In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d ≥ 4 and number of observations n ≥ d + 2. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d but n → ∞ and n, d → ∞ but n/d → q ≤ ∞ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.JEL classification: C13, G11.Keywords: Covariance matrix estimation, Minimum-variance portfolio, JamesStein estimation, Naive diversification, Shrinkage estimator. * We would like to thank André Güttler, Alexander Kempf, Julia Nasev, and Michael Wolf for their helpful comments on the manuscript. Special thanks are due to Arndt Musche, who conducted the first run of the moving-blocks bootstrap which is presented at the end of this work. The opinions expressed in this paper are those of the authors and do not necessarily reflect the opinions of the Deutsche Bundesbank. † Phone: +49 221 470-4267, email: frahm@statistik.uni-koeln.de. ‡ Phone: +49 69 9566-8531, email: christoph.memmel@bundesbank.de. 1 A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT MotivationWhen implementing portfolio optimization according to Markowitz (1952), one needs to estimate the expected asset returns as well as the corresponding variances and covariances. If the parameter estimates are based only on time series information, the suggested portfolio tends to be far removed from the optimum. For this reason, there is a broad literature which addresses the question of how to reduce estimation risk in portfolio optimization. In a recent study, DeMiguel et al. (2009) compare portfolio strategies which differ in the treatment of estimation risk. It turns out that none of the strategies suggested in the literature is significantly better than naive diversification, i.e., taking the equally weighted portfolio. Further, the study conducted by DeMiguel et al. (2009) confirms that the considered strategies perform better than the traditional "plug-in" implementation of Markowitz optimization, which means replacing the unknown parameters by their sample counterparts.Constrained and unconstrained minimum-variance portfolios have been frequently advocated in the literature (Frahm, 2008;Jagannathan and Ma, 2003;Kempf and Memmel, 2006;Ledoit and Wolf, 2003) because they are completely independent of the expected asset returns, which have been found to be the principal source of estimation risk (Chopra and Ziemba, 1993;Merton, 1980). In fact, many empirical studies indicate that minimumvariance portfolios in general lead to a better out-of-sample performance than stock index portfolios (Haugen, 1990;Haugen and Baker, 1991, 1993;Winston, 1993).The portfolio which minimizes the portfolio return variance only with respect to the budget const...
Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractIn recent publications standard methods of random matrix theory were applied to principal components analysis of high-dimensional financial data. We discuss the fundamental results and potential shortcomings of random matrix theory in the light of the stylized facts of empirical finance. Especially, our arguments are based on the impact of nonlinear dependencies such as tail dependence. After a brief discussion of the stylized facts we present the class of multivariate generalized elliptical distributions. This class allows for the modeling of various anomalies frequently observed in financial data. Thus it will serve as a general model for the investigation of standard methods of random matrix theory. It is shown that the Marčenko-Pastur law generally fails when analyzing the empirical distribution function of the eigenvalues given by the sample covariance matrix of generalized elliptically distributed data. As an alternative we derive a random matrix referred to as the spectral estimator which is distributionfree within the class of generalized elliptical distributions. Moreover, we show that the spectral estimator corresponds to Tyler's M-estimator and many important properties of the spectral estimator can be obtained from the corresponding literature. Substituting the sample covariance matrix by the spectral estimator resolves the problems which are due to the stylized facts and the Marčenko-Pastur law remains valid. This holds even if the data are not generalized elliptically distributed but mutually independent.AMS Subject Classification: Primary 15A52, Secondary 62H25. AbstractIn recent publications standard methods of random matrix theory were applied to principal components analysis of high-dimensional financial data. We discuss the fundamental results and potential shortcomings of random matrix theory in the light of the stylized facts of empirical finance. Especially, our arguments are based on the impact of nonlinear dependencies such as tail dependence. After a brief discussion of the stylized facts we present the class of multivariate generalized elliptical distributions. This class allows for the modeling of various anomalies frequently observed in financial data. Thus it will serve as a general model for the investigation of standard methods of random matrix theory. It is shown that the Marčenko-Pastur law generally fails when analyzing the empirical distribution function of the eigenvalues given by th...
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