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Solving cardinality constrained mean-variance portfolio problems via MILP

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Abstract

Controlling the number of active assets (cardinality of the portfolio) in a mean-variance portfolio problem is practically important but computationally demanding. Such task is ordinarily a mixed integer quadratic programming (MIQP) problem. We propose a novel approach to reformulate the problem as a mixed integer linear programming (MILP) problem for which computer codes are readily available. For numerical tests, we find cardinality constrained minimum variance portfolios of stocks in S&P500. A significant gain in robustness and computational effort by our MILP approach relative to MIQP is reported. Similarly, our MILP approach also competes favorably against cardinality constrained portfolio optimization with risk measures CVaR and MASD. For illustrations, we depict portfolios in a portfolio map where cardinality provides a third criterion in addition to risk and return. Fast solution allows an interactive search for a desired portfolio.

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Notes

  1. A drawback of the exact methods is that usually their computer codes are not publicly available. Moreover, programming languages provide various features for optimizing a code without modifying the algorithm and the performance of the method may depend on skills of the user. Hence, it is impractical to test the methods on new data sets and compare with competing methods.

  2. If CPLEX is used to solve MILP problems, it is possible to use “indicator constraints” and avoid using big M. In this approach, constraint \(h_{i}\ge -M(1-y_{i})\) is written as “if \(y_{i}=1\) then \(h_{i}\ge 0\) ”.

  3. By inexact methods we mean, for instance, heuristic methods or exact methods which are terminated due to loose tolerances or before optimality is confirmed.

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Acknowledgements

The authors would like to express their gratitude to the anonymous referees and the Editor-in-Chief of the journal for the helpful comments and suggestions on this paper. The first author would like to thank the Jenny and Antti Wihuri Foundation for their support in funding this research.

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Authors and Affiliations

  1. Department of Information and Service Economy, Aalto University School of Business, Helsinki, Finland

    Nasim Dehghan Hardoroudi, Abolfazl Keshvari, Markku Kallio & Pekka Korhonen

Authors
  1. Nasim Dehghan Hardoroudi
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  2. Abolfazl Keshvari
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  3. Markku Kallio
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  4. Pekka Korhonen
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Correspondence to Abolfazl Keshvari.

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The first author is supported by the Jenny and Antti Wihuri Foundation.

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Dehghan Hardoroudi, N., Keshvari, A., Kallio, M. et al. Solving cardinality constrained mean-variance portfolio problems via MILP. Ann Oper Res 254, 47–59 (2017). https://doi.org/10.1007/s10479-017-2447-x

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  • DOI: https://doi.org/10.1007/s10479-017-2447-x

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