هراس منطقی و سقوط بازار سهام

This paper offers an explanation for stock market crashes which focuses on the role of rational but uninformed traders. We show that uninformed traders can precipitate a price crash because as prices decline, they surmise that informed traders received negative information, which leads them to reduce their demand for assets and drive the price of stocks even lower. The model yields several implications, such as that crashes can occur even when the fundamentals are strong, and that the magnitude of the crash depends on the fraction of uninformed investors and the amount of unsophisticated passive investing present in the market.

Investors are often preoccupied with the possibility that the stocks they own will arbitrarily and unexpectedly plummet in value. These fears may not be entirely unwarranted given recent historical episodes such as the October 1987 crash, when the Dow Jones index declined by 23% in a single day without any obvious corresponding changes in the underlying economic fundamentals. Not surprisingly, then, economists have devoted considerable effort to understand what forces might cause stock prices to change so dramatically in the face of seemingly small changes in fundamentals, and whether such forces are still present in markets today. At least with regard to the 1987 crash, the consensus among both policymakers and academics has converged on the culpability of then-popular hedging strategies advanced by portfolio insurance models, which dictated selling stocks when their prices began to decline. This was the theme of the Brady Report [4] commissioned to investigate the causes of the crash, and was later formalized in work by Gennotte and Leland [8]. The latter develop an equilibrium model in which crashes are impossible in the absence of portfolio insurance. They then demonstrate that a relatively small amount of portfolio insurance gives rise to a discontinuous equilibrium price function in which the equilibrium price of the risky asset jumps discretely as the underlying fundamentals change continuously.1 Since the direct use of stocks for portfolio insurance purposes has all but disappeared, some have concluded from this analysis that markets today are less vulnerable to price crashes of similar magnitudes as in the 1987 crash. The purpose of this paper is to take issue with this conclusion by demonstrating that the phenomenon described by Gennotte and Leland does not require the presence of exogenous portfolio insurance. We consider a market that is populated both by agents who are informed about the underlying fundamentals as well as agents who are uninformed. This asymmetric information plays a crucial role in our setup, in contrast with Gennotte and Leland's model where such asymmetry plays no essential role given the presence of exogenous portfolio insurance. In particular, uninformed agents in our model can end up acting in the same way as portfolio insurers are assumed to act in Gennotte and Leland's model: they are willing to buy stocks at relatively high prices but avoid them at relatively low prices. The reason for this is that uninformed agents are aware that there are informed agents operating in the market, which gives them incentive to use prices to make inference about the underlying fundamentals. In what follows, we assume there are many informed agents active in the market, so no informed agent has incentive to hide his information. This avoids situations as in [14] or [15] where informed traders take into account the information their actions reveal and accordingly act to conceal it. Since informed traders fail to act strategically and bid up the price of stocks when they learn fundamentals are favorable, uninformed traders correctly infer that high stock prices are more likely when fundamentals are favorable. If this effect is strong enough, uninformed agents could end up avoiding stocks altogether when stock prices are too low but not when prices are high. Thus, the demand of uninformed agents will be upward-sloping. However, the demand of uninformed traders will eventually assume a downward sloping shape at high prices, both because these traders cannot afford to buy as many assets as prices continue to rise as well as because high prices eventually lead to a low rate of return. The demand of uninformed traders will therefore be upward-sloping in our framework only locally. When we add the backwards-bending demand curve of uninformed traders to the demand of informed agents, we end up with an aggregate demand curve that folds over itself, forming an inverted-S shape: at low prices, only informed and noise traders purchase assets, and their demand for the asset is downward sloping; at higher prices, the aggregate demand curve is upward sloping because of the upward-sloping demand of uninformed traders; and at still higher prices, when the demand of uninformed traders turns downward sloping again, aggregate demand will be downward sloping as well. This folding implies that as we trace along values of the fundamentals, we will inevitably be forced to jump from the upper branch of the demand curve, where the demand of uninformed agents is high, to the lower branch, where the demand of the uninformed agents is low. Hence, the equilibrium price must change discontinuously at some point. This discontinuity loosely captures the notion of rational panic among uninformed traders: a small decline in price leads uninformed agents to no longer view stocks favorably and to withdraw from the market, causing the price of the stock to fall even further. The folding property is identical to the one that Gennotte and Leland describe, except that rather than assuming exogenous hedging demand to generate this folding, our result arises endogenously under asymmetric information. While it is well-known that demand for a good can be upward sloping in asymmetric information models—Wilson [22] and [23], for example, derives an upward-sloping demand in a market for goods whose quality is known only to sellers—previous work has failed to point out its role in generating stock market crashes. For one thing, simply constructing a tractable model with upward-sloping demand in a market for financial assets proves to be a difficult task, in contrast with markets for goods where heterogeneous taste for quality across consumers greatly facilitates the construction of upward-sloping demand schedules. In fact, Grossman [9] is able to definitively rule out upward-sloping demand for financial assets under weak conditions when asset returns are normally distributed and all traders share a common utility function. Grossman's result does not extend to models that allow for noise trading, but the strong assumptions that are often invoked to yield analytically tractable models that allow for noise do not naturally yield upward-sloping demand curves either. Still, several papers have been able to generate upward-sloping demand schedules for financial assets. For example, Admati [1] uses multiple assets in a static model to generate upward-sloping demand for individual assets. Wang [20] shows numerically that in a dynamic model with noise, changes in the quantity demanded by uninformed traders can be positively correlated with price changes even if there is only one risky asset, although he finds this case to be unlikely. But while both of these models give rise to upward-sloping demand, they only consider linear demand schedules, so that demand for an asset never folds back as it does in our model, and the price function in both models remains continuous. By contrast, in this paper we are able to generate discontinuous price functions. This is because we adopt the alternative framework we developed in [3], which allows for non-linear equilibrium price functions and demand schedules. This non-linearity allows us to generate locally upward-sloping demand as opposed to globally upward-sloping demand, and thus to generate price crashes as an equilibrium phenomenon. Although the model we use is essentially the same as the one studied in [3], that paper focuses on parameter values for which upward-sloping demand curves do not occur, and thus abstracts from most of the issues we consider here. We should note that ours is not the first paper to model prices crashes in the absence of portfolio insurance. However, we believe existing models of price crashes, which focus on very different mechanisms than the one we consider, are not as convincing in explaining price crashes in equity markets. For example, Bulow and Klemperer [5] generate price crashes in a sequential auction of a finite number of commodities in which goods are sold one at a time. A crash occurs when the valuation between two agents is sufficiently far apart so that the price of the next commodity being auctioned has to fall before another agent is willing to buy it. This sequential auctioning of goods one at a time is essential for generating a crash, but it is not the way in which assets are typically traded. Madrigal and Scheinkman [17] generate a price crash that is due to strategic manipulation by a fully informed market maker who finds it optimal to coarsen the information set for potential buyers in bad states of the world, causing a discrete jump down in prices when bad outcomes occur. This would imply a stock crash is intentionally caused by a market maker, a scenario about which Madrigal and Scheinkman are themselves skeptical. A third explanation argues that small events can reveal substantial information to partially informed agents, causing traders to significantly reallocate their portfolios in response to small changes in the underlying environment, which in turn leads to large changes in stock prices. This explanation is pursued by Kraus and Smith [13], Romer [18], Caplin and Leahy [6], Lee [16], Zeira [25], and Hong and Stein [11] among others. An unattractive feature of these models is that they explain crashes as episodes in which agents learn about some underlying fundamental they were previously uncertain about and react to this information. But in practice, episodes where stock prices fall substantially appear to involve confusion and uncertainty rather than transparency and clarity. The advantage of our explanation is that it predicts crashes are driven by uninformed investors who are uncertain about the fundamentals of the asset, not by uncertain agents who suddenly become more informed about them. The paper is organized as follows. Section 1 describes the basic layout of the model. Section 2 establishes the existence of an equilibrium in which the price is discontinuous in the underlying fundamentals, so a small change in fundamentals can be associated with an abrupt jump in the price. Section 3 discusses some implications and potential generalizations of our model. Section 4 concludes.

This paper constructs a model that formalizes the notion that crashes—abrupt changes in the price of a stock that occur despite the absence of any corresponding change in the economic fundamentals of the underlying asset—could arise because of the behavior of uninformed traders who panic and cause the price of stocks to plummet. The mechanism we describe is identical to the one developed by Gennotte and Leland [8] in which exogenous portfolio insurers precipitate price declines by selling stocks when prices fall, thus magnifying small price changes into large, discontinuous jumps. In our framework, it is uninformed traders who sell stocks at low prices, but this is rational for them since they try to read information they know is available to other traders who are active in the market. Thus, there is no need to appeal to exogenous trading motives or systematic misperception among traders about the motives of others in order to sustain this mechanism for crashes. While we view our model as a theoretical contribution of how this explanation can be formally modelled, we believe our story has some empirical plausibility for actual stock trading, and so we conclude with short remarks why our explanation can accord with actual crashes. Previous authors have ascribed as an important role to uninformed traders in precipitating price crashes. For example, White [21] writes During the tulipmania, Garber notes that the middle classes and even monied workers began to speculate in the market for tulips, which previously had been the province of specialists. In their enthusiasm to participate, new investors seem incredibly incautious... In 1873, new German investors played a prominent role. They busily acquired new, untested foreign securities for their portfolios. In the 1920s the shift in business financing from short-term commercial bank loans to bonds and stocks meant that instead of commercial banks who had considerable experience in evaluating firms, the general investing public became the chief creditors of corporations. Americans who had never owned stocks before were now buying. Given the increased difficulty of evaluating fundamentals and the general optimism from the decade of prosperity, it is not surprising that prices were pushed above fundamentals. Similarly in the 1980s, new financial instruments drew in new investors from both at home and abroad. (p. 238) The role of nervous investors was even noted in the original Brady report [4] as one of the causes of the 1987 crash. As quoted by Wyatt [24], the report notes that “to the market, [mutual fund investors’] behavior looked much like that of the portfolio insurers, that is, selling without primary regard to price.” Our paper is the first to our knowledge to provide a formal model that rationalizes these observations. Given that stock market participation has increased over the past decade, particularly among small investors who are more likely to be uninformed about economic fundamentals in real time, our analysis suggests that stock markets today may be vulnerable to crashes, despite the fact that the use of stocks for portfolio insurance has largely disappeared. Moreover, the recent shift from passive investing strategies to more aggressive trading practices such as day-trading, which are more reactive to prices, might very well serve to magnify the magnitude of price crashes if they were to occur, since passive traders tend to mitigate price crashes by blindly buying up the stocks from those who panic and sell them.