The Statistical Limit of Arbitrage

In the context of a linear asset pricing model, we document a statistical limit to arbitrage due to the fact that arbitrageurs are incapable of learning a large cross-section of alphas with sufficient precision given a limited time span of data. Consequently, the optimal Sharpe ratio of arbitrage portfolios developed under rational expectation in the classical arbitrage pricing theory (APT) is overly exaggerated, even as the sample size increases and the investment opportunity set expands. We derive the optimal Sharpe ratio achievable by any feasible arbitrage strategy, and illustrate in a simple model how this Sharpe ratio varies with the strength and sparsity of alpha signals, which characterize the difficulty of arbitrageurs' learning problem. Furthermore, we design an ``all-weather'' arbitrage strategy that achieves this optimal Sharpe ratio regardless of the conditions of alpha signals. We also show how arbitrageurs can adopt multiple-testing, LASSO, and Ridge methods to achieve optimality under distinct conditions of alpha signals, respectively. Our empirical analysis of more than 50 years of monthly US individual equity returns shows that all strategies we consider achieve a moderately low Sharpe ratio out of sample, in spite of a considerably higher yet infeasible one, suggesting the empirical relevance of the statistical limit of arbitrage and the empirical success of APT.

Jointly written with Rui Da and Stefan Nagel