Is the Empirical Out-of-Sample Variance an Informative Risk Measure for the High-Dimensional Portfolios?

The main contribution of this paper is the derivation of the asymptotic behaviour of the out-of-sample variance, relative loss, and of their empirical counterparts in the high-dimensional setting. The results are obtained for the traditional estimator of the global minimum variance portfolio, for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018), and for the equally weighted portfolio. We show that the behaviour of the empirical out-of-sample variance may be misleading is many practical situations and that its relative loss provides a natural normalization in the high-dimensional setup