Accounting for Risk in a Linearized Solution: How to Approximate the Risky Steady State and Around It

Abstract

We propose a novel approximation of the risky steady state and construct first-order perturbations around it for a general class of dynamic equilibrium models with time-varying and non-Gaussian risk. We offer analytical formulas and conditions for their local existence and uniqueness. We apply this approximation technique to models featuring Campbell-Cochrane habits, recursive preferences, and time-varying disaster risk, and show how the proposed approximation represents the implications of the model similarly to global solution methods. We show that our approximation of the risky steady state cannot be generically replicated by higher-order perturbations around the deterministic steady state, which cannot account well for the effects of risk in our applications even up to third order. Finally, we argue that our perturbation can be viewed as a generalized version of the heuristic loglinear-lognormal approximations commonly used in the macro-finance literature.

Keywords: Perturbation methods; Risky steady state; Macroeconomic uncertainty; Solving dynamic equilibrium models; Time-varying risk premia

JEL Classification: C63, G12, E32, E44