Improving portfolios global performance using a cleaned and robust covariance matrix estimate

This paper presents how the use of a cleaned and robust covariance matrix estimate can improve significantly the overall performance of maximum variety and minimum variance portfolios. We assume that the asset returns are modelled through a multi-factor model where the error term is a multivariate and correlated elliptical symmetric noise extending the classical Gaussian assumptions. The factors are supposed to be unobservable and we focus on a recent method of model order selection, based on the random matrix theory to identify the most informative subspace and then to obtain a cleaned (or de-noised) covariance matrix estimate to be used in the maximum variety and minimum variance portfolio allocation processes. We apply our methodology on real market data and show the improvements it brings if compared with other techniques especially for non-homogeneous asset returns.

[1]  Matthew R. McKay,et al.  A Robust Statistics Approach to Minimum Variance Portfolio Optimization , 2015, IEEE Transactions on Signal Processing.

[2]  Douglas Kelker,et al.  DISTRIBUTION THEORY OF SPHERICAL DISTRIBUTIONS AND A LOCATION-SCALE PARAMETER GENERALIZATION , 2016 .

[3]  Frédéric Pascal,et al.  Improving Portfolios Global Performance with Robust Covariance Matrix Estimation: Application to the Maximum Variety Portfolio , 2018, 2018 26th European Signal Processing Conference (EUSIPCO).

[4]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[5]  Walid Hachem,et al.  Statistical Inference in Large Antenna Arrays Under Unknown Noise Pattern , 2013, IEEE Transactions on Signal Processing.

[6]  Philippe Forster,et al.  Asymptotic Properties of Robust Complex Covariance Matrix Estimates , 2012, IEEE Transactions on Signal Processing.

[7]  E. Fama,et al.  A Five-Factor Asset Pricing Model , 2014 .

[8]  Christian Gourieroux,et al.  Robust Portfolio Allocation with Systematic Risk Contribution Restrictions , 2012 .

[9]  Antonio J. Plaza,et al.  Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches , 2012, IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing.

[10]  Matthew R. McKay,et al.  Minimum variance portfolio optimization in the spiked covariance model , 2015, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[11]  S. Péché,et al.  Eigenvectors of some large sample covariance matrix ensembles , 2009 .

[12]  G. Simons,et al.  On the theory of elliptically contoured distributions , 1981 .

[13]  W. Sharpe.  CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK* , 1964 .

[14]  Jean-Philippe Bouchaud,et al.  Financial Applications of Random Matrix Theory: Old Laces and New Pieces , 2005 .

[15]  William F. Rentz,et al.  Testing Fama–French’s new five-factor asset pricing model: evidence from robust instruments , 2015 .

[16]  J. Bouchaud,et al.  RANDOM MATRIX THEORY AND FINANCIAL CORRELATIONS , 2000 .

[17]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[18]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[19]  Philippe Forster,et al.  Performance Analysis of Covariance Matrix Estimates in Impulsive Noise , 2008, IEEE Transactions on Signal Processing.

[20]  Olivier Ledoit,et al.  Improved estimation of the covariance matrix of stock returns with an application to portfolio selection , 2003 .

[21]  Jean-Philippe Bouchaud,et al.  Rotational Invariant Estimator for General Noisy Matrices , 2015, IEEE Transactions on Information Theory.

[22]  Patrick Duvaut,et al.  Multifactor Models : Examining the potential of signal processing techniques , 2011 .

[23]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[24]  Antoine Mandel,et al.  Agent-based dynamics in disaggregated growth models , 2010 .

[25]  Barr Rosenberg,et al.  Extra-Market Components of Covariance in Security Returns , 1974, Journal of Financial and Quantitative Analysis.

[26]  B. Rosenow,et al.  Collective behavior of stock price movements—a random matrix theory approach , 2001 .

[27]  R. Couillet,et al.  Random Matrix Methods for Wireless Communications , 2011 .

[28]  Philippe Forster,et al.  Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis , 2008, IEEE Transactions on Signal Processing.

[29]  J. Bouchaud,et al.  Financial Applications of Random Matrix Theory: a short review , 2009, 0910.1205.

[30]  T. Roncalli,et al.  The properties of equally-weighted risk contributions portfolios , 2010 .

[31]  Katrien van Driessen,et al.  A Fast Algorithm for the Minimum Covariance Determinant Estimator , 1999, Technometrics.

[32]  Romain Couillet,et al.  Robust Estimates of Covariance Matrices in the Large Dimensional Regime , 2012, IEEE Transactions on Information Theory.

[33]  D. Melas,et al.  Efficient Replication of Factor Returns, June 2009 , 2009 .

[34]  David E. Tyler.  A Distribution-Free $M$-Estimator of Multivariate Scatter , 1987 .

[35]  Frédéric Pascal,et al.  A Toeplitz-Tyler Estimation of the Model Order in Large Dimensional Regime , 2018, 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[36]  Philippe Loubaton,et al.  A subspace estimator for fixed rank perturbations of large random matrices , 2011, J. Multivar. Anal..

[37]  Patrick Duvaut,et al.  Multi-factor Models and Signal Processing Techniques: Application to Quantitative Finance , 2013 .

[38]  H. Vincent Poor,et al.  Complex Elliptically Symmetric Distributions: Survey, New Results and Applications , 2012, IEEE Transactions on Signal Processing.

[39]  Yves Choueifaty,et al.  Toward Maximum Diversification , 2008, The Journal Of Portfolio Management.

[40]  Frédéric Pascal,et al.  New model order selection in large dimension regime for complex elliptically symmetric noise , 2017, 2017 25th European Signal Processing Conference (EUSIPCO).

[41]  J. H. Ward.  Hierarchical Grouping to Optimize an Objective Function , 1963 .

[42]  Boaz Nadler,et al.  Non-Parametric Detection of the Number of Signals: Hypothesis Testing and Random Matrix Theory , 2009, IEEE Transactions on Signal Processing.

[43]  Olivier Ledoit,et al.  A well-conditioned estimator for large-dimensional covariance matrices , 2004 .

[44]  Romain Couillet.  Robust spiked random matrices and a robust G-MUSIC estimator , 2015, J. Multivar. Anal..

[45]  Jean-Philippe Ovarlez,et al.  Robust Model Order Selection in Large Dimensional Elliptically Symmetric Noise , 2017 .

[46]  Jean-Philippe Bouchaud,et al.  Cleaning large correlation matrices: tools from random matrix theory , 2016, 1610.08104.

[47]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[48]  Yuri I. Abramovich,et al.  Diagonally Loaded Normalised Sample Matrix Inversion (LNSMI) for Outlier-Resistant Adaptive Filtering , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[49]  Julien Reynier,et al.  Properties of the Most Diversified Portfolio , 2011 .

[50]  Yacine Chitour,et al.  Generalized Robust Shrinkage Estimator and Its Application to STAP Detection Problem , 2013, IEEE Transactions on Signal Processing.

[51]  Alfred O. Hero,et al.  Robust Shrinkage Estimation of High-Dimensional Covariance Matrices , 2010, IEEE Transactions on Signal Processing.

[52]  Olivier Ledoit,et al.  Eigenvectors of some large sample covariance matrix ensembles , 2009, 0911.3010.

[53]  Harindra de Silva,et al.  Minimum Variance , Maximum Diversification , and Risk Parity : An Analytic Perspective , 2012 .

[54]  Romain Couillet,et al.  RMT for whitening space correlation and applications to radar detection , 2015, 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[55]  Anton Zabrodin,et al.  Financial applications of random matrix theory: a short review , 2018 .