Recursive preferences have become a common-place feature of the modern consumption based asset pricing literature. Impoeting this ingredient into the production based models allows us to study the joint behaviour of real and financial variables along the business cycle. As the analysis of risk premia naturally require computing risk adjustments simple log-linearization techniques become insuffiecient for solving such models. In this paper we describe how log-linear dynamics and correct risk adjustments based on these dynamics can be computed relying on log-normality of the typical RBC model. We compare our suggested method with existing alternatives and show the log-linear dynamics plus risk adjustment are much more tractable mathematically, while preserving the accuracy of the approximation compared to second and higher order perturbation methods.
Alternative methods employed to solve production based asset pricing models such as projection methods and value function iteration are typically computationally intensive and ill-suited for problems with large number of state variables. Our method is closest to higher-order perturbation methods. Indeed we show how the two are exactly related and that the resulting approximations are very close numerically for the examples we consider. The linearity and analytical form of the solution is more convenient to deal with than the software based output of second-order perturbation methods (that captures risk adjustments).
Another advantage compared to perturbation methods is that our method allows us to deal with stochastic volatility very easily. We will denostrate this in an example. When using perturbation methods, on the other hand, at least a third order expansion is typically necessary, whereas our method captures the dynamics of stochastic volatility in first-order terms as well as capturing the risk adjustment. We compare the results of our.
Our method is close in spirit to the techiques used in consumption based asset pricing. We refer to our method as “log-normal risk adjustment”. The reason is that it relies heavily on the log-normality of shocks and an affine structure of (log) state-variables in those shocks.
Contents
I. Introduction
II. Setup
- A. Affine Shocks
B. Stationarised Version, Equilibrium Conditions and the Solution
C. Log-normal Risk Adjustment: An Approximation Technique
III. Example 1: RBC with Recursive Preferences
- A. Quantities and Prices
B. Log-normal Risk
C. Comparison with 2nd order perturbation methods
D. Numerical Comparison of the two
IV. Example 2: Stochastic volatility in a simple growth model
- A. Comparison with perturbation methods
B. Numerical Comparison of the two
C. Stochastic volatility and implications for asset prices
V. Conclusion
References
Appendices
- A. Equilibrium Conditions
B. Non-Stochastic Steady State
C. Difference Equation Solution
D. Alternative Solution: Method of Unknown Coefficients
E. Alternative Solution: Method of Unknown Coefficients
F. Relation to perturbation methods
