Abstract

Stochastic discount factor (SDF) models are the dominant framework for modern asset pricing. The Hansen-Jagannathan bound is a characterization of the admissible set of SDFs, given a vector of asset returns. The admissible set provides (i) a test of the asset-pricing model and (ii) information on how to modify the SDF to be consistent with asset returns, neither of which requires solving the model. In this article we use the Hansen-Jagannathan bound to examine asset-pricing implications and to test specific asset-pricing models using bootstrap experiments.

Christopher Otrok is the Sam B. Cook Professor of Economics at the University of Missouri–Columbia and a research fellow at the Federal Reserve Bank of St. Louis. B. Ravikumar is an economist, a senior vice president, and the deputy director of research at the Federal Reserve Bank of St. Louis.

INTRODUCTION

An asset-pricing model is typically defined by its stochastic discount factor (SDF). For instance, Mehra and Prescott (1985) used constant-relative-risk-aversion (CRRA) preferences and the SDF in their model was a function of consumption growth. The validity of an SDF is determined by its ability to match the observed asset returns. An early test of an asset-pricing model with CRRA preferences was the Hansen and Singleton (1982) J-test. For U.S. stock and bond returns data, this test typically rejects the model. The J-test tells us whether or not an asset-pricing model has statistically significant pricing errors. It does not provide information on how to modify the SDF to improve the fit. Hansen and Jagannathan (1991) derive a volatility bound (HJ bound) that is based on necessary conditions that an asset-pricing model must satisfy. The HJ bound characterizes the admissible set of SDFs that is consistent with the observed asset returns.

The HJ bound exploits two conditions: (i) the intertemporal Euler equation that connects the price of an asset to the covariance of the asset's payoff with the SDF and (ii) the implication from linear pricing that the SDF be a linear function of payoffs. The asset-pricing model is said to be consistent with the data if the volatility of the proposed SDF (evaluated at the mean SDF) is greater than the volatility implied by the HJ bound. The HJ bound is a lower bound and, hence, is a necessary but not sufficient condition that an asset-pricing model must satisfy. In other words, the HJ bound provides a "test" of an asset-pricing model based solely on necessary conditions implied by the model.

The HJ bound approach in a sense works backward: Instead of writing down a model, solving it, and then testing it, the HJ bound asks what a valid SDF should look like in the mean-­variance space. The HJ bound approach has several advantages. First, the bound is model-free; that is, it is constructed using only observed asset returns. Second, one does not need to solve the nonlinear asset-pricing model. Specifically, there is no need to find a partial equilibrium or a general equilibrium solution to the model. Third, there is no limit on the number of assets used in the construction of the bound. Fourth, the bound is informative on how to modify the SDF in order to be consistent with the data.

In this article we provide a derivation of the HJ bound and then apply the bound to examine a few popular SDFs. The results provide an illustration of the equity premium puzzle. We then check the robustness of the resolutions of the puzzle with a bootstrap experiment. Our bootstrap results indicate that minor variations in asset return moments and consumption moments can yield large variations in the distance between an SDF's volatility and the HJ bound. We conclude with some implications for business cycle models.

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