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

Taras Bodnar, Nestor Parolya*, Erik Thorsén

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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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.

Original languageEnglish
Article number103807
Number of pages11
JournalFinance Research Letters
Publication statusPublished - 2023


  • Shrinkage estimator
  • High-dimensional covariance matrix
  • Random matrix theory
  • Minimum variance portfolio
  • Parameter uncertainty


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