The Equity Premium and the Term Structure of Interest Rates with Stochastic Differential Utility

Mark E. Fisher and Christian Gilles - Federal Reserve Bank

We solve for the equity premium and the term structure of interest rates in a stochastic differential utility framework of Duffie and Epstein (1992). We model the economy with a single state variable, which we alternatively take to be (1) the expected return on capital, (2) the expected growth rate of consumption, and (3) the interest rate. In each case, the first-order condition is a second-order, nonlinear, non-autonomous ODE similar to that of Foldes (1996a,b) for which no closed-form solution is available. "Solving the equation" amounts to finding the unique initial conditions that produce a solution over the entire real line. We show that the instantaneous variance of the capital stock can be many times larger than the instantaneous variance of consumption when preferences allow the separation of risk aversion and intertemporal substitution. We are able to simultaneously address the equity premium puzzle of Mehra and Prescott (1985), the risk-free rate puzzle of Weil (1989), and the term premium puzzle of Backus, Gregory, and Zin (1989).

Scheduled for Session 4.1 Times Series Analysis Of Asset Pricing