I construct risk-corrected approximations of the policy functions of DSGEmodels around the stochastic steady state and ergodic mean that are linear in the state variables. The resulting approximations are uniformly more accurate than standard linear approximations and capture the dynamics of asset pricing variables such as the expected risk premium missed by standard linear approximations. The algorithm is fast and reliable, requiring only the solution of linear equations using standard perturbation output. I examine the joint macroeconomic and asset pricing implications of a real business cycle model with stochastic trends and recursive preferences. The method is able to estimate risk aversion under these preferences using the Kalman filter, where a standard linear approximation provides no information and alternative methods require computationally intensive particle filters subject to sampling variation.