Mathematics and Economics

Bubbles and Crashes: A Survey of the Efficient Market Hypothesis, Part I

In Asset Pricing Reading Group, Bubbles and Crashes, Economics, Finance on November 29, 2009 at 2:58 am


“A study of economics usually reveals that the best time to buy anything is last year.”

-Marty Allen

1. Introduction

A thorough study of asset price bubbles and market crashes necessitates the study of the efficient market hypothesis, one of the most controversial theories in modern finance and economics. In the wake of the 2008-2009 financial crisis, the efficient market hypothesis (EMH) has faced both opprobrium and found defense. In various editorials and journal articles, economists have criticized the financial community for what they argue was unreasonable and nearsighted adherence to the pronouncements of the hypothesis (Thaler, 2008). Others, however, hold that government interference, specifically its attempts to manage inefficiencies, have created the conditions necessary to promote price bubbles (Thompson, 2006).

In an ideal capital market, prices incorporate all available information necessary for the proper allocation of resources. However, the existence of asset bubbles and crashes may suggest that the allocation of resources is often improper. Indeed much of the discussion about the current financial crisis implies that efficient markets preclude asset price bubbles. However, definitions of the EMH vary and are implicitly based on general equilibrium principles that may be consistent with asset price bubbles. Therefore one must determine the extent to which the forms of the efficient market hypothesis preclude various assertions about bubbles. Of related importance are the relationships between financial efficiency, loosely defined as the absence of arbitrage opportunities, and other standards of efficiency (e.g. informational and welfare efficiency). The latter are the frameworks of normative economics, and our discussion of how resources should be allocated.

The literature of efficient markets is replete with theory and empirical study. Empirical work constituted much of the early development of the field. As early as 1863, Jules Regnault was aware that stock price deviations were proportional to the square root of time. Eventually a theoretic treatment developed and the EMH was defined rigorously.

In this survey, we review Eugene Fama’s seminal 1970 paper, its discussions of influential papers, and recent developments in efficient market literature related to the study of asset price bubbles and crashes. We begin by reviewing the various definitions of the efficient market hypothesis and by briefly discussing the joint hypothesis problem, a challenge in the empirical study. Then, we discuss various tests of these and more complex models. Finally, we discuss the empirical results of these tests, paying particular attention to papers relevant to the study of asset price bubbles and crashes.

2. Definitions and Models

The efficient market hypothesis has often been defined in literary terms. The first discussion of efficient markets arose in the Frenchman Louis Bachelier’s work in 1900, in which he stated “the mathematical expectation of the speculator is zero.” John Maynard Keynes elaborated on this concept, a consequence of the EMH, in 1923, arguing that investors in financial markets are compensated not because of superior knowledge, but because of a willingness to bare greater risk than other investors. Later, Alfred Cowles conducted empirical research demonstrating that, on average, financial professionals do not perform better than the market. Further econometric evidence seemed to support the notion that assets in the American stock market were properly priced. Through the implementation of time series analysis, Maurice Kendall concluded that the change in stock prices could be modeled by a symmetric random walk. Preceeding the work of Samuelson and Fama, Paul Cootner provided an explanation for this behavior by arguing that investors’ beliefs about the state of asset prices would affect those same prices to the extent that the expectation of “tomorrow’s price, given today’s price, is today’s price” (Cootner 1964).

This piecemeal construction of a financial theme culminated in the work of Eugene Fama and Paul Samuelson in 1965 and 1970. Samuelson, by applying probability theory and other tools of modern economics, proved that properly anticipated prices fluctuate randomly. This was a critical element in the development of the modern theory. At around the same time, Fama defined the EMH in his seminal 1970 survey by stating: “A market in which prices ‘fully reflect’ available information is called ‘efficient'” (Fama, 1970). In other words, Fama defined an efficient market as one whose agents operate under rational expectation and incorporate all available information to price assets properly. He also mentions three types of efficient markets based on the information sets incorporated into asset prices.

Definition 1 Suppose {\Phi} is an information set. Then a market is called

  1. Weak form efficient if {\Phi} is equal to the set of prices of an asset,
  2. Semi-strong form efficient if {\Phi} is equal to the set of all publicly available information, and
  3. Strong form efficient if {\Phi} includes all public and private information.

This statement became the starting point for the efficient market literature, yet, over time, many other definitions were posited. One of the most important variations of the efficient market hypothesis was stated by Michael Jensen in his 1978 paper, “Some Anomalous Evidence Regarding Market Efficiency.” Jensen defines an efficient market as one in which “Prices reflect information to the point where the marginal benefits of acting on information do not exceed the marginal costs” (Jensen, 1978). We will primarily use this version of the EMH because Jensen chooses to include the manner in which information is incorporated into prices as well as the extent to which that incorporation occurs. In general, the field of behavioral economics strives to understand this issue as well and its conclusions will also be critical in our study.

Before continuing, we note a structural problem in the study of efficiency. Market efficiency per se is, as suggested earlier, a vague notion that requires greater precision to permit empirical study. Any attempt to examine its nature requires that we prescribe some type of additional structure. It follows that any empirical test of market efficiency also tests this structure; this is known as the joint hypothesis problem. This also implies that the efficient market hypothesis itself is not a well defined hypothesis. One can test only the specific models chosen, not the general description of efficiency. If we are to determine the degree of efficiency in real markets, it becomes necessary to define what it means for a real market to “properly” reflect all “relevant” information. We must specify an equilibrium model of efficiency, investor behavior and information structure.

Suppose {H_{0}} is the statement “market prices properly reflect all relevant information,” the literary definition of efficiency. Let {H_{1}} be the statement “the words ‘properly’ and ‘relevant’ are described by model {X}.” The joint hypothesis problem states that it is difficult, if not impossible, to test {H_{0}} without testing {H_{0} \vee H_{1}} for some hypothesis {H_{1}} dependent on model {X}. Clearly, results in which the efficient market hypothesis is implicit are dependent on some model, {X}; any discussion of asset bubbles will therefore be dependent on the choice of model {X}.

Expected Return Models

Early studies were based on the assumption that market equilibrium could be described through the use of expected return models. These models suppose that the equilibrium expected return, conditional upon the information set, is some function of the risk inherent in a specific asset or transaction. For a general expected return model, one states

\displaystyle  E(\tilde{p}_{j,t+1}\mid \Phi_{t}) = \left[1+E(\tilde{r}_{j,t+1}\mid\Phi_{t})\right]p_{j,t}

where {E} is the expected value operator, {p_{j,t}} is the price of asset {j} at time {t}, {\Phi_{t}} is the information set {\Phi} at time {t}, and {r_{j,t+1}} is the return percentage at time {t+1}, i.e. {r_{j, t+1} = \frac{p_{j,t+1}-p_{j,t}}{p_{j,t}}}, or, in other words, {p_{j,t}\left(1+r_{j, t+1}\right) = p_{j,t+1}}. Finally, the tildes above {p, r} indicate that these elements are random variables. It is important to note that the expected value is simply one of many descriptions of the distribution of returns, hence any empirical studies that implement this type of expected return model assume that expected value is the most valid description. The notion that empirical study assumes a surprisingly deep validity of a specific model is a theme we will see repeated.

Definition 2 Under the assumption that market equilibrium can be described by expected returns and that expected returns are based on the information set {\Phi} at time {t}, trading systems that operate based on {\Phi_{t}} must have expected returns equal to equilibrium expected returns. It follows that if {\left\{x_{j,t}\right\}} is the sequence of excess returns, that is,

\displaystyle  x_{j,t+1} = p_{j,t+1} - E(\tilde{p}_{j,t+1}\mid \Phi_{t}),

we must have

\displaystyle  E(\tilde{x}_{j,t+1} \mid \Phi_{t}) = 0.

In this case, we say the sequence {\left\{x_{j,t}\right\}} is fair game with respect to the sequence of information sets {\Phi_{t}}. We can equivalently replace the prices {p} above with percentage returns, {r}.

We can formalize even further. Let {\alpha(\Phi_{t}) = \left[ \alpha_{1}(\Phi_{t}), \dots \alpha_{n}(\Phi_{t})\right]} be a sequence of suggested transactions by trading system {\alpha} where {\alpha_{j}(\Phi_{t})} denotes the amount and type of transaction of asset {j}. In addition, let {V_{t+1}} be the excess value produced by the trading system at time {t+1}. We find that

\displaystyle  V_{t+1} = \sum_{j=1}^{n}\alpha_{j}(\Phi_{t})\left[r_{j,t+1} - E(\tilde{r}_{j, t+1}\mid\Phi_{t})\right].

An application of the fair game hypothesis demonstrates that the expectation of this value is

\displaystyle  E(\tilde{V}_{t+1} \mid \Phi_{t}) = \sum_{j=1}^{n}\alpha_{j}(\Phi_{t})E(\tilde{z}_{j,t+1}\mid\Phi_{t}) = 0.

The Submartingale Model

Definition 3 Suppose that

\displaystyle  E(\tilde{p}_{j,t+1}\mid \Phi_{t}) \geq p_{j,t},

or equivalently for {r}. Then the sequence of prices {p_{j,1}, p_{j,2}, \dots, p_{j,t}, \dots} follows a submartingale with respect to the information sequence {\left\{\Phi_{t}\right\}}. If above relationship is equality, we say the sequence of prices follows a martingale.

The submartingale model leads to an important empirical result. Consider a market with one asset, {j=1}. If the submartingale model holds, then there exists no trading strategy better than that of buying asset {j=1} and holding it. Although a market with only one asset is unreasonably simplistic, the model provides us with an important starting point. Empirical study will often use this test to conclude efficiency of a real market if no asset in the market obeys the property above unless in the case of equality. Asset price bubbles may function conversely.

The Random Walk Model

The random walk model comprised most of the early literature of the efficient market hypothesis. Empirical research suggested that successive one-period returns were random, and independent of the information set at time {t}; this model was more general than the fair game model proposed above.

Definition 4 Suppose {f} is a probability density function on a probability space {(X, \Omega, \mu)}. The sequence of price changes, or returns, follows a random walk (with drift) if

\displaystyle  f(r_{j,t+1}\mid\Phi_{t}) = f(r_{j,t+1}),

that is, if successive price changes are independent of the information set {\Phi_{t}} at time {t}, and are identically distributed and if {\Phi_{t}} contains the history of all previous returns. Note that we say “with drift” because prices may not follow a random walk; successive changes are independent of each other and are independently distributed but the distribution of price changes may depend on the price level.

Generally speaking, empirical evidence suggesting such a model is somewhat weak. Instead, we use a more general model,

\displaystyle  E(\tilde{r}_{j,t+1}\mid\Phi_{t}) = E(\tilde{r}_{j,t+1}),

which says that the expectation of returns is independent of the information set {\Phi_{t}}.

While the fair game hypothesis proposed earlier assumes that market equilibrium can be described by expected returns of an asset without describing the stochastic process through which this occurs. The random walk model, on the other hand, makes a much stronger assumption: it proposes a distribution function and assumes that the entire distribution of returns is independent.

With these models in hand, we have developed the basic framework necessary to discuss empirical studies of the efficient market hypothesis. While one may easily determine sufficient conditions for the hypothesis, necessary conditions are usually more difficult to determine and are often debated. If all the information in the set {\Phi} is freely available, the market is free of transaction costs, and all participants come to the same conclusions with the information in {\Phi}, it seems evident that fair game market efficiency will hold. Fama’s results, as well as those of many others, seem to indicate that even in the presence of transaction costs, moderately costly information, and a variety of opinions based on {\Phi}, as long as there exist sufficient numbers of investors with access to information, markets may be efficient. In particular, efficiency is possible as long as there are no individuals that consistently “make better evaluations of available information than are implicit in prices.” (Fama, 1970)

Investor inconsistency, transaction costs, and unavailable information may all be sources of market inefficiency; studying their impact, as well as the influence of other conditions, on the development of prices is the primary goal in the empirical literature. Reassessment of the original models has, of course, accompanied the progression of the field. In some cases, the definition of an efficient market was modified to incorporate a potential source of inefficiency. Other results demonstrate that efficiency is impossible under certain conditions. Hence, it will be necessary to determine the conditions under which asset price bubbles may form, and deduce which, if any, are inconsistent with an appropriate model of market efficiency.

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