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Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation

Author

Listed:
  • Francisco Fernández-Navarro

    (Department of Quantitative Methods, Universidad Loyola Andalucía, 41704 Sevilla, Spain)

  • Luisa Martínez-Nieto

    (Department of Financial Economics and Accounting, Universidad Loyola Andalucía, 41704 Sevilla, Spain)

  • Mariano Carbonero-Ruz

    (Department of Quantitative Methods, Universidad Loyola Andalucía, 41704 Sevilla, Spain)

  • Teresa Montero-Romero

    (Department of Financial Economics and Accounting, Universidad Loyola Andalucía, 41704 Sevilla, Spain)

Abstract

The mean-variance (MV) portfolio is typically formulated as a quadratic programming (QP) problem that linearly combines the conflicting objectives of minimizing the risk and maximizing the expected return through a risk aversion profile parameter. In this formulation, the two objectives are expressed in different units, an issue that could definitely hamper obtaining a more competitive set of portfolio weights. For example, a modification in the scale in which returns are expressed (by one or percent) in the MV portfolio, implies a modification in the solution of the problem. Motivated by this issue, a novel mean squared variance (MSV) portfolio is proposed in this paper. The associated optimization problem of the proposed strategy is very similar to the Markowitz optimization, with the exception of the portfolio mean, which is presented in squared form in our formulation. The resulting portfolio model is a non-convex QP problem, which has been reformulated as a mixed-integer linear programming (MILP) problem. The reformulation of the initial non-convex QP problem into an MILP allows for future researchers and practitioners to obtain the global solution of the problem via the use of current state-of-the-art MILP solvers. Additionally, a novel purely data-driven method for determining the optimal value of the hyper-parameter that is associated with the MV and MSV approaches is also proposed in this paper. The MSV portfolio has been empirically tested on eight portfolio time series problems with three different estimation windows (composing a total of 24 datasets), showing very competitive performance in most of the problems.

Suggested Citation

  • Francisco Fernández-Navarro & Luisa Martínez-Nieto & Mariano Carbonero-Ruz & Teresa Montero-Romero, 2021. "Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation," Mathematics, MDPI, vol. 9(3), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:223-:d:485804
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    References listed on IDEAS

    as
    1. Ron Bird & Mark Tippett, 1986. "Note---Naive Diversification and Portfolio Risk---A Note," Management Science, INFORMS, vol. 32(2), pages 244-251, February.
    2. Tu, Jun & Zhou, Guofu, 2011. "Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies," Journal of Financial Economics, Elsevier, vol. 99(1), pages 204-215, January.
    3. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," The Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    4. Guillaume Coqueret, 2015. "Diversified minimum-variance portfolios," Annals of Finance, Springer, vol. 11(2), pages 221-241, May.
    5. Maillet, Bertrand & Tokpavi, Sessi & Vaucher, Benoit, 2015. "Global minimum variance portfolio optimisation under some model risk: A robust regression-based approach," European Journal of Operational Research, Elsevier, vol. 244(1), pages 289-299.
    6. Guillaume Coqueret, 2015. "Diversified minimum-variance portfolios," Post-Print hal-02312223, HAL.
    7. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    8. Angela J. Black & David G. McMillan, 2004. "Non‐linear Predictability of Value and Growth Stocks and Economic Activity," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 31(3‐4), pages 439-474, April.
    9. repec:dau:papers:123456789/14735 is not listed on IDEAS
    10. Apostolos Kourtis, 2015. "A Stability Approach to Mean-Variance Optimization," The Financial Review, Eastern Finance Association, vol. 50(3), pages 301-330, August.
    11. Anatoly B. Schmidt, 2019. "Managing portfolio diversity within the mean variance theory," Annals of Operations Research, Springer, vol. 282(1), pages 315-329, November.
    12. Best, Michael J & Grauer, Robert R, 1991. "On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results," The Review of Financial Studies, Society for Financial Studies, vol. 4(2), pages 315-342.
    13. Vijay K. Chopra & William T. Ziemba, 2013. "The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 21, pages 365-373, World Scientific Publishing Co. Pte. Ltd..
    14. Laurens Swinkels & Liam Tjong-A-Tjoe, 2007. "Can mutual funds time investment styles?," Journal of Asset Management, Palgrave Macmillan, vol. 8(2), pages 123-132, July.
    15. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    16. Haotian Cai & Anatoly B. Schmidt, 2020. "Comparing mean–variance portfolios and equal-weight portfolios for major US equity indexes," Journal of Asset Management, Palgrave Macmillan, vol. 21(4), pages 326-332, July.
    17. Bertrand Maillet & Sessi Tokpavi & Benoît Vaucher, 2015. "Global minimum variance portfolio optimisation under some model risk : A robust regression-based approach," Post-Print hal-02312329, HAL.
    18. Liesiö, Juuso & Xu, Peng & Kuosmanen, Timo, 2020. "Portfolio diversification based on stochastic dominance under incomplete probability information," European Journal of Operational Research, Elsevier, vol. 286(2), pages 755-768.
    19. Wei Xia & Juan C. Vera & Luis F. Zuluaga, 2020. "Globally Solving Nonconvex Quadratic Programs via Linear Integer Programming Techniques," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 40-56, January.
    20. Guillaume Coqueret, 2015. "Diversified minimum-variance portfolios," Post-Print hal-02009587, HAL.
    21. Haotian Cai & Anatoly B. Schmidt, 0. "Comparing mean–variance portfolios and equal-weight portfolios for major US equity indexes," Journal of Asset Management, Palgrave Macmillan, vol. 0, pages 1-7.
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