Asymptotic analysis of different covariance matrices estimation for minimum variance portfolio

In dynamic minimum variance portfolio, we study the impact of the sequence of covariance matrices taken in inputs, on the realized variance of the portfolio computed along a sample market path. The allocation of the portfolio is adjusted on a regular basis (every H days) using an updated covariance matrix estimator. In a modelling framework where the covariance matrix of the asset returns evolves as an ergodic process, we quantify the probability of observing an underperformance of the optimal dynamic covariance matrix compared to any other choice. The bounds depend on the tails of the returns, on the adjustment period H, and on the total number of rebalancing times N. These results provide asset managers with new insights into the optimality of their choice of covariance matrix estimators, depending on the depth of the backtest N H and the investment period H. Experiments based on GARCH modelling support our theoretical results.