This paper discusses the practical aspects of working with high-dimensional shrinkage portfolios. It presents the R package HDShOP which provides a comprehensive framework for such work. In particular, we cover the construction of portfolios using shrinkage-based estimators for the mean vector, covariance matrix, and precision matrix of asset returns, as well as the shrinkage estimators derived directly for the weights of optimal portfolios. Moreover, shrinkage-based tests on the mean-variance efficiency of a given portfolio are discussed. Aspects related to programming, such as classes and methods used in the construction of optimal portfolios, are described. The description of the software is preceded by underlying theory and it is accompanied by several empirical illustrations based on the data consisting of returns on stocks from the S&P 500 index.

In this paper, we analyze the asymptotic behavior of the main characteristics of the mean-variance efficient frontier employing random matrix theory. Our particular interest covers the case when the dimension p and the sample size n tend to infinity simultaneously and their ratio p/n tends to a positive constant (Formula presented.). We neither impose any distributional nor structural assumptions on the asset returns. For the developed theoretical framework, some regularity conditions, like the existence of the 4th moments, are needed. It is shown that two out of three quantities of interest are biased and overestimated by their sample counterparts under the high-dimensional asymptotic regime. This becomes evident based on the asymptotic deterministic equivalents of the sample plug-in estimators. Using them we construct consistent estimators of the three characteristics of the efficient frontier. Furthermore, the asymptotic normality of the considered estimators of the parameters of the efficient frontier is proved. Verifying the theoretical results based on an extensive simulation study we show that the proposed estimator for the efficient frontier is a valuable alternative to the sample estimator for high dimensional data. Finally, we present an empirical application, where we estimate the efficient frontier based on stocks from the S&P 500 index.

In this paper, using the shrinkage-based approach for portfolio weights and modern results from random matrix theory we construct an effective procedure for testing the efficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior of the proposed test statistic under the high-dimensional asymptotic regime, namely when the number of assets p increases at the same rate as the sample size n such that their ratio p/n approaches a positive constant cin (0,1) as nto infty. We provide an extensive simulation study where the power function and receiver operating characteristic curves of the test are analyzed. In the empirical study, the methodology is applied to the returns of S&P 500 constituents.

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 distribution-free and is optimal in the sense of maximizing with probability one 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.