Optimal Shrinkage-Based Portfolio Selection in High Dimensions

Abstract

In this article, 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 distributionfree and is optimal in the sense of maximizing with probability 1 the asymptotic out-of-sample expected utility, that is, 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 → c ∈ (0, +∞). The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely the existence of the 4 + ε 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.

Dynamic Shrinkage Estimation of the High-Dimensional Minimum-Variance Portfolio

“…The statement of Lemma 5.1 is derived as Lemma 5.3 in Bodnar et al (2021c) Lemma 5.1. Let ξ and θ be two nonrandom vectors with bounded Euclidean norms.…”

“…Shrinkage-type estimators were first proposed by Stein (1956) with the aim to reduce the estimation error present in the sample mean vector computed for a sample from a multivariate normal distribution. Recently, this procedure has also been applied in the construction of the improved estimators of the high-dimensional mean vector (cf, Chételat and Wells (2012), Wang et al (2014), Bodnar et al (2019b)), covariance matrix (see, e.g., Ledoit and Wolf (2004), Ledoit and Wolf (2012), Bodnar et al (2014)), inverse of the covariance matrix (see, e.g., Wang et al (2015), Bodnar et al (2016)), as well as of the optimal portfolio weights (see, Golosnoy and Okhrin (2007), Frahm and Memmel (2010), Ledoit and Wolf (2017), Bodnar et al (2018), Bodnar et al (2021c)). Interval shrinkage estimators of optimal portfolio weights have recently been derived by Bodnar et al (2019a), Bodnar et al (2021b).…”

Statistical inference for the EU portfolio in high dimensions

“…As a solution to this challenging problem, we suggest a new approach for testing the structure of the EU portfolio by a single test. The new procedure is based on the shrinkage estimator of the EU portfolio weights as suggested by Bodnar et al (2020) obtained for the GMV portfolio in Bodnar, Dmytriv, Parolya and Schmid (2019), which is a very special case of the EU portfolio. In contrast to the EU portfolio, the weights of the GMV portfolio do not depend on the mean vector.…”

“…This approach might introduce a bias in the estimator, but on the other side it reduces the variability of the sample estimator considerably. Bodnar et al (2020) determine the optimal shrinkage intensity α * n as the solution of the maximization problem based on the mean-variance objective function. It is given by…”

“…This can be achieved by taking their functional dependence on the mean vector and of covariance matrix. Following this approach Bodnar et al (2018) suggest the optimal shrinkage estimator for the GMV portfolio weights, while Bodnar et al (2020) propose the optimal shrinkage estimator for the EU portfolio weights. Both estimators are derived by using recent results in random matrix theory and appear to be feasible even in the case of c > 1 .…”