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Sentiment
-ln(g) = neoclassical log-SDF
Function
-20%
-30%
-40%
-50%
-60%
ln(g)
Fig. 5.2 Illustration of the impact of less underconfidence by pessimists in Figure 5.1.
1%
3%
4%
6%
-4%
-3%
-1%
88 Behavioralizing Asset Pricing Theory
5.2 How Markets Aggregate Investor Attributes
The empirical evidence strongly indicates that there is considerable
heterogeneity in respect to investors beliefs, degree of risk aversion,
and rate of impatience. Some of this evidence is direct, meaning that
it stems from surveys concerning these variables. Other evidence is
indirect, being based on investors decisions, such as the composition
of their portfolios.3 Therefore, a core issue in asset pricing theory is to
understand how markets aggregate investors beliefs and preferences.
In regard to preferences, recall from the discussion in Section 3 that the
change of measure technique applies both to preferences and to beliefs.
5.2.1 Aggregating Beliefs, Risk Tolerance, and
Rates of Impatience
The study of the aggregation question dates back at least as far as Lint-
ner (1969). Shefrin (2008a) presents an equilibrium aggregation result
for a complete markets model featuring power utility. He demonstrates
that the equilibrium market pdf is a generalized weighted Hölder aver-
age of the individual investors pdfs, with the weights reflecting rel-
ative wealth or consumption, and the exponents being coefficients of
relative risk aversion. Shefrin (2008a) also analyzes aggregation under
the assumption of constant absolute risk aversion (CRRA), although
he suggests that CARA is unrealistic from a behavioral perspective.
Jouini and Napp (2006, 2007) establish similar results. They focus on
the special case when investors share the same coefficient of relative risk
aversion. Jouini and Napp s analysis of aggregation under the assump-
tion of constant absolute risk aversion is more general.
For a detailed discussion of the aggregation theorems, readers are
referred to Shefrin (2008a) and the works of Jouini and Napp (2007).
For the purpose of the current discussion, it is sufficient to describe
the general features of the representative investor for the case of
power utility (CRRA). The log-SDF for the case when investors have
3
In a series of papers, Barber et al. (2005), Barber et al. (2009a), and Barber et al. (2009b)
provide evidence that individual investors as a group influence prices, but lose money to
professional investors. However, they also document considerable heterogeneity, demon-
strating that some individual investors possess superior skills.
5.2 How Markets Aggregate Investor Attributes 89
CARA utility functions is similar in form, except that all averages are
unweighted, rather than being consumption-weighted as is the case with
CRRA utility (Shefrin, 2008a).
Heterogeneity leads the representative investor s beliefs to be differ-
ent from those of the individual investors. For example, suppose that
all investors have log-utility, share the same intertemporal discount
factor ´, and have normally distributed beliefs with different means. In
this case, the generalized weighted Hölder average is a simple wealth-
weighted average of probability density functions. If the means are suf-
ficiently different across investors, such a pdf can be multi-modal, and
therefore non-normal. If investors differ in their intertemporal discount
factors, the representative investor s stochastic process will typically
violate Bayes rule.
At any moment in time, the representative investor s degree of
risk tolerance will be a consumption weighted convex combination of
the individual investors degree of risk tolerance. Because consump-
tion shares are time varying, heterogeneity leads the representative
investor s utility exponent ³ to be state dependent. This means that
³ will not measure the representative investor s degree of relative risk
aversion.
The representative investor s intertemporal discount rate ´ is a non-
linear function of the individual investors discount rates. However,
because the aggregation process relates to discount factors, not dis-
count rates, the representative investor s discount factor will typically
be non-exponential, even though every investor has an exponential dis-
count factor. Moreover, because Hölder averages do not always sum
to unity, the representative investor s discount rate will also reflect
probability aggregation. Stated somewhat differently, the representa-
tive investor might exhibit aversion to ambiguity, even if none of the
individual investors do.
The characteristics of the aggregation process hold clear lessons
for asset pricing theorists of all stripes, neoclassical and behavioral.
Typically, the market does not aggregate heterogeneous beliefs and
preferences in a way that leads to the existence of a typical neoclassical
representative investor, or a representative investor whose preferences
and beliefs feature typical behavioral characteristics.
90 Behavioralizing Asset Pricing Theory
5.2.2 Liquidity Premium and Asymmetric Volatility
The preceding discussion describes the manner in which sentiment
reflects the aggregation of investors beliefs and preferences, while
Equation (5.3) reflects the manner in which the SDF incorporates
sentiment. In this respect, consider how market prices reflect a liquidity
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