Abstract
In this paper, we investigate the distributional properties of the estimated tangency portfolio (TP) weights assuming that the asset returns follow a matrix variate closed skew-normal distribution. We establish a stochastic representation of the linear combination of the estimated TP weights that fully characterizes its distribution. Using the stochastic representation we derive the mean and variance of the estimated weights of TP which are of key importance in portfolio analysis. Furthermore, we provide the asymptotic distribution of the linear combination of the estimated TP weights under the high-dimensional asymptotic regime, i.e., the dimension of the portfolio p and the sample size n tend to infinity such that 
A good performance of the theoretical findings is documented in the simulation study. In an empirical study, we apply the theoretical results to real data of the stocks included in the S&P 500 index.
Acknowledgment
The authors are thankful to Prof. Zhe George Zhang, the Associate Editor, and two anonymous Reviewers for the careful reading of the manuscript and for their suggestions that have improved an earlier version of this paper The authors acknowledge financial support from the project” Models for macro and financial economics after the financial crisis” (Dnr: P18-0201) funded by the Jan Wallander and Tom Hedelius Foundation. Farrukh Javed and Stepan Mazur also acknowledge financial support from the internal research grants of Örebro University.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 In the Bayesian framework, the posterior distribution of the TP weights is proportional to the product of (singular) Wishart matrix and (singular) normal vector under the assumption of normally distributed data. The distributional properties of these products are well studied by Bodnar et al. (Citation2013, Citation2016), Bodnar et al. (Citation2020), Bodnar et al. (Citation2019).
2 Additionally assuming that rank () = Bodnar et al. (Citation2016, Citation2017) and Bodnar et al. (Citation2019) employed the Moore-Penrose inverse in the portfolio context. One can also make use of different regularization methods such as the ridge-type approach (Tikhonov & Arsenin, Citation1977), the Landweber-Fridman algorithm (Kress, Citation1999), the spectral cut-off approach (Chernousova & Golubev, Citation2014), the Lasso-type method (Brodie et al., Citation2009), and an iterative method based on a second order damped dynamical systems (Gulliksson et al., Citation2023; Gulliksson & Mazur, Citation2020).
Bodnar, T., Mazur, S., & Okhrin, Y. (2013). On the exact and approximate distributions of the product of a Wishart matrix with a normal vector. Journal of Multivariate Analysis, 122, 70–81. https://doi.org/10.1016/j.jmva.2013.07.007 Bodnar, T., Mazur, S., & Okhrin, Y. (2016). Distribution of the product of singular Wishart matrix and normal vector. Theory of Probability and Mathematical Statistics, 91, 1–15. https://doi.org/10.1090/tpms/962 Bodnar, T., Mazur, S., Muhinyuza, S., & Parolya, N. (2020). On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension. Theory of Probability and Mathematical Statistics, 99(2), 39–52. https://doi.org/10.1090/tpms/1078 Bodnar, T., Mazur, S., & Parolya, N. (2019). Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions. Scandinavian Journal of Statistics, 46(2), 636–660. https://doi.org/10.1111/sjos.12383 Bodnar, T., Mazur, S., & Okhrin, Y. (2016). Distribution of the product of singular Wishart matrix and normal vector. Theory of Probability and Mathematical Statistics, 91, 1–15. https://doi.org/10.1090/tpms/962 Bodnar, T., Mazur, S., & Podgórski, K. (2017). A test for the global minimum variance portfolio for small sample and singular covariance. AStA Advances in Statistical Analysis, 101(3), 253–265. https://doi.org/10.1007/s10182-016-0282-z Bodnar, T., Mazur, S., & Parolya, N. (2019). Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix-variate location mixture of normal distributions. Scandinavian Journal of Statistics, 46(2), 636–660. https://doi.org/10.1111/sjos.12383 Tikhonov, A., & Arsenin, V. (1977). Solutions of ill-posed problems. Winston. Kress, R. (1999). Linear integral equations. Springer. Chernousova, E., & Golubev, Y. (2014). Spectral cut-off regularizations for ill-posed linear models. Mathematical Methods of Statistics, 23(2), 116–131. https://doi.org/10.3103/S1066530714020033 Brodie, J., Daubechies, I., De Mol, C., Giannone, D., & Loris, I. (2009). Sparse and stable markowitz portfolios. Proceedings of the National Academy of Sciences of the United States of America, 106(30), 12267–12272. https://doi.org/10.1073/pnas.0904287106 Gulliksson, M., Oleynik, A., & Mazur, S. (2023). Portfolio selection with a rank-deficient covariance matrix. Computational Economics, 104(4), 4. https://doi.org/10.1007/s10614-023-10404-4 Gulliksson, M., & Mazur, S. (2020). An iterative approach to ill-conditioned optimal portfolio selection. Computational Economics, 56(4), 773–794. https://doi.org/10.1007/s10614-019-09943-6 Additional information
Funding
This work was supported by the Jan Wallanders och Tom Hedelius Stiftelse samt Tore Browaldhs Stiftelse.
https://orcid.org/0000-0002-1395-9427
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