  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 with known mean vector and the 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 three conditions: (1) bounded
covariance spectrum, (2) arbitrary number of diverging spikes when $c < 1$, and
(3) fixed number of diverging spikes 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 powerful tool for
portfolio optimization in high dimensional settings.
