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On Kernels and Sentiment

Conclusion and Future Directions

There is a debate underway about whether the dominant paradigm in finance will continue to be market efficiency-based, or whether it will shift to being behaviorally-based. I suggest that the weight of the evidence has been moving in favor of a behaviorally-based paradigm, and discuss the role of this paper against that backdrop. Traditional asset pricing theory is centered around theories of risk and return (such as the capital asset pricing model), the term structure of interest rate models (such as Cox-Ingersoll-Ross), and option pricing models (such as Black-Scholes). The efficient market hypothesis is common to all these models in that all assume that prices correctly reflect underlying fundmentals, as if the representative trader uses the objectively correct probabilities. In contrast, behaviorally-based asset pricing models assume that traders are overconfident in their judgements and commit systematic errors such as representativeness.

Examples are of the latter are Barberis, Shleifer, Vishny (1998), and Daniel, Hirshleifer, and Subrahmanyan (1998, 2001). I suggest that in the future asset pricing theory will combine the best of both approaches. Traditional asset pricing theory, centered on the pricing kernel (SDF), is both powerful and elegant. At the moment, its chief weakness is that it assumes that prices are objective. In this respect, many traditional asset pricing models begin by assuming that prices are set by a representative expected utility maximizing trader who holds objectively correct beliefs.

I argue instead, that relative to objectively correct beliefs, the presence of heterogeneous beliefs leads the representative trader to hold beliefs that are multi-modal and fat-tailed. Recent behavioral asset pricing models suffer from a different weakness. Although based on observations about individual errors and biases that are well documented in the behavioral decision literature, many of the recent behavioral papers use these observations in an ad hoc manner.

For example, Edwards (1965) argues that people routinely violate Bayes rule, and react to new information in an overly conservative manner. Barberis, Shleifer, and Vishny (1998) justify their assumptions about conservatism by appeal to Ed-wards’ work. However, they also assume that traders update their beliefs in accordance with Bayes rule, a feature clearly at odds with Edwards’ main point. Daniel, Hirshleifer, and Subrhamanyan (1998, 2001) develop a model where traders suffer from a collection of behavioral biases that have been documented in several studies. These biases include overconfidence and self-attribution error. Daniel, Hirshleifer, and Subrhamanyan argue that these behavioral features lead investors to treat private signals differently from public signals.

This assumption is intriguing, but has no empirical basis in the behavioral decision literature.47 Both Barberis, Shleifer, and Vishny (1998), and Daniel, Hirshleifer, and Subrhamanyan (1998, 2001) ignore heterogeneity in traders’ errors. Barberis, Shleifer, and Vishny assume a representative trader who commits particular behavioral errors. Daniel, Hirshleifer, and Subrahmanyan assume that all noise traders make the same errors, a feature that leads prices in their model to be set by a representative trader who makes the same types of errors as the individual traders. However, the analysis in this paper, and the earlier work of Shefrin-Statman (1994), suggests that the representative trader’s errors reflect not only the mean trader error, but the error-wealth covariance. The error-wealth covariance is stochastic, and can cause the representative trader’s errors to look nothing like the individual traders’ errors.

Recall that the representative trader can hold beliefs that are multi-modal and fat-tailed, even though the beliefs of the individual traders are unimodal without fat tails. As I noted earlier, one of the most pronounced biases identified in the behavioral decision literature stems from the heuristic known as representativeness.

Tversky and Kahneman (1974) document a wide variety of errors that people make because they rely on intuitive notions of representativeness to make judgements about distributional features of a population. I suggest that through their reliance on traditional representative trader-based arguments, financial economists leave themselves vulnerable to errors and biases associated with the representativeness heuristic.

Traditional asset pricing theorists do so when they assume that the representative trader holds objectively correct beliefs. Behavioral asset pricing theorists do so when they assume that the representative trader commits errors that have been attributed to individuals in the behavioral decision literature. Both groups of theorists fail to account for the impact of the error-wealth covariance on the beliefs of the representative trader.

My goal is provide traditional asset pricing theorists with a sense of how they can adapt traditional asset pricing theory to accommodate behavioral features. In this respect, the log-SDF decomposition theorem is central. Because the log-SDF decomposes into a sentiment component and a fundamental component, the core issue is to characterize the stochastic process by which sentiment evolves. In combination with the process governing fundamentals, the associated SDF underlies the pricing of all assets. I conclude by describing my sense of the future direction of asset pricing theory theory, a future that combines the best of the traditional approach and the behavioral approach. In this respect, I focus on three dimensions.

The first dimension involves a theory with the SDF at its core. The second dimension is heterogeneity. I think that instead of taking a traditional representative trader as a premise, asset pricing models need begin from first principles and postulate heterogeneity and specific error processes in respect to traders. The third dimension is to build behaviorally-based models of sentiment. In this respect, I think that theorists need to build models of the error process in the representative trader’s beliefs. Notably, these errors must reflect not only the mean trader error, but the error-wealth covariance as well.

In respect to the third dimension, I mention that Shefrin and Statman (1994) propose two classes of quasi-Bayesian behaviorally-motivated learning structures that underlie beliefs Ph(xt).[48] Notably, they do not assume that one group of traders is exogenously optimistic and the other exognenously pessimistic. Instead, optimism and pessimism are endogenously determined as outcomes of the learning processes. One group of traders underweights base rates in applying Bayes’ rule to update conditional probabilities. This error induces traders to predict the continuation of recent trends.

The second group succumbs to the “law of small numbers,” and tends to predict the reversal of recent trends. Notably, both types of quasi-Bayesian traders are implicitly overconfident. Their priors are excessively tight, relative to true Bayesians.[49] If the objective process Π is Markovian and Ergodic, true Bayesians will eventually learn the true probabilities. However, quasi-Bayesians will not, since their beliefs do not typically converge. There is a view in traditional finance that traders’ errors are temporary, and will disappear with learning. In contrast, the literature in behavioral decision making contains many studies showing that people learn very slowly, and that errors persist in the face of experience.[50]

Sentiment Λ is the aggregate reflection of traders’ errors in the market. The degree to which an individual trader’s error process affects market sentiment depends on the size of the trader’s trades, a point stressed by Linter (1969). Λ depends on risk tolerance and wealth because these variables provide the weights used to aggregate trader errors. Traders who are wealthier and more tolerant of risk take larger positions than traders who are less wealthy and less tolerant of risk.

Sentiment is time varying, and reflects the evolution of the error-wealth covariance. The magnitude of Λ varies as wealth shifts between traders who have taken the opposite sides of trades. This is because the weight attached to a trader’s beliefs is an increasing function of his trading success. Along a high consumption growth sequence, base rate underweighters will become unduly optimistic, and sentiment will move in the direction of excess optimism. Along a low consumption growth sequence, the reverse will occur. In a volatile segment, with frequent alternation between high and low consumption growth, weight will shift to the traders who believe in the law of small numbers. These traders overestimate the degree of volatility. During those segments when consumption growth is volatile, the kernel will accord their beliefs greater weight, and returns will amplify the volatility in consumption growth. What determines the relative occurrence of the different types of segments? The objective process {Π, g}.

In theory, the relative contribution of sentiment on the pricing kernel can be much larger than that of consumption growth. Fundamentals can get short shrift, at least along particular segments of the realized path. Think about theorems 3 and 4, which characterize the term structure in terms of aggregate consumption growth. These theorems suggest that information releases about consumption growth should be one of the most important pieces of news bond traders receive. Nevertheless, Balduzzi, Elton, and Green (1997) find that consumption growth is one of the least important variables influencing the Treasury market. Why? I suggest that the answer concerns the contribution of sentiment relative to fundamentals in determining returns in the short-run.

The process governing the evolution of sentiment Λ is based on the beliefs of a representative trader. In a prolonged period of favorable fundamentals, the representative trader will move in the direction of irrational exuberance. In a prolonged period featuring unfavorable fundamentals, the reverse will be true. In a period of rapidly fluctuating fundamentals, the representative trader will display frequent changes of opinion. Market returns will tend to become more positively autocorrelated along long runs, and move to being negatively autocorrelated during periods that feature short runs. In other words, the process governing Λ will have a stochastic autocorrelation structure that depends on the history.

There are additional behavioral elements concerning preferences, most notably loss aversion and reference point-based mental accounting. In sprit, these are related to habit formation models, especially in connection with reference point issues. Thus far, I have ignored behavioral features involving preferences, and have instead concentrated on beliefs. However, behavioral preferences can be accommodated within the present framework, with minor modifications. Shefrin and Statman (1989) explore the implications for the pricing kernel that stem from the introduction of prospect theory preferences (Kahneman and Tversky, 1979). A prospect theory utility function is S-shaped, and is defined over consumption changes (i.e., gains and losses) relative to a reference point. The S-shape depicts risk aversion in the domain of gains, and risk seeking in the domain of losses.[51]

Shefrin and Statman (1989) demonstrate that the introduction of prospect theory traders tends to flatten the graph of the pricing kernel. Prospect theory traders tend to shun claims that pay off in loss states, typically those for which m(xt) is high, in exchange for claims that pay off in gains states, typically those for which m(xt) is low.[52] As the relative proportion, and wealth, of prospect theory traders increases, the graph of the pricing kernel begins to flatten in the loss states segment. If it flattens completely, turning horizontal, then risk averse agents will choose to hold portfolio insurance in equilibrium. They will do so because the conditional risk premium vanishes, where the conditioning is for states in which prospect theory traders register losses.

The pricing kernel equation (13) includes time varying stochastic terms for both risk tolerance θR and time preference δR. Notably, during periods where wealth shifts from those who are less risk-tolerant to those who are more risk-tolerant, the representative trader becomes more risk tolerant. This phenomenon tends to make the representative trader more risk tolerant during up markets, a point made by Benninga and Mayshar (1993, 2000). A similar remark applies to time discounting. Recently, Barberis, Huang, and Santos (2000, forthcoming) have developed a prospect theory motivated model with the same time varying, stochastic risk tolerance properties described above. Specifically, they explain the equity premium puzzle by establishing that prior gains lead to an increase in the market’s tolerance for risk, whereas prior losses lead to the reverse.

In conclusion, I propose that in the future asset pricing theory will develop around models that feature a behavioral SDF that incorporates both sentiment and behaviorallybased risk preferences. Notably, the representative trader’s errors and preferences will typically have a different character than those of individual traders. For errors especially, the representative trader’s errors will reflect the error-wealth covariance, not just the mean trader error.

Prof. Hersh Shefrin

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