«Abstract We estimate a dynamic asset pricing model characterized by heterogeneous boundedly rational agents. The fundamental value of the risky asset ...»
Behavioral Heterogeneity in Stock Prices
H. Peter Boswijk Cars H. Hommes Sebastiano Manzan
CeNDEF, University of Amsterdam, the Netherlands
We estimate a dynamic asset pricing model characterized by heterogeneous boundedly
rational agents. The fundamental value of the risky asset is publicly available to all
agents, but they have diﬀerent beliefs about the persistence of deviations of stock
prices from the fundamental benchmark. An evolutionary selection mechanism based on relative past proﬁts governs the dynamics of the fractions and switching of agents between diﬀerent beliefs or forecasting strategies. A strategy attracts more agents if it performed relatively well in the recent past compared to other strategies. We estimate the model to annual US stock price data from 1871 until 2003. The estimation results support the existence of two expectation regimes. One regime can be characterized as a fundamentalists regime, because agents believe in mean reversion of stock prices toward the benchmark fundamental value. The second regime can be characterized as a chartist, trend following regime because agents expect the deviations from the fundamental to trend. The fractions of agents using the fundamentalists and trend following forecasting rules show substantial time variation and switching between predictors. The model oﬀers an explanation for the recent stock prices run-up. Before the 90s the trend following regime was active only occasionally. However, in the late 90s the trend following regime persisted and created an extraordinary deviation of stock prices from the fundamentals. Recently, the activation of the mean reversion regime has contributed to drive stock prices back towards their fundamental valuation.
Acknowledgments. Earlier versions of this paper were presented at the IFAC symposium Computational Economics, Finance and Engineering, Seattle, June 28-30, 2003 and at seminars at Tilburg University, the University of Amsterdam, the University of Udine and the University of Warwick. Stimulating discussions and helpful comments are gratefully acknowledged. This research has been supported by the Netherlands Organization for Scientiﬁc Research (NWO) under a NWO-MaG Pionier grant.
Corresponding author: Sebastiano Manzan, Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, NL-1018 WB Amsterdam, The Netherlands; e-mail: email@example.com, webpage: http://www.fee.uva.nl/cendef/.
Introduction Historical evidence indicates that stock prices ﬂuctuate heavily compared to indicators of fundamental value. For example, the price to earnings ratio of the S&P500 was around 5 at the beginning of the 20s, but more than 25 about nine years later to fall back to about 5 again by 1933. In 1995 the price/earnings ratio of the S&P500 was close to 20, went up to more than 40 at the beginning of 2000 and then quickly declined again to about 20 by the end of 2003. Why do prices ﬂuctuate so much compared to economic fundamentals?
This question has been heavily debated in ﬁnancial economics. At the beginning of the 80s, Shiller (1981) and LeRoy and Porter (1981) claimed that the stock market exhibits excess volatility, that is, stock price ﬂuctuations are signiﬁcantly larger than movements in underlying economic fundamentals. The debate evolved in two directions.
On the one hand, supporters of rational expectations and market eﬃciency proposed modiﬁcations and extensions of the standard theory. In contrast, another part of the literature focused on providing further empirical evidence against the eﬃciency of stock prices and behavioral models to explain these phenomena. The debate has recently been revived by the extraordinary surge of stock prices in the late 90s. The internet sector was the main driving force behind the unprecedented increase in market valuations. Ofek and Richardson (2002, 2003) estimated that in 1999 the average price-earnings ratio for internet stocks was more than 600.
A recent overview of rational explanations based on economic fundamentals for the increase in stock prices in the late 90s is e.g. given by Heaton and Lucas (1999). They oﬀer three reasons for the decrease of the equity premium, i.e. the diﬀerence between expected returns on the market portfolio of risky stocks and riskless bonds. A ﬁrst reason is the observed increase of households’ participation in the stock market. This implies spreading of equity risk among a larger population, which could explain a decrease of the risk premium required by investors. Secondly, there is evidence that investors hold more diversiﬁed portfolios compared to the past. In the 70s a large majority of investors concentrated their equity holdings on one or two stocks. More recently households have invested a large proportion of their wealth in mutual funds achieving a much better diversiﬁcation of risk. Both facts justify a decrease of the required risk premium by investors.
Although the wider participation seems unlikely to play an important role in the surge of stock prices in the 90s, the increased portfolio diversiﬁcation could at least partly account for the decrease in the equity premium and the unprecedented increase in market valuations. A third, fundamental explanation for the surge of the stock market that has been proposed is a shift in corporate practice from paying dividends to repurchasing shares as an alternative measure to distribute cash to shareholders. In this case dividends do not measure appropriately the proﬁtability of the asset and such a shift in corporate practice explains, at least partly, an increase in price-earnings or price-dividend ratios or equivalently a decrease of the risk premium. Further evidence on this issue is provided by Fama and French (2001).
Some recent papers attempt a quantitative evaluation of the decrease in the equity premium. Fama and French (2002) argue that, based on average dividend growth, the real risk premium has signiﬁcantly decreased from 4.17% in the period 1872-1950 to 2.5% after
1950. Jagannathan, McGrattan, and Scherbina (2000) go even further and, comparing the equity yield to a long-term bonds yield, reach the conclusion that the risk premium from 1970 onwards was approximately 0.7%. That is, investors require almost the same return to invest in stocks and in 20 years government bonds. The explanations above indicate structural, fundamental reasons for a long-horizon tendency of the risk premium to decrease, or equivalently for an increase of the valuation of the aggregate stock market.
However, to quantify the decrease in the equity premium is diﬃcult and the estimates provided earlier are questionable. Although fundamental reasons may partly explain an increase of stock prices, the dramatic movements e.g. in the nineties are hard to interpret as an adjustment of stock prices toward a new fundamental value.
Another strand of recent literature has provided empirical evidence on market ineﬃciencies and proposed a behavioral explanation. Hirshleifer (2001) and Barberis and Thaler (2003) contain extensive surveys of behavioral ﬁnance and empirical results both for the cross-section of returns and for the aggregate stock market. Much attention has been paid to the continuation of short-term returns and their reversal in the long-run. This was documented both for the cross-section of returns by de Bondt and Thaler (1985), and Jegadeesh and Titman (1993) and for the aggregate market by Cutler, Poterba, and Summers (1991). At short run horizons of 6-12 months, past winners outperform past losers, whereas at longer horizons of e.g. 3-5 years, past losers outperform past winners.
A behavioral explanation of this phenomenon is that at horizons from 3 months to a year, investors underreact to news about fundamentals of a company or the economy. They slowly adjust their valuations to incorporate the news and create positive serial correlation in returns. However, in the adjustment process they drive prices too far from what is warranted by the fundamental news. This shows up in returns as negative correlation at longer horizons. Several behavioral models have been developed to explain the empirical evidence. Barberis, Shleifer, and Vishny (1998), henceforth BSV, assume that agents are aﬀected by psychological biases in forming expectations about future cash ﬂows. BSV consider a model with a representative risk-neutral investor in which the true earnings process is a random walk, but investors believe that earnings are generated by one of two regimes, a mean-reverting regime and a trend regime. When confronted with positive fundamental news investors are too conservative in extrapolating the appropriate implication for the immediate asset valuation. However, they overreact to a stream of positive fundamental news because they interpret it as representative of a new regime of higher growth.
The model is able to replicate the empirical observation of continuation and reversal of stock returns. Another behavioral model that aims at explaining the same stylized facts is Daniel, Hirshleifer, and Subrahmanyam (1998), henceforth DHS. Their model stresses the importance of biases in the interpretation of private information. DHS assume that investors are overconﬁdent and overestimate the precision of the private signal they receive about the asset pay-oﬀ. The overconﬁdence increases if the private signal is conﬁrmed by public information, but decreases slowly if the private signal contrasts with public information. The model of BSV assumes that all information is public and that investors misinterpret fundamental news. In contrast, DHS emphasize overconﬁdence concerning private information compared to what is warranted by the public signal. These models aim to explain the continuation and reversal in the cross-section of returns. However, as suggested by Barberis and Thaler (2003), both models are also suitable to explain the aggregate market dynamics.
In this paper we consider an asset pricing model with behavioral heterogeneity and estimate the model using yearly S&P 500 data from 1871 to 2003. The model is a reformulation, in terms of price-to-cash ﬂow ratios, of the asset pricing model with heterogeneous beliefs introduced by Brock and Hommes (1997, 1998). Agents are boundedly rational and have heterogeneous beliefs about future pay-oﬀs of a risky asset. Beliefs about future cash ﬂows are homogeneous and correct, but agents disagree on the speed the asset price will mean-revert back towards its fundamental value. A key feature of the model is the endogenous, evolutionary selection of beliefs or expectation rules based upon their relative past performance, as proposed by Brock and Hommes (1997). The estimation of our model on yearly S&P 500 data suggests that behavioral heterogeneity is signiﬁcant
and that there are two diﬀerent regimes, a “mean reversion” regime and a “trend following” regime. To each regime, there corresponds a diﬀerent (class of) investor types:
fundamentalists and trend followers. These two investor types co-exist and their fractions show considerable ﬂuctuations over time. The mean-reversion regime corresponds to the situation when the market is dominated by fundamentalists, who recognize a mispricing of the asset and expect the stock price to move back towards its fundamental value. The other trend following regime represents a situation when the market is dominated by trend followers, expecting continuation of say good news in the (near) future and expect positive stock returns. Before the 90s, the trend regime is activated only occasionally and never persisted for more than two consecutive years. However, in the late 90s the fraction of investors believing in a trend increased close to one and persisted for a number of years.
The prediction of an explosive growth of the stock market by trend followers was conﬁrmed by annual returns of more than 20% for four consecutive years. These high realized yearly returns convinced many investors to also adopt the trend following belief thus reenforcing an unprecedented deviation of stock prices from their fundamental value.
The outline of the paper is as follows. Section I discusses some closely related literature.
Section II describes the asset pricing model with heterogeneous beliefs and endogenous switching, while Section III presents the estimation results. Section IV discusses empirical implications of our model, in particular the impulse response to a permanent positive shock to the fundamental and a simulation based prediction of how likely or unlikely high valuation ratios are in the future. Finally, Section V concludes.
I Related Literature
Our model is closely related to other work in behavioral ﬁnance, and it is useful to discuss some similarities and diﬀerences with other behavioral models in the recent literature. We also refer the reader to the extensive surveys on behavioral ﬁnance by Barberis and Thaler (2003) and Hirshleifer (2001) and the recent survey on dynamic heterogeneous agent models in economics and ﬁnance in Hommes (2005). Behavioral heterogeneity diﬀerentiates our model from Barberis, Shleifer, and Vishny (1998) and Daniel, Hirshleifer, and Subrahmanyam (1998) who both assume a representative agent. In contrast, we allow for the coexistence of diﬀerent types of investors with heterogeneous expectations about future payoﬀs and evolutionary switching between diﬀerent investment strategies. Disagreement in asset pricing models can arise because of two assumptions: diﬀerential information and diﬀerential interpretation. In the ﬁrst case, there is an information asymmetry between one group of agents that observes a private signal and the rest of the population that has to learn the fundamental value from public information such as prices. Asymmetric information causes heterogeneous expectations among agents. Recent models of this type are Grundy and Kim (2002) and Biais and Bossaerts (2003). The second assumption is based on the fact that a public signal can be interpreted in diﬀerent ways by investors.