Expected Returns, Firm Characteristics, and Cardinality Constraints

Abstract

I propose estimating stochastic discount factors for the cross-section of expected returns under the assumption of worst-case differences between the population moments and sample moments of characteristic-based factor portfolios’ excess returns. I show that a stochastic dis- count factor estimated under the worst-case difference assumption has a population and out-of- sample Sharpe ratio greater than or equal to a lower bound that can be computed ex-ante with model residuals. I also show that l1 regularization, in the context of SDF estimation, is hard thresholding on the factors’ sample Sharpe ratios, and propose using l0 regularization to estimate sparse SDFs instead. I find that the paper’s worst-case robustness approach to stochastic discount factor estimation works well using both out-of-sample and bootstrapped Sharpe ratios. The characteristic-based l0-sparse SDFs also perform much better than the l1-sparse SDFs and have very similar performance to sparse SDFs built with latent factors from the factor portfolios’ expected returns.

Keywords: Factor model, stochastic discount factor, cross-section, sparsity, machine learning, variable selection, firm characteristics, expected returns, robust regression, minmax optimization

JEL Classification: C13, C52, G11, G12