Professor Markose: Notes on Bubbles and Crashes
1. Introduction Historical evidence indicates large fluctuations of stock prices 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? Some slides from Markus K. Brunnermeier Princeton University
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Internet bubble?
NASDAQ Combined Composite Index
-
1990’s
NEMAX All Share Index (German Neuer Markt)
Chart (Jan. 98 - Dec. 00) 38 day average
Chart (Jan. 98 - Dec. 00) in Euro 38 day average
Loss of ca. 60 % from high of $ 5,132
Loss of ca. 85 % from high of Euro 8,583
Why do bubbles persist? Do professional traders ride the bubble or attack the bubble (go short)? What happened in March 2000?
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Do (rational) professional ride the bubble? South Sea Bubble (1710 - 1720)
Isaac Newton
04/20/1720 sold shares at £7,000 profiting £3,500 re-entered the market later - ended up losing £20,000 “I can calculate the motions of the heavenly bodies, but not the madness of people”
Internet Bubble (1992 - 2000)
Druckenmiller of Soros‟ Quantum Fund didn‟t think that the party would end so quickly.
“We thought it was the eighth inning, and it was the ninth.”
Julian Robertson of Tiger Fund refused to invest in internet stocks
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Pros’ dilemma
“The moral of this story is that irrational market can kill you … Julian said „This is irrational and I won‟t play‟ and they carried him out feet first. Druckenmiller said „This is irrational and I will play‟ and they carried him out feet first.” Quote of a financial analyst, New York Times
April, 29 2000
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Timing Game - Synchronization
(When) will behavioral traders be overwhelmed by rational arbitrageurs? Collective selling pressure of arbitrageurs more than suffices to burst the bubble. Rational arbitrageurs understand that an eventual collapse is inevitable. But when? Delicate, difficult, dangerous TIMING GAME !
This question has been strongly debated in financial economics. At the beginning of the 80s, Shiller and LeRoy claimed that the stock market exhibits excess volatility, that is, stock price fluctuations are significantly larger than movements in underlying economic fundamentals. Historical stock price data incorporate positive and negative bubbles when the trend growth in prices deviate from the fundamental values for extensive periods of time. The 1980s bull market was followed by a precipitous and sudden collapse in a day or two in 1987, while in the late 1990’s, the so called Internet bubble gave way to a more gradual collapse over 2 years starting in 2000. This provides evidence for ‘periodically collapsing bubbles’, a term coined by Blanchard and Watson (1982). The papers of Fama and French ( 2002) and others, have contributed to the growing consensus on the empirical validity of mean reversion in stock returns with negative correlation in long horizon returns and trend growth with positive correlation in short term returns. The length of time involved and the size of some of the recent bubbles before reverting to fundamental values has stimulated extensive research (see, a recent discussion in Brooks and Katsaris, 2005).
Figure 1 Stylized depiction of Mean Reversion and Trend Regimes:
Mean Reversion Trend Price
Fundamental value V*
If EPt+1 > V * overvalued If E Pt+1 > V*
: Sell if you are a fundamentalist – the stock is : Buy if you’re a trend follower.
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. It has been estimated that in 1999 the average price-earnings ratio for internet stocks was more than 600.
Another strand of recent literature has provided empirical evidence on market inefficiencies and proposed a it behavioral explanation. (Hirshleifer and Barberis, Thaler) 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 Bond and Thaler ), and and for the aggregate market . 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 Vishny), 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 one that 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 Thaler, both models are also suitable to explain the aggregate market dynamics. The Non-computability models and endogenous heterogeneity models:
It appears that the jury is still out as to what causes fat tailed returns distributions in a generative sense of trader behaviour. In a recent survey, Aoki and Yoshikawa (2006) have shown how the class of statistical models for asset returns which have a random multiplicative structure can produce the fat tailed distributions. They also conjectured how this can be generated in heterogeneous interacting ACE models. Since the pioneering artificial stock market models of Arthur et. al ( 1997) it is well understood that a market of only technical traders whose strategies follow trends in the historical prices will create one way markets where prices simply go up or down and that fundamental traders are needed to contribute the necessary contrarian strategies to produce the stylized dynamics for stock prices. Though there is survey evidence that such classes of investors coexist in the market, the problem in so called heterogeneous agent based models (HAMs) has been how to endogenize the fractions of each of these groups of investors. In the original Santa Fe ASM the persistence of periodically collapsing bubble phenomena was obtained by the modeller only by exogenously increasing the frequency of updating of forecast models in the form of ‘retraining’ of the adaptive trading agents. In contrast, Markose et. al. (2004) utilize the random multiplicative statistical model and include a Red Queen style endogenous retraining of adaptive investors which requires investors to ‘retrain’ when their wealth falls below the average (aggregate) wealth to produce a more appropriate power law distribution in investor wealth than what is implied by the SFI model. The Red Queen principle characterizes competitive co-evolution in that it is only when traders seek to beat the market (return) by evolving strategies that can seek out profitable trades there is a point at which none can do this systematically. However, some traders over time accumulate wealth with the well known Pareto law tail index for skewed investor wealth distribution close to 2.1.
CARS HOMMES, BOSWIJK MODEL The Boswijk et. al. assume that agents have common knowledge of the fundamental value for the stock, however, as in the HAMs framework the agents differ in that some of them believe that the price will revert to the true value in a longer period and hence the follow the trend and others, the fundamentalists, believe that the correction is in the short term. Hence, the two groups differ only in terms of one parameter, , that represent their beliefs about the persistence of the bubble. With 1 <1, it characterizes those of Type 1 who believe that reversion to the fundamentals is imminent while as 2 >1 will exacerbate the deviations from the bubble making , Type 2 investors are chartists.
In order to model how a fixed number of traders switch between these strategies, Boswijk et. al. follow Brock and Hommes (1998) in the use of the Anderson et. al. (1993) discrete choice model with multinomial logit probabilities which enable agents to implement a boundedly rational choice over finite (2 in their stock market model) alternatives in a dynamic way. The proportions using each strategy /belief type alters according to the most recent profits arising from the investment strategy. This paper is one of the first to attempt an econometric estimation of the non-linear model of the stock returns based on the two belief types as well as the dynamic switching between the two types of investors. They use annual US S&P 500 index from 1871 – 2003 for the estimation and find that the null hypothesis of linearity versus the non-linear stock returns model being proposed can be rejected.
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 reinforcing an unprecedented deviation of stock prices from their fundamental value. The Model: Rt+1 = Pt+1 + Yt+1 - (1 + r )Pt Y is dividend or cash flow. We assume heterogenous beliefs about future payoffs , say 2 types, h=1,2
(1)
Agents have myopic mean variance demand:
Zht =
E h Rt 1 a 2
(2)
a ; Risk Aversion parameter ; identical for all agents : Variance of returns : identical for all agents Denote fraction of investors using predictor h as nht . Under assumption of zero net supply of risky asset, the market clearing equation is:
nht
h
ERht 0 a 2
(3)
Substitute (1) and solve for equilibrium price Pt
Pt
1 nht Eht ( Pt 1 Yt 1 ) 1 r h
(4)
● It is clear the price is high today if the optimists outnumber the pessimists
etc. All agents have correct beliefs on cash flow at t+1 Eht Yt+1 = (1+g) Yt g: constant growth rate (5)
Reformulate pricing equation (4) in terms of the price to dividend ratio t = Pt /Yt .
t =
1 1 R*
n
h
n ,t
Eh,t t 1
R*
1 r 1 g
(6)
Heterogeneous belief: Eh,tt+1 = m + fh (xt-1, ……. Xt-L )
m: is rational expectation P/ Y Deviations from the fundamental P/Y
(7)
Deviation from fundamental P/Y ratio xt = t - * With only two belief types the dynamic asset pricing model can be written as: R*xt = nt Φ1 xt-1 + (1- nt) Φ2 xt-1 + t (8) The value of Φh can be interpreted as follows: Φh > 1 : Deviations from trend will persist hence those with this are trend followers. Φh < 1: Deviations from trend are damped and hence those with this are those who believe that price will revert to mean.
Realized Profits for type h trader:
h,t-1 = Rt-1 zht-2 How to agents switch beliefs: (8)
The fractions nh,t evolve according to a discrete choice model with a multinomial logit probability:
n h ,t
exp (
k h
exp ( h,t 1 )
k ,t 1
)
> 0 : is the intensity of choice parameter. The proportions in each belief type is simply a ratio of type h’s profit relative to profits attained by other belief types.
Estimation Results of belief coefficients: xt is deviations from fundamental P/Y ratio.
R*xt = nt Φ1 xt-1 + Φ1 = .76 Φ2
(1- nt) Φ2 xt-1 + t
= 1.35 when nt = 0 , the P/Y ratios are explosive
They obtain the estimated dynamics of nt. Conclusions:
The Boswijk et.al. paper has had more success in relating bubble formation and collapses directly to investor strategies and beliefs when compared to other recent non-linear econometric models, see, Brooks and Katsaris (2005). However, the latter paper has been more effective in tracking actual occurrence of booms and busts than have Boswiijk et.al. by incorporating the abnormal volume of trades as an important ingredient in the anatomy of a boom and bust. There is considerable evidence that switching to chartism from within a fixed number of stock market investors alone cannot explain the large deviations from fundamentals and the increase in trading volume observed in the final stages of a stock market boom and in the subsequent bust. As the bubble grows, new entrants (who have never before owned stock) and new capital via disintermediation from other investment classes have to be included in the stock market model. Presumably, the estimated coefficients 1 and 2 which are assumed to be constant in the current version of the
Boswijk et. al. have to be also to be made time varying with the trade volumes to proxy for the disintermediation effects.