Minimum variance portfolio optimization in the spiked covariance model

International audience—We study the design of minimum variance portfolio when asset returns follow a low rank factor model. Using results from random matrix theory, an optimal shrinkage approach for the isolated eigenvalues of the covariance matrix is developed. The proposed portfolio optimization strategy is shown to have good performance on synthetic data but not always on real data sets. This leads us to refine the data model by considering time correlation between samples. By updating the shrinkage of the isolated eigenvalues accounting for the unknown time correlation, our portfolio optimization method is shown to have improved performance and achieves lower risk values than competing methods on real financial data sets

Spectrally-Corrected and Regularized Global Minimum Variance Portfolio for Spiked Model

Considering the shortcomings of the traditional sample covariance matrix estimation, this paper proposes an improved global minimum variance portfolio model and named spectral corrected and regularized global minimum variance portfolio (SCRGMVP), which is better than the traditional risk model. The key of this method is that under the assumption that the population covariance matrix follows the spiked model and the method combines the design idea of the sample spectrally-corrected covariance matrix and regularized. The simulation of real and synthetic data shows that our method is not only better than the performance of traditional sample covariance matrix estimation (SCME), shrinkage estimation (SHRE), weighted shrinkage estimation (WSHRE) and simple spectral correction estimation (SCE), but also has lower computational complexity

A Robust Statistics Approach to Minimum Variance Portfolio Optimization

We study the design of portfolios under a minimum risk criterion. The performance of the optimized portfolio relies on the accuracy of the estimated covariance matrix of the portfolio asset returns. For large portfolios, the number of available market returns is often of similar order to the number of assets, so that the sample covariance matrix performs poorly as a covariance estimator. Additionally, financial market data often contain outliers which, if not correctly handled, may further corrupt the covariance estimation. We address these shortcomings by studying the performance of a hybrid covariance matrix estimator based on Tyler's robust M-estimator and on Ledoit-Wolf's shrinkage estimator while assuming samples with heavy-tailed distribution. Employing recent results from random matrix theory, we develop a consistent estimator of (a scaled version of) the realized portfolio risk, which is minimized by optimizing online the shrinkage intensity. Our portfolio optimization method is shown via simulations to outperform existing methods both for synthetic and real market data

Cleaning large correlation matrices: tools from random matrix theory

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.Comment: 165 pages, article submitted to Physics Report