Comparison of several covariance matrix estimators for portfolio optimization

Modern portfolio theory dates back to a seminal 1952 paper by H. Markowitz and has been very influential both in academic finance and among practitioners in the financial industry. Given a set of assets, the theory can be used to compute the amount to be invested in each asset in order to construct an optimally diversified portfolio. One of the parameters required in this calculation is the covariance matrix of asset returns which, in any practical application, is unknown and must be estimated from historical data. Due to the fact that financial data is often nonstationary, basing the estimates on historical data over a very long time period may not be advisable. This renders the problem of covariance estimation difficult, especially for large portfolios. A large body of literature exists proposing different covariance estimators. We focus on one frequently cited paper by Ledoit and Wolf which proposes a covariance estimation method and purports to show that this method leads to statistically significant improvements over several other methods. We show that this is not the case: in fact, their method does not exhibit statistically significant differences from three other methods.