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

Abstract

<p>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 paper, 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 <img src="https://:0/" alt=""><img src="https://:0/" alt="">p/n→c∈(0,∞), where <em>p</em> is the portfolio dimension and <em>n</em> 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: (1) bounded covariance spectrum, (2) arbitrary number of diverging spikes when <img src="https://:0/" alt=""><img src="https://:0/" alt="">c<1, and (3) fixed number of diverging spikes with weak requirement on their diverging speed when <img src="https://:0/" alt=""><img src="https://:0/" alt="">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.<br></p>