Expected Shortfall Factor Models: Common Tail Losses and Expected Returns

Abstract

We develop an expected shortfall factor model (ESFM) to estimate and price common variation in the severity of lower-tail losses in large panels of asset returns. Mean factor models describe common variation in average returns, while quantile factor models describe common movements in tail thresholds. ESFM instead captures common variation in the average severity of losses below those thresholds. The model combines observed risk exposures with latent common factors. We estimate ESFM using an orthogonalized two-step procedure under which first-stage quantile estimation error has no first-order effect on the ES coefficient estimates. We establish nonasymptotic error bounds for the ES coefficients, a finite-sample Gaussian approximation, and consistent selection of the number of latent factors. Applied to a large panel of equities, ESFM uncovers common factors that react sharply to market stress and contain information not captured by mean and quantile factors. Average returns increase across portfolios sorted on ESFM exposure; high-minus-low portfolios earn annualized returns of 8.0%--11.7% and Fama--French five-factor alphas of 10.3%--15.0%. These spreads remain positive and statistically significant after conditioning separately and jointly on mean- and quantile-factor exposures. Tail-by-tail spanning tests show that ESFM factors retain significant alphas after controlling for standard traded factors and the corresponding mean and quantile factors. Adding ESFM to these benchmark factor sets increases the maximum attainable Sharpe ratio. These findings identify common loss severity as a distinct and priced dimension of downside risk.