Estimation of the global minimum variance portfolio in high dimensions

Abstract

We estimate the global minimum variance (GMV) portfolio in the high-dimensional case using results from random matrix theory. This approach leads to a shrinkage-type estimator which is distribution-free and it is optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets p depends on the sample size n such that p n → c ∈ (0, +∞) as n tends to infinity. The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely it is only required the fourth moments existence. Furthermore, we make no assumption on the upper bound of the spectrum of the covariance matrix. As a result, the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in financial literature nowadays. This is also well-documented in a numerical study where the smalland large-sample behavior of the derived estimator are compared with existing estimators of the GMV portfolio. The resulting estimator shows significant improvements and it turns out to be robust to the deviations from normality. JEL Classification: G11, C13, C14, C58, C65

Tests for the Weights of the Global Minimum Variance Portfolio in a High-Dimensional Setting

“…First, we study its behaviour under the high-dimensional asymptotics, and, after that, we propose an alternative test that makes use of the shrinkage estimator for the portfolio weights (cf. Bodnar, Parolya and Schmid (2018)).…”

“…Glombeck (2014) formulated tests for the portfolio weights, variances of the excess returns, and Sharpe ratios of the GMVP for c ∈ (0, 1). Bodnar, Parolya and Schmid (2018) and Bodnar, Okhrin and Parolya (2019) derived the shrinkage estimators for the GMVP and for the mean-variance portfolio, respectively, under the Kolmogorov asymptotics for c ∈ (0, ∞). Bodnar and Schmid (2008) proposed a test for a general linear hypothesis of the weights of the global minimum variance portfolio.…”

Bayesian Inference for the Tangent Portfolio

“…Jorion (1986), Kandel and Stambaugh (1996), Barberis (2000), Pastor (2000) used the Bayesian framework to analyze the impact of the underlying asset pricing or predictive model for asset returns on the optimal portfolio choice. Wang (2005), Kan and Zhou (2007), Golosnoy and Okhrin (2007), Golosnoy and Okhrin (2008), Bodnar et al (2017c) concentrated on shrinkage estimation, which allows to shift the portfolio weights to prespecified values, which reflect the prior beliefs of investors. Brandt (2010) gives a state of the art review of the modern portfolio selection techniques, paying a particular attention to Bayesian approaches.…”

An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection

“…The assumption of short selling might be not fulfilled on some capital markets. On the other side, the assumption of short selling is a common assumption in the literature (see, for example, Britten-Jones 1999;Kan and Smith 2008;Bodnar et al 2018bBodnar et al , 2019a and is a practically reliable assumption.…”