Annals of Operations Research

, Volume 251, Issue 1–2, pp 161–177 | Cite as

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

  • Yue Qi
  • Ralph E. Steuer
  • Maximilian Wimmer


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.


Multiple criteria optimization Tri-criterion portfolio selection Minimum-variance frontier e-Constraint method Efficient surface Paraboloids 



The authors are thankful to Markus Hirschberger for comments and to the software package PGFPLOTS by Feuersänger (2014) for use in constructing the graphs. The first author acknowledges support from the Ministry of Education of China (Grant No. 14JJD630007), the National Natural Science Foundation of China (Grant No. 71132001), and the Program for Changjiang Scholars and Innovative Research Team in University, IRT0926.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Financial Management, Business SchoolNankai UniversityTianjinChina
  2. 2.Department of FinanceUniversity of GeorgiaAthensUSA
  3. 3.Department of FinanceUniversity of RegensburgRegensburgGermany

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