Pricing Kernels and Factor Models
WHAT IS THE RELATIONSHIP between the DAPMs that start from an expression like[see, e.g., (10.1) in Chapter 10] and those that focus on beta relations that express expected returns in terms of covariances of these returns with a benchmark return? Such a beta relation is implied by the celebrated static capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), where the return on a “market portfolio” serves as a benchmark. However, from Merton (1973) and Long (1974), we know that the market return is not in general a benchmark return in intertemporal asset pricing models. This chapter explores in depth the nature of beta or factor models for excess returns implied by DAPMs.1
Starting from a general DAPM with pricing in terms of a pricing kernel q* (see Chapter 8), we derive a “single-beta” representation of expected excess returns on traded assets. Among the aims of this chapter are: (1) linking the beta relations with time-varying betas studied in the literature to the preference-based DAPMs reviewed in Chapter 10, (2) characterizing the set of returns that can serve as benchmarks for beta relations, (3) identifying, where possible, observable members of this set, and (4) assessing the empirical support for factor models.
In addressing these issues, we answer the questions: (1) What is the link between conditional mean-variance efficiency and conditional singlebeta models? (2) How does reducing the conditioning information set from agents’ set to that of an econometrician affect the set of admissible benchmark returns for beta representations? (3) Do single-beta representations capture all of the restrictions on returns implied by DAPMs? (4) What are the implications of the answers to these questions for recent econometric studies of CAPMs with time-varying conditional moments? The answer to the second question, in particular, is central to the feasibility of econometric
1 The approach to beta models taken in this chapter is based on Hansen et al. (1982).