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## On Kernels and Sentiment

In this section, I provide a theorem about the existence and nonuniqueness of a representative trader. This theorem is utilitarian, in that it lays the foundation for the analysis of how sentiment impacts the pricing kernel. [14]

In theorem 1 below, I make use of the notion of objective prices, which is a particular form of market efficiency. The equilibrium state prices v are defined by the condition

, with numeraire x0. These prices determine the equilibrium prices of all securities. When prices are objective, the price of every security coincides with its fundamental value. To make this notion precise, suppose P h = II for all h, so that all traders’ subjective probabilities are objectively correct. Let the equilibrium state prices associated with this case be denoted v Π. I call v Π the objective state prices. That is, I say that prices are objective when they are set as if all traders hold objectively correct beliefs.

In this case, every security is priced in terms of the objectively correct distribution of its future cash flow payoffs. If P h≠ II for some h, and as a result equilibrium prices are not equal to vΠ, then I say that state prices are not objective. When P h ≠ IIfor some trader h, then trader h holds erroneous beliefs. I use the term sentiment to refer to the collective
error in the market.

Theorem 1 below establishes that the equilibrium v can be characterized as if there were a single representative trader in the market. Indeed, imagine that there is only one trader, having risk aversion parameter θR and discounted probability weights

Define the cumulative growth rate of aggregate consumption g(xt) as:

In this case, (3) together with the equilibrium condition imply that v(xt) takes the form:

I characterize sentiment through the beliefs of a representative trader. Theorem 1 below shows how the parameters of a representative trader can be defined in terms of the parameters of the individual traders, market portfolio, and equilibrium prices. I emphasize that this theorem is utilitarian, in that its purpose is to provide a vehicle for characterizing state prices. Under plausible conditions, equilbrium prices are unique.

See the appendix. Uniqueness of equilibrium is important because I establish that the representative trader characterization is not unique, and describe how different representative traders relate to one another. In particular, any representative trader can be chosen for the purpose of characterizing state prices. State prices in no way depend on which particular representative trader is selected. However, I would point out that the choice of representative trader affects the ease with which the structure of state prices can be elucidated. Only in this sense is the choice of representative trader critical.

Lintner (1969) first analyzed the manner in which the market aggregates the heterogeneity across traders. In this respect, I echo his observation that the representative traders’ beliefs and preferences are not as well behaved as those of the individual traders. For example, the risk aversion parameter of the representative trader may depend on xt, rather than being constant. Likewise, the discount term & may vary with t. This variability occurs when risk tolerance or the discount factor is not uniform across the trading population. In this case, θR(xt) reflects the curvature of the utility function of xt-consumption, but loses its interpretation as a measure of the degree of relative risk aversion. In addition, the representative trader’s probability beliefs may violate the principle of conditional probability.

Let v Π be the equilibrium price vector v, and C h,Π be the equilibrium value of C h for the case when P h = II for all h. These are objective benchmark values. To set the stage for the theorem, I define two variables used in the construction of the representative trader. The first variable plays a key role in the approach that Benninga-Mayshar (1993) develop to define the representative trader’s coefficient θR.

I note that theorem 1 below extends the Benninga-Mayshar aggregation to accommodate heterogeneous beliefs as well.15 The second variable is:

Theorem 1 Let v be an equilibrium state price vector. (1) v satisfies

(7)

where θR, R, and P R have the structure described below:

(8)

(9)

where the summation in (9) is over all traders and xt-events at date t, and

(10)

(2) The representative trader is not unique. Equation (5) implies that any two representative traders, denoted R, 1 and R, 2, are related together through the expression:

(11)

In order to illustrate the character of the results in the paper, I develop two examples, both motivated by Browning’s discussion of Wall Street strategists’ market predictions (see section 2). In the first example, there are two traders with equal initial wealth. Both have log-utility, so that θh = 1 for h = 1, 2. To keep the example simple, I let T = 1, and focus attention on the heterogeneous beliefs about the probabilities with which various states occur. In the example, the set S contains 65 states. Each state is associated with a rate g of consumption growth, with g ranging from 0.80 to 1.44, in increments of 0.01. I note that because θh = 1 for all h, (8) implies that θr = 1 [16] I assume that the beliefs of the two traders are approximately lognormal, but feature different means. As in Browning’s description, one trader is overly optimistic about future consumption growth (bullish) and the other is overly pessimistic (bearish). The true mean is 5.28%. However, the bull believes that the mean value of log-consumption growth (ln(g)) is 12.01%, while the bear believes it to be −1.00%. Both believe the standard deviation to be 3.76%, and I assume this to be the objectively correct value.

Consider theorem 1, as it relates to the preceding example. How does the probability density of the representative trader compare to the probability densities of the individual traders? Being approximately lognormal, the density functions of the individual traders are single peaked. But because of the divergence in their viewpoints, the modes (peaks) are widely separated. Theorem 1 implies that for this example, the representative trader’s beliefs are a convex combination of the individual traders’ beliefs. See the discussion in the appendix.

figure 1

Figure 1 displays the impact of these divergent viewpoints on the shape of the representative trader’s density function. The figure displays the objective density function, the individual traders’ density functions, and the representative trader’s density function. Compared to the objective density, notice that representative trader’s density is bimodal, and fat-tailed. [17]

Prof. Hersh Shefrin

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