Higher-Order Effects in Asset-Pricing Models with Long-Run Risks

This paper analyzes both the existence of solutions to long-run risk asset pricing models as well as the practicality of approximating these solutions by the Campbell-Shiller log-linearization. We prove a simple relative existence result that is sufficient to show that the original Bansal-Yaron model has a solution. Log-linearization fares less well: we find that for very persistent processes the approximation errors in model moments can be as large as 50%, and can get such basic facts wrong as the direction of the yield curve. The increasing complexity of state-of-the-art asset-pricing models can lead to complex nonlinear solutions with considerable curvature, which in turn can have sizable economic implications. Therefore, these models require numerical solution methods, such as the projection methods employed in this paper, that can adequately describe the higher-order equilibrium features.