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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.

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  • 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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    4. Armin Varmaz & Christian Fieberg & Thorsten Poddig, 2024. "Portfolio optimization for sustainable investments," Annals of Operations Research, Springer, vol. 341(2), pages 1151-1176, October.
    5. Yue Qi, 2018. "On outperforming social-screening-indexing by multiple-objective portfolio selection," Annals of Operations Research, Springer, vol. 267(1), pages 493-513, August.
    6. A. Garcia-Bernabeu & J. V. Salcedo & A. Hilario & D. Pla-Santamaria & Juan M. Herrero, 2019. "Computing the Mean-Variance-Sustainability Nondominated Surface by ev-MOGA," Complexity, Hindawi, vol. 2019, pages 1-12, December.
    7. Yue Qi, 2022. "Classifying the minimum-variance surface of multiple-objective portfolio selection for capital asset pricing models," Annals of Operations Research, Springer, vol. 311(2), pages 1203-1227, April.
    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).
    9. Yue Qi & Xiaolin Li, 2020. "On Imposing ESG Constraints of Portfolio Selection for Sustainable Investment and Comparing the Efficient Frontiers in the Weight Space," SAGE Open, , vol. 10(4), pages 21582440209, December.
    10. Kerstin Dächert & Ria Grindel & Elisabeth Leoff & Jonas Mahnkopp & Florian Schirra & Jörg Wenzel, 2022. "Multicriteria asset allocation in practice," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(2), pages 349-373, June.
    11. S., Glogger & S., Heiden & D., Schneller, 2019. "Bearing the bear: Sentiment-based disagreement in multi-criteria portfolio optimization," Finance Research Letters, Elsevier, vol. 31(C), pages 47-53.
    12. Maciej Nowak & Tadeusz Trzaskalik, 2022. "A trade-off multiobjective dynamic programming procedure and its application to project portfolio selection," Annals of Operations Research, Springer, vol. 311(2), pages 1155-1181, April.
    13. Adrian Gepp & Geoff Harris & Bruce Vanstone, 2020. "Financial applications of semidefinite programming: a review and call for interdisciplinary research," Accounting and Finance, Accounting and Finance Association of Australia and New Zealand, vol. 60(4), pages 3527-3555, December.
    14. Salo, Ahti & Doumpos, Michalis & Liesiö, Juuso & Zopounidis, Constantin, 2024. "Fifty years of portfolio optimization," European Journal of Operational Research, Elsevier, vol. 318(1), pages 1-18.
    15. Yue Qi & Ralph E. Steuer, 2020. "On the analytical derivation of efficient sets in quad-and-higher criterion portfolio selection," Annals of Operations Research, Springer, vol. 293(2), pages 521-538, October.
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