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On Kernels and Sentiment

Literature Review of Heterogeneity

In this section, I review two different strands of literature. The first involves the treatment of heterogeneity in the finance literature. The second involves heterogeneity in the behavioral decision literature.

There is a longstanding literature in finance about the manner in which financial markets aggregate the beliefs and risk preferences of individual traders. Lintner (1969) was the first to address the question in a systematic manner. He uses a general equilibrium framework in which the returns to all securities are normally distributed, and the preferences of traders feature constant absolute risk aversion (CARA). Lintner permits his traders to disagree about security return means and the return covariance matrix. He establishes that securities can be priced as if there is a representative trader whose mean forecast of security value is a wealth-weighted convex combination of the individual traders’ means. Notably, the representative trader’s risk premia do not exactly conform with the wealthweighted convex combination property. As to risk preferences, the risk tolerance parameter in Lintner’s framework is the mean of the risk tolerance parameters of the individual traders.

Lintner’s article, although seminal, seems to have faded from view. Rubinstein (1974) develops an aggregation approach along the lines of Gorman (1953). Instead of assuming normality, as Lintner had done, Rubinstein uses a discrete state space, and arbitrary probabilities. He establishes an aggregation theorem for the case in which the Arrow-Pratt measure of risk aversion for individual traders, and most importantly for the representative trader, has a particular form. That form is 1/(A + Bc), where A and B are parameters, and c is the level of consumption. I note that utility functions which exhibit either constant absolute risk aversion (CARA) or constant relative risk aversion (CRRA) satisfy the 1/(A + Bc)− condition. Rubinstein’s work deals with sufficient conditions for aggregation and was later extended by Brennan and Kraus (1978) to establish necessary as well as sufficient conditions. In Lintner (1969), market prices are a function of the underlying wealth distribution.

However, Gorman aggregation requires that prices be independent of the wealth distribution. As a result, the aggregation conditions in Rubinstein/Brennan-Kraus are stringent. Roughly speaking, the Rubinstein/Brennan-Kraus conditions require either that traders possess homogeneous beliefs and a common B−parameter, or that their utility functions feature constant absolute risk aversion (B = 0). Subsequent treatments of heterogeneity have been varied. Jaffee-Winkler (1976), Figlewski (1978), Feiger (1978), and Shefrin (1984) study partial equilibrium models in which traders hold heterogeneous beliefs, but share the same tolerance for risk. Mayshar (1983) and Dumas (1989) focus on a two-trader general equilibrium model where the traders

hold the same beliefs, but have different tolerances for risk. Benninga and Mayshar (1993, 2000) extend the Dumas approach to many traders, and focus on how the representative trader serves to aggregate the risk tolerance parameters of the individual traders. Their 2000 paper analyzes the impact of heterogeneity, especially in respect to risk tolerance, on option prices. Similar to Dumas, Wang (1996) uses a model with heterogeneous traders to analyze the term structure of interest rates.

Detemple and Murthy (1994) develop a continuous time, incomplete market, logutility model in which traders hold differential beliefs about diffusion process parameters. Shefrin and Statman (1994) analyze a model similar to that of Detemple and Murthy, but where time is discrete. At the heart of their analysis is a representative trader whose beliefs are a wealth-weighted combination of the individual traders’ probability beliefs. Cuoco and He (1994a, 1994b) also use the notion of a representative trader in their analysis of a continuous time dynamic equilibrium with heterogeneous traders.

Kurz (1997) develops a model in which trader disagreements conform to a rational framework. Basak (2000) builds on Detemple-Murthy (1994) in analyzing a model with both fundamental risk and nonfundamental risk. His framework has several features that are similar to the features in this paper.3 Treynor (1998, 2001) develops a model composed of both bullish traders and bearish traders to explain speculative bubbles.

There is a parallel literature in noisy rational expectations models that also features heterogeneus beliefs. These models are based on the Diamond-Verecchia (1981) model, whereby traders form their beliefs by combining their own private signals with partially
revealing prices. For the purpose of tractability, these models eliminate the impact of wealth distribution on prices, by assuming CARA-utility. As in Lintner (1969), these models tend to impose normality, so that the heterogeneity is in respect to the value of the underlying distribution parameters. Consider next the literature in behavioral decision making. One of the most robust empirical findings in the behavioral decision literature is that individuals are heterogeneous in respect to both beliefs and preferences.Grether (1980)documents three important find-ings about beliefs.First,he finds that even in the face of financial incentives,individuals are irrational:they consistently violate Bayes rule. Second,Grether finds widespread hetero-geneity that cannot be traced to differential priors.Third,he shows that in the aggregate, people rely on a particular heuristic known as representativeness.

Representativeness in-volves overreliance on stereotypes.In consequence,errors are systematic,not randomly distributed around rational beliefs.As to preferences, Barsky, Kimball,Miles,and Juster (1997) document that individuals systematically differ in respect to their degree of risk tolerance.Indeed,the relevant section in their article is titled “Heterogeneity in Individu-als’ Risk Tolerance”(p.547).I should note that although the general framework features heterogeneity in respect to both beliefs and preferences, the focus of the paper is on the impact of heterogeneous beliefs.Later in the paper,I discuss how the two differ in so far as the character of asset pricing is concerned.

Studies by Edwards (1965,reprinted in 1982)first documented systematic viola-tions of Bayes rule.Tversky and Kahneman (1974)were the first to propose that violations of Bayes rule stem from representativeness.Grether (1980)initially hypothesized that the violations of Bayes rule due to representativeness would disappear in the face of financial incentives, and he designed an experiment to test this hypothesis. Despite his initial skepticism, Grether found that financial incentives do not elim-inate violations of Bayes rule stemming from reliance on the representativeness heuristic.

As to heterogeneity,Grether conducted his study using seven groups at six different univer-sities, where he paid his subjects by how well they predicted the source of drawings from bingo cages.Interestingly,the responses by his subjects feature considerable heterogeneity across venue,with the coefficient of variation in their responses equal to 10.9 percent.4 Be-cause Bayesian priors are a part of Grether’s experiment,and his subjects received common information,the heterogeneity in his subjects’responses stems from their errors,not from different information or different priors.In addition,many of Grether’s subjects were stu-dents at the California Institute of Technology,were well acquainted with Bayes rule,and yet relied on representativeness.

Grether’s experiment involved drawings from bingo cages, rather than financial variables. One study involving financial variables is De Bondt (1993), who analyzed trend extrapolation and overconfidence in predictions of the future value of the S&P 500.5 In table 2 (p. 359) of his paper, De Bondt documents the extent to which his subjects differ from each other in their predictions, with the extent of disagreement being stronger in bear markets than in bull markets. One way of measuring the extent of heterogeneity is by measuring the coefficient of variation across forecasts. 6 I have replicated De Bondt’s study with two separate groups of MBA students and one group of investment professionals.

7 For students the coefficient of variation is about 12.8 percent, and for investment professionals it is a little lower at 11.1 percent. Hence there is considerable heterogeneity within groups, but little difference in the degree of heterogeneity between groups. A real world analogue to De Bondt’s experiment is the annual survey conducted on the television program Wall $treet Week with Louis Rukeyser.

In 1983 Rukeyser began collecting twelve-month forecasts from his panelists for the year-end closing value of the Dow Jones Industrial Average. I have found that the mean annual coefficient of variation over the period 1983 through 2000 is 8.9 percent.8 The intertemporal standard deviation, which indicates the degree of variation in the dispersion of forecasts over time, is 2.6 percent. This is especially interesting, given that the composition of panelists is quite stable from year to year.

The highest value for the coefficient of variation is 16 percent, and occurred in connection with the forecast for year-end 1988. In line with De Bondt’s finding about disagreement being higher after bear markets, disagreement among Wall $treet Week panelists was highest for the prediction made two months after the 1987 crash. Depending on the year, 17 to 27 panelists participated in the Wall $treet Week survey. In summary, the panelists’ prediction exhibit both significant intrayear heterogeneity and significant interyear volatility. These are the two main features upon which I focus. In the paper, I provide a series of examples, all motivated by findings about the heterogeneity and volatility of beliefs that are documented in the behavioral decision literature and mirrored in media reports.

My examples loosely reflect the features described in an article by Browning, about strategists’ opinions, that appeared in the January 16, 2001 issue of the Wall Street Journal. Browning states:9 “In fact, as many as three differing scenarios can be found among market experts, ranging from nasty bear market to a strong bull market.”10
Along the lines of Treynor (1998), my examples deal with the nature of asset
pricing when there are two dominant views in the market, one strongly bullish and the other strongly bearish. 11 In developing these examples, I seek to establish the manner in which such heterogeneity affects the nature of sentiment, the SDF, option prices, the term structure of interest rates, and the return distribution of mean-variance efficient portfolios. I should add that my formal model places very few restrictions on traders’ subjective probability density functions. The idea is to develop a set of theorems to identify the broad framework, and then use examples to illucidate specific features.

Prof. Hersh Shefrin

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