American Finance Association Assessing Specification Errors in Stochastic Discount Factor Models

In this article we develop alternative ways to compare asset pricing models when it is understood that their implied stochastic discount factors do not price all portfolios correctly. Unlike comparisons based on x2 statistics associated with null hypotheses that models are correct, our measures of model performance do not reward variability of discount factor proxies. One of our measures is designed to exploit fully the implications of arbitrage-free pricing of derivative claims. We demonstrate empirically the usefulness of our methods in assessing some alternative stochastic factor models that have been proposed in asset pricing literature. IN THEORIES OF ASSET PRICING, portfolio payoffs are modeled as bundled contingent claims to a numeraire consumption good. When asset markets are frictionless, portfolio prices can be characterized as a linear valuation functional that assigns prices to the portfolio payoffs (e.g., see Ross (1978), Harrison and Kreps (1979), Kreps (1981), Chamberlain and Rothschild (1983), Hansen and Richard (1987), and Clark (1993)). These valuation functionals are typically represented as inner products of payoffs with pricing kernels or stochastic discount factors. As argued by Hansen and Richard (1987), observable implications of candidate models of asset markets are summarized conveniently in terms of their implied stochastic discount factors.' WVhile formal statistical testing of such asset pricing models can yield insights, it is also of interest to evaluate the performance of these models even when it is understood that the implied stochastic discount factors do not correctly price all portfolios. Pricing errors may occur either because the model * Hansen is from the University of Chicago, The National Opinion Research Center, and National Bureau of Economic Research. Jagannathan is from the Carlson School of Management and Visitor, Research Department, Federal Reserve Bank of Minneapolis. Comments by Renee Adams, Edward Allen, Fischer Black, John Cochrane, John Heaton, Peter Knez, Erzo Luttmer, Marc Roston, S. Viswanathan, two referees, and the editor of the journal are gratefully acknowledged. Expert research assistance was provided by Marc Roston and Amir Yaron. Hansen received funding from the National Science Foundation in support of this research. Jagannathan acknowledges partial financial support from the National Science Foundation, grant SBR-9409824. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. ' We use the term stochastic discount factor as a label for a state-contingent discount factor. The market value today of an uncertain payoff tomorrow is represented by multiplying the payoff by the discount factor and adding across states of nature using the underlying probabilities. The discount factor is stochastic because it varies across states of nature. This variation captures corrections for risk as argued by Rubinstein (1976).