TY - JOUR

T1 - Is the empirical out-of-sample variance an informative risk measure for the high-dimensional portfolios?

AU - Bodnar, Taras

AU - Parolya, Nestor

AU - Thorsén, Erik

PY - 2023

Y1 - 2023

N2 - The main contribution of this paper is the derivation of the asymptotic behavior of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios p/n and p/m tend to some positive constants as m→∞ and n→∞, where p is the portfolio dimension, while n and m are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio and for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018). We show that the behavior of the empirical out-of-sample variance may be misleading in many practical situations, leading, for example, to a comparison of zeros. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if the out-of-sample variance can safely be used in practice for portfolios constructed from a large asset universe.

AB - The main contribution of this paper is the derivation of the asymptotic behavior of the out-of-sample variance, the out-of-sample relative loss, and of their empirical counterparts in the high-dimensional setting, i.e., when both ratios p/n and p/m tend to some positive constants as m→∞ and n→∞, where p is the portfolio dimension, while n and m are the sample sizes from the in-sample and out-of-sample periods, respectively. The results are obtained for the traditional estimator of the global minimum variance (GMV) portfolio and for the two shrinkage estimators introduced by Frahm and Memmel (2010) and Bodnar et al. (2018). We show that the behavior of the empirical out-of-sample variance may be misleading in many practical situations, leading, for example, to a comparison of zeros. On the other hand, this will never happen with the empirical out-of-sample relative loss, which seems to provide a natural normalization of the out-of-sample variance in the high-dimensional setup. As a result, an important question arises if the out-of-sample variance can safely be used in practice for portfolios constructed from a large asset universe.

KW - Shrinkage estimator

KW - High-dimensional covariance matrix

KW - Random matrix theory

KW - Minimum variance portfolio

KW - Parameter uncertainty

UR - http://www.scopus.com/inward/record.url?scp=85150762027&partnerID=8YFLogxK

U2 - 10.1016/j.frl.2023.103807

DO - 10.1016/j.frl.2023.103807

M3 - Article

SN - 1544-6123

VL - 54

JO - Finance Research Letters

JF - Finance Research Letters

M1 - 103807

ER -