Pricing Kernels and DAPMs
VIRTUALLY ALL OF the DAPMs that we explore empirically can be characterized in terms of the restrictions they impose on the joint distribution of a pricing kernel q*, a vector of state variables or “risk factors” Y, and the securities with payoffs P that are being priced. In some cases q* has a natural “structural” interpretation, in that it can be linked directly to agents’ preferences or available technologies. In other cases, q* is given as a “reducedform” function of Y, in a manner that rules out arbitrage opportunities, but does not provide a direct link to preferences. This chapter outlines these approaches to modeling q* in more detail, thereby setting the stage for indepth analyses of the empirical fits of DAPMs.
We begin by expanding on the notion of a pricing kernel q* introduced in Chapter 1 and, following Hansen and Richard (1987), present quite general conditions under which a payoff qt+1 is priced asfor an information set It. Section 8.2 relates q* to agents’ marginal rate of substitution of consumption and, thereby, presents the conceptual foundations for the consumption-based models examined in Chapter 10. This is followed by a discussion of conditional “beta” and factor models in Chapter 11. Though we derive a beta representation of asset returns starting from an arbitrary admissible pricing kernel q*, we treat beta and factor models as part of our discussion of preference-based models, because this facilitates economic interpretation of the “benchmark” returns in beta/factor models. Finally, Section 8.3 presents an overview of the “no-arbitrage” approach to pricing risky securities, in which a reduced-form representation of q* is combined with the assumption of no arbitrage opportunities to restrict the dynamic properties of security prices. This formulation of the pricing kernel underlies the models covered in Chapters 12 through 16.
To introduce the concept of a pricing kernel more formally, it is helpful to be more precise about the contents of the payoff spaces and information