show that there is a value premium in the subsample of U.S. data that pre-
cedes the data set used in Fama and French (1992), while Fama and French
(1998) document a value premium in international stock markets.
Rouwenhorst (1998) shows that the momentum effect is alive and well in
international stock market data.
If the empirical results are taken at face value, then the challenge to the
rational paradigm is to show that the above cross-sectional evidence
emerges naturally from a model with fully rational investors. In special
cases, models of this form reduce to the CAPM, and we know that this does
not explain the evidence. More generally, rational models predict a multi-
factor pricing structure,
(16)where the factors proxy for marginal utility growth and where the loadings
βi,kcome from a time series regression of excess stock returns on excess fac-
tor returns,
ri,t−rf,t=αi+βi,1(F1,t−rf,t)+...+βi,K(FK,t−rf,t)+εi,t. (17)To date, it has proved difficult to derive a multifactor model that explains
the cross-sectional evidence, although this remains a major research direc-
tion.
Alternatively, one can skip the step of derivinga factor model, and sim-
ply try a specific model to see how it does. This is the approach of Fama
and French (1993, 1996). They show that a certain three-factor model does
a good job explaining the average returns of portfolios formed on size and
B/M rankings. Put differently, the αiintercepts in regression (17) are typi-
cally close to zero for these portfolios and for their choice of factors. The
specific factors they use are the return on the market portfolio, the return
on a portfolio of small stocks minus the return on a portfolio of large
stocks—the “size” factor—and the return on a portfolio of value stocks
minus the return on a portfolio of growth stocks—the “book-to-market”
factor. By constructing these last two factors, Fama and French are isolat-
ing common factors in the returns of small stocks and value stocks, and
their three factor model can be loosely motivated by the idea that this co-
movement is a systematic risk that is priced in equilibrium.
The low αiintercepts obtained by Fama and French (1993, 1996) are not
necessarily cause for celebration. After all, as Roll (1977) emphasizes, in
any specific sample, it is always possible to mechanically construct a one
factor model that prices average returns exactly.^27 This sounds a cautionary
rrif i−=ββ,, 11 ()Fr− +f⋅⋅⋅+ iKKf( ),F r−A SURVEY OF BEHAVIORAL FINANCE 39(^27) For any sample of observations on individual returns, choose any one of the ex-post
mean-variance efficient portfolios. Roll (1977) shows that there is an exact linear relationship
between the sample mean returns of the individual assets and their betas, computed with re-
spect to the mean-variance efficient portfolio.