The Rodney L . White Center for Financial Research Solving Models with External Habit

Habit utility has been the focus of a large and growing body of literature in financial economics. This study investigates ways of accurately and efficiently solving the Campbell and Cochrane (1999) external habit model. Solutions for this model based on a grid of values for the state variable are shown to converge as the grid becomes increasingly fine. Convergence is substantially faster if the price-dividend ratio is computed as a series of “zero-coupon equity” claims rather than as the fixed-point of the Euler equation. Fitting the model to the term structure as well as to equity moments (as in Wachter (2005)) also results in faster convergence. I am grateful to John Campbell, Thomas Cosimano, Motohiro Yogo, and Harold Zhang (the reviewer) for helpful comments. Address: The Wharton School, University of Pennsylvania, 3620 Locust Walk, Philadelphia, PA 19104; Tel: (215) 898-7634; Email: [email protected]; http://finance.wharton.upenn.edu/ ̃ jwachter/ Introduction Habit utility has been the focus of a large and growing body of literature in financial economics. Constantinides (1990) and Sundaresan (1989) show that habit preferences, which assume an agent’s previous consumption affects his utility from current consumption, can help explain the high equity premium documented by Mehra and Prescott (1985). Abel (1990) shows that preferences where the agent evaluates consumption relative to past aggregate consumption (“catching up with the Joneses”) can also help resolve the equity premium puzzle. In both types of models, consumption is evaluated relative to a time-varying reference point, however, in the Abel model this reference point is external in that the agent’s current consumption choice does not affect future utility. For this reason, “catching up with the Joneses” preferences are sometimes referred to as external habit formation. Building on these contributions, Campbell and Cochrane (1999) show that a model where utility is a function of consumption minus external habit is capable of reconciling the low standard deviation of consumption growth with a high equity premium, high volatility of returns, and a low and smooth riskfree rate. Recently, external habit models have been extended to address a broad range of phenomena (see, e.g., Abel (1999), Brandt and Wang (2003), Buraschi and Jiltsov (2003), Campbell and Cochrane (2000), Chan and Kogan (2002), Dai (2000), Lettau and Uhlig (2000), Menzly, Santos, and Veronesi (2004), Pastor and Veronesi (2004), Wachter (2005)), and tested in a variety of ways (see, e.g., Chen and Ludvigson (2003), Duffee (2004), Gomes and Michaelides (2003), Korniotis (2005), Li (2001), Tallarini and Zhang (2004)). Given the enduring interest in external habit models, it is important to investigate ways of solving such models accurately and efficiently. This study focuses on the external habit model of Campbell and Cochrane (1999) and its extension in Wachter (2005). Both papers solve for the price-dividend ratio by iterating on a grid of values for the state variable. While choosing a grid that is too coarse can lead to inaccuracies, this study shows that for both calibrations of the model (Campbell and Cochrane; Wachter), the solution for the price-dividend ratio converges as the grid becomes finer. Convergence is substantially faster if the price-dividend ratio is computed as a series of “zero-coupon equity” claims, rather than as the fixed-point of the Euler equation. Fitting the model to the term structure as well as to equity moments (as in Wachter (2005)) also results in faster convergence. The remainder of this paper is organized as follows. Section 1 briefly describes the model for the representative agent and the aggregate endowment. Section 2 describes the solution techniques explored in this paper, Section 3 the calibration, and Section 4 the results. Extensions to the basic