We show that higher-order perturbation solutions to rational expectations models exhibit a certainty correspondence property: they can be obtained from the derivatives of the model’s perfect foresight solution. This extends the well-known certainty equivalence result beyond first order. For heterogeneous-agent models with large idiosyncratic state spaces, our approach is significantly faster than existing state-space perturbation or global solution methods. We apply our method to solve a HANK model and a menu cost model to third order in aggregate risk, quantifying the time variation in the fiscal multiplier and the responsiveness of inflation to costs, as well as the effect of perceived aggregate risk on the steady state and impulse responses.