Chapter 9

Factor Models

An economist is someone who sees something in the real world and wonders if it would work in theory.

—Ronald Reagan

The CAPM can be interpreted as a single index factor model, and since we are already familiar with it, I will use the CAPM to introduce this important class of pricing models.

Factor models can be thought of as models of the conditional mean return to an asset (or portfolio of assets). We used unconditional mean return estimates as inputs to the optimal mean-variance efficient portfolios solved for in Chapters 6 and 7. To understand the difference, let r_it be the return on asset i at time t. Then the asset's unconditional mean return is simply the average return, that is, E(r_i) = . Now, think of the CAPM model:

For ease of exposition, drop the risk-free rate (that is, assume that cash earns no return) and take expectations (see the statistical review in the appendix to Chapter 5) noting that E(ε_i) = 0 by definition. Then,

Simply stated, this says that the expected return is now conditional on the market return. The market return is therefore the single factor and beta is the return (sometimes called the factor loading) to that factor.

There are several attractive features of factor models that are developed in this chapter, but the first and most obvious feature is that factors help attribute the asset's return to movements in specific underlying sources of risk and return. The Fama-French three-factor model introduced in Chapter 8, for example, decomposes the asset's return into three component parts—the return attributable to the value portfolio (HML), the return attributable to market capitalization or size (SMB), and the market return:

Since the factors themselves are returns, then the factor returns (the betas) measure the return attributable to the asset's exposure to each of these factors. Thus, a 1 percent change in the factor is expected to move the asset's return by the amount of its beta with respect to that factor. The parameters of factor models are generally estimated by ordinary least squares available in virtually all spreadsheet and statistical software packages. Thus, collecting a time series of returns on the asset and the factors (the latter are available at French's website) then regressing r_i on the factors will generate estimates of the parameters. We explore some applications further on.

See Kenneth French's website at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/index.html.