Investment and profitability versus value and momentum: The price of residual risk

Barillas and Shanken (2017) show that a traded factor is redundant if the intercept (alpha) in the regression of the factor on other factors is zero. I find that covariance-based asset pricing models imply that the alpha is proportional to the residual risk of the traded factor. Empirical estimation of the price of the residual risk suggests that the q-factor model of Hou et al. (2015) subsumes the roles of not only the value and momentum factors but also the investment and profitability factors of Fama and French (2015). However, the Fama-French (2015) model fails to subsume the roles of the momentum factor and the factors in the q-factor model. The momentum factor, along with the profitability factor in the Fama-French (2015) or Novy-Marx (2013) model does not subsume the role of the profitability factor in the q-factor model.