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 $p/n \rightarrow c\in (0,+\infty)$. The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely the existence of the $4+\varepsilon$ 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.