Skip to main content
SpringerLink
Log in

A new method for mean-variance portfolio optimization with cardinality constraints

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM) model, where the assets are limited with the introduction of quantity and cardinality constraints.

We propose a completely new approach for solving the LAM model based on a reformulation as a Standard Quadratic Program, on a new lower bound that we establish, and on other recent theoretical and computational results for such problem. These results lead to an exact algorithm for solving the LAM model for small size problems. For larger problems, such algorithm can be relaxed to an efficient and accurate heuristic procedure that is able to find the optimal or the best-known solutions for problems based on some standard financial data sets that are used by several other authors. We also test our method on five new data sets involving real-world capital market indices from major stock markets. We compare our results with those of CPLEX and with those obtained with very recent heuristic approaches in order to illustrate the effectiveness of our method in terms of solution quality and of computation time. All our data sets and results are publicly available for use by other researchers.

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

Similar content being viewed by others

References

  • Anagnostopoulos, K. P., & Mamanis, G. (2010). A portfolio optimization model with three objectives and discrete variables. Computers & Operations Research, 37, 1285–1297.

    Article  Google Scholar 

  • Anagnostopoulos, K. P., & Mamanis, G. (2011). The mean-variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications, 38, 14208–14217.

    Google Scholar 

  • Armañanzas, R., & Lozano, J. A. (2005). A multiobjective approach to the portfolio optimization problem. In Proceedings of the 2005 IEEE congress on evolutionary computation (CEC 2005) (Vol. 2, pp. 1388–1395). New York: IEEE Press. doi:10.1109/CEC.2005.1554852.

    Chapter  Google Scholar 

  • Beasley, J. E. (1990). Or-library: distributing test problems by electronic mail. Journal of the Operational Research Society, 41, 1069–1072.

    Google Scholar 

  • Bertsimas, D., & Shioda, R. (2009). Algorithms for cardinality-constrained quadratic optimization. Computational Optimization and Applications, 43, 1–22.

    Article  Google Scholar 

  • Bienstock, D. (1996). Computational study of a family of mixed-integer quadratic programming problems. Mathematical Programming, 74, 121–140.

    Google Scholar 

  • Bomze, I. M. (1998). On standard quadratic optimization problems. Journal of Global Optimization, 13, 369–387.

    Article  Google Scholar 

  • Bomze, I. M., Locatelli, M., & Tardella, F. (2008). New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Mathematical Programming, 115, 31–64.

    Article  Google Scholar 

  • Borchers, B., & Mitchell, J. E. (1994). An improved branch and bound algorithm for mixed integer nonlinear programs. Computers & Operations Research, 21, 359–367.

    Article  Google Scholar 

  • Branke, J., Scheckenbach, B., Stein, M., Deb, K., & Schmeck, H. (2009). Portfolio optimization with an envelope-based multi-objective evolutionary algorithm. European Journal of Operational Research, 199, 684–693.

    Article  Google Scholar 

  • Canakgoz, N. A., & Beasley, J. E. (2008). Mixed-integer programming approaches for index tracking and enhanced indexation. European Journal of Operational Research, 196, 384–399.

    Article  Google Scholar 

  • Cesarone, F., Scozzari, A., & Tardella, F. (2008). Efficient algorithms for mean-variance portfolio optimization with hard real-world constraints. In Proceedings of the 18th AFIR colloquium: financial risk in a changing world, Rome.

  • Chang, T.-J., Meade, M., Beasley, J. E., & Sharaiha, Y. M. (2000). Heuristics for cardinality constrained portfolio optimization. Computers & Operations Research, 27, 1271–1302.

    Article  Google Scholar 

  • Chiam, S. C., Tan, K. C., & Al Mamum, A. (2008). Evolutionary multi-objective portfolio optimization in practical context. International Journal of Automation and Computing, 5, 67–80.

    Article  Google Scholar 

  • Crama, Y., & Schyns, M. (2003). Simulated annealing for complex portfolio selection problems. European Journal of Operational Research, 150, 546–571.

    Article  Google Scholar 

  • Di Gaspero, L., Di Tollo, G., Roli, A., & Schaerf, A. (2007). Hybrid local search for constrained financial portfolio selection problems. In P. Van Hentenryck & L. Wolsey (Eds.), Lecture notes in computer science: Vol. 4510. Integration of AI and OR techniques in constraint programming for combinatorial optimization problems, fourth international conference, CPAIOR 2007 (pp. 44–58). Berlin: Springer.

    Chapter  Google Scholar 

  • Di Gaspero, L., Di Tollo, G., Roli, A., & Schaerf, A. (2010). Hybrid metaheuristics for constrained portfolio selection problems. Quantitative Finance. doi:10.1080/14697680903460168.

    Google Scholar 

  • Ehrgott, M., Klamroth, K., & Schwehm, C. (2004). An MCDM approach to portfolio optimization. European Journal of Operational Research, 155, 752–770.

    Article  Google Scholar 

  • Elton, E. J., & Gruber, M. J. (1995). Modern portfolio theory and investment analysis (5th ed.). New York: Wiley.

    Google Scholar 

  • Fernández, A., & Gómez, S. (2007). Portfolio selection using neural networks. Computers & Operations Research, 34, 1177–1191.

    Article  Google Scholar 

  • Fiacco, A. V., & Kyparisis, J. (1986). Convexity and concavity properties of the optimal value function in parametric nonlinear programming. Journal of Optimization Theory and Applications, 48, 95–126.

    Google Scholar 

  • Fieldsend, J. E., Matatko, J., & Peng, M. (2004). Cardinality constrained portfolio optimisation. In Lecture notes in computer science (Vol. 3177, pp. 788–793).

    Google Scholar 

  • Frangioni, A., & Gentile, C. (2006). Perspective cuts for a class of convex 0–1 mixed integer programs. Mathematical Programming, 106, 225–236.

    Article  Google Scholar 

  • Gomez, M. A., Flores, C. X., & Osorio, M. A. (2006). Hybrid search for cardinality constrained portfolio optimization. In Proceedings of the 8th annual conference on genetic and evolutionary computation (pp. 1865–1866). New York: ACM.

    Chapter  Google Scholar 

  • Gulpinar, N., An, L. T. H., & Moeini, M. (2010). Robust investment strategies with discrete asset choice constraints using DC programming. Optimization, 59, 46–62.

    Article  Google Scholar 

  • Holmstrom, K., Goran, A. O., & Edvall, M. M. (2007). Users guide for TOMLAB.

  • Jobst, N. J., Horniman, M. D., Lucas, C. A., & Mitra, G. (2001). Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints. Quantitative Finance, 1, 489–501.

    Article  Google Scholar 

  • King, A. J. (1993). Asymmetric risk measures and tracking models for portfolio optimization under uncertainty. Annals of Operations Research, 45, 165–177.

    Article  Google Scholar 

  • Konno, H., & Yamazaki, H. (1991). Mean-absolute deviation portfolio optimization model and its application to Tokyo stock exchange. Management Science, 37, 519–531.

    Article  Google Scholar 

  • Lee, E. K., & Mitchell, J. E. (1997). Computational experience of an interior-point SQP algorithm in a parallel branch-and-bound framework. In Proceedings of high performance optimization technique. Berlin: Springer.

    Google Scholar 

  • Lemke, C. E., & Howson, J. T. (1964). Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics, 12, 414–423.

    Article  Google Scholar 

  • Li, D., Sun, X., & Wang, J. (2006). Optimal lot solution to cardinality constrained mean-variance formulation for portfolio selection. Mathematical Finance, 16, 83–101.

    Article  Google Scholar 

  • Maringer, D., & Kellerer, H. (2003). Optimization of cardinality constrained portfolios with a hybrid local search algorithm. OR-Spektrum, 25, 481–495.

    Article  Google Scholar 

  • Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77–91.

    Google Scholar 

  • Markowitz, H. M. (1959). Cowles foundation for research in economics at Yale university, monograph: Vol. 16. Portfolio selection: efficient diversification of investments. New York: Wiley.

    Google Scholar 

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

    Google Scholar 

  • Martinjak, I. (2009). Cardinality constrained portfolio optimization by means of genetic algorithms. In Proceedings of the 20th central European conference on information and intelligent systems, Varazin, Croatia.

  • Mills, T. C. (1997). Stylized facts on the temporal and distributional properties of daily FT-SE returns. Applied Financial Economics, 7, 599–604.

    Article  Google Scholar 

  • Mitra, G., Kriakis, T., Lucas, C., & Pirbhai, M. (2003). Advances in portfolio construction and implementation. In S. E. Stachell & A. Scowcroft (Eds.), A review of portfolio planning: models and systems. Oxford: Butterworth-Heinemann.

    Google Scholar 

  • Moral-Escudero, R., Ruiz-Torrubiano, R., & Suárez, A. (2006). Selection of optimal investment with cardinality constraints. In Proceedings of the IEEE world congress on evolutionary computation, CEC 2006 (pp. 2382–2388). New York: IEEE Press.

    Google Scholar 

  • Morgan, J. P. (1996). Riskmetrics-technical document (Tech. report, 4th ed.). New York: Morgan Guaranty Trust Company of New York.

  • Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–42.

    Google Scholar 

  • Ruiz-Torrubiano, R., & Suarez, A. (2010). Hybrid approaches and dimensionality reduction for portfolio selection with cardinality constrains. IEEE Computational Intelligence Magazine, 5, 92–107.

    Article  Google Scholar 

  • Schaerf, A. (2002). Local search techniques for constrained portfolio selection problems. Computational Economics, 20, 177–190.

    Article  Google Scholar 

  • Scozzari, A., & Tardella, F. (2008). A clique algorithm for standard quadratic programming. Discrete Applied Mathematics, 156, 2439–2448.

    Article  Google Scholar 

  • Shaw, D. X., Liu, S., & Kopman, L. (2008). Lagrangian relaxation procedure for cardinality-constrained portfolio optimization. Optimization Methods & Software, 23, 411–420.

    Article  Google Scholar 

  • Streichert, F., & Tanaka-Yamawaki, M. (2006). The effect of local search on the constrained portfolio selection problem. In IEEE international congress on evolutionary computing, proceedings, CEC (pp. 2368–2374). New York: ACM.

    Google Scholar 

  • Streichert, F., Ulmer, H., & Zell, A. (2003). Evolutionary algorithms and the cardinality constrained portfolio selection problem. In D. Ahr, R. Fahrion, M. Oswald & G. Reinelt (Eds.), Selected papers of the international conference on operations research. Berlin: Springer.

    Google Scholar 

  • Streichert, F., Ulmer, H., & Zell, A. (2004). Comparing discrete and continuous genotypes on the constrained portfolio selection problem. In Genetic and evolutionary computation conference, proceedings, Part II, GECCO (pp. 1239–1250). Berlin: Springer.

    Google Scholar 

  • Tardella, F. (2004). Connections between continuous and combinatorial optimization problems through an extension of the fundamental theorem of linear programming. Electronic Notes in Discrete Mathematics, 17, 257–262.

    Article  Google Scholar 

  • Tardella, F. (2011). The fundamental theorem of linear programming: extensions and application. Optimization, 60, 283–301.

    Article  Google Scholar 

  • Vielma, J. P., Ahmed, S., & Nemhauser, G. L. (2008). A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS Journal on Computing, 20, 438–450.

    Article  Google Scholar 

  • Woodside-Oriakhi, M., Lucas, C., & Beasley, J. E. (2011). Heuristic algorithms for the cardinality constrained efficient frontier. European Journal of Operational Research, 213, 538–550.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Economia, Università degli Studi Roma Tre, Via Silvio D’Amico, 77, 00145, Roma, Italy

    Francesco Cesarone

  2. Facoltà di Economia, Università degli Studi “Niccolò Cusano”, Via Casalmonferrato, 2/B, 00182, Roma, Italy

    Andrea Scozzari

  3. Dipartimento di Metodi e Modelli per l’Economia, il Territorio e la Finanza, Sapienza Università di Roma, Via del Castro Laurenziano, 9, 00161, Roma, Italy

    Fabio Tardella

Authors
  1. Francesco Cesarone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Scozzari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fabio Tardella
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Scozzari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cesarone, F., Scozzari, A. & Tardella, F. A new method for mean-variance portfolio optimization with cardinality constraints. Ann Oper Res 205, 213–234 (2013). https://doi.org/10.1007/s10479-012-1165-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1165-7

Keywords

Search

Navigation