An analytical derivation of the efficient surface in portfolio selection with three criteria
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.
KeywordsMultiple 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.
- Bana e Costa, C. A., & Soares, J. O. (2001). Multicriteria approaches for portfolio selection: An overview. Review of Financial Markets, 4(1), 19–26.Google Scholar
- Calvo, C., Ivorra, C., & Liern, V. (2014). Fuzzy portfolio selection with non-financial goals: Exploring the efficient frontier. Annals of Operations Research. doi: 10.1007/s10479-014-1561-2.
- Chow, G. (1995). Portfolio selection based on return, risk, and relative performance. Financial Analysts Journal, 51(2), 54–60.Google Scholar
- Ehrgott, M. (2005). Multicriteria optimization (2nd ed.). Berlin: Springer.Google Scholar
- Feuersänger, C. (2014). Manual for package PGFPLOTS, version 1.11, Universität Bonn.Google Scholar
- Guerard, J. B., & Mark, A. (2004). Data show importance of company R&D in picking stocks. Pensions & Investments, 32(25), 30–31.Google Scholar
- Huang, C., & Litzenberger, R. H. (1988). Foundations for financial economics. Englewood Cliffs, NJ: Prentice Hall.Google Scholar
- Konno, H., & Suzuki, K. I. (1995). A mean-variance-skewness portfolio optimization model. Journal of the Operations Research Society of Japan, 38(2), 173–187.Google Scholar
- Lee, S. M. (1972). Goal programming for decision analysis. Philadelphia: Auerbach Publishers.Google Scholar
- Lo, A. W., Petrov, C., & Wierzbicki, M. (2003). It’s 11pm—do you know where your liquidity is? The mean-variance-liquidity frontier. Journal of Investment Management, 1(1), 55–93.Google Scholar
- Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7(1), 77–91.Google Scholar
- Miettinen, K. (1999). Nonlinear multiobjective optimization. Boston: Kluwer.Google Scholar
- Speranza, M. G. (1996). A heuristic algorithm for a portfolio optimization model applied to the Milan stock market. Computers & Operations Research, 23(5), 431–441.Google Scholar
- Spronk, J., & Zambruno, G. M. (1981). A multiple-criteria approach to portfolio selection. In Göppl, H., Henn, R. (Eds.) Geld, Banken und Versicherungen, Band 1, Athenum, pp. 451–459.Google Scholar
- Utz, S., Wimmer, M., & Steuer, R. E. (2015). Tri-criterion modeling for creating more-sustainable mutual funds. European Journal of Operational Research. doi: 10.1016/j.ejor.2015.04.035.
- Ziemba, W. (2006). Personal communication at 21st European Conference on Operational Research, Reykjavik, Iceland, July 3.Google Scholar