Abstract
The global minimum-variance portfolio is a typical choice for investors because of its simplicity and broad applicability. Although it requires only one input, namely the covariance matrix of asset returns, estimating the optimal solution remains a challenge. In the presence of high dimensionality in the data, the sample covariance estimator becomes ill-conditioned and leads to suboptimal portfolios out-of-sample. To address this issue, we review recently proposed efficient estimation methods for the covariance matrix and extend the literature by suggesting a multifold cross-validation technique for selecting the necessary tuning parameters within each method. Conducting an extensive empirical analysis with three datasets based on the Russell 3000, we show that choosing the specific tuning parameters with the proposed cross-validation improves the out-of-sample performance of the global minimum-variance portfolio. In addition, we identify estimators that are strongly influenced by the choice of the tuning parameter and detect a clear relationship between the selection criterion within the cross-validation and the evaluated performance measure.
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Notes
DeMiguel et al. (2009b) additionally show that the mean-variance portfolio is outperformed out-of-sample by the minimum-variance portfolio not only in terms of risk, but as well in respect to the return-risk ratio.
This class of estimators was first introduced by Stein (1986).
Section 2.4 offers more details on this type of estimators.
Following this definition and assuming K common factors with \(K<n\), a covariance matrix estimator based on factor models only needs to estimate \(K(K+1)/2\) covariance entries and is thus more stable.
For the operational use of POET, the threshold value c needs to be determined, so that the positive-definiteness of \(\widehat{{\varSigma }}^{c}_{u,K}\) is assured in finite samples. The choice of c can therefore occur from a set, for which the respective minimal eigenvalue of the errors’ covariance matrix after thresholding is positive. The minimal constant c that guarantees positive-definiteness is then chosen. For more details, see Fan et al. (2013).
This idea was first proposed by Dempster (1972) with the so-called covariance selection model.
This ensures that no penalty is applied to the asset returns’ sample variances.
Goto and Xu (2015) induce sparsity to enhance robustness and lower the estimation error within portfolio hedging strategies, Brownlees et al. (2018) develop a procedure called “realized network” by applying GLASSO as a regularization procedure for realized covariance estimators, and Torri et al. (2019) analyze the out-of-sample performance of a minimum-variance portfolio, estimated with GLASSO.
For clarity in the notation, we do not differentiate between covariance estimators. The procedure is applied to all methods equally.
The price history originates from http://www.kibot.com/.
As a reference, De Nard et al. (2019) consider a study setup with concentration ratios \(q\approx \left\{ 0.08, 0.4, 0.8\right\} \).
To account for possible differences in the results due to randomization, we performed the study with various random seeds and reached similar results.
For the sake of completeness, we have also performed a block bootstrap as in Ledoit and Wolf (2011). The corresponding significant values are comparable to those from the HAC test and are therefore not reported.
The other datasets produce similar results. For reference, see Fig. 4, “Appendix A”.
“Appendix B” compares further datasets. Overall, the results are similar in tendency, but are less pronounced due to a lower dimensionality in the data.
Similar reduction in turnover takes place in the case of the \(\hbox {LW}^{\mathrm{CC}}\) estimator, as well.
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We would like to thank the editor Markus Schmid and the anonymous referees for their constructive comments and recommendations, which helped to improve the quality of this paper.
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Appendices
A covariance parameters
See Fig. 4.
Optimally selected parameters with original, CV1, CV21, and CV22 covariance estimation methods for the 100RUA and 250RUA datasets
B GMV
Relative differences in the annualized SD of GMV portfolios with the 100RUA and 250RUA datasets across the efficient covariance estimation methods
C GMV-130/30
Relative differences in the annualized SD of GMV-130-30 portfolios with the 100RUA and 250RUA datasets across the efficient covariance estimation methods
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Husmann, S., Shivarova, A. & Steinert, R. Cross-validated covariance estimators for high-dimensional minimum-variance portfolios. Financ Mark Portf Manag 35, 309–352 (2021). https://doi.org/10.1007/s11408-020-00376-y
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DOI: https://doi.org/10.1007/s11408-020-00376-y