00Thaler_FM i-xxvi.qxd

approximate J-factor structure describes the variance-covariance matrix of
returns.

εi,t

(0, σ^2 ei), fj,t

(0, 1). (4)

However, in contrast to the previous models, factor loadings do not de-
scribe expected returns. Instead, we assume expected returns are a function
of the observable, slowly varying firm attribute θ

~

i,t:

(5)

As in Model 2, the innovations in θare negatively correlated with the re-
turns on the stock, but θis not directly related to the loadings on the dis-
tressed factors. What is unique about Model 3 is that firms exist that load
on the distressed factors but which are not themselves distressed, and there-
fore have a low θand commensurately low return (and vice-versa). If
Model 3 is true, then following a string of negative shocks to the oil factor
there may be some stocks that, despite their high loadings on the oil factor,
are still not distressed. Model 2 suggests that these firms should still earn
the distress premium, because they behave like other distressed firms. In
contrast, Model 3 suggests their returns behavior does not matter: if they
are not distressed they will not earn the premium. Note also that Model 3
implies that a clever investor can earn the B/M return premium without
loading on any common factors.

D. Empirical Implications of the Models

The empirical evidence in Fama and French (1993) can be summarized
with two empirical facts: (1) the stocks in the high B/M portfolio strongly
covary with one another; and (2) high B/M stocks have high returns. The
conclusion conventionally drawn from this evidence is that the firms in the
high B/M portfolio are all loading on a factor that has a high premium; this
is indeed the intuition suggested by the first two models.
Models 2 and 3 illustrate why this conclusion need not follow from the
evidence. It is true that since the distressed firms covary with one another,
on average these firms must load on the same factor, which we can call the
distressed factor. Of course, a firm will become distressed when a factor on
which it loads has a strong negative realization. Using Bayesian reasoning,
it therefore follows that firms that are distressed will, on average, load on
the same factor. In both Model 2 and 3 (the characteristic-based model),
this is why distressed firms covary with one another, not because of the
presence of a separate distress factor. One way to discriminate between
Model 1 and Models 2 and 3 is to see whether the return standard deviation

Er abtit−−^111 [ =+ ⋅it

̃,,] θ ̃.

rEr ̃it,,,,,t [ ̃it] i jf ̃ ̃

j

J

=+ +− jt it

=

1 ∑

1

βε

326 DANIEL AND TITMAN