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Information diffusion and the boundary of market efficiency

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Information diffusion and the boundary of market efficiency

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Content INFORMATION DIFFUSION AND THE BOUNDARY OF MARKET EFFICIENCY by Hai Lu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) August 2004 Copyright 2004 Hai Lu Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UM I Number: 3145239 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3145239 Copyright 2004 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I owe an irreducible debt to my dissertation chair, Randy Beatty, for his close guidance in every stage of the development of the thesis over the last two years; for many hours of his time which he made available for discussions of the thesis and other working papers; for allowing me to work as his research assistant though which I have learned how to think as a financial economist. He solidly supported me and paid as much attention and care as if this project had been his own. I am very grateful to K.R. Subramanyam for his many insightful comments and encouragements. His expertise has helped me shape this thesis into the current version. I would like to thank Raffi Indjejikian for his insightful comments and steering my theoretical work. I would especially thank Chuck Swenson for his continuous support and advice while I was studying at USC. I would also thank my econometrics professor, Cheng Hsiao, for helpful suggestions. Thanks go to all the above professors for serving on my thesis committee. I would like to thank my friend, Qingzhong Ma, a Ph.D. candidate in finance, who showed continuing patience to listen to and comment on my work. I have benefited from many discussions with him. I thank the Marshall School of Business, the Leventhal School of Accounting, at the University of Southern California, and the SEC Financial Reporting Institute for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the doctoral fellowships which allowed me to have maximum time to pursue my research. Finally, this dissertation would never have been possible without the spiritual support from my family members. Their continuing support makes it possible for me to stay in the United States to finish my degree. I dedicate this thesis to them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGEMENTS....................................................................................... ii LIST OF TABLES.........................................................................................................vi LIST OF FIGURES....................................................................................................... vii ABSTRACT.................................................................................................................. viii CHAPTER 1 INTRODUCTION................................................................................. 1 CHAPTER 2 A MODEL OF INFORMATION DIFFUSION AND THE PRICE DISCOVERY PROCESS.......................................................... 13 2.1 Introduction................................................................................................ 13 2.2 An Intuitive Example of Information Diffusion....................................... 15 2.3 The Information Diffusion M odel........................................................... 17 2.3.1 A Diffusion Equation Derived in a Discrete Time Setting 18 2.3.2 A Diffusion Equation Derived in a Continuous Time Setting. 19 2.3.3 Interpretation of the Diffusion Equation................................... 21 2.3.4 Comparative Statics.................................................................. 25 2.4 Price Mechanism.........................................................................................27 2.5 Information Diffusion and Earnings Announcements............................. 31 2.5.1 Information Content.................................................................... 32 2.5.2 Information Diffusion Environment.......................................... 34 2.5.3 Information Processing Impedance............................................36 2.5.4 Information Shock Absorption Capacity............................... ,..38 2.6 Conclusions..................................................................................................39 CHAPTER 3 INFORMATION DIFFUSION AND THE BOUNDARY OF MARKET EFFICIENCY........................................................................ 41 3.1 Introduction.............................................................................................41 3.2 Hypothesis Development.......................................................................45 3.2.1 Three Hypotheses.....................................................................46 3.2.2 Proxies for the Constructs........................................................51 3.3 Sample Selection and Variable Measurement......................................53 3.3.1 Sample Selection...................................................................... 53 3.3.2 Variable Measurement.............................................................55 3.4 Empirical Results....................................................................................61 3.4.1 Descriptive Statistics................................................................62 3.4.2 Portfolio Tests of Hypothesis 1 ...............................................63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V 3.4.3 Earnings Surprise, AV0L3D, and AVAR3D....................... 69 3.4.4 Regression Analysis Tests of Hypotheses 2 ........................ 73 3.4.5 Portfolio Tests of Hypothesis 3 ............................................ 80 3.5 Sensitivity T ests.................................................................................... 82 3.5.1 Quarterly vs. Annual Earnings Announcements...................82 3.5.2 NYSE&AMEX vs. NASDAQ Firm s.................................. 84 3.5.3 News Classification with Earnings Surprise...................... 84 3.5.4 Persistence of Information Diffusion.................................. 85 3.5.5 Risk vs. Diffusion................................................................. 86 3.5.6 Momentum Effect.................................................................. 88 3.6 Conclusions........................................................................................... 88 CHAPTER 4 CONCLUDING REMARKS.................................................................91 BIBLIOGRAPHY..........................................................................................................93 APPENDIX 1 ENTROPY AS THE MEASURE OF INFORMATION..................102 APPENDIX 2 SOLUTION TO THE PDE WITH SINGLE SIGNAL SETTING............................................................................. 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table 3.1 Sample Selection Criteria......................................................................54 Table 3.2 Six Benchmark Models for Calculating Expected Returns 58 Table 3.3 Descriptive Statistics............................................................................. 59 Table 3.4 The Difference in CARs between Good and Bad News Groups in Each of the Abnormal Trading Volume Quartiles........................69 Table 3.5 Two-Way Sort: Earnings Surprise and Abnormal Trading Volume.................................................................................... 72 Table 3.6 Regression Tests of Hypotheses 1 and 2 ..............................................78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure 1.1 A Schematic Illustration of Information Diffusion...........................5 Figure 2.1 Entropy Change Following Two Different Information Events 17 Figure 3.1 The Projected Price Movement for a Single Diffusion Process........ 48 Figure 3.2 The Distribution of Abnormal Trading Volume and Return Volatility Within a Three-day Window [-1, 1] around Quarterly Earnings Announcements.....................................................................63 Figure 3.3 Cumulative Abnormal Returns for Good and Bad News Firms in Each Abnormal Trading Volume Quartile..................................... 66 Figure 3.4 Cumulative Abnormal Returns during the Second Quarter after Two Consecutive Quarterly Announcements with High Information Content.............................................................................. 81 Figure 3.5 Cumulative Abnormal Returns for Good and Bad News, Quarterly and Annual Announcements that Belong to the Fourth Quartiles of AVOL3D and AVAR3D.................................................83 Figure 3.6 Cumulative Abnormal Returns for Good and Bad News Firms (Classified by SUE) in Each Abnormal Trading Volume Quartile....................................................................................85 Figure 3.7 The Returns from the Zero-investment Portfolio Holding for 50 days Built on the Good/Bad News Classification, AVOL3D, and AVAR3D Each Year from 1973 to 2001 ................................... 87 Figure A2.1 The Relations between I(x, t) / H0 (Entropy Reduction) and the VariableX = x/(2-jD-t) When Both Drift and Diffusion Forces Exist...................................................................................................... 110 Figure A2.1 The Relations between I(x,t)/H0 (Entropy Reduction) and the Variable A = x/(2-jD-t) When Only Diffusion Force E xist.............. 110 Figure A2.3 The Relation between H(x,t), x and t ................................................. I ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT This dissertation investigates the boundary of market efficiency theoretically and empirically. It suggests that a diffusion mechanism exists in capital markets. A stylized information diffusion model is presented to describe the process by which investors gradually assimilate the information following an information event in capital markets. The model suggests that a diffusion process depends on both a drift and a diffusion force. If prices have converged before the drift force disappears, the diffusion phenomenon is difficult to observe. Otherwise, the diffusion force dominates in the longer horizon and imposes a boundary on market efficiency. The model indicates that the speed of information diffusion depends on information content, information conductivity, and information shock absorption capacity. Each signal has a different degree of information content, and each firm has a unique information conductivity. Consequently, the information on different firms diffuses across investors at different rates. Several implications from this diffusion model are tested with quarterly earnings announcement data. Post-announcement drift is documented only for earnings announcements that have high information uncertainty (content), measured by high abnormal trading volume and return volatility surrounding the announcement. The results are robust after controlling for the magnitude of earnings surprise. Preliminary evidence also suggests that the magnitude of drift for good (bad) news Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. firms is positively (negatively) related to analyst following and institutional ownership. Moreover, price movement after a second earnings announcement is affected by the information contained in the previous announcement. These findings are consistent with an information diffusion process at work in capital markets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 Introduction Asset pricing models typically assume both that the diffusion of every type of publicly available information takes place instantaneously among all investors and that investors act on the information as soon as it is received (Merton, 1987). These are the fundamental assumptions in existing asset pricing theories. These theories usually examine prices at equilibrium states, but largely ignore the process to reach equilibrium. Such an analysis simply prescribes how much new information is required to change investors’ beliefs about firms’ future cash flows for the market to move from one price equilibrium to another, but it does not acknowledge that information diffusion is inherently a non-equilibrium process. It considers neither the information diffusion mechanism nor the theoretical methods for computing how quickly price moves from one equilibrium to another. In a standard asset pricing setting, the supply and demand equilibrate quickly enough to be considered at instantaneous equilibrium at all times, so investigation on the adjustment dynamics is not necessary. In a perfectly efficient financial market, such an assumption is reasonable because information is impounded into prices instantaneously (Fama 1970). However, there are a number of reasons why this assumption is too restrictive. For example, Grossman and Stiglitz (1980) argue that the number of informed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 investors is endogenous in a capital market with a noisy rational expectations equilibrium. Information search is costly and time-consuming. If search costs exceed the marginal benefits of becoming informed, an investor has no incentive to become informed. On the other hand, if no one is informed, the price will not reveal all public information. Intuitively, the process through which information is impounded into prices is dynamic and should take time. Is the price adjustment period so short that it is negligible after the market receives new information? Early empirical studies on price response following various public events seem to give an affirmative answer. Such studies as Dann, Mayers, and Raab (1977), Lloyd-Davies and Canes (1978), Epps (1979), Hillmer and Yu (1979), Patell and Wolfson (1984), Jennings and Starks (1985), Barclay and Litzenberger (1988), Kim, Lin, and Slovin (1997), and Busse and Green (2001) show that the price adjustment time ranges from a few minutes to days after large trades, analysts’ recommendations, and corporate announcements including dividend, earnings, and seasoned equity offerings. This empirical evidence is consistent with the views of many financial economists that stock returns can not be predicted based on publicly available information alone because arbitrage removes any predictable patterns quickly and efficiently (e.g., Samuelson, 1965; Fama, 1970, 1991). Others, however, believe that capital markets are inefficient (e.g., Thaler, 1993; Lo and Mackinlay, 1999; Shleifer, 2000; Lee, 2001). They argue that the strong and systematic market anomalies that have been identified, e.g., overreaction and underreaction phenomena, are the results of the systematic cognitive biases of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. investors or bounded rationality (e.g., Daniel, Hirsheleifer, and Subramanyam, 1998, 2001; Barberis, Shleifer, and Vishny, 1998; Hong and Stein, 1999) and that real-world arbitrage is risky and therefore limited (Shleifer and Vishny, 1997). A growing body of empirical literature seems consistent with this belief that information is impounded into price gradually. Bernard and Thomas (1990) conclude that investors do not fully understand the implications of current earnings for future earnings and that this leads to a delayed price response to earnings announcements. This delayed price response has been further investigated in numerous other accounting studies (e.g., Rangan and Sloan, 1998; Brown and Han, 2000). Also, the momentum effect identified in the finance literature implies that stock prices respond to information slowly. For example, Chan, Jegadeesh, and Lakonishok (1996) find a delayed price response to both the information in past returns and past earnings. Hong and Stein (1999) apply the notion of gradual information diffusion as one key assumption in developing an equilibrium model to explain the return under- and overreaction puzzle. Their approach is based on the bounded rationality notion and is different from the approach of other studies based on the behavioral biases such as overconfidence (e.g., Daniel, Hirsheleifer, and Subramanyam, 1998, 2001). However, Hong and Stein (1999) do not address the issue of when and where the diffusion would occur because in their equilibrium model the diffusion is defined as the fact that one of two groups of traders receives the different pieces of information in rotation. What are the mechanisms and the factors that drive the information delay and the subsequent price response? Given the fact that the diffusion is a non-equilibrium Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 phenomenon, it is appropriate to investigate the mechanism and its determinants with a non-equilibrium model. Recalling the motivation of his pioneering study on the diffusion of hybrid com technology, Griliches pointed out that “we never have had a good theory of transition. And the field, by and large, moved toward an interpretation where everything was in equilibrium, all the time” (Krueger and Taylor, 2000). In financial market equilibrium analysis, it is assumed that agents understand the full state of affairs in all markets simultaneously. They have infinite computation power and foresight ability to calculate market clearing prices and offer the general equilibrium prices. The possibility that agents may want to explore the profitable opportunities in a non-equilibrium state is also eliminated. These assumptions are questionable. When information becomes publicly available, some investors respond to the information earlier than others. First, the asymmetry when information is assessed could be due to imperfections in the information environment, information searching costs, and investors’ limited attention. Second, even if all investors receive the information simultaneously, it may take them differential amounts of time to understand the implications contained in the information due to bounded rationality (Sargent 1993) and information processing capacity (Sims 2001; Hirshleifer and Teoh, 2002). Third, the investors who have received and understood the information may not immediately take action due to investment constraints or uncertainty about the information. Finally, late information receivers may need time to extract information from the trades of earlier information receivers, i.e., uninformed investors use the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 market price to back out the information of informed investors (Grossman and Stiglitz 1980). Figure 1.1 provides a schematic illustration on the factors affecting the information diffusion process and delayed price response. From one price equilibrium to another, information has to travel through several major components of impedance as shown in the Figure. These components fundamentally address the following questions: (1) How much time is needed for all investors to receive the signal? (2) how would investors respond to the information they receive? and (3) how would the market absorb the reactions from investors? A careful consideration on these issues may provide insight into delayed price response to new information. Figure 1.1 A Schematic Illustration of Information Diffusion Event with information content Information revealed in price Imperfection o f diffusion environment Info processing impedance Market absorption capacity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The purpose of this dissertation is to propose a single-signal non-equilibrium information diffusion model and test the implications of this model. The dissertation has two major sections. The first section (chapter 2) presents an information diffusion model and discusses the price discovery process. The second section (chapter 3) tests implications of the model with the stock market data and discusses the boundaries of market efficiency. Information diffusion is defined as the process through which information is gradually assimilated by market participants. Following an information event, an information diffusion process and a transient price discovery process occur. Although it is difficult to model each investor’s behavior, it is possible to describe a general pattern of information revelation at the macro level. A partial differential equation (PDE) is developed to model the diffusion process leading to symmetric information across all investors. The model implies that the path to reaching information symmetry is influenced by (1) information content (uncertainty) contained in the signal, (2) information conductivity, and (3) information shock absorption capacity. The diffusion model assumes that information content is a necessary determinant of investors’ reactions. Each firm has a unique information conductivity ranging from near zero to infinity. Information conductivity of a firm depends on the degree of imperfection of the environment and information processing limitations. Infinite information conductivity implies that the market is perfectly efficient for the specific firm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 The essence of the diffusion model can be demonstrated by the following thought experiment. A firm that is traded at $40/share in the market announces that annual earnings exceed forecasted earnings by one dollar per share. Investors realize that at $40 the stock is under-priced and react to the good news by purchasing more shares. Under the efficient market hypothesis, one concludes that price will impound the new information contained in the earnings signal and converge to a new level instantaneously. However, what is the new equilibrium price? If the $1 earnings surprise is completely transitory, the equilibrium price is $41; if it is a permanent increase, then the new equilibrium price should be $50, assuming a 10% discount rate. Because the $1 earnings surprise may consist of both transitory and permanent components, the true new equilibrium price falls between $41 and $50. Can investors determine the relative weight of the transitory and permanent earnings immediately upon receipt of the information? It is very difficult, if not impossible, for the most sophisticated investors. Investors need time to understand the information, so price may not impound all the information instantaneously. The time needed for the information to diffuse throughout the market depends on various frictions in the market. In the above experiment, price may have converged to an equilibrium price (e.g., $45), but investors will continue to investigate the implications of the earnings signal (ongoing diffusion process) before they understand the full implications of the information. In contrast to the traditional view, this study argues that, because prices do not impound all public information instantaneously, information diffusion and price Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 convergence take place concurrently when information arrives, but they evolve on separate paths. Information may continue to diffuse while price has converged to a new equilibrium. In other words, price adjustment is likely to be complete earlier than information diffusion, though the two processes are indistinguishable in a perfectly efficient market. The information diffusion perspective contemplates two important information processing phases: the drift and diffusion phases. The drift phase, which is caused by the incentives of investors to explore the information, decays quickly over time while the diffusion phase persists. If prices have already converged before the drift phase ends, the diffusion phenomenon is hard to detect. Otherwise, the diffusion phase dominates in the longer horizon and thus imposes a boundary on market efficiency. The major implications of the theory are tested with the setting of earnings announcements. The earnings announcement event is chosen for its unique nature as well as its importance. The earnings announcement is unique in the sense that the information content is determined by the accounting system and complex, regular earnings signals. Complexity, the first feature, captures the notion that it may not be easy for investors to draw inferences from earnings surprises. Moreover, the information in an earnings announcement may be incomplete because the detailed financial statements are not available until several weeks after the announcement. This time lag may help distinguish informed traders from noise traders because the former, who have the ability to analyze and draw inferences from the financial statements, have the greatest impact on the pricing after detailed financial statements become Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 available. The second feature, regularity (earnings are announced every three months) can be used to test a superposition effect of information diffusion. A diffusion process following the first event may not be complete when a subsequent event arrives. With a sufficiently long information diffusion process, the price adjustment after a subsequent announcement should be jointly determined by the information contained in both events. The information content of the earnings announcement is measured by the abnormal trading volume and abnormal return volatility in the short window surrounding the earnings announcement (Beaver, 1968; Landsman and Maydew, 2002). Consistent with the information diffusion model, post-announcement drift is proportional to the magnitude of the information content of the signal. Furthermore, after controlling for information content of the earnings announcement, the magnitude of the drift is positively associated with the information conductivity proxied by the number of analysts following and the institutional ownership. This evidence indicates that future stock returns can be predicted by past trading volume and return data. In addition, the empirical results show that there is a superposition effect of the information diffusion, i.e., in the setting of multiple signals, if the second information diffusion starts while the first diffusion is not complete, the residual information contained in the first signal will affect the price movement after the second signal. These findings challenge the weak-form market efficiency notion. Consequently, this dissertation empirically suggests the possibility of a boundary of market efficiency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 This study investigates a fundamental question of the impact of accounting information on markets with a more general perspective, both theoretically and empirically. If the market is not perfectly efficient all the time, what are the implications for capital market research? Should we reexamine some of the existing findings in accounting and notions in the law drawn from the efficient market hypothesis? If diffusion time is not negligible, information diffusion theory may explain some anomalies in the accounting and finance literatures, e.g., post-eamings- announcement drift (Ball and Brown, 1968; Bernard and Thomas, 1990), short-term underreaction, and long-term overreaction (DeBondt and Thaler, 1985). A fully developed understanding of information diffusion may better explain the consequences of introducing new accounting standards. It is arguable that conventional academic research does not conform very well to the characteristics of policy-related research. Schipper (1994) attributes this to the fact that academic accounting research largely focuses on examining what are the relations among things and why things are as they are. This positive perspective does not necessarily fit with the normative questions faced by standard setters. This research provides directional guidance for policy relevant questions. The following two examples demonstrate what should be considered in a capital market with bounded efficiency. The first example concerns the choice of recognition vs. disclosure. Standard setters often have to decide whether, when and how a given accounting item should be disclosed or recognized. Under an efficient market hypothesis, the question of whether an item should be disclosed/recognized is more important than when and how it Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 should be disclosed, because the market will not be fooled by the presentation format and location. However, in a bounded-efficiency market, the latter question may be as important as the former, because how information is disclosed may affect the speed at which the information is impounded into price. As a result, the impact of a new disclosure/recognition standard on different firms is different. The second example concerns regulation fair disclosure (FD). The purpose of regulation FD imposed by the SEC is to provide equal access to information disclosures across all investors. The policy was designed to improve the information environment and protect some investors (e.g., individual investors) who traditionally access information later than others (e.g., financial analysts). Consequently, it is believed that regulation FD should expedite price adjustment. However, a recent survey (NIRS, 2001) reveals that some firms may have reduced the disclosure level of information that may help investors understand earnings. If this survey captures actual practice, the changing direction of the speed of price adjustment before and after regulation FD is not predictable from an information diffusion perspective. The improvement in information environment makes the price adjustment process shorter but the increase in information content of earnings measured by uncertainty level, in contrast, makes the price adjustment process longer. Within the diffusion framework, one can expect that disentangling the impact of information content and information conductivity will help us to understand the likely impact of a change in regulation in a capital market. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 In short, the investigation of the diffusion mechanism and understanding the boundaries of market efficiency will make contributions to the academic literature. Regulators and practitioners will also be better able to understand likely impacts of changing regulations on markets. In the long run, one hopes that this deeper understanding of the impact of accounting disclosures will aid in designing better market mechanisms to improve market efficiency. The remainder of the dissertation is organized as follows: Chapter 2 presents the model of information diffusion and discusses the determinants of the diffusion process. Chapter 3 provides empirical evidence about the price discovery process following earnings announcements. The evidence is consistent with the predictions from information diffusion theory. Chapter 4 summarizes the results and discusses the implications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Chapter 2 A M odel of Inform ation Diffusion and the Price Discovery Process 2.1 Introduction In the physical sciences, many elegant theories have been built on rigorous mathematical models and then tested empirically or experimentally. At the same time, some other theories have been built with an inductive approach. They first come from empirical observations but survive many rounds of falsifications, for example, the three laws of thermodynamics. Many systems consisting of interacting units obey universal laws that show a mean effect and that are independent of microscopic details of each individual behavior. The finding of universal properties that do not depend on the specific form of interactions gives rise to the intriguing hypothesis that universal laws may be applicable in economic and social systems (Mandelbrot, 1963; Mantegna and Stanley, 1999). Diffusion is a widely observed phenomenon in nature and has been studied in physics, chemistry, biology, sociology, and even economics. A pioneering work on the diffusion problem in economics was done by Griliches (1957). He used logistic curves to analyze the diffusion of hybrid corn technology and showed that this technology diffuses slowly across farmers, space, and time. Despite some later work (e.g., Mansfield, 1961; Gort and Klepper, 1982), it is not clear yet why diffusion would take Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 place in an economic system. Are the agents slow to react because they face high adjustment costs? Are they slow to react because they are uncertain about the information available to them and are waiting for more evidence? Or are they only apparently slow because the models leave out important forces affecting their behavior which change only slowly? (Griliches, 1998) More importantly, in a competitive market, why do some agents not take advantage by acting as first movers? These observations prompt us to examine the possibility that information diffusion across different investors may be present in financial markets. Information diffusion is a process through which information moves among different investors. At the conceptual level, we define information as the signal that might change the expectations of investors, so information diffusion is a process through which investors revise their expectations. At a concrete level, information is defined in various ways in different fields. Information is sometimes defined as the inverse of the variance of a noisy signal (e.g., Verrechia, 1982) while sometimes it is defined as the correlation between the noise signal and the underlying variable (Epstein and Turnbull, 1980). Shannon (1948) introduces entropy as a measure of information. These measures all reflect the elements of uncertainty. To facilitate the discussion, I use entropy as the information measure. Appendix 1 provides an elementary discussion of the definition and properties of entropy and on how entropy is related to this study and the traditional information measure - variance - in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 economics literature.1 The entropy of an information signal is proportional to the number of choices and the inequality of the probabilities of these choices. Intuitively, the variance of the signal is large when the entropy is large. For a normally distributed random variable it can be shown that the entropy is equal to the logarithm of variance plus a constant. In the following model, the information variable is captured by the entropy measure. 2.2 An Intuitive Example of Information Diffusion The standard economics assumption is that price should impound all information when investors in the market receive the information. The traditional efficient market view is that prices react to information as soon as it becomes public. This characteristic may be true for some information events in which entropy is small. However, investors may have great difficulty in understanding the implications of some high-entropy information events in a short period. Consider the following thought experiment. A firm being traded at $40/share in the market announces its annual earnings. It exceeds the consensus earnings forecasts by one dollar. Given the information, rational investors realize that at $40 the stock is underpriced. They should react to the good news and purchase more shares. What is the new equilibrium price? If the $1 earnings surprise is completely 1 Entropy is the most basic concept in thermodynamics, information theory, and signal processing in communication, but it is not paid as much serious attention as variance in information economics literature. Lev’s PhD thesis (Lev 1969) uses entropy to measure the information loss of accounting information aggregation. I suggest here that the information is hidden instead of lost in the index measure of income. 2 $ 1 is a hypothetical number to illustrate the basic ideas. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 transitory, the equilibrium price is $41; however, if the earnings increase is permanent, then the new equilibrium price should be $50 assuming a 10% discount rate. Because the earnings surprise may consist of both a transitory and a permanent component, the new equilibrium price should be somewhere between $41 and $50. Can investors figure out the relative weight of the persistent and transitory earnings immediately after they receive the information? They must investigate financial statements and other disclosed information and observe how other investors react to that information. Initially investors may interpret the information contained in the signal differently; their beliefs will converge gradually on a path toward the new equilibrium. Figure 2.1 illustrates how entropy changes following the information event. Event A has low entropy (panel A) and event B has high entropy (panel B). The Y axis represents the entropy level, while the X axis represents investors X) to Xn. Initially, all investors hold the same entropy level, H 0. When the information is announced, let us assume it may reduce the investors’ entropy level from H0 to Hx if they fully understand the information. For event A (e.g., an interest rate cut announcement), the entropy level of all investors will immediately adjust from H 0 to H x. The information trajectory follows the line S-01-02-A as shown in Figure 2.1a. For event B (e.g., earnings announcements), the entropy level for each investor is different after the announcement. The uncertainty does not fully adjust to the//, level. The information trajectory follows the line S-01-02-B as shown in Figure 2.1b. Although both announcements lead to a price change, we do not expect to observe the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 diffusion phenomenon in event A, while we may observe the diffusion in event B. The price adjustment process will not be complete in a short period after event B. Figure 2.1 Entropy Change Following Two Different Information Events a. Information Event A Information Announcement X,. X2 , .... Xn, XhX2 , .... Xn, b. Information Event B Ol 02 Information Announcement X,, X2 , ..., Xn, Xh X2 , ..., Xn, 2.3 The Information Diffusion Model The information diffusion model focuses on a single security. It assumes that all investors have the same level of information uncertainty about the security value before a new information signal arrives. The arrival of new information creates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 information asymmetry among the investors because investors understand the information at different times and at different precisions (entropy levels). The entropy, H, contained in the signal is used as the measure of information uncertainty that each investor seeks to resolve. We derive the model from both a discrete time setting and a continuous time setting. 2.3.1 A Diffusion Equation Derived from a Discrete Time Setting Investors are assumed to locate at a specific point along the X axis. Each investor is j*h units away from the original information source, where j — 0, 1, 2, 3, ..., N. h is an equal distance between investors, and N is the total number of investors. It is further assumed that the update of information of investor j is a weighted average of the information of its closest neighbor points. We divide time into n steps. Since the information level H(j) is only affected only by its neighbors’ information level H(j-1) and H(j+1), we write the following balance equation: H n + ] (j) = p -H n{j~\) + q-Hn (j +1), (2-1) where p and q are the relative weight that investor j places on the information of investors j-1 and j+1. Since we assume the neighboring investors are the only investors influencing investor j, p + q = 1. Equation (2-1) leads to the following difference equation: 3 It is likely that j ’s information is also affected by many other investors in the market. Such a scenario is discussed in the continuous setting. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 Hn + X U) - Hn U) = P-[H„ u -I )-H„ (j)] + q ■ [Hn (j + Y)-Hn (j )] = \ W. U +1) - 2' H, O') + H„ ( / -1)] - [H, (j +1) - H, (j - 1)] ( * We can rewrite equation (2-2) as Hn+\(j) ~ Hn(j) h2 ( Hn(j + Y)-2- Hn(j) + Hn(j - 1) At 2- At v h2 (p-q)-h ( H nU + \ ) - H n( j ~ ^ (2-3) At 2-h The left-hand side of equation (2-3) is the finite difference approximation to dH / dt. The first and second terms in parentheses on the right-hand side of the equation represent a finite difference approximation to the second- and first-order derivative of H with respect to the distance variable x, respectively. In the limited case while At 0,h -» 0 , equation (2-3) becomes dH(x,t) D d2H(x,t) dH(x,t) dt dx dx (2-4) where D = h212At , and v -[(p-q)-h]/ At. Both D and v should approach finite limits. Equation (2-4) is the Fokker-Planck equation or Kolmogorov’s forward equation in mathematics (Lifshitz and Pitaevskii, 1981). The intuition for this equation will be discussed after it is derived in a continuous time setting in the next subsection. 2.3.2 A Diffusion Equation Derived in a Continuous Time Setting Since investor i could be affected by other investors j, the information uncertainty for i at time t + x measured by entropy, Ht(x,t + z) , is conditioned Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 onHj(x,t), where i * j and x is the distance between the investor and the information source. Assume that investor i at time t locates at position x*. The relative weights of the contributions from all other investors are assumed to follow a probability distribution f(8,r,xb), where 8 is the distance between investors i and j. The function H(x,t + r)can be expressed as follows: H(x,t + r) = Y,H(xb,t) • f(8,T,x b) . (2-5) a llS Expand/ ( 8,r,xh), H(xb,t), and H(x,t + r)at position x and time t, respectively, ■c/s - \ ft 2 _ s? Sf(S,T,Xb) 1 2 ^ f ( d ’T’Xb) , /n/'r3\ D J(d,T,xb) = j ( d , T , x ) - d --------- + - d ----------— ------+G {d ), (2-0) ox 2 ox H(xb, t) = H(x - 8, t) = H(x, t)-8 ■ dH^ + }- S 2 8 + 0 ( S 3), (2-7) H(x, t + r) = H(x, t) + t • dH(- X’ $ + 0 (t2). . (2-8) dt Substituting (2-6) through (2-8) into (2-5) and keeping only terms with the first order in x and the second order in 8,4 we get H(x, t) + r ■ ^ H(x, t) ■ f{8, r, x) ot § _Y§ d(H(x,t) ■ f (8,r,xb)) +]_y j2 d\H(x,t)-f(S,T,xb)) ( 2 ‘ 9 ) x dx 2 x dx2 4 All derivatives have finite values at point x for time t. When % is very small, terms with a higher order become negligible. Also, because of the nature of the diffusion process, the function f(8,t,xb) becomes more and more localized at 8 = 0 when x becomes small. So the terms with higher order 8 are negligible as well. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 At time t, each investor must be at some position x, so the sum of the probabilities / ( 8,t,xb) over all 8 must equal unity, i.e., ^ f ( S , T , x b) = 1. s The location of investor x and the distance between investors 6 are independent, so equation (2-9) can be reorganized as r. MjM = I iL. [ H ( x , ,)£ y . m r > * ) ] * 2& ' (2-10) ax ^ T-» /* o 2 \ rp / o»\ Let D = —-— - , v = . Equation (2.10) can then be rewritten as 2 T T dH(x,t) =D d2H(x, t) ^ gj/(x,Q (2_n . dt dx2 dx Equation (2-11) is identical to (2-4).5 In deriving equation (2-4) in a discrete time setting, we assume that the information held by one investor equals the weighted average of the information held by two neighbor investors. In a continuous time setting, the assumption is replaced by the one that the probability distribution function / ( 5 , t, xh) becomes more localized when time interval x becomes smaller. 2.3.3 Interpretation of the Diffusion Equation Equation (2-11) is a partial differential equation (PDE) describing how information flows in capital markets. D is defined as information conductivity; v is the 5 The equation is also similar to the one that leads to the Black-Scholes formula in option pricing. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 2 drift velocity, which is driven by arbitrage forces in the capital market; and H is the entropy, which reflects the level of uncertainty about the security value in the market. The equation has two terms. The first term, called the diffusion term, is due to information asymmetry in the market. Some investors have more information than others, but information tends to become more homogeneous across investors over time. The second term, called the drift term, arises because investors have incentives to explore information immediately after its arrival and because information has a preferred direction of movement across investors. For example, information may be more likely to flow from a sophisticated investor to a naive investor than the other way around. dH/dx is the information gradient; the minus sign is necessary because information diffuses in the direction of decreasing information uncertainty. The speed of information diffusion depends on both the diffusion and drift terms. The parameter D, information conductivity, is an intrinsic characteristic of a firm. It is the main parameter determining the information flow across different investors when the diffusion force dominates the drift force. The value of information conductivity determines how rapidly the information gets to investors and how quickly the investors make trading decisions. In the absence of arbitrage opportunities, the drift term disappears (v = 0). In such a case, the information flow is contributed by the diffusion force only. Bachelier (1900) describes the law of Brownian motion with stock price data in Paris while Einstein (1905, 1906) develops the law independently and discusses the determinants of the Brownian motion in physics. The general model of Brownian motion is used as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 a standard assumption in continuous-time finance literature starting from Merton (1972). PDE (2-11) asserts that information variable (entropy) rather than price follows a Brownian motion process. If price impounds information instantaneously after an event, price follows a Brownian motion with a jump. However, if price impounds information gradually, price can not simply be modeled as a random walk with a jump. In the diffusion model, information diffusion starts immediately following an information announcement and ends when the market fully understands the information. In an efficient market, the value of parameters D and v are so large that the speed of information diffusion is close to infinite. Conversely, if the parameter values are very small, no diffusion or drift will occur. The speed of information diffusion is close to zero, so price impounds no new information over a brief period. It is likely that the market behaves somewhere between these two extreme situations. PDE (2-11) provides a governing law depicting how information level depends on time and space. To focus on a single signal setting in capital markets, we must introduce several additional assumptions. First, the information level jumps at x = 0 and t = 0. Before an information event, information across investors is assumed to be uniformly distributed. There is no other information source during the period we are studying. Other information events will either lead to an additional source term in the PDE or create a superposition effect. Second, the number of investors who understand and react to the information is assumed to be large enough that the diffusion process can be treated as continuous and infinite. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 These assumptions lead to the following initial and boundary conditions: Initial condition: H(x, t = 0) = H0. All investors possess identical information about the firm at the beginning but before the announcement. Boundary condition 1: H ( x - 0 , t ) - H x. When information arrives, the first receiver is assumed to understand the new information.6 Without loss of generality, Hx is set to zero. Hence, H0 represents the possible information that investors can gain from the signal. Boundary condition 2: H(x - co, t) = H0. At any specific point in time after an information event, an investor located at the position with an infinite distance from the information source does not extract any information from the event. The incentive for investors to arbitrage based on the information signal decreases quickly as time elapses, so the drift force decreases quickly. I assume that the drift velocity is inversely proportional to the square root of time, i.e., v - k / 4 t , where k is a constant.7 Hence, PDE (2-11) has the following analytical solution: 6 In reality, it is likely that no single investor understands earnings information instantaneously and perfectly. The insiders know the information, but they are bounded not to trade. This assumption is imposed in order to get an analytical solution. The intuition behind the assumption is that the expected distance between the investors decreases sharply as time elapses. (2-12) r 1 + erf v ■ V Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 where erf (.) is a Gaussian error function. Equation (2-12) indicates that the information being understood, H0 depends upon a set of variables: H 0, D, v , x, and t. Appendix 2 shows how to get this solution and provides several remarks on the features of this solution. The next section discusses the comparative statics related to the empirical tests in the subsequent chapter. Information uncertainty ( H 0) and information conductivity (D) are highlighted, as they are the focus of the tests. 2.3.4 Comparative Statics 2.3.4.1 The Drift Force vs. the Diffusion Force The drift force dominates the diffusion force for a short period following an information event. Most of the information contained in the signal may be impounded into price in this turbulent phase, and the incentive for investors to arbitrage with the information fades away. In the subsequent smooth phase, the diffusion force dominates, so the second term on the right-hand side of equation (2-11) equals zero approximately, which indicates that the information variable (entropy) follows a random walk. If price impounds information instantaneously after an event, price follows a Brownian motion (Bachelier, 1900; Osborne, 1959), which is established and used in the finance literature. Without the drift force (v = 0), equation (2-12) is reduced to the following form: H(x,t) = H0-erf{— ? = ) (2-13) 2 sIDt 8 To avoid confusion, the drift phase is called turbulent phase as in the empirical tests in Chapter 3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 2.3.4.2 Information Content (Uncertainty) Equation (2-13) disentangles the effects of information uncertainty (H0) contained in a signal and information conductivity (D) of a firm in the smooth phase. It suggests that the amount of information being understood is proportional to the information content (uncertainty). Consequently, for a signal with small uncertainty, the information diffusion process is too short to be detected. Empirically, the diffusion mechanism can be observed only for information shocks with high information uncertainty. One may suspect that earnings announcements have such characteristics because a simple earnings number is generated from a very complicated accounting system. The fact that financial statements related to the earnings number are not available until several weeks after the earnings announcement and that many items in the statements could contribute to earnings surprise may make the diffusion observable. 2.3.4.3 Information Conductivity As an intrinsic firm characteristic, information conductivity (D) measures how rapidly the information gets to investors and how quickly the investors process the information after the drift force disappears. It is the main parameter determining the information flow across different investors in the smooth phase. A large information conductivity implies that a firm has a better information environment and its investors Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 have less difficulty in processing information, so it implies that the market is quite efficient for this specific firm. In summary, when H0 is small but the values of D and v are very large, the speed of diffusion is close to infinity, which means that prices impound information promptly. Conversely, little diffusion and drift occur if the values of parameters D and v are very small. 2.4 Price Mechanism Equation (2-12) provides a structure that helps us understand the diffusion mechanism. Information per se is not measurable. To observe diffusion, we need to infer the effect from trades. Specifically, we can trace the changes in stock prices and volume after an information event. The following simple derivation with the traditional utility maximization approach illustrates how price changes over time, based on the gradual information diffusion perspective.9 Assume investors trade only when the uncertainty level decreases to a threshold. The initial equilibrium stock price is Po. Investors will receive (Pi + e) at a new equilibrium, s is normally distributed with mean zero and variance related to the uncertainty regarding the information event, and s becomes smaller over time. The demand from investor i is zt, and the net demand from all uninformed traders (below the information threshold) is assumed to be zero. Investors have identical negative 9 No strategic behavior is considered in the setting. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 exponential utility with risk aversion coefficient y, i.e., Ui - -e~y W ' , where Wj ={P^+e-P)- z,. Price change is assumed to be proportional to excess demand.1 0 Maximizing expected utility for an informed investor, i.e., max [ ) ], we have = (p\ ~ p ) !\r • varO )] • (2-14) where /],y,var(£) are exogenous. When investors are more risk averse and information uncertainty is large, they are less likely to trade. Meanwhile, when the difference between the current price and the expected new equilibrium price is larger, investors will place more orders. Rewrite (2-14) as z . — a —b - P , (2-15) where a = Pl /[y • var(f)], b = l/[y ■ v ar(f)]. The drift force is zero (v = 0) in the smooth phase. In such a case, equation (2-12) implies that the number of investors understanding information is proportional to the square root of information conductivity D and time t and is inversely proportional to the information content Ho, i.e., x x 4 D l / H 0. Taking the derivative with respect to time t, we have: dx x s[d /(H0 -*Jt)-dt. (2-16) 1 0 If news is negative, the excess demand becomes the excess supply. Other assumptions remain unchanged. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 Equation (2-15) suggests that net excess demand equals (a - b - P ) - d x i within a specific time interval. We assume a linear relationship between price change and net excess demand, hence dP = c -(a -b -P ) i=-dt. (2-17) H 0-dt where c is a constant. Solving the differential equation (2-17), we get Pi - P = r . var (s) • exp(---- - — . (2-18) yvar(s)-H 0 Equation (2-18) provides several interesting implications for our empirical tests. First, Pi - P is larger when H0 is higher. The larger the information content contained in a signal, the larger the price change in the smooth phase. Second, it takes a longer time for the price to converge to a new equilibrium when the information content is higher. Third, when H0 is constant, the speed of price adjustment is positively associated with the value of information conductivity, D. The equation explains why abnormal returns decay slowly after an event with high entropy, and it also predicts that any factors increasing the value of D would speed up such a decay process. Finally, the decay process of abnormal returns is associated with other factors, such as c and y, which measure information shock absorption capacity and investor characteristics, respectively. In addition, equation (2-18) implies that information diffusion and price convergence may take place concurrently when the information arrives, but they Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 evolve on separate paths. Information may continue to diffuse while price has converged to a new equilibrium. When Pi - P is sufficiently small, the price convergence process has been completed but the diffusion process continues. In other words, price adjustment is likely to be completed earlier than information diffusion, even though in a perfectly efficient market the two processes are indistinguishable. The above sections depict a picture of the information diffusion process in a capital market. The magnitude and speed of diffusion depend on the information content (Ho), drift velocity (v), information conductivity (D), and the distance of the investors from the information source (x). Although the model is applicable to all information events, empirically, the diffusion process may be only observable for some events with high entropy (uncertainty). For example, for a salient common shock, such as a change in short-term interest rates, the price adjustment process may be so short that information diffusion is not detectable. For some firm-specific information shocks, the implications of the information event are not easy to extract, so the diffusion process is long enough to be observable. An earnings announcement is a classic information event from which investors evaluate the firm’s performance and revise their expectations. A simple earnings number is generated from a very complicated accounting system, so an earnings announcement may have high entropy. In the remainder of this chapter, I discuss the implications of the diffusion model in the setting of earnings announcements and integrate the existing findings in the literature into the diffusion framework by analyzing the parameters of the solution to the PDE. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 2.5 Information Diffusion and Earnings Announcements Earnings contain new information when there is a discrepancy between expected earnings and realized earnings. The market reactions to earnings surprises can be observed from price movements and volatility changes. Ball and Brown (1968) document that positive (negative) earnings surprises are accompanied by positive (negative) abnormal stock returns. Deciphering the information content of an earnings surprise is especially complicated. Investors must estimate the probabilities that the earnings surprise is transitory and/or persistent. Financial statements contain a multitude of data items, and all items and their combinations could contribute to an earnings surprise.1 1 It is likely that, the larger the earnings surprise, the more the potential choices contributing to the surprise, and the higher the information content for the earnings announcement. In our earlier thought experiment, the first investor receiving the information will investigate the causes and partially reduce the uncertainty in the market. She absorbs the information and accordingly her information level rises as she now knows the information with better precision. She then decides whether to trade. Then, the second investor receives the information. She can also infer some of the information from the price movement caused by the trading of the first investor. For example, observing that the price is moving up from a positive surprise, the second investor 1 1 Notice that current financial statements are issued a few weeks after the earnings announcements. Investors may obtain the explanations of earnings surprise from other sources or wait until the statements become publicly available. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 infers that the first investor perceived the earnings surprise to be due to a persistent increase in earnings. Upon analysis of the information, she may increase her estimate of the probability that earnings are persistent. If the prior probabilities for various choices are equal, moving away from the prior probabilities will decrease the entropy value and uncertainty. Then, the second investor decides whether she should trade. The process continues from the third to the fourth, and so on through N investors. This process contributes to the diffusion of earnings information. In contrast, some other information events may not have such a complicated origin. Their choice may be much simpler, and accordingly, their information content, measured by the entropy, is smaller. The diffusion for these types of information events may be limited to a few minutes or hours. 2.5.1 Information Content Equation (2-12) shows that the value of information understood is proportional to the new information content contained in the earnings signal. This result implies, ceteris paribus, that the greater the information content, the longer the price adjustment process. Interestingly, some prior empirical evidence is consistent with this expectation. Jennings and Starks (1985) find that the price adjustment process is longer for earnings announcements with higher information content. They measure the information content by the revision of analysts’ forecasts of year-end earnings in response to quarterly earnings. Pincus (1983) concludes that price adjustment is faster Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 for firms with higher earnings predictability. Since he uses the magnitude of earnings surprise (the difference between realized and expected earnings), his proxy for earnings predictability might be a proxy for the earnings information content. Kross and Schroeder (1984) show that quarterly earnings announcements with smaller time lag have larger abnormal returns after the sign and magnitude of earnings forecast errors and firm size are controlled. This study assumes that the financial statements issued sooner after the year-end are perceived to contain more information because, as time goes on, other information available may decrease the information contained in the financial statements. How would one reconcile the findings that relatively complex disclosures, such as nonperforming loans and pension obligations, are apparently reflected in stock prices, yet more “visible” variables such as earnings and book value are not fully reflected in prices (Beaver 2002)? If we address the question with the diffusion framework, it is likely that the entropy contained in earnings and book value is much larger than that in nonperforming loan and pension obligations. It is not surprising to see that it takes more time for the information in earnings to be impounded into the prices. To facilitate our discussion, we further decompose the information conductivity, D, into two components, Di and D2, where Di reflects how quickly the information reaches investors (information diffusion environment) and D2 reflects how long it takes for investors to process the information (information processing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 impedance). Without loss of generality we assume the drift force is zero. So from equation (2-13), we know that the information understood can be written as 2.5.2 Information Diffusion Environment (Di) The parameter Di reflects the efficiency of the information environment. Taking the derivative with respect to Di in equation (2-19), we get: The derivative is always positive, so the speed of receiving information for investors is faster when the parameter value increases. Consequently, the price convergence process is shorter the better the information diffusion environment. It leads to the following proposition: Ceteris paribus, the higher the diffusion parameter component Di, the more efficient the information environment, and the shorter the 1 9 price adjustment process. Intuitively, the information diffusion environment for each firm is directly associated with the attention which the firm gets in the market. Accordingly, it is an endogenous variable that is affected by firm size, the number of analysts following, media coverage, and other potential unidentified variables. Firm size is an important factor affecting the diffusion environment because large firms get more attention. If 1 2 The conclusion can be drawn from equation (2-18) directly. I(x,t) = H0- H(x,t) = H0 ■ erfc( X ) 2 V(D,+D2)-t (2-19) dl(x,t) (2-20) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 investors face fixed costs of information acquisition, they may choose to devote more effort to learning about those stocks in which they can take large positions. Financial analysts fill an economic niche by providing investors with interpretation and commentary on financial statements and financial reporting practices. Investors consider financial analysts’ forecasts and recommendations as an important information source. The number of analysts following the firm is thus an important determinant of the price adjustment process. For example, Hong, Lim and Stein (2000) show that low analyst coverage results in a sluggish price response after controlling for firm size. Other potential variables may affect the diffusion environment as well. For example, it is found that stocks with listed options have a more rapid price adjustment process (Jennings and Starks, 1985) and option traders participate generally in the price discovery process (Amin and Lee, 1997). Regulation FD is designed to provide fair information disclosure across all investors so that the investors can obtain the information simultaneously. In theory, the policy would increase the parameter Di and increase the speed of price adjustment. However, if some firms reduce the disclosure of information that may help investors to understand earnings, the uncertainty contained in earnings will increase. As a result, the changing direction of the speed of price adjustment before and after regulation FD is not predictable. An increase in Di makes the price adjustment process shorter, but the increase in uncertainty, in contrast, makes the process longer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 2.5.3 Information Processing Impedance (D2) The duration between the time that an investor receives information and the time that the investor understands and responds to the information is determined by the parameter D2 . We call it information processing impedance. All investors have budget constraints and limited computing power to process information. No matter how efficient the information diffusion environment is, information processing impedance imposes a limit on the speed of processing and understanding information. The first-order derivative of information understood with respect to D2 is always positive. Accordingly, ceteris paribus, the higher the parameter component D2, the smaller the information processing impedance, and the shorter the price adjustment process. Each firm has different management, business, and investors. Investors have different ways of processing information. The public information available in the market that helps investors digest the earnings information is also different across firms. As a result, one would expect that disclosure level and investors’ sophistication would contribute to the value of the parameter D2. It is unrealistic to suppose that a stock price is equal to the fundamental value of the firm at all times. Most likely, the price fluctuates around the firm’s value (Lee, 2001). When public information about a firm increases, the realized stock price should be closer to the fundamental value of the firm. Understanding the implications of the earnings number requires investors to refer to the information disclosed in the financial statements or elsewhere. Conversely, more disclosure increases the complexity of the financial statements and increases the time of processing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 information. It depends on the extent to which the disclosure adds new information to what is already known from the financial statements. Botosan (1997) and Sengupta (1998) find that greater corporate disclosure is associated with a lower cost of equity capital and debt, respectively. Both studies suggest that the disclosure level is important in reducing capital costs when there is greater uncertainty about the firm. So the evidence indicates that a higher disclosure level will lead to a quicker elimination of information uncertainty. Large institutional investors having better resources can process information more quickly and accurately than small firms and individual investors. In contrast, unsophisticated investors may primarily trade on intuition or information received from the financial press and trading activities of sophisticated investors. Information processing speed is expected to be increasing in the degree of investor sophistication. Several recent studies examine the impact of investors’ information capacity constraints and how such a constraint may affect asset prices (Peng, 2002; Peng and Xiong, 2002; Hirshleifer and Teoh, 2002; Hong, Torous, and Valkanos, 2002). Since investors have limited time and resources, the information produced is incomplete, so they can generate only a partial inference about the fundamental variables after they receive the fundamental shock. As a sequence of the capacity allocation decision, assets may have heterogeneous information environments. Their prices incorporate fundamental shocks at different speeds and respond differently to new information. Empirical evidence shows that the market responds to negative information slowly (Busse and Green, 2001; Hong, Lim, and Stein, 2000). The sluggishness with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 which bad news information is impounded into price may be due to the small value of parameter D2 . An alternative is that the market absorbs the excess supply slower than excess demand.1 3 2.5.4 Information Shock Absorption Capacity In economic analysis, price is determined by the demand and supply of products, but the slope of demand curves is different across products. The same change in the number of products may move prices to different levels for different products. Analogous to this scenario, similar information shocks may cause different price movements. On the other hand, the number of shares needed to make the same price movement from one equilibrium to another is not identical. We call the measure information shock absorption capacity. This measure is closely related to market depth but not identical to it. The shock absorption capacity is the total number of orders needed to move the price to a new equilibrium while market depth is the minimum orders needed to make the price move. They are identical only when the market is always in equilibrium. Xn is the last investor who trades to complete the price convergence process, though information may still continue to diffuse across Xn+i, X n+2, and so on. The price convergence period is inversely related to the capacity. The larger the absorption capacity the longer the price convergence process. This information absorption capacity is proportional to firm size. However, the magnitude of the impact on the 1 3 See the discussions in market microstructure reference, e.g., Harris (2002). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 diffusion time will be mitigated by the value of size-dependent parameter Di. Defeo (1986) shows response duration is longer for large firms, which suggests that the shock absorption capacity based on firm size may dominate the environment effect (Di) contributed by firm size in his sample. In the above discussions, we focused only on the behavior of rational investors. We assumed a zero net demand from those investors who do not trade on information, so their trades make no predictable price impact. However, noise traders may be an important proportion of the market (Black 1986). Their existence may pose a challenge to the design of empirical tests. 2.6 Conclusions The model presented in this chapter depicts the information diffusion mechanism and the price discovery process in capital markets. The price adjustment on the path toward equilibrium has received limited attention in traditional asset pricing theories. The information diffusion process is determined by both a drift and a diffusion force. The drift force is the result of investors’ incentives to explore for information. Diffusion occurs after a specific information event because (1) an information shock reaches investors at different times, (2) investors process information at different rates, and (3) later investors rationally infer information from the trades of earlier information processors. The diffusion effect can be detected if price has not been adjusted to a new equilibrium but the drift force has disappeared. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 The existence of the diffusion force in the long horizon thus imposes a boundary on market efficiency. The theory predicts that four major factors affect the information diffusion and price discovery process: information content, information diffusion environment, information processing impedance, and information shock absorption capacity. Some prior studies provide empirical evidence consistent with this conceptual theory. Nevertheless, each of these studies focuses on a single factor. The theory reconciles these empirical findings and provides a framework to illustrate why the price convergence process occurs in a predictable way. Some factors, such as information content and information shock absorption capacity, are assumed to be exogenous, but other factors are endogenous, which could be affected by the standard-setting process. In contrast to other studies, we distinguish the information diffusion process from the price convergence process. Information continues to diffuse regardless of whether trades occur. There are fluctuations of information levels among investors, but it is difficult to detect the fluctuations with trading data due to market frictions (transaction costs). The theory provides useful implications for short-window market studies. If the price does not fully reflect the fundamental value of the information immediately after an information event, we should not simply use the market price to proxy for the firm value because stock prices undergo a discovery process. This problem may be mitigated in long-window studies. The next chapter describes tests of these implications of diffusion theory in a long-window setting. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 Chapter 3 Inform ation Diffusion and the Boundary o f M arket Efficiency 3.1 Introduction In chapter 2, I propose a model of information diffusion describing the transition of price from an initial equilibrium to a subsequent equilibrium. The model is built on the belief that an information diffusion mechanism exists in capital markets and that it can be observed at the macro level. The theory predicts that diffusion depends upon the information content (uncertainty) contained in a signal (e.g., earnings announcement) and the information conductivity of the firm.1 4 While information uncertainty is a characteristic of the signal that changes market beliefs about firm value, information conductivity is a characteristic that captures how quickly investors receive, process, and respond to information through trading. Information conductivity depends upon the degree of imperfection in the information 1 4 An earnings announcement has information content if it leads to a change in investors’ assessment of the probability distribution of future price (Beaver, 1968). Thus, information content has both mean and variance dimensions. Price change measures the information content. However, the existence of uncertainty may prevent the price from fully adjusting to a new level implied by the signal in the announcement period, so part of the information will be processed continuously beyond the announcement period (e.g., Holthausen and Verrecchia, 1990; Penno, 1996; Subramanyam, 1996). According to Theil (1967), uncertainty and expected information are dual concepts. The uncertainty resolution process results in information gain (see discussion in Appendix 1). Under this framework, the total information contained in a signal has two components: the information content realized within the announcement window and the expected information realized afterwards. In the diffusion framework, it is the second component of information content that is diffusing in capital markets. In the traditional accounting literature, information content is measured by abnormal return volatility and trading volume in short windows. Both measures are the noisy proxies for “information uncertainty” or “expected information.” To avoid ambiguity, I hereafter simply use “information uncertainty” rather than the term “information content” to characterize this effect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 environment and the difficulty of information processing. Because the model assumes that information diffuses through the market, information uncertainty and information conductivity are directly related to the magnitude and the timing of post announcement abnormal returns. The purpose of this chapter is to test the implications of this diffusion theory in the quarterly earnings announcement setting. I examine three research questions: First, does empirical evidence support an information diffusion process? Second, in what ways does the diffusion process depend upon information uncertainty (proxied by three-day abnormal trading volume and return volatility) and information conductivity (proxied by institutional ownership and analyst following)? Finally, if a diffusion process is not yet complete when a new information event arrives, how does the residual information from the first announcement affect prices after the second announcement (superposition effect)? The findings in this chapter are consistent with the information diffusion theory. First, the cumulative abnormal returns (CAR) realized within three months after earnings announcements are proportional to the magnitude of abnormal trading volume and return volatility within the three-day window surrounding earnings announcements. CARs move upward (downward) in response to good (bad) news announcements. Both portfolio and regression analysis demonstrate that the conclusions are robust after controlling for the magnitude of earnings surprise. Second, regression results show that the number of analysts following and institutional ownership are positively (negatively) associated with the three-month CARs for good Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 (bad) news.1 5 Third, a superposition effect is documented in a two-period setting for good news. Price movement after the second announcement is affected by information in the first announcement when the second announcement is good news. The study presented in this chapter makes three main contributions to the literature. First, it provides empirical evidence supporting an information diffusion mechanism in capital markets. Prior studies consistent with a delayed price response to information do not focus on predicting when and where the delay will occur. Also, other studies assume that investors receive information at different times or gradually in pricing models but do not examine explicitly when and where the diffusion occurs (e.g., Hirshleifer, Subrahmanyam, and Titman, 1994; Hong and Stein, 1999). This study adds to the literature by documenting that a delay in response will be detectable when an event with high information uncertainty takes place in a firm with low information conductivity. Second, this chapter brings together two related strands of literature originated with Ball and Brown (1968) and Beaver (1968), namely, the post-eamings- announcement drift (PEAD) and information content literatures, which examine the market response to earnings for long and short windows, respectively. This chapter presents evidence that the market responses in long and short windows are interrelated. When news has high information uncertainty, the market responds dramatically to the news in the short term (e.g., Subramanyam, 1996). It will then 1 5 As discussed in Section 3.4.2, the abnormal returns accumulate from trading day 15 to day 64, with day 0 being the earnings announcement date. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 adjust continuously in the long term through the diffusion process. Traditional PEAD studies use earnings surprise as a classification variable and observe abnormal returns post earnings announcement (see Kothari, 2001 for a summary). This study innovates by documenting the PEAD using price change as a classification variable and abnormal trading volume or returns volatility as a measure of information uncertainty. Interestingly, the results show that there is no significant PEAD when there is no uncertainty about the information signal even if the earnings surprise is large; in contrast, there is a large PEAD when an announcement has significant uncertainty, even if the earnings surprise is small. In this fashion, this study provides a structure for understanding the PEAD, an open puzzle in the accounting and finance literature (Fama, 1998). Finally, the collective evidence presented in this chapter also speaks to the notion that weak-form market efficiency is likely to be bounded. Under weak-form efficiency, future returns cannot be predicted with past price, returns, or volume information. However, if the capital market is weak-form efficient at all times, the documented diffusion pattern after the first announcement should not exist, and the information contained in the first announcement should not impact the price movement after a subsequent announcement. The boundary of market efficiency is imposed by the nature of an information signal, the imperfection of the information environment, and the investors’ difficulty in processing information. The results imply that resolving information uncertainty is costly and may be time-consuming. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 3.2 Hypothesis Development As discussed in Chapter 2, the diffusion model focuses on a single security. The model assumes that a number of investors initially have the same information about security value but that the level of information becomes unequal after a new information event. The update of information of one investor is assumed to be a weighted average of the information of other investors. The arrival of new information creates information asymmetry among the investors because some investors understand and respond to information more quickly than others. As time approaches infinity, the information asymmetry is assumed to disappear. These basic assumptions result in a partial differential equation (PDE) describing how information flows in capital markets. The model implies that the process is determined by the initial information uncertainty and information conductivity associated with the security in a single-signal setting. An important characteristic of information diffusion is that the period following an information event can be classified into two phases — a “turbulent” phase and a “smooth” phase — based on the relative weight of a drift force and a diffusion force.1 6 The turbulent phase is typically a very short period after an information event, and the smooth phase is the period following the turbulent phase in 1 6 The meaning of the word “drift” here is different from the “drift” used in the PEAD studies. I use the term “drift” to be consistent with the “drift term” and “diffusion term” in the model, but I replace “drift and diffusion phases” with “turbulent and smooth phases” to avoid confusion. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 which price movement is more stable.1 7 The drift force, which is largely driven by the incentives of speculators to exploit the information, dominates in the turbulent phase but decays quickly toward the smooth phase when price movements are dominated by information diffusion. The diffusion force persists in the smooth phase, during which rational investors expend resources to understand and act upon the information. Most or all of the information contained in the signal may be quickly revealed in price due to the drift force. If price has already converged to the new equilibrium before the drift force disappears, the diffusion phenomenon is difficult to observe. Otherwise, the diffusion force dominates and can be observed through slow price movement. 3.2.1 Three Hypotheses The information diffusion model implies that there should be no diffusion when there is no information uncertainty about the signal and that the time of information diffusion is positively associated with the magnitude of information uncertainty. The larger the information uncertainty of the signal, the more abnormal returns will be realized in the diffusion process. I thus posit the first hypothesis in alternative form as follows: Hypothesis 1: Information diffusion is observable when a signal contains significant information uncertainty. The abnormal returns realized within the diffusion 1 7 One may think about throwing a pebble into the water. At the moment the pebble enters the water, it is hard to predict the ripples immediately. However, one can find a governing equation to describe the movement of wave and its determinants after a few seconds. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 period are positively associated with the information uncertainty contained in the signal. Panel A of Figure 3.1 illustrates how hypothesis 1 will be tested in the portfolio analysis. The information event is followed by a short turbulent phase, which in turn is followed by a smooth phase. The trajectory of price change in the turbulent phase may not follow the diffusion model because of information deficiencies in interpreting earnings and the price fluctuations caused by speculators. I examine the price patterns in both the turbulent and smooth phases, although I expect to detect the information diffusion only in the smooth phase. I use abnormal trading volume and return volatility to proxy for information uncertainty in this study, so Hypothesis 1 posits that for good (bad) news events with high abnormal trading volume and returns volatility the price continues to move upward (downward) following the announcement. Information conductivity is a result of two factors: the imperfection of the information environment and the difficulty of information processing on the part of investors. While the first factor affects when investors receive the information, the second affects how much time and effort it takes for investors to understand and respond to the information. When the time is negligible, the information will be impounded in price promptly. Otherwise, the price will continue to drift toward a new equilibrium. Hypothesis 2 is thus stated as follows: Hypothesis 2: Ceteris paribus, the speed o f information diffusion is positively associated with information conductivity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Figure 3.1 The Projected Price Movement for a Single Diffusion Process Panel A: Cumulative Abnormal Return Turbulent Smooth Good N ew s, High Info Uncertainty Good News, Low Info Uncertainty N o Info Uncertainty Bad News, Low Info Uncertainty Bad News, High Info Uncertainty Time (Days) t = 0 Panel B: Cumulative Abnormal Return Y : Low Info Conductivity X: High Info Conductivity tl t2 t = 0 Time (Days) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Panel B of Figure 3.1 illustrates a good news scenario1 8 in which two announcements have identical information content but the information conductivity of X is higher than that of Y, i.e., investors receive and process information faster for announcement X than for Y. Accordingly, the abnormal return accumulating from the beginning of the diffusion to a point in time before the diffusion completes, such as tl- t2 in Panel B, is higher for X than for Y. When both diffusion processes complete, price moves from the original price Po to the new equilibrium price Pi, so total abnormal return in Y is equal to that in X. Considering the possibility that diffusion lasts more than a calendar quarter, as shown in other PEAD studies (e.g., Bernard and Thomas, 1989), I expect that the abnormal returns realized within a quarter are higher for the announcements with larger analyst following and higher institutional ownership.1 9 The previous hypotheses examine implications of the diffusion perspective for a single earnings announcement event. Since earnings announcements occur regularly, one may consider the impact of multiple signals on price formation. In other words, if a new diffusion process starts while an earlier diffusion continues, how does the price move after the second announcement? The Efficient Market Hypothesis predicts that the information contained in the first announcement should no longer affect price movements after the second announcement. In contrast, the diffusion theory predicts 1 8 The bad news analysis is a perfect reflection of the good news analysis. 1 9 Findings in related work (Bartov, Radhakrishnan, and Krinsky, 2000; Mikhail, Walther, and Willis, 2003) that the total PEAD is negatively related to institutional ownership and analyst experience seemingly contradict this prediction. I discuss the difference in footnote 33 of the results section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 that as long as the information uncertainties in the first and second announcements have any orthogonal components, i.e., the second information announcement does not fully resolve the remaining uncertainty in the first announcement, the price movement in the second period will be jointly affected by the two diffusion processes. Thus, the third hypothesis is stated as follows: Hypothesis 3: Information diffusion has a superposition effect. The information uncertainty o f the first announcement will affect the price movement after the second announcement. Those announcements with high information uncertainty for two consecutive quarters are classified into the following portfolios: Good-Good, Good-Bad, Bad- Good, and Bad-Bad.2 0 If hypothesis 3 is descriptive, the second-period abnormal returns of the Good-Good and Good-Bad portfolios will be larger than the returns for the Bad-Good and Bad-Bad portfolios. In other words, the second-period abnormal returns of the portfolios with the good news in the first period will be greater than the abnormal returns of the portfolios with the bad news in the first period. The interaction of two consecutive announcements has been examined in two related studies but from a different perspective. Freeman and Tse (1989) and Bartov (1992) show that most of the PEAD associated with unexpected earnings after the first announcement disappears after controlling for the unexpected earnings in the second announcement. Hypothesis 3 differs from these studies by focusing on the price 2 0 The first and second words in each portfolio refer to the first and second announcements, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 movement after the second announcement. In addition, I use price change rather than earnings surprise as a classification variable, and use abnormal trading volume and return volatility to capture the information signal. 3.2.2 Proxies for the Constructs Information Uncertainty Information uncertainty contained in the earnings announcement is proxied by the abnormal trading volume and return volatility in the short window surrounding the announcement. Prior theories suggest that trading volume is associated with investors’ heterogeneous interpretation of a signal (e.g., Holthausen and Verrecchia, 1990; Kim and Verrecchia, 1991; Harris and Raviv, 1993; Kandel and Pearson, 1995; Bamber, Barron, and Stober, 1999). Higher abnormal trading volume reflects greater uncertainty of the information signal and a greater degree of heterogeneity of investors’ beliefs. Under conditions of greater heterogeneity, there are strong incentives for market participants to process, interpret, and generate more information. Thus, a signal creating high abnormal trading volume provides a good setting to investigate the diffusion process. Since I estimate return volatility within a three-day window (Landsman and Maydew, 2002), the measure is likely to be high when abnormal returns are high. However, the volatility may also be high even if the three-day abnormal returns are close to zero because daily abnormal returns fluctuate. Hence, return volatility Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 captures both the magnitude of price change and the uncertainty about the security’s price. Despite this limitation, it is expected that return volatility will capture some elements of information uncertainty. Modeling earnings expectations is a challenging task. In previous studies, earnings expectations are based on either a seasonal random walk with trend or a consensus analyst forecast. While the former gives a noisy proxy of expectations, the latter assumes that the sample mean is equal to the population mean in the market. The difficulty in estimating earnings surprise further motivates the choice of abnormal trading volume and return volatility. Even though these measures are correlated with earnings surprise (e.g., Bamber, 1986, 1987), they are preferable measures for the concepts underlying the diffusion model. Information Conductivity Since information conductivity is related to both the imperfection o f the information environment and the difficulty o f information processing on the part o f investors, I use institutional ownership and the number o f analysts following to proxy for this concept. High institutional ownership implies high information conductivity because sophisticated institutional investors have better resources to facilitate information processing, which allow them to understand information better and more quickly than unsophisticated investors (Hand, 1990; Kim and Verrecchia, 1994). Financial analysts fill an economic niche by providing investors with interpretations o f financial reports and commentary on firm s’ financial performance. Investors consider Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 financial analysts’ forecasts and recommendations as an important information source (e.g., Barber, Lehavy, McNichols, and Trueman, 2000). Brennan and Subrahmanyam (1995) find that firms with more analyst coverage have lower trading costs due to the reduction of information asymmetry. Hong, Lim, and Stein (2000) suggest that low analyst coverage results in a sluggish price response after controlling for firm size. Hence, the evidence is consistent with the perception that information will reach investors faster when analysts’ coverage is higher. 3.3 Sample Selection and Variable Measurement 3.3.1 Sample Selection This study uses four databases: COMPUSTAT, CRSP, TFN (CDA/ Spectrum), and IBES. Earnings announcement dates and other financial accounting information are obtained from the combined Industrial Quarterly, Full Coverage Quarterly, and Research files in COMPUSTAT from 1972 to 2002.2 1 The sample consists of both active and inactive firms listed on the NYSE, AMEX, and NASDAQ. Stock price, daily return, trading volume, shares outstanding, decile returns, and equal- and value- weighted market returns are extracted from the CRSP daily securities and index files. The information on analysts and institutional ownership is extracted from the IBES and TFN institutional ownership databases, respectively. 2 1 The sample consists of the data in COMPUSTAT updated by WRDS up to June 2003 at the time when the study started, so the data for fiscal year 2002 may be incomplete. TFN and IBES start from 1980 and 1983, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 Table 3.1 Sample Selection Criteria Selection Criteria Observations Earnings announcement dates available in Compustat from 1972-2002 567,241 Minimum 100 normal trading days in preceding year, trading volume and return data available in days [-1,1], and no consecutive earnings announcements within 20-day window. 439,229 NYSE, AMEX, and NASDAQ Firms 355,650 Price (closing) changes from day -2 to day 1 321,131 Sample 1 321,131 Sample 2: Observations with standardized earnings surprise estimated from SRW model 212,152 Sample 3: Observations with earnings surprise based on the analyst’s forecasts in IBES 90,817 To estimate the normal level of trading volume and return volatility, I require that there be at least 100 trading days in the year before the earnings announcements. Trading volume and prices within the three-day window surrounding earnings announcements must be available. To avoid information contamination, I require that there be no other earnings announcements within 20 trading days of the earnings O ' ) announcement of interest. In addition, the price change from day -2 to day +1 0 " \ relative to the earnings announcement date should be nonzero. These criteria lead to a full sample consisting of 321,131 earnings announcements (see Table 3.1, Sample 2 2 It is noticed that occasionally two announcements reported in COMPUSTAT are very close. The irregularity of announcements leads to a number of firms having more than four announcements within a calendar year. For example, 2.63% of the firms have five announcements in 1975 and 1.15% have six or more announcements in 2000. 2 3 The price change around an earnings announcement will be used to categorize good and bad news. This constraint is relaxed when I conduct a sensitivity test using the direction of three-day abnormal return to classify news. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 1). Sample 1 is used in the subsequent portfolio tests that do not control for earnings surprises. When earnings surprise information is needed, the sample size is reduced to 212,152 (Sample 2) and 90,817 (Sample 3) corresponding to earnings expectations based on a seasonal random walk model and consensus analyst forecasts, respectively. 3.3.2 Variable Measurement Abnormal Trading Volume (AVOL3D) I follow Landsman and Maydew (2002) and define abnormal trading volume, AVOL3D, as the sum of daily abnormal volume over the three-day announcement window [t = -1, 0, 1], with day 0 being the announcement date: AVOL3D, = Y.AVOL,, = £ (V„-V,)/<J„ (3-1) t = - l t= -1 where Vit is the number of shares of firm i traded during day t divided by shares outstanding of the firm on day t. V , and cr, are the mean and standard deviation in daily trading volume for firm i in the estimation window [-253, -10], excluding 10 trading days surrounding prior earnings announcements. Landsman and Maydew (2002) modify the original abnormal volume metric in Beaver (1968) as they argue that trading volume has increased approximately tenfold during the last three decades. Scaling by cr, , they argue that the formula controls for changes in normal, non- announcement-period volume across time. I estimate the normal trading volume with the data in the preceding year only because the measure will be used to forecast future Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 price movements. I exclude 10 days rather than 20 days surrounding prior earnings announcements to retain as many days as possible. Abnormal Return Volatility The abnormal return volatility, AVAR3D, is the sum of daily abnormal volatility over the three-day announcement window [t = -1, 0, 1] (Beaver, 1968; Landsman and Maydew, 2002): i r=i A VAR3Dt = YaA VAR, s = u 2 / a? , (3-2) ( = - i t= - i where uu = Rit ~{at + /?iRm t), Rit is the raw return of firm i for day t, and Rm t is the equal-weighted market return for day t. a, and /?, are firm i’s market model parameter estimates, and cr2 is the variance of firm i’s market model adjusted returns. The parameters are estimated with the window [-253, -10], excluding 10 trading days surrounding other prior earnings announcements. News Classification I employ the direction of price change surrounding earnings announcements to classify the earnings announcements into three types of news: good, bad, and neutral. Price change is utilized because it does not introduce the problem of estimating appropriate earnings expectations. Price change is computed as the difference between the price on day 1 and day -2. The earnings announcement is considered good, bad, or neutral news if the price change is positive, negative, or zero, respectively.2 4 The 2 4 Price may not be fully adjusted in the short window, but price change is a valid classification criterion as long as the direction of initial price response is correct. Penman (1987) uses the daily abnormal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 choice of a three-day window assures that almost all announcements fall into this 9 < window. For the firms with earnings announcement dates recorded in both COMPUSTAT and IBES, I examine whether there is any discrepancy between these two. The cross database verification shows that only 70.1% of the announcements have the exact same reporting dates in both databases. IBES leads one day for 12.9% of the announcements and lags one day for 10.4% of the announcements. It is not clear which one has more precise announcement dates, but at least 94% of the announcements fall within the three-day window in both databases. Abnormal Returns Abnormal returns are equal to daily raw returns minus expected returns. I use six different expected-retum models simultaneously to test the robustness of the results in the portfolio analysis. These six benchmark models are: equal- and value- weighted market returns (EW and VW), decile returns (DEC), CAPM models from equal- or value-weighted market returns (EWM and VWM), and the Fama-French three-factor model (3F). The expected returns from these models are listed in Table 3.2. returns from equal-weighted market returns to classify news. I use the sign of cumulative abnormal returns over the three-day window [-1,1] as a sensitivity test. The results are robust. Not surprisingly, the correlation between the two dummies of price change and abnormal return change is as high as 0.87. 251 use day -2 instead of day -1 because the daily price reported in CRSP is the closing price. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Table 3.2 Six Benchmark Models for Calculating Expected Returns Model Formula Explanations Equal-weighted market return (EW) E(Rn ) is the expected return of firm i at day t, and Rm t represents the equal-weighted market returns for the firms in NYSE, AMEX, and NASDAQ. Value-weighted market return (VW) E(Rn ) is the expected return of firm i at day t, and Rm t represents the value-weighted market returns for the firms in NYSE, AMEX, and NASDAQ. Decile returns (DEC) E(R,j) = Rdecf E( Rn ) is the expected return of firm i at day t, Rd e c t represents the daily decile returns for the firms in NYSE, AMEX, and NASDAQ at day t. The deciles are classified in CRSP based on the firm market value at the beginning of the calendar year. CAPM with equal- weighted market model (EWM) E(Rit) = cci + p r Rm t E(Rn ) is the expected return of firm i at day t, and Rm t represents the equal-weighted returns for the firms in NYSE, AMEX, and NASDAQ. The risk-free rate is not adjusted because the model is typically applied to nominal returns when daily data are used (Campbell, Lo, and MacKinlay, 1997). a , and are estimated with the data from the previous year. The estimation window excludes 5 days surrounding the previous earnings announcements (normally 3-5 announcements), so the estimation for each firm- announcement has approximately 240 observations. CAPM with value- weighted market model (VWM) E(Rll) = ai + ft- R m t As in EWM, except that Rm t represents the value-weighted returns for the firms in NYSE, AMEX, and NASDAQ. Fama-French three- factor model (3- factor) E(R,t) - R Lt = a i + + P2smbl + /?3 hmlt E(RU) is the expected return of firm i at day t, Rm t is the value-weighted market return at day t, and Rj t is the risk-free rate at day t. smbt and hmlt are the size and book-to- market factors at day t, respectively (Fama and French, 1993). The estimation window for the three-factor model is identical to the window used in market models EWM and VWM. 00 Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Table 3.3 Descriptive Statistics Panel A. Descriptive Information for Eight Portfolios Formed on Good/Bad News and AVOL3D quartiles AVOL3D range Q1 (min, -0.814) Q2 [-0..814,0.148) Q3 [0.148.,2.195) Q4 [2. 195, max) Pooled Price Change AP = 1 AP = 0 AP = 1 AP = 0 AP = 1 AP = 0 AP = 1 AP = 0 No. of Observations 36,283 43,997 39,623 40,662 42,940 37,342 46,610 33,674 321,131 % of NYSE&AMEX 65.3% 65.4% 63.6% 62.9% 65.0% 64.8% 63.7% 64.0% 64.3% Market Value ($mil) 1,009 912 975 881 1,683 1,442 1,888 2,224 1,371 Stock Price ($) 25.68 26.11 25.31 15.09 30.02 27.33 25.66 26.15 26.44 Panel B. Descriptive Information for Eight Portfolios Formed on Good/Bad News and AVAR3D quartiles AVAR3D range Q1 (0,1.139) Q2 [1.139,2.762) Q3 [2.762, 6.334) Q4 [6.334, max) Price Change AP = 1 A = 0 AP = 1 > I I © AP = 1 AP = 0 AP = 1 AP = 0 No. of Observations 38,878 40,490 38,352 40,975 41,348 38,074 45,071 34,379 % of NYSE&AMEX 63.7% 64.6% 64.9% 65.0% 64.8% 64.7% 64.0% 62.5% Market Value ($mil) 1,100 994 1,399 1,210 1,577 1,407 1,522 1,650 Stock Price ($) 29.01 29.14 24.68 26.55 26.12 26.96 26.99 21.23 V O 60 The daily abnormal returns, ARi t , are calculated as follows: (3-3) where Rt, is the daily raw return of firm i at day t and E(Rl t) is the expected return for firm i at day t calculated from each of the benchmark models. Cumulative abnormal return (CAR) is a simple sum of the daily abnormal returns in the period investigated. The robustness tests show that the results from buy-and-hold returns are identical to those reported here. Earnings Surprise I estimate earnings surprises using two approaches. The first one, standardized earnings surprise (SUE), is estimated based on the expectations from the seasonal random walk model (Foster, Olsen, and Shevlin, 1984; Bernard and Thomas, 1989), i.e., where eit is the actual earnings before extraordinary items reported in COMPUSTAT for firm i at announcement t, the expected earnings at t is E(ei t) = ej t_4 + ei, and cr, is The second measure, UE, is based on the analyst forecasts of quarterly earnings. That is, (3-4) • . 2 6 the standard deviation of earnings surprise from the prior six years. 2 6 Foster, Olsen, and Shevlin (1984) show that there is essentially no difference between the results from a seasonal random walk model and a time-series model. I require a minimum of 10 quarters of data in the prior six years when estimating cr,. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where et, and E(ejt) are the actual earnings and the consensus analyst forecasts reported in IBES, respectively. The consensus forecasts are the average of the analyst forecasts in the most recent month before the announcement. 3.4 Empirical Results This section presents the results of testing the three hypotheses. Section 3.4.1 describes the sample characteristics. Section 3.4.2 provides the portfolio analysis showing that the cumulative abnormal returns are significant only in the portfolios that consist of the announcements with high information uncertainty (hypothesis 1). Section 3.4.3 addresses whether the price movement documented in Section 3.4.2 simply captures the PEAD effect based on earnings surprise. The return analysis based on the two-way sorting portfolios not only indicates that they are different but also suggests that the PEAD based on an earnings surprise strategy becomes much smaller and even insignificant when the abnormal trading volume is small. In Section 3.4.4,1 discuss the regression results that support hypotheses 1 and 2. Section 3.4.5 presents preliminary evidence consistent with hypothesis 3, and Section 3.4.6 includes the results from additional sensitivity tests. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 3.4.1. Descriptive Statistics Among the 321,131 earnings announcements from 1972 to 2002, 206,530 announcements are from NYSE and AMEX firms while 114,601 are from NASDAQ firms. This sample is divided into eight portfolios based on a good/bad news classification and quartiles of abnormal trading volume or return volatility. Table 3.3 provides descriptive information on firm size, price, and exchange listing for each of these portfolios. The results show no apparent sample bias related to small firm or market structure effects (e.g., Stoll, 2000). Figure 3.2 presents the distribution of abnormal trading volume and return volatility for the sample. Panel A of Figure 3.2 shows that for most of the earnings announcements AVOL3D fluctuates around zero. The distribution suggests that only about a quarter of the announcements exhibit significant increases in trading volume. Panel B of Figure 3.2 confirms that only about a quarter of the announcements deviate substantially from the mean (3.0) for the AVAR3D distribution. These observations suggest that the tests of information diffusion should emphasize the earnings announcements that generate the largest abnormal volume and volatility. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 Figure 3.2 The Distribution of Abnormal Trading Volume and Return Volatility within a Three-day Window [-1,1] aroundQuarterly Earnings Announcements Panel A: Panel B: £ £ o > 3 o r < u 30 25 20 15 10 5 0 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 M id-point V alue of AVOL3D £ 30 5. 20 10 - 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 M id-point v a lu e of AVAR3D 3.4.2 Portfolio Tests of Hypothesis 1 I test hypothesis 1 by comparing post-eamings-announcement CARs for good (bad) news firms across four quartiles (Q1-Q4) of AVOL3D and AVAR3D. Hypothesis 1 suggests that price movement in response to news is most notable for the earnings announcements with the greatest information uncertainty, Q3 and Q4. Since Q4 has higher information content than Q3, the trajectory of price movement for Q4 should deviate furthest of all quartiles from zero. Since Q1 and Q2 have little Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 information uncertainty, price movement for these two portfolios should show little upward or downward trend. An important issue is the choice of an appropriate starting date for accumulating abnormal returns in order to generate a powerful test of the diffusion effect. Traditionally, we accumulate abnormal returns immediately after an information event. As discussed in Section 3.2, in the period immediately following an earnings announcement traders may lack information to interpret the announcement because detailed financial statements are not immediately available. It seems reasonable to begin the test when financial statements become publicly available. However, because it is difficult to obtain the exact dates that firms filed the statements with the SEC for such a comprehensive sample from 1972 to 2002, I estimate the availability of financial statements from earnings announcement dates and SEC disclosure rules. I analyze the average number of days between the dates of fiscal quarter end and earnings announcement dates recorded by IBES. The sample pooling all four quarters suggests that, on average, earnings are announced about 30 days after the fiscal quarter ends, and the mean reporting lag is very stable over the period from 1984 to 2001.2 7 The SEC requires that quarterly statements must be filed within 45 days after the end of the quarter and annual statements within 90 days after the end of 2 7 The mean reporting lag is 25 and 41 days for quarterly and annual reports, respectively. The reporting lag is consistent with those documented in several earlier studies (Chambers and Penman, 1984; Kross and Schroeder, 1984). Several other studies find that good news, in general, is announced slightly earlier than bad news (e.g., Penman, 1987; Lu and Ma, 2003), but this study does not make that distinction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 the fiscal year. Financial statements should be available somewhere between the announcement date and the due dates. I infer that, on average, the quarterly financial statements become available about 15 trading days after the announcement (see footnote 27). Thus, the accumulation period for abnormal returns is from day 15 to day 64. One may argue that information starts to diffuse even before the financial statements become available due to other corporate disclosure practices or analyst activities. Nevertheless, the argument is biased against finding a diffusion effect. If I still capture the diffusion effect with the current window, the true diffusion, in fact, is stronger than what I measure. Softer and Lys (1999) suggest that no information has been impounded into the price in the first 15 days in their PEAD study. It is not clear whether their finding is coincidently due to the delay of filing of financial statements. In order to investigate price patterns between day 2 and day 14, I examine the price movement over this period as well. The four quartiles of AVOL3D are divided into eight portfolios: P-Ql, P-Q2, P-Q3, P-Q4, N-Ql, N-Q2, N-Q3, and N-Q4. The Good-Bad news group in Q1 is labeled as P-Ql and N-Ql respectively, and so forth. Panel A and B of Figure 3.3 show the CARs from equal-weighted market returns for these eight portfolios. Panel A reveals that if we accumulate abnormal returns from day 2, the overall trajectories for the eight portfolios are difficult to predict primarily because responses immediately following the announcements are quite diverse. For example, let us examine the announcements for quartile 4. Although price for the good news group is moving Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 upward gradually as the diffusion model would predict, price for the bad news group surprisingly also goes up in the first two weeks and then declines over the next 50 days. The evidence is consistent with the notion that this initial period is a turbulent phase in which price movements may not be predictable from the diffusion model. However, when abnormal returns are accumulated from day 15, the pattern becomes much clearer, as shown in Panel B of Figure 3.3. The results support hypothesis 1. First, there is no obvious drift among low abnormal trading volume portfolios: P-Ql, N-Ql, P-Q2, and N-Q2. Second, a significant drift exists only in the portfolios P-Q4 and N-Q4. The drift for P-Q3 and N-Q3 is only about one-third of the drift for P-Q4 and N-Q4. The results based on the abnormal returns from other benchmark models are qualitatively similar. Figure 3.3 Cumulative Abnormal Returns for Good and Bad News Firms in Each Abnormal Trading Volume Quartile Panel A: CAR (2,64) 2 1.5 1 0.5 0 -0.5 ■ 1 -1.5 Trading Days P-Q4 N-Q4 P-Q1 N-Q1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Figure 3.3 (Continued) Panel B: CAR (15,64) 0.5 '3S O -0.5 -1.5 T rading Days P-Q1 -P-Q2 - - - P-Q3 --------------------------P-Q4 N-Q1 -N-Q2 - - - N-Q3 --------------------------N-Q4 Table 3.4 reports the difference of CARs between good and bad news announcements in each of the AVOL3D quartiles at different dates after the earnings announcements. Panel A shows that the CARs from equal-weighted market returns for good news firms in quartile 4 are 2.21% higher than the CARs for the bad news firms in the same quartile. The t-statistic of testing the null hypothesis that the difference in the CARs at day 60 is zero is 15.39. Row 4 of Panel A also shows that the CARs from the equal-weighted market returns for good news firms in quartiles 3, 2, and 1 are only 0.69%, 0.31%, and 0.30% higher than the CARs for bad news firms in the same quartiles, respectively. The magnitude of the difference in the CARs decreases monotonically with the AVOL3D value. Although the t-statistics of testing the significance of the CARs difference are all statistically significant when the CARs are calculated using equal-weighted market returns, the t-statistics for Q1 and Q2 are not Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 robust across other benchmark models, as shown in the other panels of Table 3.4. Importantly, the t-statistics for Q3 and Q4 are statistically significant across all six benchmark models. The CAR difference between good and bad news firms in quartile 4 is larger than the CARs in the other three quartiles. Thus, the results in Table 3.4 support hypothesis 1. Information diffusion can be observed only among the announcements with high uncertainty measured by high abnormal trading volume. The results are also robust when I use two subsamples. Each consists of quarterly and annual announcements only. They are also robust across NYSE/AMEX and NASDAQ firms, although the drift is larger for NASDAQ firms than for NYSE/AMEX firms. This conclusion is robust for the eight portfolios formed on the good/bad news classification and for the AVAR3D quartiles as well. It is interesting to revisit the pattern that price drifts up for both good and bad news for quartile 4 in the first 14 trading days. This pattern is present for all of the other benchmark models used to calculate CAR. One possible explanation for the early price increases is an information risk hypothesis. Brown, Harlow, and Tinic (1988) suggest that, following a dramatic financial news event, both the risk and expected return of the affected firms increase systematically, so the price will drift upward in response to both good news and bad news. Due to information deficiencies, information risk may dominate information diffusion immediately after earnings announcements. However, after the financial statements become available, information diffusion eventually dominates the risk effect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Table 3.4 The Difference in CARs between Good and Bad News Groups in Each of the Abnormal Trading Volume Quartiles Quartile 1 Quartile2 Quartile 3 Quartile 4 Day CARdiff (%) t-stats CARdiff (%) t-stats CARdiff (%) t-stats CARdiff (%) t-stats Panel A. EW 15 0.039 1.54 0.001 0.06 0.050 2.14 -0.009 -0.36 30 0.097 1.15 0.034 0.42 0.244 3.06 0.703 8.28 45 0.159 1.36 0.113 1.00 0.372 3.30 1.434 12.04 60 0.299 2.14 0.312 2.29 0.685 5.05 2.211 15.39 Panel B. VW 15 0.053 2.09 0.020 0.81 0.070 2.94 0.007 0.27 30 0.270 3.18 0.214 2.63 0.418 5.19 0.906 10.53 45 0.374 3.14 0.337 2.92 0.589 5.14 1.739 14.38 60 0.691 4.85 0.588 4.23 0.929 6.73 2.406 16.47 Panel C. DEC 15 0.039 1.52 0.006 0.25 0.056 2.38 -0.002 -0.07 30 0.154 1.82 0.110 1.36 0.312 3.91 0.822 9.68 45 0.241 2.05 0.210 1.84 0.476 4.21 1.628 13.67 60 0.452 3.21 0.451 3.29 0.824 6.06 2.336 16.28 Panel D. EWM 15 0.017 0.67 -0.004 -0.15 0.045 1.91 -0.022 -0.90 30 -0.117 -1.36 -0.031 -0.38 0.141 1.73 0.403 4.63 45 -0.095 -0.78 0.011 0.09 0.191 1.62 0.971 7.73 60 -0.026 -0.17 0.216 1.46 0.406 2.75 1.673 10.63 Panel E. VWM 15 0.039 1.57 0.022 0.91 0.070 2.98 0.004 0.17 30 0.189 2.18 0.234 2.80 0.427 5.17 0.808 9.17 45 0.229 1.84 0.337 2.78 0.563 4.68 1.511 11.87 60 0.522 3.38 0.633 4.18 0.910 6.06 2.117 13.33 Panel F. Three-factor 15 0.034 1.37 -0.002 -0.068 0.048 2.06 -0.014 -0.56 30 0.073 0.85 -0.009 -0.105 0.237 2.89 0.629 7.28 45 0.088 0.71 0.045 0.373 0.316 2.68 1.306 10.49 60 0.264 1.72 0.283 1.900 0.594 4.04 1.913 12.29 3.4.3 Earnings Surprise, AVOL3D, and AVAR3D Does the price movement documented in the previous section simply capture the well-known PEAD anomaly first observed by Ball and Brown (1968)? The diffusion theory asserts that it is the information uncertainty rather than the magnitude Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 of surprise that drives PEAD. An important implication of this perspective is that there would be no PEAD when an announcement does not have significant information uncertainty even if the earnings surprise is large. On the other hand, there would be a significant PEAD when the announcement has high information uncertainty, even if the earnings surprise is relatively small. Specifically, a simple one-dollar earnings surprise may not cause a drift, whereas a complicated one-cent earnings surprise may 98 lead to a long-lasting price drift. This intuition from the information diffusion perspective is tested using a two-way sorted portfolio analysis. Ten-by-ten portfolios are formed on earnings surprise and AVOL3D deciles. Panel A of Table 3.5 shows the CARs from equal-weighted market returns for each portfolio formed on SUE deciles and AVOL3D deciles. Decile 1 has the smallest SUE or AVOL3D values, while decile 10 has the largest. Within AVOL3D deciles 1 to 3, the difference in CAR between SUE decile 10 and decile 1 is 0.94%, 0.42%, and 1.11%, respectively, insignificantly different from zero for a p-value of 0.05. However, the difference becomes significantly different from zero for AVOL3D deciles 4 to 10. The difference ranges from 1.81% within AVOL3D decile 4 to 3.94% within AVOL3D decile 10. On the other hand, within SUE decile 1 (the most negative earnings surprise), the difference in CAR between AYOL3D deciles 10 and 1 is about -1.30%, significantly different from zero. When SUE becomes positive (SUE deciles 5 to 10), the difference in CAR between AVOL3D decile 10 and 1 becomes 2 8 The anomaly documented in the previous section and PEAD can be reconciled only if earnings surprise also captures a large portion of information uncertainty. If the number of the possible causes of earnings surprise is positively related to the magnitude of the surprise, the discussion in Appendix 1 does indicate that larger earnings surprise implies higher entropy (uncertainty). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 significantly positive regardless of the magnitude of SUE. These results suggest that some portion of the information contained in AVOL3D is orthogonal to the information in SUE. Furthermore, the preliminary evidence supports the belief that it is information uncertainty rather than the magnitude of surprise per se that drives the post-earnings-announcement drift. In other words, the PEAD may not exist for 9 Q announcements that have large earnings surprise but induce little uncertainty. This inference is striking and has important implications, and therefore it is worthy of further examination. I use the earnings surprise (UE) based on IBES analyst forecasts to repeat the previous tests. Panel B of Table 3.5 presents the results on the portfolios formed on the UE and AVOL3D deciles. Although the number of observations is substantially reduced from 212,164 to 90,817, the conclusions are robust for this alternative specification. 2 9 The same tests are extended to the 100 portfolios based on SUE deciles and AVAR3D deciles. The analysis reveals that the difference between the CARs of SUE deciles 10 and 1 increases, in general, with AVAR3D. However, the difference is statistically different from zero for the announcements in the low deciles of AVAR3D. This is not quite surprising. A substantial portion of the information in AVAR3D and SUE may overlap. Large SUE causes a large change in price from day -2 to 1, which in turn leads to a large AVAR3D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Table 3.5 Two-Way Sort: Earnings Surprise and Abnormal Trading Volume Panel A: CARs for 100 Portfolios Formed on SUE and AVOL3D Deciles (In Percentage) AVOL Deciles (Mean AVOL3D in parenthesis) SUE Deciles (mean SUE) D1 (-1.87) D2 (-1.18) D3 (-0.78) D4 (-0.41) D5 (0.02) D6 (0.58) D7 (1.34) D8 (2.46) D9 (4.36) D10 (12.57) D10-D1 T-Stats D1 (-4.45) -1.02 -0.80 -0.73 -1.56 -1.11 -1.97 -1.84 -1.48 -2.15 -2.32 -1.30 -1.89 D2 (-1.01) -1.65 -0.93 -1.16 -0.74 -1.49 -1.54 -1.12 -1.51 -1.23 -2.35 -0.71 -1.08 D3 (-0.39) -2.26 -0.94 -1.29 -1.43 -1.21 -0.59 -0.81 -1.49 -1.96 -2.17 0.08 0.13 D4 (-0.06) -1.02 -0.47 -0.06 -0.12 -0.66 -0.09 -1.31 -0.65 -1.59 -1.30 -0.28 -0.42 D5 (0.12) -0.93 -0.10 -0.23 -0.16 0.44 -0.37 -0.91 0.13 -0.70 1.09 2.02 3.34 D6 (0.33) -0.59 -0.22 -0.60 -0.05 0.11 0.12 0.04 0.04 0.37 0.59 1.18 2.05 D7 (0.62) -0.82 -0.30 0.47 0.68 0.63 0.18 -0.37 0.45 0.13 1.04 1.86 3.33 D8 (1.04) -0.36 0.21 0.08 -0.02 -0.20 -0.60 0.11 0.58 0.60 1.02 1.37 2.42 D9 (1.74) -0.80 -0.10 0.25 0.46 0.30 0.16 0.26 0.29 0.63 1.45 2.24 4.22 D10 (3.86) -0.09 -0.38 0.38 0.25 0.36 0.86 0.03 0.92 0.32 1.61 1.70 2.97 D10-D1 0.94 0.42 1.11 1.81 1.47 2.83 1.87 2.40 2.47 3.94 T-Stats 1.48 0.69 1.72 3.07 2.47 4.92 3.22 4.03 4.19 6.23 Panel B: CARs for 100 Portfolios Formed on UE and AVOL3D Deciles (%) AVOL Deciles (Mean AVOL3D in parenthesis) UE Deciles D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 Mean UE (%) (-1.82) (-1.09) (-0.67) (-0.27) (0.21) (0.81) (1.61) (2.75) (4.70) (12.49) D10-D1 T-Stats D1 (-226.58) 0.42 0.85 0.22 -0.85 0.85 -1.59 -1.58 -0.58 -0.39 -1.70 -2.11 -1.76 D2 (-26.43) -1.58 -0.51 -1.00 -0.30 -2.03 -2.04 -1.91 -2.28 -2.34 -1.75 -0.17 -0.16 D3 (-9.54) -2.27 -2.07 -1.18 -1.49 -0.99 -2.35 -3.02 -2.44 -3.07 -2.60 -0.32 -0.33 D4 (-2.19) -1.89 -1.10 -2.67 -1.88 -2.44 -2.86 -2.37 -1.51 -2.10 -2.75 -0.87 -0.93 D5 (0) -1.57 -1.85 -2.55 -1.86 -2.43 -1.97 -2.17 -1.33 -3.02 -3.08 -1.51 -1.32 D6 (2.41) -2.12 -2.26 -1.96 -1.90 -2.25 -2.78 -2.04 -1.95 -1.61 -1.10 1.02 1.09 D7 (6.03) -1.69 -0.47 -1.11 -1.87 -1.17 -1.63 -0.28 -1.72 -1.48 -0.91 0.78 0.76 D8 (11.64) -2.70 0.57 -1.38 0.37 0.81 -0.28 0.53 0.38 -0.55 -0.76 1.94 1.72 D9 (24.18) 0.06 -0.36 0.20 -0.54 0.41 1.38 1.27 2.57 1.29 1.40 1.34 1.09 D10 (197.60) -1.12 -0.56 0.38 1.29 2.97 1.46 1.41 2.03 3.04 3.71 4.84 3.89 D10-D1 -1.54 -1.41 0.16 2.15 2.12 3.05 2.99 2.61 3.43 5.41 T-Stats -1.26 -1.19 0.15 1.71 1.82 2.59 2.40 2.18 2.85 4.42 -J to 73 In summary, the empirical results support hypothesis 1. That is, information diffusion and the subsequent price drift can be observed only when the information uncertainty of a signal is high. The traditional earnings surprise measure appears to capture some portion of this information uncertainty (Bamber, 1987; Subramanyam, 1996). These results suggest that other measures of information uncertainty may also be related to post-announcement drift. The diffusion model predicts that information content (uncertainty) and information conductivity jointly determine the trajectory of price movement. We now turn to regression analyses to examine hypotheses 1 and 2. 3.4.4 Regression Analysis Tests of Hypotheses 1 and 2 Hypothesis 2 posits that the speed of information diffusion is affected by information conductivity. In the tests that follow, information conductivity for a firm is proxied by the number of analysts following and institutional ownership. However, when there is very little information uncertainty, information diffusion is unlikely to be detectable, and information conductivity will not impact pricing. In addition, the number of analysts following and institutional ownership are highly correlated with firm size.3 0 Therefore, I use a regression analysis to test hypothesis 2. The full regression equation is as follows: 3 0 The earnings from a large firm may be more difficult to understand. Meanwhile, large firms usually get more attention from analysts and institutional investors. Consequently, the diffusion is affected simultaneously by the information uncertainty about the signal, information conductivity, and firm size. It is hard to construct portfolios to test Hypothesis 2 without controlling for information uncertainty and firm size appropriately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 CARt (15,64) = /?(,+/?!• SUE, + P2AVOLl>Di + &AVAR3D, + pA ANALYST, ■ IDUMi + pJNST, • IDUM, + /?6 S/Z£, +£, , (3-6) where CAR (15, 64) represents the abnormal returns accumulating from day 15 to day 64 and the subscript i represents each firm-announcement. The daily abnormal returns are calculated from equal-weighted market returns.3 1 AVOL3D and AVAR3D are the same as defined previously, SIZE is the logarithm of the firm’s market value at earnings announcement dates, and INST is the percentage of institutional ownership measured as the number of shares held by institutional investors divided by total shares outstanding during the month of the earnings announcement. ANALYST equals the logarithm of (1+ number of analysts following the firm). The number of analysts following the firm is the number of IBES analysts who provide earnings estimates in the month of earnings announcements. While some studies use the raw number of analysts (e.g., Bhushan, 1989; Brennan and Hughes, 1991), I follow Hong, Lim, and Stein (2000) and use log (1 + number of analysts) to capture the intuition that one extra analyst should matter much more if a firm has few analysts than if it has many. IDUM is a dummy variable capturing the information content. For a good news announcement, IDUM is “1” if the announcement is in SUE decile 8, 9, or 10 and “0” in all other cases. For a bad news announcement, IDUM is “1” if the announcement is in SUE decile 1, 2, or 3 and “0” in all other cases. The interaction terms ANALYST • IDUM and INST ■ IDUM capture the model implication that the number 3 1 Prior PEAD studies usually use the decile-adjusted abnormal returns (e.g., Bernard and Thomas, 1989). For comparison, I use the decile-adjusted abnormal returns to check the conclusions from the regression analysis. The main results are robust. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 of analysts following and institutional ownership matter only when the information content of the earnings signal is salient. Panel A of Table 3.6 provides signed predictions for each variable. The theoretical construct “information uncertainty” is proxied by AVOL3D, AVAR3d, and SUE. The information diffusion model predicts that the magnitude of total drift is positively associated with an earnings announcement’s information uncertainty. Thus, the signs for AYOL3D and AVAR3D will be positive (negative) for good (bad) news earnings announcements. The sign for SUE is expected to be positive for both good and bad news. (The sign for bad news is true when CAR and SUE are both negative for bad news classified by price change.) The significance of these coefficients provides further evidence of whether these three variables measure the same information set. The theoretical construct “information conductivity” is proxied by ANALYST and INST. The sign is predicted to be positive (negative) for good (bad) news for both variables.3 2 These predictions imply that the CARs from day 15 to day 64 are higher if a firm has a larger number of analysts following and higher institutional ownership after controlling for information uncertainty. It is important, however, to reiterate the reason (see Panel B of Figure 3.1). The diffusion process is sufficiently long (more than one quarter), and the abnormal returns accumulate from the very beginning of the diffusion process (i.e., from day 15). SIZE is used as a control variable, and the sign on SIZE should be negative for both good and bad news. 3 2 Although the correlation between ANALYST and INST is 0.46, the correlation between ANALYST*IDUM and INST*IDUM is as high as about 0.9 for both bad and good news groups. Two interaction terms are thus not used in the regression simultaneously. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 This size variable attempts to control for the size premium of small firms, i.e., for good (bad) news, CARs are more (less) positive (negative) with small firms. Panel B of Table 3.6 presents the results of regressing CAR (15, 64) on various subsets of the above variables for each of the bad and good news groups. Several general findings are revealed from the regressions. First, for good news, the sign for each variable is significant and consistent with predictions. AVOL3D, AVAR3D, and SUE are all significantly positive. This indicates that CAR is positively correlated with the magnitude of information uncertainty and that no single variable dominates the other two, suggesting that each information variable consists of some portion of information that is orthogonal to the others. The sign for the number of analysts following and institutional ownership is also positive, suggesting that larger information conductivity leads to larger abnormal returns from day 15 to day 64. Second, for bad news, the sign for each variable is, in general, also consistent with the prediction. The only exceptions are that, among the three information content variables, SUE and AVOL3D become marginally significant or insignificant depending upon the model specification, while the significance of AVAR3D is robust in all models. The results support the hypothesis that CAR decreases (becomes more negative) for negative news when information uncertainty is high. Again, the negative sign for analysts following and institutional ownership suggests that they affect the • 7-3 speed of information diffusion in the predicted direction. 3 3 It is not appropriate to directly compare the positive relation between CAR (15, 64) and analysts following and institutional ownership identified here with the findings in Bartov et al. (2000) and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 The above specification uses IDUM to capture the information content. Alternatively, the results are robust for good news group when IDUM is replaced with VDUM. VDUM is a dummy variable. It is “1” if the AVOL3D is positive and “0” otherwise. For bad news group, the coefficients on the interactive terms become insignificant. In summary, the regression analysis in this section provides evidence supporting both hypotheses 1 and 2.3 4 It refines our understanding of the PEAD anomaly and market efficiency. Using the earnings announcement setting as a specific example, we may argue that even if the time of diffusion can be shortened with large number of analysts following and institutional investors, the continuous price movement following an information event with high information uncertainty cannot be completely eliminated in capital markets. In this sense, the existence of the information diffusion mechanism imposes a boundary on market efficiency. Mikhail et al. (2003) that the total PEAD is negatively associated with institutional ownership and the experience of analysts. This paper focuses on the diffusion period (smooth phase), so the initial market response is beyond the scope here, while the traditional PEAD captures the greatest portion of the market response in the short window surrounding the announcements. The seeming inconsistency between this study and the others may be reconciled if there is a large number of analysts following and institutional investors help to mitigate the initial market response. 3 4 The regression yields similar conclusions when SUE is replaced with the UE variable. The results are generally robust when the regression is conducted with quarterly and annual announcements only, except that the coefficients on analysts and institutional ownership are insignificant for bad news in the regressions with the annual announcement sample. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced w ith permission o f th e copyright owner. Further reproduction prohibited without permission. Table 3.6 Regression Tests of Hypotheses 1 and 2 Panel A. Summary of Sign Prediction Theoretical Construct Proxies Good News Cumulative Abnormal Return Bad News Information Content Abnormal Trading Volume Positive Negative (uncertainty) Abnormal Return Volatility Positive Negative Earnings Surprise Positive Positive Information Conductivity Number of Analysts Following Positive 4 Negative4 Institutional Ownership Positive 4 Negative4 Control Variables Size Negative Negative A These matter only when information content is high. Panel B. Regression Results (DV: CARs from Equal -W eighted M arket Returns) G ood N ews Group Variables \Model 1 2 3 4 5 6 7 8 9 10 Intercept 0.0389 0.216 0.343 -0.140 -0.030 0.203 -0.157 14.319 15.685 11.036 (0.66) (4.03)*** (6.28)*** (-2.24)** (-0.48) (3.71)*” (-2.48)’* (34.38)*** (30.75)**’ (15.50)*’* SUE 0.147 0.139 0.146 0.139 0.179 0.130 0.072 (5.74)’" (5.47)*** (5-71)*** (5.49)*’* (6.68)“ * (3.93)’* ’ (1.82)’ AVOL3D 0.010 0.091 0.093 0.081 0.066 0.072 0.062 (7.60)*** (7.22)’* * (6.75)’* * (6.15)*** (5.07)*** (4.39)*** (2.88)’" AVAR3D 0.009 0..010 0.004 0.005 0.005 0.009 0.010 (2.39)** (2.99)*** (1.27) (1.78)* (1.58) (4.83)*** (2.16)" Analyst* IDUM 0.452 (4.17)’* * Inst*IDUM 1.863 (5.89)*** Size -1.163 -1.275 -0.925 (-36.7)*** (-32.62)*** (-17.7)*" N 108,194 163,244 163,244 108,194 108,194 163,244 108,194 108,194 80,980 49,776 Adj. R2 (%) 0.04 0.09 0.02 0.13 0.08 0.09 0.13 1.62 1.67 0.73 o o CD ■ o - s o Q. I " O CD — i 3 c / ) c /j o' o o CD a. Table 3.6 (Continued) Bad News Group o o CD —s Variables\ Model 1 2 3 4 5 6 7 8 9 10 Intercept -0.911 -0.642 -0.603 -0.834 -0.760 -0.583 -0.743 8.840 9.030 3.544 T 1 (- (- (- (- (- (- ( - 3. 3" 15.25)*** 11.79)** 10.58)*** 13.49)*** 11.75)*** 10.2)*** 11.42)” (20.50)*” (16.94)*” (4.82)*** CD SUE 0.002 0.002 0.002 0.002 0.003 0.002 0.047 CD (1.76)* (1.78)* (1.84)* (1.83)*’ (1.70)’ (2.45)** (1.23) ■ o AV0L3D -0.060 -0.056 -0.042 -0.030 -0.004 -0.003 -0.030 O Q . (- (- (- C & 5.03)’* * (— 4.31)*** 3.38)*** 2.12)** (-0.28) (-0.17) (-1.43) o o AVAR3D -0.019 -0.026 -0.014 -0.022 -0.021 -0.019 -0.018 ■ O (- (- o S ’ (-4.80)’* * (-5.48)*** 3.33)*’* 4.43)*** (-4.27)**’ (-3.16)“ * (-2.54)” & Analyst* CD Q _ IDUM -0.212 $ (-1.66)* l-H Inst* o c IDUM -1.766 l-H ■ O /Tv (-5.26)*” v U - 5 - T Size -0.780 -0.764 -0.368 B w' (-23.4)” * (-18.8)” * (-6.78)*** c/> o' N 101,576 154,253 154,253 101,576 101,576 154,253 101,576 101,576 75,029 44,947 3 Adjusted R2 (%) 0.00 0.02 0.02 0.02 0.04 0.03 0.04 0.67 0.64 0.18 VO 80 3.4.5 Portfolio Tests of Hypothesis 3 The test of the superposition effect of diffusion requires two consecutive earnings announcements with both having significant information content. The observations in the fourth quartile of AVOL3D and AVAR3D in both current and prior fiscal quarters are employed to test hypothesis 3. This sample consists of 6,492 announcements from 1972 to 2002. The observations are then sorted into four portfolios: Good-Good (P1P2), Bad-Good (N1P2), Good-Bad (P1N2), and Bad-Bad (N1N2), where P1P2 (N1N2) consists of the observations with good (bad) news followed by good (bad) news, and N1P2 (P1N2) consists of the observations with bad (good) news followed by good (bad) news. If hypothesis 3 is descriptive and price movement captures the superposition effect of information diffusion, the abnormal returns between portfolios P1P2 and N1P2 and between P1N2 and N1N2 will be different. In contrast, if the information in the first announcement is completely impounded in the price before the second announcement (i.e., the diffusion process has completed in the first quarter), there should be no difference in portfolio returns Figure 3.4 shows the CARs from day 15 to day 64 after the second announcement. CARs are accumulated for the daily abnormal returns from equal- weighted market returns.3 5 First, the magnitude of cumulative abnormal returns at day 64 generally has the following order: CAR(P1P2) > CAR(N1P2) > CAR(P1N2) > CAR(N1N2), consistent with the residual information from the first announcement 3 5 Using other expected-retums benchmark models yields the same results. When I estimate the market model and the three-factor model, I use the window (-365,-10) relative to the first announcement in order to avoid the systematic biases caused by good and bad news in the first period. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 impacting the price movement in the second period. Second, the difference in abnormal returns between P1P2 and N1P2 is an economically significant 3% at day 64. For bad news in the second period, there is no detectable difference in CARs between P1N2 and N1N2. This suggests that the residual information from the first announcement, if any, is no longer important when the second announcement is bad news. Figure 3.4 Cumulative Abnormal Returns during the Second Quarter after Two Consecutive Quarterly Announcements with High Information Content 3 » o of 5 3 P1N2 N1N2 2 1 0 1 2 3 ■ 4 Trading Days The result that prior period information affects the current period price movement is interesting, although the empirical evidence reveals that the first diffusion process affects the second one only when the subsequent announcement is good news. One possible explanation is that the market responds differently to bad news and good news (e.g., Hayn, 1995). If bad news diffuses fast and impounds into Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 the price more quickly, then it is possible that there is not much residual information carried over to the subsequent period. 3.5 Sensitivity Tests The results reported in the above sections are robust to the following alternative measures: (1) use the direction of abnormal return (raw returns adjusted for equal-weighted market returns) to classify news; (2) use earnings surprise calculated from IBES data instead of the standardized earnings surprise based on seasonal random walk model; (3) use buy-and-hold returns from day 15 to 64 instead of the simple summation of daily abnormal returns; (4) exclude the observations with price lower than $3 from the sample. This section presents additional discussions on several other issues: quarterly vs. annual earnings announcements; NYSE&AMEX vs. NASDAQ firms; news classification with earnings surprises (SUE); persistence of information diffusion; risk vs. diffusion; and momentum effect. 3.5.1 Quarterly vs. Annual Earnings Announcements Prior accounting literature shows that the market responds to both quarterly and annual earnings and post-earnings-announcement drift exists for both types of announcements (Kothari, 2001). Therefore, the main tests in this study pool quarterly and annual announcements together. Although annual announcements mainly contain the information in the fourth fiscal quarter, the time allowed by SEC to file annual financial statements after the fourth quarter is twice as long (90 days) as the time Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 allowed to file quarterly statements (45 days). Consequently, the starting time of information diffusion may be different for the two types of financial statements. Figure 3.5 and unreported results reveal that the conclusion on hypothesis 1 is, in general, insensitive to the choice of quarterly or annual announcements. Figure 3.5 Cumulative Abnormal Returns for Good and Bad News, Quarterly and Annual Announcements that Belong to the Fourth Quartiles of AVOL3D and AVAR3D. P(N) represents good (bad) news. An announcement is classified as good (bad) news if the price change from day -2 to day 1 is positive (negative). Quarter (Annual) represents quarterly (annual) announcements. Regression analysis on either quarterly or annual announcements indicates that hypothesis 2 is generally supported for both. However, there are two differences between quarterly and annual announcements in regression analysis: First, the coefficient for AVOL3D is insignificant after controlling SUE and AVAR3D for -3 Days Quarter-P Quarter-N ■Annual-P Annual-N Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 annual announcements, while it is highly significant for quarterly announcements; Second, the coefficient on analysts and institutional ownership variables becomes insignificant for bad news in the regressions for the sample with annual announcements only. 3.5.2 NYSE & AMEX vs. NASDAQ Firms NYSE & AMEX are the exchanges with specialists, while NASDAQ is a dealer market. This difference in market structure may affect the speed of information diffusion. The data shows that the difference in the CARs from day 15 to 64 between good and bad news portfolios in NASDAQ market is twice as big as that in NYSE & AMEX. The difference is about 3% for NYSE & AMEX firms, while it is about 6% for NASDAQ firms. Exploring the reason why there is such a big difference is not the focus of this study. The result is consistent with either diffusion theory (that information about NASDAQ firms diffuses faster) or transaction costs argument (that the transaction costs for firms traded on NASDAQ is higher (Stoll, 2000; Huang and Stoll, 1996). 3.5.3 News Classification with Earnings Surprise In all of the above tests, good and bad news is classified by the change of price in the announcement window. As a result, we do not have to justify the true expectations on earnings in the market. One may naturally ask: Is the conclusion robust if the news is classified by the sign of earnings surprises? Again, the four Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 quartiles of AYOL3D are divided into eight portfolios, but good and bad news are classified with the sign of SUE. Positive (negative) earnings surprises stand for good (bad) news. Figure 3.6 shows the CARS from the equal-weighted market returns for these eight new portfolios. It appears that hypothesis 1 is supported. Comparing Figure 3.6 with the Panel B of Figure 3.3, we see that the eight curves are very similar. An additional analysis reveals that many announcements with positive price change are identified as announcements with negative earnings surprises, and vice versa. It indicates that the eight portfolios are different. Figure 3.6 Cumulative Abnormal Returns for Good and Bad News Firms (Classified by SUE) in Each Abnormal Trading Volume Quartile 0.5 3 l O 45 ^ -0.5 -1.5 Trading Days N -Q l -N-Q2 N-Q3----------------------- N-Q4I P -Q 1 .P -Q 2 P-Q3 - - - P-Q4] 3.5.4 Persistence of Information Diffusion Schwert (2002) concludes that many anomalies tend to decrease over time since they were discovered. So one may ask how persistent is this diffusion process Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 documented? I explore the issue by separating NASDAQ and NYSE firms because there are no NASDAQ firms in the 1970s. Only those announcements in quartile 4 of AVOL3D and AVAR3D are included in my sample. For NYSE & AMEX firms, the CAR (15, 64) is 3.0%, 3.6%, and 2.0% in the 1970s, 1980s, and 1990s, respectively. However, for NASDAQ firms, the CAR (15, 64) is 4.9% and 6.2% in the 1980s and 1990s, respectively. Pooling the sample together, the CARs (15, 64) in the 1980s and 1990s are all about the same. The results show that the magnitude of abnormal returns following earnings announcements with high information content does not decrease over the last thirty years. 3.5.5 Risk vs. Diffusion Are good news firms riskier than bad news firms? Bernard and Thomas (1989) justify that risk misspecification is not plausible. One reason is that the change in beta (a proxy for risk) can only explain a small portion of PEAD. I examine the beta from market models and the loading on the market factor in the Fama and French three- factor model. When I classify the sample with SUE deciles, I find the beta (loading) in the top decile is about 10% larger than the beta (loading) in the bottom decile. It confirms the findings in BT (1989) and Ball, Kothari, and Watts (1990). However, when these same announcements are sorted into quartile 4 of AVOL3D and AVAR3D, I find that the beta (loading) is smaller for good news firms than for bad news firms. In addition, the price to book value (P/B) ratio is 11.47 for the good news Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 portfolio and 13.96 for the bad news portfolio. The firm size of the two portfolios is similar. This evidence is inconsistent with the risk argument. Furthermore, I construct a zero-investment portfolio based on the good (bad) news categorization and the AVOL3D and AVAR3D fourth quartile. Figure 3.7 shows 50-day buy-and-hold returns each year from 1973 to 2001. The difference in CAR between good and bad news portfolios has been nonnegative for thirty years and shows no particular trend. The average 50-day return is 4.23% from 1973 to 2001. Figure 3.7 The Returns from the Zero-Investment Portfolio Flolding for 50 Days Built on the Good/Bad News Classification, AVOL3D, and AVAR3D Each Year from 1973 to 2001 10 973 1976 1979 1982 1985 1988 1991 1994 1997 2000 -2 J --------------------------------------------- Year 3 6 In Bernard and Thomas’s study (1989), each $1 long position in top SUE portfolio is always offset by a $1 short position in bottom SUE portfolio with similar size. Rebalancing sometimes requires waiting after earnings announcements until an offsetting match is available. When I calculate the return of zero- investment portfolio, I simply subtract the abnormal returns of bad news portfolio from the abnormal returns of good news portfolio (both abnormal returns are calculated by subtracting equal-weighted market returns from raw returns). To implement such a strategy, simple imagine that an investor has an intermediate equal-weighted market index account. When good (bad) news appears, she buys (sells) the good (bad) news stocks and sells (buys) the market index. In this way she does not have to wait for the match coming, though the transaction costs increase. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 3.5.6 Momentum Effect One may think that the separation of price movement may simply document a momentum effect. However, further tests indicate that it seems unlikely that the price pattern identified in this study can be explained by the momentum effect, i.e. good (bad) news firms having been drifting upward (downward) before earnings announcements. I examine the trajectory of abnormal returns in the quarter before the earnings announcements with high information uncertainty for both good and bad news portfolios. While there is weak indication that the difference of CARs from -64 to -5 between the two portfolios is nonzero (1%), the magnitude is only about one fourth of the difference in the subsequent quarter (4%). In addition, the abnormal returns from equal-weighted market returns for good news firms do not move upward before the announcements. 3.6 Conclusions This study finds results consistent with the implications from the information diffusion theory. Stock price continues to move upward (downward) after good (bad) news when earnings announcements have high information uncertainty, as measured by high abnormal trading volume and return volatility. The speed of information diffusion is positively correlated with analyst following and institutional ownership. Consequently, the magnitude of post-eamings-announcement abnormal returns is higher with greater analyst following and institutional ownership. The study also finds Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 that, in the setting of two consecutive announcements with high information uncertainty, the price movement after the second earnings good news announcement is affected by the information contained in the first earnings announcement. This finding is consistent with the superposition effect of two diffusion processes. This chapter also shows that the market responses in long and short windows are interrelated. Thus, the study brings together two related strands of literature originated with Ball and Brown (1968) and Beaver (1968). Prior studies have used earnings surprise to classify good/bad news and to explore the existence and determinants of PEAD. As an implication of the diffusion theory, the finding that there is (no) significant drift when there is (no) uncertainty contained in the earnings signal, regardless of the magnitude of earnings surprise broadens our understanding of the PEAD. If the magnitude of earnings surprise is correlated with the information content (uncertainty) of the earnings signal, information diffusion theory may well provide a rational explanation for this long-lasting puzzle. A full understanding of information diffusion has policy implications for corporate disclosure and accounting regulations. Recent studies have found that abnormal trading volume and return volatility in response to earnings announcements has increased over time (e.g., Landsman and May dew, 2002). Information diffusion is jointly determined by information content and information conductivity. While improvement of the information environment increases the speed of information diffusion, an increase in information uncertainty decreases the speed of information Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 diffusion. Under this framework, the issues of where and how some specific accounting information is disclosed may affect the price formation process. Finally, this study explores the notions of bounded efficiency in capital markets. The boundaries are imposed by the nature of information signals, the imperfection of information environments, and investors’ difficulty in processing information. The empirical findings indicate that future returns are predictable with past price and volume information when earnings announcements have high uncertainty. Traditional asset pricing theories typically examine prices at equilibrium states and do not focus on the information assimilation process. They assume that investors understand the information instantaneously upon the arrival of the information. The findings in this chapter seem to suggest that the limited speed of information diffusion may make a weak-form efficient market approachable only under certain circumstances. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Chapter 4 Concluding Remarks If, as Paul Samuelson has suggested, financial economics is the crown jewel of the social sciences, then the Efficient Market Hypothesis (EMH) must account for half the facets (Lo, 1997). The EMH has been hotly debated in the research community for the last decade. Some researchers hold the belief that the market is efficient (e.g. Fama 1970, 1990, 1998) while others have the view that the market is inefficient (e.g., Lo and Mackinlay, 1999; Shleifer, 2000; Lee, 2001). The scenario reminds us of the debate on the nature of light in the history of science. Despite Young’s beautiful double-slit experiment on interference, it took physicists a hundred years to reconcile the different views between Thomas Young and Isaac Newton. We eventually learned that the nature of light has wave-particle duality nature. The EMH is likely to be debated for quite some time. The mounting empirical evidence certainly raises the possibility of market inefficiency, and a few recent economics models attempt to approach this puzzle by incorporating the biases in the decision making of human beings. The central argument of this thesis is that market efficiency is bounded. Based on the conceptual diffusion framework outlined, this dissertation indicates that the market is efficient if drift force is very large or information conductivity is close to infinite. It explains why most of the time we observe an efficient market. On the other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 hand, if prices do not converge to a new equilibrium before the drift force disappears, the diffusion force imposes a boundary on market efficiency. The diffusion framework suggests: (1) Each information signal in the capital markets has a different degree of information content (uncertainty); (2) Each firm has a unique information conductivity, just as each material has its own electrical conductivity. Accordingly, the information on different firms diffuses across investors at different rates after an information event. The price movements consist of many information diffusion processes. The inefficiency can be observed only when the uncertainty is high and information conductivity is low. The arguments are seemingly supported by the limited empirical results presented in this thesis. The approach is different from the derivation of other existing behavioral finance models. The diffusion model is a process model based on the belief that nature and our social systems demonstrate that the general movement of a macro system is independent of the microscopic details. To identify the mechanism at the macro level, we need to turn to the inductive approach in science. In other words, when a theory is developed, the theory should survive many falsifications, e.g., the three laws of thermodynamics. The diffusion mechanism in capital markets will also rely on whether it can survive many future falsifications. In short, diffusion exists because information signals have high information content, and investors have bounded rationality. It persists for such a long time because of limited arbitrage. The existence of a diffusion mechanism leads to a boundary of market efficiency. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 BIBLIOGRAPHY Amin, K., and C. Lee, 1997, Option Trading and Earnings News Dissemination, Contemporary Accounting Research, 14, 153-192. Bachelier, L., 1900, Theorie de la speculation, Ann. Sci. Ecole Norm Sup. 17, 21-86. 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Willis, 2003, Security Analyst Experience and Post- Earnings Announcement Drift, Journal o f Accounting, Auditing and Finance/KPMG conference. Osborne, M., 1959, Brownian M otion in the Stock Market, Operations Research, 7, 145-173. Patell, J., 1976, Corporate Forecasts of Earnings Per Share and Stock Price Behavior: Empirical Test, Journal o f Accounting Research, 14, 2, 246-276. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Patell, J., and M. Wolfson, 1984, The Intraday Speed of Adjustment of Stock Prices to Earnings and Divend Announcemnts, Journal o f Financial Economics, 13, 223-252. Peng, L., 2002, Learning with Information Capacity Constraints, Working Paper, Duke University. Peng, L., and W. Xiong, Capacity Constraint Learning and Asset Price Co-movement, Working Paper, Duke University and Princeton University. 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Sengupta, P., 1998, Corporate Disclosure Quality and the Cost of Debt, The Accounting Review, 73, 4, 459-474. Shannon, C., 1948, A Mathematical Theory o f Communication, The University of Chicago Press. (Originally published in Bell System Technical Journal, 27: 379-423, 623-656, 1948). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 Shleifer, A., 2000, Inefficient Markets: An Introduction to Behavioral Finance, Oxford University Press. Shleifer, A., and Vishny, R., 1997, The Limits of Arbitrage, Journal o f Finance, 47, 1343-66. Shores, D., 1990, The Association between Interim Information and Security Returns Surrounding Earnings Announcements, Journal o f Accounting Research, 28, 1, 164-181. Sims, C., 2001, Implications of Rational Inattention, Working Paper, Princeton University. Soffer, L., and T. Lys, 1999, Post-eamings Announcement Drift and the Dissemination of Predictable Information, Contemporary Accounting Research, 16,2, 305-331. Stoll, H., 2000, Friction, Journal o f Finance, 55, 1479-1514. Subramanyam, K. R., 1996, Uncertain Precision and Price Reactions to Information, The Accounting Review, 71,2, 207-220. Thaler, R., 1993, Advances in Behavioral Finance, Russell Sage Foundation, New York. Theil, H., 1967, Economics and Information Theory, North-Holland Publishing Company, Amsterdam; Rand McNally & Company, Chicago. Verrecchia, R., 1982, Information Acquisition in a Noisy Rational Expectations Economy, Econometrica, 50, 1415-1430. White, H., 1980, A Heteroskedasticity-consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity, Econometrica, 48, 817-838. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 APPENDIX 1 ENTROPY AS THE MEASURE OF INFORMATION Entropy was introduced and well established in thermodynamics and statistical physics by Clausius, Boltzman, and Gibbs a century and half ago. Shannon introduces entropy as the measure of information in his seminal paper “A Mathematical Theory of Communication” and lays out the foundation of modem information theory (Shannon, 1948). As a theoretical construct of information content (uncertainty), entropy is also applied to economics (e.g., Theil, 1967; Sims, 2001) and accounting (e.g., Lev, 1969). As Theil (1967) points out: The reason information theory is nevertheless important in economics is that it is more than a theory dealing with information concepts. It is actually a general partitioning theory in the sense that it presents measures for the way in which some set is divided into subsets. This may amount to dividing certainty (probability) into various possibilities none o f which is certain. (Economics and Information Theory, page 19). ■jn The entropy H(X) of a discrete random variable is defined by : m H(.X) = -J ^ P ,lo g P , x = 1 3 7 This appendix only discusses some basics of entropy. More detailed analysis can be found in other references (e.g., Theil, 1967). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 where X is a random variable with m alternative choices, m e [2,oo). p x = p {X = x} is the probability mass function for X, so the probability of each choice is p i,p 2,...,pm. Three basic properties can be derived easily: First, the value of entropy is proportional to the number of choices. Second, entropy depends upon the distribution of the probabilities. With a constant number of choices, entropy is maximized when the probabilities for all choices are equal, which intuitively corresponds to the most uncertain situation. Third, as one choice becomes more probable than the others, the entropy value decreases. Entropy is zero when the probability of one choice becomes 1, i.e., there is no uncertainty at all. The entropy of a random variable X with a continuous probability density function f(x) is defined as follows: H (X)= - J / ( x ) log / (x)dx It is the negative value of the expectation of the logarithm of a random variable's density function. The continuous format shares most properties with the entropy of a discrete random variable. It is easy to prove that, if X has a normal distribution N ( p ,a 2) , then entropy equals the logarithm of variance plus a constant. Proof: p(x) = 1 e x p (-^ -y ), and - log/?(x) = log(2^)1 /2 a + (2k) (7 2cr 2u Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 H (x) = - 1 p(x) log p(x)dx - ^p{x)\og(2K)xncrdx + |/? ( x ) " ~ y dx log(2;r)1/2cr + G l a 2 = log < t + 0.5 log(2^e) The properties of entropy lead us to believe it is a measurable physical quantity, such as density or mass. The physical meaning of entropy is the degree of randomness in a physical system. In an economic system, entropy can be interpreted as the degree of uncertainty. The following simple example illustrates the properties mentioned. Suppose that investors are investigating three cases related to earnings surprise. Each case has an identical $1 earnings surprise but different possibilities for the cause of the surprise. The distribution of the causes and probabilities is: Case A: (0.99,0.01); Case B: (0.5, 0.5); CaseC: (0.1, 0.1, ...,0 .1 ) Both case A and case B have two possible causes, but the initial probability for A is 0.99 and 0.01, and for B it is 0.5 and 0.5. Case C is a very complicated one, with ten equally probable causes. Intuitively, the uncertainty for case C is higher than the uncertainty for case B, which is, in turn, higher than the uncertainty for case A. The entropy for cases A, B, and C is 0.056, 0.693, and 2.303, respectively. Investors gain more information from eliminating uncertainty in case C than case A, so case C has Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 more information content than the other two. In this example, the difference in uncertainty across three cases is due to both the difference in the number of possible causes and the inequality of the probabilities. Applying the concept to earnings announcements, a large earnings surprise may imply, in general, more possible causes so that the entropy of the earnings surprise is large. On the other hand, the distribution of probability may also affect the entropy value. If post-earnings-announcement drift (PEAD) is caused by the resolution of uncertainty (an eventual information gain process), the diffusion model predicts that PEAD should be observed in the circumstances in which the event is classified by the magnitude of the earnings surprise as well as other uncertainty measures. Importantly, the drift should be most significant when both the magnitude of surprises and other uncertainty measures are large. In Chapter 3, I adopt abnormal trading volume and abnormal return volatility surrounding the earnings announcements as the measures of information uncertainty. The diffusion model predicts that high trading volume and return volatility would lead to a significant price movement after the initial market response to the information contained in earnings. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 APPENDIX 2 SOLUTION TO THE PDE WITH SINGLE SIGNAL SETTING This appendix solves PDE (2-11) with the following initial and boundary conditions: dH n d2H dH . . . . D — -—t v -----= 0 (A2-1) dt dx dx IC: H (x,t = 0) = H 0; BC: H (x = 0,0 = 0 ; H (x = oo,t) = H 0 The solution to the PDE may be obtained by recognizing the existence of a similarity variable X through which the PDE may be transformed to an ordinary X differential equation (ODE). Let X = ----- — , hence, 2-[D-t] dH _ dH dl _ 1 dH dx dX dx 2 -(D-t)U 2 dX d2H d ,dH^ dX 1 d 2H (A2-2) = (A2-3) (A2-4) dx2 dX dx dx A D t dX dH dH dX x dH dt dX dt A-t-(D-t) dX t - Defining a transformation variable co = v ■ (—) 2 and substituting (A2-2), (A2- 3), and (A2-4) into equation (A2-1), we can rewrite equation (A2-1) as 3 8 To simplify the expressions, H (x,t) is written as H . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The initial and boundary conditions are transformed as follows: H(x,t = 0) = H 0, H(x = oo,t) H0 ^>H{/1 = 0 0) = H0; and H(x = 0,1) = 0 => H(X = 0) = 0. To get the analytical solution to the transformed ordinary differential equation (A2-5), we treat information conductivity (D) as a constant. Furthermore, we assume the drift velocity (v) is inversely proportional to the square root of time, i.e. v = k / 4 t , where k is a constant, so the variable c o is constant. This assumption captures the features that drift velocity should decay with time quickly. The more time that elapses after the information is announced, the less incentive the investors will have to use the information to seek out mispricing, and, accordingly, the smaller the force that drives the drift. The ODE (A2-5) is solved by a separating variable approach. Rewrite (A2-5) as integrating twice, we get the general solution Applying the boundary and initial conditions to the equation (A2-7), we obtain (A2-7) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 H J1 exp[-(/l* - (of ]dX Ho exp[-(/T - co)2]dX erf (X - co) + erf (co) erf (A - co) + erf (w) erf (oo) + erf (a?) 1 + erf (co) (A2-8) Hence, H = H0- f i — /• x t erf -----, ------ v • J — [ 2 - f D f \D / t + erf v -J — 4 • D / v / / 1 + erf t - \ v • j — V (A2-9) erf (o ) is the Gaussian error function, which is defined as erf (X) = -j= |* exp (~y2)dy Remarks on the Solution Remark 1: Define the information that has been understood as the change in entropy, e.g., I(x,t) = H0 - H(x,t) . Equation (A2-9) demonstrates that the speed of the information understood by different investors at different times depends on the magnitude of the parameters for the diffusion (D) and the drift term (v) as well as the relative weights of the two terms. Figure A2.1 shows how the drift and diffusion terms interact and influence the speed of information diffusion. When the drift term is zero, the ratio of information understood to total information content I(x,t)/ H0 is a complementary error function, i.e., Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The percentage of information understood (reduction in entropy) is described by curve 1 in Figure A2.1 When the drift term exists, the curve shifts to the right. The more the weight on the drift term, the more the resulting curve deviates from curve 1. Curve 2 and 3 are the cases when c o equals 1 and 3, respectively. It indicates that a large drift term greatly speeds up the information diffusion process. Remark 2: The drift force may dominate the diffusion force in the short term. However, it decays with time and afterwards information diffusion is solely determined by the diffusion force. Figure A2.2 shows two scenarios with different information conductivity when there is no drift term. Curve 1 is the same as curve 1 in Figure A2.1. Curve 2 is the scenario that information conductivity is increased by one order. If price converges before the drift force disappears, it will be difficult to observe diffusion. If the price does not reach a new equilibrium when the drift force becomes negligible, diffusion will be observable. Thus, the diffusion force and information conductivity impose a boundary on market efficiency. Remark 3: When only diffusion force exists, the time required to reach a certain uncertainty reduction is proportional to the square of its distance from the information shock, and the time required at the given location to attain a level of uncertainty reduction varies inversely with the magnitude of the information conductivity. Figure A2.3 shows the relation among H(x,t), x, and t when D is fixed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 Figure A2.1. The Relation between I(x,t)/H0 and the Variable A - xl(2^JD-t ) When Both Drift and Diffusion Forces Exist C u rv e 3: D iffusion + Drift (w = 3) 0.8 * . 0 .7 \ C urve 2: \ D iffusion + Drift (w = 1) 0 .4 0.2 C u rv e 1: D iffusion 0 0 .5 1 .5 2 2 .5 Lam da = x/2(D*t)1/2 3 3 .5 1 4 Figure A2.2 The Relation between I(x,t)/H0 and the VariableX = x/(2«jD-t) When Only Diffusion Force Exists i 0 .7 C u rv e 2 0 .4 £ 0.2 C u rv e 1 3 ,5 o 0 .5 1 1 .5 2 2 .5 3 4 L am da = x/2{D*t)1/2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Figure A2.3 The Relation between H(x,t), x and t. t= 1 t= 5 x I t= 10 0 ) > 4 > -J t=20 < v 0 .4 c 3 >. a 2 c L U 0 0.5 1.5 2 .5 3 3.5 4 1 2 Remark 4: The change rate of entropy with location and time is = ----------- H > . • exp(---- — — ), (A2-11) dx [1 + 71 ■ D t 4'D-I = ^ _ . exp(. _ £ l _ )i (A2-12) dt 1 + erf(co) 2J x - D -t3 4-D-t The relationship between the rate of change and distance variable x is straightforward. Expression (A2-11) simply shows that the rate of change decays exponentially with the increase of distance x. The relation between time and the rate of change in information level is more complicated. Using the second derivative of I(x,t) with respect to time t, we can verify the following results: When t is smaller than x2 /(6 • D), the rate of change increases with time t (the first-order derivative is convex). Afterwards, the rate of change decreases with time t (the first-order derivative is concave). It indicates that, after an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 information shock, a peak of the change rate of information level is moving along the investors. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Lu, Hai (author)

Core Title Information diffusion and the boundary of market efficiency

School Graduate School

Degree Doctor of Philosophy

Degree Program Business Administration

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag business administration, accounting,Economics, Finance,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Beatty, Randy (committee chair), Hsiao, Cheng (committee member), Indjejikian, Raffi (committee member), Subramanyam, K.R. (committee member), Swenson, Charles (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-408264

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