Complexity in Factor Pricing Models

We theoretically characterize the behavior of machine learning asset pricing models. We prove that expected out-of-sample model performance—in terms of SDF Sharpe ratio and average pricing errors—is improving in model parameterization (or “complexity”). Our results predict that the best asset pricing models (in terms of expected out-of-sample performance) have an extremely large number of factors (more than the number of training observations or base assets). Our empirical findings verify the theoretically predicted “virtue of complexity” in the cross-section of stock returns and find that the best model combines tens of thousands of factors. We also derive the feasible Hansen- Jagannathan (HJ) bound: The maximal Sharpe ratio achievable by a feasible portfolio strategy. The infeasible HJ bound massively overstates the achievable maximal Sharpe ratio due to a complexity wedge that we characterize.