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 $p/n \to c \in (0,\infty)$, where $p$ is the portfolio dimension and $n$ 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 $c < 1$, and (3) fixed number of diverging spikes with weak requirement on their diverging speed when $c \ge 1$. 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.

翻译：投资组合优化的目标在于构建具有显著样本外表现的现实投资组合，其表现通常通过样本外夏普比率来衡量。然而，由于样本内乐观偏差，使用样本内估计的协方差矩阵来评估样本外夏普比率是不合适的，尤其是在高维设定下。本文基于随机矩阵理论，提出了一种仅利用样本内数据估计样本外夏普比率的新方法。此外，投资组合管理者可将估计的样本外夏普比率作为选择最优调参准则以构建投资组合的依据。具体而言，我们考虑经典的马科维茨均值-方差投资组合优化框架在高维体系 $p/n \to c \in (0,\infty)$ 下的情形，其中 $p$ 为投资组合维度，$n$ 为样本数或时间点数。我们提出通过正则化矩阵修正样本协方差矩阵，并给出其夏普比率的一致性估计量。该新估计量在以下任一条件下均表现良好：(1) 协方差谱有界，(2) 当 $c < 1$ 时存在任意数量的发散尖峰，以及 (3) 当 $c \ge 1$ 时存在固定数量的发散尖峰且对其发散速度要求较弱。我们还可将结果扩展至构建全局最小方差投资组合及修正样本外有效前沿。通过综合仿真与真实数据实验，我们验证了所提方法的有效性。研究结果凸显了该方法作为高维设定下投资组合优化实用工具的潜力。