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# SDF and Sentiment

In this section I provide a formal definition of sentiment that measures the collective error in the market. Having defined sentiment, I establish the central result in the paper. This result, theorem 2 below, states that the pricing kernel can be decomposed into two components, a fundamental component associated with zero market error, and a sentiment component that reflects the market error. Let r(Z) denote the (gross) return vector for security Z. In general, a pricing kernel M t is a stochastic process that satisfies

The state price vector v provides the present value, at date 0, of a contingent claim to one xt−dollar. In a discrete time, discrete state model, a pricing kernel M t restates this present value, in terms of per unit probability, v/II. [18] Now M(Xt) is more correctly written M(Xt|x0). To obtain the stochastic process for the kernel, define

where M t is a random variable. For this reason, I focus on M(X1) below, as the prototypical case. Using (7), obtain:

where I have suppressed the notation indicating that both ΔR and θr are time and statedependent respectively. I now turn to the formal definition of a sentiment variable Λ. This variable is basedon two terms. The first term, and the more important of the two is the likelihood ratio

that appears in (12). The second term involves the value of ln(ΔR) that arises when all traders hold objectively correct beliefs. Call this value ln(ΔR,II). Define sentiment by the variable

In the discussion below, I present an example that illustrates how optimism and pessimism are reflected in the properties of Λ. The impact of sentiment on the pricing kernel is most easily summarized through the log-kernel process m Ξ ln(M) and sentiment process Λ.[19] Equation (12) implies:

Theorem 2

where m, Λ, θr, and g are functions of x 1.

Theorem 2 states that the log-SDF is the sum of two stochastic processes, a sentiment process and a fundamental process based on aggregate consumption growth. Note that prices are objective when the sentiment variable  is uniformly zero, meaning its value is zero at every node in the tree. Hence, when prices are objective there is no aggregate belief distortion, in which case there is only one effective driver in (13), the fundamental process.

Theorem 2 implies that prices are objective if and only if Λ is the zero-function. In other words, prices are objective if and only if sentiment is zero. This result generalizes a result in Shefrin-Statman (1994) about when market aggregation leads the errors of traders to be self-cancelling. Shefrin-Statman (1994) assume log-utility. They define the discounted trader error by

(14)

where

Note that εh measures the difference between h’s discount-weighted subjective probability and the corresponding objective probability. Let W h denote h’s relative wealth share

and define the covariance

by:

where

The Shefrin-Statman (1994) efficiency condition is:

This condition states that prices are objective if and only if the sum of the error-wealth covariance and product of the average error and mean wealth is equal to zero.[20] The condition can also be expressed in the following dot product form:

Using the example developed below, it will be straightforward to see that the preceding dot product function (15) and the sentiment function Λ convey identical information. Figure 2 consists of three panels that illustrate the main concepts discussed in this section. The top panel depicts the graph of the dot product of relative wealth and trader errors against the gross consumption growth rate g. This illustrates the Shefrin-Statman (1994) efficiency condition. Notice that the dot product is negative in the range 1.01 to 1.09, and nonnegative elsewhere. This pattern corresponds to the relationship between the objective probability density and representative trader’s probability density in figure 1. Notice that in that figure, the representative trader attaches too low a probability togrowth rates between 1.01 and 1.09.

The middle panel in figure 2 depicts the graph of sentiment Λ against g. Note that in this example, the graph takes the shape of a smile. To understand how the graph of Λ portrays the nature of sentiment, consider a series of special cases. In the first case, prices are objective.21 Hence, Λ = 0 for all g, meaning that the graph of Λ against g is flat. In the second case the market is dominated by bulls. Bulls attach too high a probability to high consumption growth states, and too low a probability to low consumption growth states. Hence, in this case the graph of Λ is positively sloped.

In particular, Λ is negative for low consumption growth states, and positive for high consumption growth states. In the third case, the market is dominated by bears. This is the reverse of the previous case. Hence, the graph of Λ is negatively sloped. The smile pattern in the middle panel of figure 2 reflects the views of both bulls and bears. The graph of Λ is negatively sloped for low consumption growth states, and positively sloped for high consumption growth states. That is, although sentiment reflects the beliefs of both bulls and bears, the beliefs of the bulls dominate for high consumption growth states, and the beliefs of bears dominate for low consumption growth states.

Note too that Λ takes values that are negative between 1.01 and 1.09, and positive elsewhere. In this regard, compare the top and middle panels of figure 2. The dot product function (15) depicted in the top panel is defined in absolute terms, and approaches zero at the extreme ends of the range. However, because sentiment Λ is defined as a log-likelihood ratio, it is expressed in relative terms, and is actually furthest away from zero at the extremes.

In relative terms, the extremes are where the errors are most severe. The bottom panel of figure 2 shows the graph of the log-SDF and its two components, as functions of g. Theorem 2 tells us that when sentiment is zero, the graph of the log-SDF (equal to −θr ln(g) where θr = 1) is downward sloping. But the theorem also shows the sentiment smile being transmitted to the graph of the log-SDF. Why is the decomposition theorem for the log-SDF important? Campbell (2000) points out the volatility of the SDF appears to be puzzling high for a variable that is bounded below by zero and whose mean is unity. His observation is based on the fact that the coefficient of variation of the SDF is bounded from below by the maximum Sharpe ratio across securities. However, the SDF that Campbell discusses is based only on fundamentals, and has the interpretation of being a ratio of marginal utilities. However, in my framework, the log-SDF is the sum of a fundamental component and sentiment. Sentiment can be extremely volatile, as I suggested in my review of strategists’ predictions in section 2.

The volatility issue also applies to the equity premium. In the Mehra-Prescott (1985) analysis of the equity premium puzzle, the puzzle hinges on the fact that the volatility of consumption growth appears to be too low to justify the magnitude of the equity premium. However, it is the SDF that serves as the channel through which consumption growth affects the equity premium. And consumption growth enters through the fundamental component described above. Mehra-Prescott implicitly assume that sentiment is
zero.

Their model does not take account of the fact that the sentiment component can add substantial volatility to the SDF. Campbell and Cochrane (1999) analyze the equity premium puzzle using a model that features habit formation and time-varying risk aversion. In this regard, I would point out that there is a striking formal similarity between the SDF that they derive and the SDF (M) in my paper. Both have the same general form, except that where I have a sentiment term, Campbell and Cochrane have a habit formation term. In addition, heterogeneity leads the coefficient of relative risk aversion to be stochastic in my model, whereas it is constant in their model. What is more important is that Campbell and Cochrane rely on their habit formation term to explain the volatility of the SDF. I find it implausible that habit formation associated with consumption variation would explain the intraday volatilites associated withthe Sharpe ratios in equities. To my mind, intraday volatility in stock prices is driven by volatility in sentiment, rather than consumption variation relative to a habitual reference point.

Like Campbell and Cochrane (1999), Whitelaw (2000) uses a representative traderbased approach to study the equity premium. He too focuses on the challenge of reconciling the volatility of stock prices with the volatility inherent in an SDF that only reflects fundamentals. Whitelaw proposes a regime-switching model, where the representative trader knows the correct regime-switching model. In the CRRA utility model, the coefficient of risk tolerance and intertemporal elasticity of substitution share the same value. Whitelaw points out that generalized models that relax this restriction feature increased volatility in the SDF, but not increased volatility in the correlation between the SDF and equity returns. I note that a feature of my model is that time varying sentiment, induced by heterogeneous beliefs, leads to both increased volatility in the SDF and a time varying correlation between the SDF and equity returns.

Prof. Hersh Shefrin

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