Dominating estimators for minimum-variance portfolios

Abstract

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...

Replica approach to mean-variance portfolio optimization

“…This, of course, does not constitute a proof, but rigorous proofs exist in similar, and even much more complicated, problems in the theory of disordered systems [60], [61], [62], which lends support to our conjecture. Furthermore, the results to be derived below agree with the rigorous results by [30], [39], [43] in the thermodynamic limit, and also with the numerical experiments by [36].…”

“…We note that in the case of the variance, finite N and finite T results also exist (e.g. [30], [39], [43]) and in the appropriate limit they coincide with our results.…”

“…One of the main results of the present calculation will be to derive the expectation value of q 0 for an arbitrary covariance matrix and verify its universality (its independence of the covariance matrix and the return), a possibility first raised on the basis of numerical experiments by [27]. As for the second moment, its behaviour can be inferred from the results in [30], [39], or [43]: in the thermodynamic limit the variance of q 0 vanishes, in the parlance of the theory of disordered systems, q 0 self-averages. In addition to the estimation error, we may even worry about whether the optimization can be carried out at all, that is whether the estimated covariance matrix σ ij preserves the positive definiteness of the true σ ij , so that to remain invertible.…”

Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks

“…The assumption c < 1 is for ease of exposition only. Unlike the proposals by Kan and Zhou (2007), Frahm and Memmel (2010) and Tu and Zhou (2011), our method also handles the case c > 1, where the sample covariance matrix is not invertible. This extension is presented in Appendix D.…”

“…Recent proposals by Wolf (2003, 2004a,b), Kan and Zhou (2007), Brandt et al (2009), DeMiguel et al (2009a, Frahm and Memmel (2010), and Tu and Zhou (2011), among others, show that this topic is currently gathering a significant amount of attention. All these articles resolve the problem by going from O(N 2 ) degrees of freedom to O(1).…”