Optimal shrinkage-based portfolio selection in high dimensions

Abstract

In this paper we estimate the mean-variance portfolio in the high-dimensional case using the recent results from the theory of random matrices. We construct a linear shrinkage estimator which is distribution-free and is optimal in the sense of maximizing with probability $1$ the asymptotic out-of-sample expected utility, i.e., mean-variance objective function for different values of risk aversion coefficient which in particular leads to the maximization of the out-of-sample expected utility and to the minimization of the out-of-sample variance. One of the main features of our estimator is the inclusion of the estimation risk related to the sample mean vector into the high-dimensional portfolio optimization. The asymptotic properties of the new estimator are investigated when the number of assets $p$ and the sample size $n$ tend simultaneously to infinity such that $p/n\to c\in (0,+\infty )$. The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely the existence of the $4+\epsilon$ moments is only required. Thereafter we perform numerical and empirical studies where the small- and large-sample behavior of the derived estimator is investigated. The suggested estimator shows significant improvements over the existent approaches including the nonlinear shrinkage estimator and the three-fund portfolio rule, especially when the portfolio dimension is larger than the sample size. Moreover, it is robust to deviations from normality.

45 pages, UPDATE3: revised version of the manuscript accepted by Journal of Business and Economic Statistics. substantially revised: Ledoit-Wolf and Kan-Zhou estimators were added, conditions weakened, proofs revised, discussion on the Moore-Penrose approximation included, mistake in the shrinkage formula for c>1 corrected (big boost in performance as a result)

References in corpus (6)

• Nonlinear shrinkage estimation of large-dimensional covariance matrices
• High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation
• Estimation of the Global Minimum Variance Portfolio in High Dimensions
• Statistical inference for the EU portfolio in high dimensions
• Sampling Distributions of Optimal Portfolio Weights and Characteristics in Low and Large Dimensions
• Spectral analysis of large reflexive generalized inverse and Moore-Penrose inverse matrices

Cited by in corpus (5)

• Statistical inference for the EU portfolio in high dimensions
• Sampling Distributions of Optimal Portfolio Weights and Characteristics in Low and Large Dimensions
• Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio
• Is the empirical out-of-sample variance an informative risk measure for the high-dimensional portfolios?
• Tests for the weights of the global minimum variance portfolio in a high-dimensional setting