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be riskier in terms of fundamentals, can feature lower expected returns
than safer stocks.
In Baker Wurgler, a stock s sensitivity to sentiment is measured in
terms of a sentiment beta. A sentiment beta explains the sentiment
component of the risk premium. Conditional on the risk-free rate, this
component effectively corresponds to the third term in the right-hand
side of Equation (5.5); unconditionally, it corresponds to the sum of
the first and third terms.
Empirically, Baker Wurgler report that returns are predictable,
conditional on the value of sentiment in the prior month. When past
sentiment has been high, subsequent returns to speculative stocks,
more difficult-to-arbitrage stocks are indeed lower than returns to safer
stocks which are easier to arbitrage. Conversely, when past sentiment
has been low, subsequent returns to speculative stocks, more difficult-
to-arbitrage stocks are higher than to safer stocks which are easier to
arbitrage.
5.5 Long-run Fitness
The neoclassical perspective has tended to be that information
investors will exploit any inefficiencies immediately, so that prices are
efficient at all times. In contrast, the behavioral position is different,
namely that in the short-run, limits to arbitrage can prevent mar-
ket prices from being efficient. The conditions for efficiency essen-
tially require that the average investor error be zero, so that investor
bias is unsystematic, and that the covariance between investor error
and investor wealth be zero, so that any errors are not concentrated
within the investor population (Shefrin and Statman, 1994). In contrast
to the neoclassical perspective, the behavioral perspective holds that
information traders, meaning investors who are free from error, will
not guarantee market efficiency, because information investors are typ-
ically reluctant to take on the risk associated with the large positions
required to eliminate inefficiencies completely.
5.5 Long-run Fitness 101
What about the long-run? Will the existence of information traders
eliminate any price inefficiencies? The neoclassical answer to this ques-
tion is yes, with the argument being that investors who are subject to
error will lose their wealth to information investors. In contrast, the
behavioral answer has been not necessarily, that investors who are sub-
ject to error can take on more risk than information investors, thereby
earning higher returns in the long-run, and surviving.
Lawrence Summers first proposed the intuition underlying the
behavioral perspective, and the idea was formally discussed by DeLong
et al. (1991). Their framework features CRRA utility functions, and
describes conditions under which investors who are prone to error sur-
vive in the long-run and impact prices.
The analysis in DeLong et al. (1991) is controversial. In particular,
the main result is at odds with the general literature on long-term
survivability or fitness. This literature has recently been surveyed by
Blume and Easley (2008). At the center of the fitness lies the concept
of entropy. Given two discrete probability density functions and P
over a series of K states, define
K
I (P ) = (k)ln( (k)/P (k)), (5.6)
k=1
and call this variable the relative entropy of P with respect to .
Consider a set of investors indexed by j. Suppose that objective
probabilities correspond to . Blume and Easley establish that the
time average of investor j s marginal utility to investor k s marginal
utility is given by:
[ln(´k) - I (Pk)] - [(ln(´j) - I (Pj)]. (5.7)
They call the difference [ln(´j) - I (Pj)] investor j s survival index.
Suppose that investor j has a lower survival index than investor k.
Then with probability one, the ratio of investor j s marginal utility
approaches + ". However, if physical resources are bounded from above
and investor utility is CRRA with risk aversion parameter ³j e" 1, then
over time investor j s consumption must go to zero.
Notice that there are two components to the survival index. The first
component relates to patience, and the second to errors in beliefs. The
102 Behavioralizing Asset Pricing Theory
survival condition suggests that long-run fitness is determined by being
sufficiently patient relative to other investors and/or being sufficiently
better informed about objective probabilities.
Setting aside impatience, the results in Blume and Easley (2008)
indicate that informed investors will dominate pricing in the long-run.
This means that there is good reason to believe that the fundamental
assertion put forth in DeLong et al. (1991) is incorrect. Indeed, Blume
and Easley (2008) state that they have difficulty following the logic
in the latter paper. In this respect, the model appears to be vaguely
specified, with the formal argument being loose.
Of course, it is entirely possible that investors with erroneous beliefs
persist in the long run, but for reasons different from those suggested
in the analysis of DeLong et al. (1991). For example, there may be no
informed investors at all, in that all investors are subject to some form
of error or bias. Alternatively, the market can feature incompleteness in
combination with overlapping generations with finite lifetimes. There-
fore, better informed investors might not live long enough to extract
most of the wealth from their less informed counterparts. Moreover,
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