Intertemporal Asset Allocation with Habit Formation in Preferences: An Approximate Analytical Solution

In this paper we derive an approximate analytical solution to the optimal con-sumption and portfolio choice problem of an in\u85nitely-lived investor with power utility de\u85ned over the di¤erence between consumption and an external habit. The investor is assumed to have access to two tradable assets: a risk free asset with constant return and a risky asset with a time-varying premium. We extend the ap-proach proposed by Campbell and Viceira (1999), which builds on log-linearizations of the Euler equation, intertemporal budget constraint, and portfolio return, to also contain the log-linearized surplus consumption ratio. The di¤erence habit model implies that the relative risk aversion is time-varying which is in line with recent ev-idence from the asset pricing literature. We show that accounting for habit a¤ects both the myopic and intertemporal hedge component of optimal asset demand, and introduces an additional component that works as a hedge against changes in the investors habit level. In an empirical application, we calibrate the model to U.S. data and show that habit formation has signi\u85cant e¤ects on both the optimal consumption and portfolio choice compared to a standard CRRA utility function