This paper presents an analysis of the higher-order dynamics of key financial quantities in asset-pricing models with recursive preferences. For this purpose, we first describe a projection-based algorithm for solving such models. The method outperforms common methods like discretization and log-linearization in terms of effciency and accuracy. Our algorithm allows us to document the presence of strong nonlinear effects in the modern long-run risks models which cannot be captured by the common methods. For example, for a prominent recent calibration of a popular long-run risks model, the log-linearization approach overstates the equity premium by 100 basis points or 22.5%. The increasing complexity of state-of-the-art asset-pricing models leads to complex nonlinear equilibrium functions with considerable curvature which in turn have sizable economic implications. Therefore, these models require numerical solution methods, such as the projection methods presented in this paper, that can adequately describe the higher-
order equilibrium features.