Equity Premia with Benchmark Levels of Consumption: Closed-Form Results

I calculate exact expressions for risk premia, term premia, and the premium on levered equity in a framework that includes habit formation, keeping/catching up with the Joneses, and possible departures from rational expectations. Closed-form expressions for the …rst and second moments of returns and for the R2 of a regression of stock returns on the dividend-price ratio are derived under lognormality for the case that includes keeping/catching up with the Joneses. Linear approximations illustrate how these moments of returns are a¤ected by parameter values and illustrate quantitatively how well the model can account for values of the equity premium, the term premium, and the standard deviations of the riskless return and the rate of return on levered equity. For empirically relevant parameter values, the linear approximations yield values of the various moments that are close to those obtained from the exact solutions. The paper was prepared as a chapter in Rajnish Mehra and Edward Prescott (eds.), Handbook of Investments: Equity Risk Premium. It previously circulated under the title “Equity Premia with Benchmark Levels of Consumption, Leverage, Imperfect Correlation of Consumption and Dividends, and Distorted Beliefs: Closed-Form Results.” I thank Martin Lettau, Sydney Ludvigson, Raj Mehra, Jessica Wachter, Amir Yaron, Jianfeng Yu, participants in the Finance Seminar at the Stern School, the European Summer Symposium in Financial Markets, Gerzensee, Switzerland and the Penn Macro Lunch Group for helpful comments. Mehra and Prescott (1985) showed that over a period of almost a century, the equity premium–the excess of the rate of return on stocks over the rate of return on riskless bills–averaged 6.18% per year. They then calibrated a general equilibrium asset-pricing model of the sort introduced by Lucas (1978) and showed that such a model, with conventional values of the coe¢ cient of relative risk aversion, cannot come close to accounting for the historically observed equity premium. In the two decades since Mehra and Prescott proclaimed an equity premium puzzle, a large body of research has been devoted to closing the gap between theoretical asset-pricing models and empirically observed asset returns. The seminal study by Mehra and Prescott used a simple general equilibrium model with a representative consumer with constant relative risk aversion transacting in frictionless asset markets to determine the prices of unlevered equity and riskless bills. Subsequent research has extended this simple model to allow for heterogeneous consumers, trading frictions, more general preferences, leverage, a richer time-series of the endowment of consumption, and departures from rational expectations. Since the equity premium puzzle is a quantitative puzzle, many of these studies provide numerical solutions rather closed-form solutions for equilibrium prices and rates of return. In this chapter, I will provide closed-form solutions for the equilibrium price and rate of return on a canonical asset. The canonical asset is general enough to include riskless bills and risky stocks, and thus allows analysis of the equity premium. I will extend the basic framework used by Mehra and Prescott to allow for preferences that display habit formation and a speci…c form of externalities. I will also allow for leverage so that I can examine the return on levered equity. Finally, I will allow for departures from rational expectations, though I will set up the framework so that rational expectations is a special–indeed focal–case. Instead of enriching the time-series structure used by Mehra and Prescott, I will simplify it by restricting attention to growth rates of consumption and dividends that are i.i.d. over time, though unlike Mehra and Prescott, I will allow consumption and dividends to di¤er from each other. As in Mehra and Prescott, I will assume that there is a representative consumer who transacts in frictionless markets. An advantage of closed-form solutions is that they provide a precise description of how equilibrium rates of return depend on various parameters. Though this description is precise, it is not always transparent. To help understand the e¤ects of various parameters on the equilibrium rates of return, I will provide linear approximations to the closed-form solutions. I will