Estimation of Out-of-Sample Sharpe Ratio for High Dimensional Portfolio Optimization

Portfolio optimization aims at constructing a realistic portfolio with significant out-of-sample performance, which is typically measured by the out-of-sample Sharpe ratio. However, due to in-sample optimism, it is inappropriate to use the in-sample estimated covariance to evaluate the out-of-sample Sharpe, especially in the high dimensional settings. In this article, we propose a novel method to estimate the out-of-sample Sharpe ratio using only in-sample data, based on random matrix theory. Furthermore, portfolio managers can use the estimated out-of-sample Sharpe as a criterion to decide the best tuning for constructing their portfolios. Specifically, we consider the classical framework of Markowits mean-variance portfolio optimization under high dimensional regime of p/n→c∈(0,∞), where <i>p</i> is the portfolio dimension and <i>n</i> is the number of samples or time points. We propose to correct the sample covariance by a regularization matrix and provide a consistent estimator of its Sharpe ratio. The new estimator works well under either of the following conditions: (a) bounded covariance spectrum, (b) arbitrary number of diverging spikes when c&lt;1, and (c) fixed number of diverging spikes with weak requirement on their diverging speed when c≥1. We can also extend the results to construct global minimum variance portfolio and correct out-of-sample efficient frontier. We demonstrate the effectiveness of our approach through comprehensive simulations and real data experiments. Our results highlight the potential of this methodology as a useful tool for portfolio optimization in high dimensional settings. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.