Perturbations Around the Risky Steady State

We obtain a novel formulation for first-order perturbations around the risky steady of a general class of dynamic equilibrium models with time-varying and non-Gaussian risk. We offer explicit formulas and conditions for their local existence and uniqueness. First-order perturbations around the risky steady state are not nested in perturbations of arbitrary order around the deterministic steady state. We apply this approximation technique to models featuring Campbell-Cochrane habits, recursive preferences, and time-varying disaster risk. In many applications the proposed approximation represents similarly to global solution methods the implications of the model for asset prices and quantities. Finally, we argue that these perturbations can be viewed as a generalized version of the heuristic loglinear-lognormal approximations commonly used in the macro-finance literature. Therefore, we unify existing theories of risk-adjusted linearizations.