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Singular inverse Wishart distribution and its application to portfolio theory

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  • Bodnar, Taras
  • Mazur, Stepan
  • Podgórski, Krzysztof

Abstract

The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean–variance portfolio weights as well as to derive its characteristic function and the moments of higher order. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:143:y:2016:i:c:p:314-326
    DOI: 10.1016/j.jmva.2015.09.021
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    References listed on IDEAS

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    3. 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.
    4. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.
    5. Taras Bodnar & Wolfgang Schmid & Taras Zabolotskyy, 2013. "Asymptotic behavior of the estimated weights and of the estimated performance measures of the minimum VaR and the minimum CVaR optimal portfolios for dependent data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(8), pages 1105-1134, November.
    6. Taras Bodnar & Wolfgang Schmid, 2009. "Econometrical analysis of the sample efficient frontier," The European Journal of Finance, Taylor & Francis Journals, vol. 15(3), pages 317-335.
    7. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    8. Taras Bodnar & Wolfgang Schmid, 2008. "A test for the weights of the global minimum variance portfolio in an elliptical model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 67(2), pages 127-143, March.
    9. Okhrin, Yarema & Schmid, Wolfgang, 2006. "Distributional properties of portfolio weights," Journal of Econometrics, Elsevier, vol. 134(1), pages 235-256, September.
    10. Giamouridis, Daniel & Vrontos, Ioannis D., 2007. "Hedge fund portfolio construction: A comparison of static and dynamic approaches," Journal of Banking & Finance, Elsevier, vol. 31(1), pages 199-217, January.
    11. Tu, Jun & Zhou, Guofu, 2004. "Data-generating process uncertainty: What difference does it make in portfolio decisions?," Journal of Financial Economics, Elsevier, vol. 72(2), pages 385-421, May.
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    Cited by:

    1. 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.
    2. Ruili Sun & Tiefeng Ma & Shuangzhe Liu & Milind Sathye, 2019. "Improved Covariance Matrix Estimation for Portfolio Risk Measurement: A Review," JRFM, MDPI, vol. 12(1), pages 1-34, March.
    3. Gulliksson, Mårten & Oleynik, Anna & Mazur, Stepan, 2021. "Portfolio Selection with a Rank-deficient Covariance Matrix," Working Papers 2021:12, Örebro University, School of Business.
    4. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof & Tyrcha, Joanna, 2018. "Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory," Working Papers 2018:1, Örebro University, School of Business.
    5. Duc Thi Luu, 2022. "Portfolio Correlations in the Bank-Firm Credit Market of Japan," Computational Economics, Springer;Society for Computational Economics, vol. 60(2), pages 529-569, August.
    6. Bauder, David & Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2020. "Bayesian inference of the multi-period optimal portfolio for an exponential utility," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    7. Kobayashi, Ken & Takano, Yuichi & Nakata, Kazuhide, 2023. "Cardinality-constrained distributionally robust portfolio optimization," European Journal of Operational Research, Elsevier, vol. 309(3), pages 1173-1182.
    8. 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.
    9. Tsukuma, Hisayuki, 2016. "Estimation of a high-dimensional covariance matrix with the Stein loss," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 1-17.
    10. Drin, Svitlana & Mazur, Stepan & Muhinyuza, Stanislas, 2023. "A test on the location of tangency portfolio for small sample size and singular covariance matrix," Working Papers 2023:11, Örebro University, School of Business.
    11. Golosnoy, Vasyl & Schmid, Wolfgang & Seifert, Miriam Isabel & Lazariv, Taras, 2020. "Statistical inferences for realized portfolio weights," Econometrics and Statistics, Elsevier, vol. 14(C), pages 49-62.
    12. 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.
    13. 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.
    14. Apostolos Chalkis & Emmanouil Christoforou & Ioannis Z. Emiris & Theodore Dalamagas, 2021. "Modeling asset allocations and a new portfolio performance score," Digital Finance, Springer, vol. 3(3), pages 333-371, December.
    15. Javed, Farrukh & Mazur, Stepan & Thorsén, Erik, 2021. "Tangency portfolio weights under a skew-normal model in small and large dimensions," Working Papers 2021:13, Örebro University, School of Business.
    16. Karlsson, Sune & Mazur, Stepan & Muhinyuza, Stanislas, 2020. "Statistical Inference for the Tangency Portfolio in High Dimension," Working Papers 2020:10, Örebro University, School of Business.
    17. 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.
    18. Kozubowski, Tomasz J. & Mazur, Stepan & Podgorski, Krysztof, 2022. "Matrix Variate Generalized Laplace Distributions," Working Papers 2022:7, Örebro University, School of Business.
    19. Don S. Poskitt, 2020. "On GMM Inference: Partial Identification, Identification Strength, and Non-Standard," Monash Econometrics and Business Statistics Working Papers 40/20, Monash University, Department of Econometrics and Business Statistics.
    20. Gulliksson, Mårten & Mazur, Stepan, 2019. "An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection," Working Papers 2019:3, Örebro University, School of Business.
    21. Ruili Sun & Tiefeng Ma & Shuangzhe Liu, 2018. "A Stein-type shrinkage estimator of the covariance matrix for portfolio selections," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(8), pages 931-952, November.
    22. Ruili Sun & Tiefeng Ma & Shuangzhe Liu, 2020. "Portfolio selection: shrinking the time-varying inverse conditional covariance matrix," Statistical Papers, Springer, vol. 61(6), pages 2583-2604, December.

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