Consistent Estimation of the High-Dimensional Efficient Frontier

TL;DR

This paper develops a high-dimensional asymptotic framework using random matrix theory to accurately estimate the efficient frontier in portfolio optimization, addressing bias issues in traditional sample estimators.

Contribution

It introduces consistent estimators for the efficient frontier characteristics in high-dimensional settings without distributional assumptions, accounting for biases related to the ratio of assets to observations.

Findings

Sample estimators are biased in high dimensions.

Proposed estimators are consistent and asymptotically normal.

Simulation and empirical analysis validate the effectiveness of the new estimators.

Abstract

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 $c\in(0,1)$. 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 $4$th 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. It it shown that the additive and/or the multiplicative biases of the sample estimates are solely functions of the concentration ratio $c$. 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 the stocks included in S\&P 500 index.