Nonlinear Factor Models and the Cross-Section of Asset Returns

We develop a nonlinear model of asset returns with latent economic factors to study theoretically how nonlinearity affects the factors recovered by principal component analysis (PCA). We derive the PCA factors and the eigenvalues of the covariance matrix of asset returns in closed form. The model generates a large leading eigenvalue, positive and approximately uniform stock weights in the corresponding factor, and positive stock betas, consistent with empirical evidence. The size of the leading eigenvalue is determined by the number of latent factors. The dominant PCA factor can absorb the nonlinear effects of many latent factors, while other PCA factors can be more directly related to individual latent factors. Small changes in asset return sensitivities to latent factors can sharply alter the number of factors selected to explain return variation, even when the number of latent factors is fixed. The calibrated model closely matches the empirical covariance spectrum of 74 characteristic-sorted portfolios.