of the Reference Point Method," European Journal of Operational Research, Elsevier, vol. 236(1), pages 313-325.
    3. Hallerbach, Winfried & Ning, Haikun & Soppe, Aloy & Spronk, Jaap, 2004. "A framework for managing a portfolio of socially responsible investments," European Journal of Operational Research, Elsevier, vol. 153(2), pages 517-529, March.
    4. Aouni, Belaid & Colapinto, Cinzia & La Torre, Davide, 2014. "Financial portfolio management through the goal programming model: Current state-of-the-art," European Journal of Operational Research, Elsevier, vol. 234(2), pages 536-545.
    5. Panos Xidonas & George Mavrotas & Theodore Krintas & John Psarras & Constantin Zopounidis, 2012. "Multicriteria Portfolio Management," Springer Optimization and Its Applications, in: Multicriteria Portfolio Management, edition 127, chapter 0, pages 5-21, Springer.
    6. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(4), pages 1851-1872, September.
    7. Abdelaziz, Fouad Ben & Aouni, Belaid & Fayedh, Rimeh El, 2007. "Multi-objective stochastic programming for portfolio selection," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1811-1823, March.
    8. Gregor Dorfleitner & Michaela Leidl & Johannes Reeder, 2012. "Theory of social returns in portfolio choice with application to microfinance," Journal of Asset Management, Palgrave Macmillan, vol. 13(6), pages 384-400, December.
    9. Ballestero, Enrique & Bravo, Mila & Pérez-Gladish, Blanca & Arenas-Parra, Mar & Plà-Santamaria, David, 2012. "Socially Responsible Investment: A multicriteria approach to portfolio selection combining ethical and financial objectives," European Journal of Operational Research, Elsevier, vol. 216(2), pages 487-494.
    10. 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.
    11. Panos Xidonas & George Mavrotas & Theodore Krintas & John Psarras & Constantin Zopounidis, 2012. "Multicriteria Portfolio Management," Springer Optimization and Its Applications, Springer, edition 127, number 978-1-4614-3670-6, December.
    12. Ralph Steuer & Yue Qi & Markus Hirschberger, 2007. "Suitable-portfolio investors, nondominated frontier sensitivity, and the effect of multiple objectives on standard portfolio selection," Annals of Operations Research, Springer, vol. 152(1), pages 297-317, July.
    13. Utz, Sebastian & Wimmer, Maximilian & Hirschberger, Markus & Steuer, Ralph E., 2014. "Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds," European Journal of Operational Research, Elsevier, vol. 234(2), pages 491-498.
    14. Stone, Bernell K., 1973. "A Linear Programming Formulation of the General Portfolio Selection Problem†," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 8(4), pages 621-636, September.
    15. Spronk, Jaap & Hallerbach, Winfried, 1997. "Financial modelling: Where to go? With an illustration for portfolio management," European Journal of Operational Research, Elsevier, vol. 99(1), pages 113-125, May.
    16. 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.
    17. Amelia Bilbao-Terol & Mar Arenas-Parra & Verónica Cañal-Fernández & Celia Bilbao-Terol, 2013. "Selection of Socially Responsible Portfolios Using Hedonic Prices," Journal of Business Ethics, Springer, vol. 115(3), pages 515-529, July.
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    8. Qi, Yue & Liao, Kezhi & Liu, Tongyang & Zhang, Yu, 2022. "Originating multiple-objective portfolio selection by counter-COVID measures and analytically instigating robust optimization by mean-parameterized nondominated paths," Operations Research Perspectives, Elsevier, vol. 9(C).
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