Skip to main content
SpringerLink
Log in

Overviewing the transition of Markowitz bi-criterion portfolio selection to tri-criterion portfolio selection

  • Original Paper
  • Published:
Journal of Business Economics Aims and scope Submit manuscript

Abstract

Over 60 years ago, Markowitz introduced the mean-variance efficient frontier to finance. While mean-variance is still the predominant model in portfolio selection, it has endured many criticisms. One serious one is that it does not allow for additional criteria. The difficulty is that the efficient frontier becomes a surface. With it now possible to compute such a surface, we provide an overview on how Markowitz’s risk-return (bi-criterion) portfolio selection can be extended to tri-criterion portfolio selection. With a focus on the geometry of the extension, many graphs are used to illustrate.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Or, in more colorful terminology, the solution is accepted simply because it is the “cleanest dirty shirt” which is a quote from an internet interview with William H. Gross, co-chief investment officer, Pacific Investment Management, on June 28, 2012.

  2. As a technical matter, slight adjustments are made to H, d, G and b in (5.2–5.3) depending upon which elements are −1 in \(\bar{{\mathbf{I}}}_n\).

References

  • Abdelaziz FB, Aouni B, El-Fayedh R (2007) Multi-objective stochastic programming for portfolio selection. Eur J Oper Res 177(3):1811–1823

    Article  Google Scholar 

  • Ballestero E, Bravo M, Perez-Gladish B, Arenas-Parra M, Plà-Santamaría D (2012) Socially responsible investment: A multicriteria approach to portfolio selection combining ethical and financial objectives. Eur J Oper Res 216(2):487–494

    Article  Google Scholar 

  • Bana e Costa CA, Soares JO (2004) A multicriteria model for portfolio management. Eur J Financ 10(3):198–211

    Article  Google Scholar 

  • Best MJ (1996) An algorithm for the solution of the parametric quadratic programming problem. In: Fischer H, Riedmüller B, Schäffler S (eds) Applied mathematics and parallel computing: festschrift for Klaus Ritter. Physica-Verlag, pp 57–76

  • Caballero R, Cerdá E, Muñoz MM, Rey L, Stancu-Minasian IM (2001) Efficient solution concepts and their relations in stochastic multiobjective programming. J Optim Theory Appl 110(1):53–74

    Article  Google Scholar 

  • Chow G (1995) Portfolio selection based on return, risk, and relative performance. Financ Anal J 52(2):54–60

    Article  Google Scholar 

  • Colson G, DeBruyn C (1989) An integrated multiobjective portfolio management system. Math Comput Model 12(10–11):1359–1381

    Article  Google Scholar 

  • Eaves BC (1971) On quadratic programming. Manage Sci 17(11):698–711

    Article  Google Scholar 

  • Ehrgott M (2005) Multicriteria Optimization, 2nd edn. Springer

  • Ehrgott M, Klamroth K, Schwehm C (2004) An MCDM approach to portfolio optimization. Eur J Oper Res 155(3):752–770

    Article  Google Scholar 

  • Feuersänger C (2010) Manual for package PGFPLOTS, version 1.4.1, Universität Bonn

  • Haimes YY, Lasdon LS, Wismer DA (1971) On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern SMC 1(3):296–297

    Article  Google Scholar 

  • He XD, Zhou XY (2011) Portfolio choice under cumulative prospect theory: an analytical treatment. Manage Sci 57(2):315–331

    Article  Google Scholar 

  • Hirschberger M, Qi Y, Steuer RE (2007) Randomly generating portfolio-selection covariance matrices with specified distributional characteristics. Eur J Oper Res 177(3):1610–1625

    Article  Google Scholar 

  • Hirschberger M, Qi Y, Steuer RE (2010) Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming. Eur J Oper Res 204(3):581–588

    Article  Google Scholar 

  • Hirschberger M, Steuer RE, Utz S, Wimmer M, Qi Y (2013) Computing the nondominated surface in tri-criterion portfolio selection. Oper Res (in press)

  • Hwang CL, Masud ASM (1979) Multiple objective decision making—methods and applications: a state-of-the-art survey. Lecture Notes in Economics and Mathematical Systems, vol 164. Springer, Berlin

  • Korhonen P, Yu GY (1998) On computing objective function values in multiple objective quadratic-linear programming. Eur J Oper Res 106(1):184–190

    Article  Google Scholar 

  • Lo AW, Petrov C, Wierzbicki M (2003) It’s 11pm—do you know where your liquidity is? The mean-variance-liquidity frontier. J Invest Manag 1(1):55–93

    Google Scholar 

  • Markowitz HM (1952) Portfolio selection. J Financ 7(1):77–91

    Google Scholar 

  • Markowitz HM (1956) The optimization of a quadratic function subject to linear constraints. Naval Res Log Q 3(1–2):111–133

    Article  Google Scholar 

  • Markowitz HM (1959) Portfolio selection: efficient diversification of investments. Wiley, Hoboken

  • Markowitz HM (1987) Mean-variance analysis in portfolio choice and capital markets. Basil Blackwell, Oxford

  • Markowitz HM, Todd P (2000) Mean-variance analysis in portfolio choice and capital markets. Frank J. Fabozzi Associates, New Hope

  • Miettinen KM (1999) Nonlinear multiobjective optimization. Kluwer, Boston

  • Niedermayer A, Niedermayer D (2010) Applying Markowitz’s critical line algorithm. In: Guerard JB (ed) Handbook of portfolio construction. Springer, Berlin, pp 383–400

  • Ogryczak W (2000) Multiple criteria linear programming model for portfolio selection. Ann Oper Res 97:143–162

    Article  Google Scholar 

  • Qi Y, Hirschberger M, Steuer RE (2009) Dotted representations of mean-variance efficient frontiers and their computation. INFOR 47(3):15–21

    Google Scholar 

  • Stein M, Branke J, Schmeck H (2008) Efficient implementation of an active set algorithm for large-scale portfolio selection. Comput Oper Res 35(12):3945–3961

    Article  Google Scholar 

  • Steuer RE (1986) Multiple criteria optimization: theory, computation and application. Wiley, New York

  • Steuer RE, Na P (2003) Multiple criteria decision making combined with finance: a categorized bibliography. Eur J Oper Res 150(3):496–515

    Article  Google Scholar 

  • Steuer RE, Qi Y, Hirschberger M (2011) Comparative issues in large-scale mean-variance efficient frontier computation. Decis Support Syst 51(2):250–255

    Article  Google Scholar 

  • Stone BK (1973) A linear programming formulation of the general portfolio selection problem. J Financ Quant Anal 8(4):621–636

    Article  Google Scholar 

  • Winston WL (2004) Operations research: applications and algorithms, 4th edn. Brooks Cole, Belmont

Download references

Author information

Authors and Affiliations

  1. Department of Finance, University of Georgia, Athens, Georgia, 30602-6253, USA

    Ralph E. Steuer

  2. Department of Finance, University of Regensburg, 93040, Regensburg, Germany

    Maximilian Wimmer

  3. Munich Re, 80802, Munich, Germany

    Markus Hirschberger

Authors
  1. Ralph E. Steuer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maximilian Wimmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Markus Hirschberger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ralph E. Steuer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Steuer, R.E., Wimmer, M. & Hirschberger, M. Overviewing the transition of Markowitz bi-criterion portfolio selection to tri-criterion portfolio selection. J Bus Econ 83, 61–85 (2013). https://doi.org/10.1007/s11573-012-0642-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11573-012-0642-4

Keywords

JEL Classification

Search

Navigation