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Portfolio Selection with a Rank-deficient Covariance Matrix

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In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz’ problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.

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  • Gulliksson, Mårten & Oleynik, Anna & Mazur, Stepan, 2021. "Portfolio Selection with a Rank-deficient Covariance Matrix," Working Papers 2021:12, Örebro University, School of Business.
    • Handle: RePEc:hhs:oruesi:2021_012
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    1. Dai, Zhifeng & Wen, Fenghua, 2018. "Some improved sparse and stable portfolio optimization problems," Finance Research Letters, Elsevier, vol. 27(C), pages 46-52.
    2. Farrukh Javed & Stepan Mazur & Edward Ngailo, 2021. "Higher order moments of the estimated tangency portfolio weights," Journal of Applied Statistics, Taylor & Francis Journals, vol. 48(3), pages 517-535, February.
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    4. Taras Bodnar & Stepan Mazur & Krzysztof Podgórski, 2017. "A test for the global minimum variance portfolio for small sample and singular covariance," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 101(3), pages 253-265, July.
    5. David Bauder & Taras Bodnar & Stepan Mazur & Yarema Okhrin, 2018. "Bayesian Inference For The Tangent Portfolio," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-27, December.
    6. Bodnar, Taras & Mazur, Stepan & Muhinyuza, Stanislas & Parolya, Nestor, 2017. "On the product of a singular Wishart matrix and a singular Gaussian vector in high dimensions," Working Papers 2017:7, Örebro University, School of Business.
    7. Alfelt, Gustav & Mazur, Stepan, 2020. "On the mean and variance of the estimated tangency portfolio weights for small samples," Working Papers 2020:8, Örebro University, School of Business.
    8. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
    9. Bodnar, Taras & Mazur, Stepan & Okhrin, Yarema, 2013. "On the exact and approximate distributions of the product of a Wishart matrix with a normal vector," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 70-81.
    10. Mårten Gulliksson & Stepan Mazur, 2020. "An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 773-794, December.
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    13. Taras Bodnar & Yarema Okhrin, 2011. "On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(2), pages 311-331, June.
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    Cited by:

    1. Farrukh Javed & Stepan Mazur & Erik Thorsén, 2024. "Tangency portfolio weights under a skew-normal model in small and large dimensions," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 75(7), pages 1395-1406, July.
    2. Bodnar, Taras & Mazur, Stepan & Nguyen, Hoang, 2022. "Estimation of optimal portfolio compositions for small sampleand singular covariance matrix," Working Papers 2022:15, Örebro University, School of Business.

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    More about this item

    Keywords

    Mean–variance portfolio; Rank-deficient covariance matrix; Linear ill-posed problems; Second order damped dynamical systems;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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