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THE ADJUSTMENT OF STOCK PRICES TO NEW INFORMATION: A TEST OF MARKET EFFICIENCY By STEWART L. BROWN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1974 ACKNOWLEDGMENTS The author would like to thank Dr. C. A. Matthews for moral and financial support while at the University of Florida. The author is indebted to Dr. Fred Arditti, Chairman of the Committee, for many helpful comments and suggestions and the rest of the Committee: Drs. John McFerrin, W. A. McCollough, Christopher Barry, and James McClave for moral support and helpful suggestions. All deficiencies and mistakes are, of course, the sole responsibility of the author. TABLE OF CONTENTS ACKNOWLEDGMENTS . . ABSTRACT . . . Chapter I. INTRODUCTION . . Purpose of the Study ... Market Efficiency with Respect to EPS Information7 The Evidence . Assumptions and Methodology . The Ball and Brown Study . The Jones and Litzenberger Study Reconciling the Two Studies . II. METHODOLOGY . . Overview . . Some General Considerations . Earnings Models . . Residual Analysis . . The Capital Asset Pricing Model, the Parameter Estimation Problem, and the Normality Assumption . The Capital Asset Pricing Model The Parameter Estimation Problem The Normality Assumption . Statistical Tests . . III. RESULTS . . Description of the Sample . An Analysis of Regression Results . Residual Analysis for Positive Forecast Error . . The Abnormal Return . . The Adjustment Process . The Speed of Adjustment .. ... Residual Analysis for Negative Forecast Error . . PreAnnouncement Movements in the CAR . . 17 . 47 . 47 S. 49 . 81 . 89 iii ii Chapter IV. SUMMARY AND CONCLUSIONS .. .. 97 Major Results . .. 97 Limitations of the Study .. 98 Some Implications of the Study .. 100 BIBLIOGRAPHY . . .. 104 BIOGRAPHICAL SKETCH. .. . .... 108 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE ADJUSTMENT OF STOCK PRICES TO NEW INFORMATION: A TEST OF MARKET EFFICIENCY By Stewart L. Brown December, 1974 Chairman: Fred D. Arditti Department: Finance and Real Estate The issue of capital market efficiency with respect to publicly available information has not been resolved. Studies examining the adjustment of stock prices to new Earnings Per Share information yield conflicting results. It seems that stock prices do not adjust instantaneously although the adjustment process has not been examined in detail. There is, however, a question as to whether or not it is possible to earn an abnormal rate of return by trading on the basis of publicly available information. The purpose of this dissertation is to examine the adjust ment of the stock market to unexpected changes in EPS and to determine if an abnormal rate of return could have been earned. Cumulative average stock price return residuals from the market model are examined on a daily basis for twenty market days before and through sixty market days after publication of annual EPS numbers in The Wall Street Journal. The market model is used to abstract stock returns from risk and general market movements. Both unexpectedly high and unexpectedly low EPS numbers are examined for a sample of 158 firms listed on the New York or American Stock Exchanges. All firms in the sample have at least a 20 percent change in annual EPS and fourth quarter results different from what could have been predicted on the basis of the first three quarters of EPS. A first order autoregressive scheme is fit to the cumulative average residual and is useful in obtaining an estimate of the confidence interval around the predicted cumulative rate of return of the securities in the sample. Statistically significant returns are observed around the publication date of the EPS number in The Wall Street Journal. The lower confidence limit of the predicted cumulative return is above estimated transactions costs for both long purchases of securities with unexpectedly high EPS and short sales of the securities with unexpectedly low EPS. The adjustment process takes about fortyfive market days after the EPS publication date. The pattern vi of adjustment is similar for both unexpectedly high and unexpectedly low EPS. There appears to be three distinct stages in the adjustment process: an initial price trend directly related to the direction of change in EPS; a second trend inversely related to the change in EPS; and a third trend directly related to the change in EPS, lasting from about day 10 to about day 45 after the earnings announcement. The price change in the third stage appears to be permanent. It is speculated that the. pattern of adjustment is caused by a lag between publica tion of the EPS number in The Wall Street Journal and availability of the more complete information in the Annual Report. It is concluded that the stock market was not efficient with respect to the securities in the sample, since an abnormal return in excess of transactions costs could have been earned by purchasing the securities on the publication date of the EPS number. The implication is that the stock market may, in general, be less efficient than has been heretofore believed. vii CHAPTER I INTRODUCTION Purpose of the Study The purpose of this paper is to test whether or not it is possible, in a statistical sense, to earn a rate of return in excess of the normal rate of return in the stock market by purchasing securities whose earnings per share (EPS) are greater than expected as soon as such information becomes publicly available. A subsidiary purpose is to examine the pattern and speed with which the stock market adjusts the prices of securities to reflect new EPS infor mation. This study is concerned with the means by which information about investments is disseminated to and used by investors. The concern with the effects if information on capital asset prices places this study in the realm of what has come to be called the "Efficient Markets Hypothesis" (EMH). This hypothesis, which would be better labeled the "Efficient Capital Markets Hypothesis," has no generally accepted formal definition other than the rather vague and basic notion that an efficient market is one that fully reflects all available information. Sharpe [33] presents a concise definition: 1 Simply put, the thesis is this: in a well functioning market, the prices of capital assets (securities) will reflect predictions based on all relevant and available information. An implication of market efficiency is that it is difficult to earn a rate of return in excess of the normal rate of return. If the stock market reflects (has discounted) all information, then an investment strategy which is based on some form of information relevant to share prices, says EPS information, will fail to produce an abnormal rate of return because such information will already have been discounted in security prices. The Efficient Markets Hypothesis is an application of the theory of perfect capital markets which, in turn, is derived from the economic theory of price formation under pure competition. The assumptions concerning the effects of information on share price is aptly pointed out by a concise definition of a perfect capital market [27]: In a "perfect capital market," no buyer or seller (or issuer) of securities is large enough for his transaction to have an apprec iable impact on the then ruling price. All traders have equal and costless access to information about the ruling price and about all other relevant characteristics of shares. No brokerage fees, transfer taxes, or other transactions costs are incurred when securities are bought, sold, or issued, and there are no tax differentials .. Thus, in a perfect capital market, with an absence of frictions and a perfect information system, the prices of capital assets are always the "correct" prices in the sense that they always fully reflect all relevant informa tion. Those who believe in the EMH contend that the condi tions in actual capital markets are sufficiently similar to the requirements of a perfect capital market that actual security prices may be thought of as the "correct" prices; i.e., they fully reflect all available information. One difference between a perfect capital market and an efficient market is that an efficient capital market may be efficient with respect to some kinds of information and not efficient with respect to other kinds. Fama [14] has established a hierarchy of information types to delineate different degrees of market efficiency. At its lowest level the stock market is deemed efficient if it reflects historical information. The stock market has been found to be quite efficient with respect to this type of information. Empirical tests at this level, labeled "weakform" tests of the EMH by Fama, include tests of the familiar random walk hypothesis (see Cootner [11]). At the second level, the market is deemed efficient if it reflects all publicly available information. Since publicly available information is also historical informa tion, tests at this level of efficiency are typically concerned with the speed at which the stock market adjusts to new information. Empirical tests at this level have been labeled "semistrong form" tests of the EMH. The hypothesis has received mixed support at this level, but the market has generally been found to be quite efficient (see Fama [14] and Downes and Dyckman [12]). At its highest level, the stock market is deemed efficient if it reflects all information, including infor mation not publicly available. Empirical tests at this level are concerned with whether or not some groups of investors have monopolistic access to sources of informa tion which allows them to reap excess returns. Tests at this level are labeled "strong form" tests of the EMH. There have been few such tests at this level, but those which have been conducted have found some market ineffi ciencies (see Jaffe [18], Niederhoffer and Osboirne [29], and Jensen [19]). Market Efficiency with Respect to EPS Information; The Evidence There are two major studies which examine market efficiency with respect to EPS information. One study, by Ball and Brown [ 3], purports to show that the market is efficient with respect to this type of information; i.e., it is impossible to make money by using EPS information as it becomes publicly available. The other study, by Jones and Litzenberger [20], purports to show that the market is inefficient; i.e., it is possible to earn an abnormal return by trading on publicly available information. Thus, the two studies arrived at opposite conclu sions about market efficiency with respect to EPS informa tion. Each of the two studies will be examined in detail, and some reasons they arrived at different conclusions will be explained and discussed. First, however, some general assumptions and methodological considerations must be discussed so as to establish an analytical framework needed to examine the articles. Assumptions and Methodology An implicit assumption of studies which examine market efficiency with respect to EPS information is that there is a direct relationship between the level of a firm's EPS and its share price (assuming all other things are equal, like risk); i.e., the higher the level of EPS, the higher the firm's share price and vice versa. Empiri cal studies have found that this assumption is warranted (see Malkial [25]). This relationship between EPS and share price allows researchers to test market efficiency with respect to EPS information. The basic method involves identifying a large sample of securities with changes in EPS and then looking at the average price change or rate of return from purchasing those securities after the arrival of the EPS information. This average price change is typically computed after abstracting from risk and general market conditions (the method of abstracting from risk will be detailed in the next chapter). If it is possible to earn an excess rate of return from purchasing the securities, then the market is said to be inefficient. This rate of return is sometimes compared to transactions costs as a further test of efficiency. The method of identifying changes in EPS and, hence, of determining the securities to be included in the sample is crucial to the analysis of the subsequent average price change observed. Different studies use different EPS models, but all have some implicit underlying assumptions. Prior to the public availability of the actual EPS number, there is assumed to exist in the market some expectation of the EPS number. This expectation of EPS is not observable but is the consensus of investors' estimates of the forth coming EPS. This consensus is reflected in the price of the security just prior to the arrival of the actual number. The change in price after the arrival of the actual number is assumed to be directly related to the difference between expected and actual EPS numbers. If the actual number is greater than the expected number, then it is assumed that the share price will rise, and vice versa. Another assump tion is that, ceteis paribus, the greater the difference between actual and expected EPS, the greater will be the resultant price change. Given the above assumptions and discussion, the Forecast Error (FE) may be defined as the difference between the actual EPS number (EPSt) in time period t and the expected EPS in time t, E(EPSt), i.e. FE = EPSt E(EPSt). (1) The expected EPS is never known with certainty but must be estimated. Different studies use different models to estimate it. This estimate of expected EPS in t will be labeled E(EPSt), whatever model is used. The estimated forecast error, FE, may then be defined as FE = EPSt E(EPSt). (2) It can be seen that the average price change of the securities in a sample after the arrival of some new EPS information will be influenced strongly by the method used to estimate E(EPSt), since this will directly influence which securities are included in the sample. The greater the difference between the actual (but unknown) E(EPSt) and the E(EPSt) estimated by some model, the smaller will be the likelihood that the average price change observed will be representative of the true market reaction. It will be useful to keep the above discussion in mind as it will bear directly on the methodology of the studies to be discussed next. The Ball and Brown Study For their main results Ball and Brown used a regression model to estimate the expected annual EPS of a sample of securities. This model attempted to take advan tage of the facts that the EPS of firms are typically correlated with one another and that the market may use this information in forecasting EPS numbers. The procedure was to regress the change in annual EPS of their sample of firms on the change in an index of EPS for all firms. This regression relationship and the change in the index in a year of interest were used to estimate the change in annual EPS of the firms in the sample. The estimated change in EPS for each firm was then used to estimate the expected EPS which was compared to the actual EPS number. If the actual number were greater than the estimated number, then the firm's earnings were classi fied as having "increased," and if the actual number were less than the estimated number, then the firm's earnings were classified as having "decreased." Once the securities in the Ball and Brown sample had been classified by the above procedure, their price behavior was examined. Ball and Brown constructed what they call an Abnormal Performance Index (API). It may be thought of as the average rate of return of the securities in the sample after abstracting from risk and general market conditions. They looked at the API for twelve months before and six months after the arrival of the new EPS information. The return was examined before the arrival of the actual number in order to scrutinize the market's anticipation of EPS. They found that the API rose throughout the year in advance of the announcement for the earnings "increased" category, and fell throughout the year for the earnings "decreased" category. They found that the drift upward and downward continued for perhaps as long as two months after the publication of the actual numbers. Although the move in the API after the arrival of the annual report was significantly different from zero in a statistical sense, it was of a very small magnitude. The API reached about 8 percent before the arrival of the actual numbers but increased only about 1 percent after the arrival. Ball and Brown concluded that it was impossible to earn an abnormal rate of return by purchasing the securities in their sample as soon as the annual EPS number became publicly available: Even if the relationship tended to persist beyond the announcement month, it is clear that unless transactions costs were within about 1 percent, there was no opportunity for abnormal profit once the income information had become generally available. Our results are thus consistent with other evidence that the market tends to react to data without bias, at least to within transactions costs. These results are taken as evidence that the market accurately anticipated all earnings changes to the point that there was little or no market adjustment for the actual information. It will be demonstrated that this conclusion may be in error. The regression model used by Ball and Brown to classify a company's earnings as having "increased" or "decreased" implies the following decision rule: When the information becomes publicly available, purchase all those securities whose actual EPS number is greater than the estimated expected EPS and sell short all of those companies whose actual EPS number is less than those pre dicted by the regression model. Ball and Brown have demon strated that the market is efficient on average to within transactions costs with respect to this decision rule. It does not demonstrate that abnormal returns cannot be earned by using other decision rules. The methodology of the Ball and Brown study actually mitigates against finding a significant average increase in stock prices after the arrival of new EPS information. Suppose that the regression model predicts that a company will earn $1, and further suppose that this is an unbiased estimate of what the market thinks the company will earn. If the company actually earns $1.01, then according to the Ball and Brown method, this company will be correctly classified as having "increased" its earnings. However, since the market had closely realized its expectation it is unlikely that there would be much price adjustment as a result of the confirmation of this expectation. Yet, this company's return would be included in each calculated average return and, in a sense, bias it downward after the receipt of the actual information. Thus, the Ball and Brown study may have consisted of many firms with little or no change in their prices and a few firms with a fairly large change in price. The resulting decision rule appears to be an unreasonable one. A second, more serious problem is that the regression model may not be a good model for estimating the EPS which the market expects. Indeed, in an earlier study Ball and Brown [2 ] found that only about 40 percent of the variance in EPS of the securities in their sample was explained by an index of all companies' EPS. It may be that investors are able to utilize other sources of infor mation to arrive at an estimate of EPS. Such sources may include quarterly or interim reports which are ignored in the Ball and Brown study. Thus, the actual expected EPS by the market may be quite different from the estimated expected EPS in the Ball and Brown regression model, which could have resulted in Ball and Brown classifying some securities as having "increased" earnings when, in fact, the market viewed these earnings as decreases. This would mean that their calculated average return for the two categories would include some securities whose actual price adjustment was in an opposite direction to the expected price change. This could have biased their observed average returns and thus may have appeared to demonstrate a more efficient market than actually existed. Another facet of the Ball and Brown study was that it used a monthly differencing interval for stock returns. The procedure was to note only the month in which an annual EPS number was reported and to ignore the actual date within the month that the report arrived. This had the effect of assuming that all reports arrived on the first day of the month. To the extent that this was not actually true, the approximate twomonth adjustment period was shortened. While this may not have been serious, the monthly differencing interval for stock prices precluded a close examination of the adjustment process, although Ball and Brown noted that "The actual income number did not appear to cause any unusual jumps in the abnormal perform ance index in the announcement month." One also has to have some doubts as to the generality of the Ball and Brown results with respect to the firms included in their sample, which were all large firms listed on the New York Stock Exchange. Since such firms tend to be well established, and their stocks are widely held, they probably receive close scrutiny from the market and the investment community. Smaller firms, perhaps OTC and AMEX firms, would be unlikely to have as efficient an information system or to be as closely watched. The Jones and Litzenberger Study Jones and Litzenberger estimate expected quarterly EPS as the projected trend in historical quarterly EPS. The study hypothesizes "that quarterly earnings reports significantly greater than anticipated by market profes sionals from historical earnings trends would cause gradual price adjustments over time and generate intermediate stock price trends." Jones and Litzenberger use quarterly earnings reports and a selection criteria which implies the following decision rule: "Purchase those common stocks whose quarterly earnings exceed their projected quarterly earnings by at least 1.5 standard errors, and sell short those securities whose quarterly earnings fall short of the projected quarterly earnings by at least 1.5 standard errors." It can be seen that this model not only classi fies earnings as increased or decreased, but also limits the sample to those companies with large increases or decreases in earnings. Jones and Litzenberger observed the sixmonth price relative of the selected securities and compared this with the S&P 425 index price relative (rate of return). Six months was used because of the preferential capital gains tax treatment involved with this period. No interim prices were observed. They found that "the average price relatives of the stocks selected in each period exceed the price relatives for the Standard and Poor's index in all ten of the periods examined (a nonparametric test gave significance at the .01 level)." They conclude that "the market may not adjust instantaneously and correctly for every item of information that becomes available." Surprisingly, Jones and Litzenberger found that this intermediate stock price trend (17 percent versus 5 percent for the market) happened only for companies with increased earnings. Those companies with lower earnings than expected did not decrease significantly. The study is not a complete test of market effi ciency, since the authors did not show that the profits from the trading rule exceed the cost of information pro cessing and transactions. "A computer is necessary to examine a large quantity of data in a reasonable period of time," and "although a small investor could use similar techniques on a more limited scale," the effectiveness of the latter approach has not been researched. Another problem is that the study does not look at intermediate price relatives. Rather, it focuses on the abnormal rate of return that could have been earned and ignores the speed of adjustment, other than the obvious implication that the time period of adjustment is less than six months. Thus, Jones and Litzenberger tell us little about their intermediate stock price trend. Reconciling the Two Studies The Ball and Brown study and the Jones and Litzenberger study arrive at completely different conclu sions regarding market efficiency. One explanation is that there are differences in the implied decision rules of the two studies. A simple model is useful as an analytical framework to more clearly understand the differences between the two decision rules. Using the same notation as before, if we subtract unity from the quotient of actual EPS divided by expected EPS, we can define this number as the Relative Forecast Error (RFE): EPSt RFE= E ) 1 (3) E (EPSt) If this number is positive, we expect a price increase, and if it is negative, we expect a price decrease. Also, by assumption, the greater the RFE, the greater is the expected price change. The assumption is reasonable, as we would expect a greater resultant price change from a RFE of 50 or 60 percent than we would from a RFE of 1 percent. Now, assume that in a certain time period we can examine a large number of securities and that we have a method by which to discern the expected EPS in the market of each with certainty. We can, therefore, calculate a RFE for each security when the actual EPS number is published. Given these assumptions, we can array all the RFE's of the securities into a distribution. Over a long period one would expect a unimodal distribution about zero. Each study used a model to estimate the expected EPSt, i.e., 2(EPSt). If this is substituted for the E(EPSt) in (3) above, it is easy to visualize a distribu tion similar to the one described above. This distribution is useful in analyzing the two studies. The Ball and Brown study treats any security to the right of zero in the distribution as an "increased earning" security. Such a procedure will treat securities with large and small forecast errors equivalently when looking at price changes. If we assume that the forecast error dis tribution is approximately normal, in random sampling, then we can expect many more securities with small forecast errors than with large forecast errors. The net result is that any price changes resulting from large forecast errors sampled in this manner are likely to be swamped by small forecast errors. Similarly, the Jones and Litzenberger study also uses a model to estimate conditional EPS, in this case quarterly EPS, and it also compares this to the actual EPS numbers. Differences exist because Jones and Litzenberger include in their sample only those companies with large relative forecast errors, the equivalent of looking only at the tails of the forecast error distribution. One expects these securities to have the largest price changes if the market is not efficient. CHAPTER II METHODOLOGY Overview This section presents some design considerations and an overview of the methodology. Following this section are other general methodological considerations, including earnings forecast models, residual analysis, and statisti cal tests. This study has been designed to draw on previous studies where appropriate, to improve on such studies where possible, and to introduce new considerations where these may add to the analysis. For instance, it was decided to examine price (return) data on a daily basis. This particular approach has not been previously used, and it allows for the close examination of the adjustment process of the stock market to new EPS information. This particu lar feature of the design is one of the major differences between this and other studies of the adjustment process. The reason that examination of price data on a daily basis has not been done previously is probably that monthly data are more widely available and easier to use. It was decided to examine the market adjustment process only for annual EPS numbers and not to examine 17 quarterly EPS adjustments, as annual numbers are probably the most important for price formation. The response of security prices to the publication of annual EPS numbers may depend in part on how the annual number compares to what could have been predicted on the basis of interim reports, most importantly, the third quarter report. The potential dependence of price changes on the third quarter report necessitated the development of an earnings pre diction model which took into account the effect of interim reports on the market expectation of annual EPS. Companies listed on both the New York and the American Stock Exchange were included in the sample. The unavailability of data precluded the inclusion of securities listed OvertheCounter. Inclusion of American Stock Exchange (AMEX) listed securities should increase the generality of this study's conclusions, as other studies have examined only New York Stock Exchange (NYSE) listed securities (see Ball and Brown [3 ], Brown and Kennelly [ 9], Fama et al. [15], etc.). Another general consideration is the necessity of incorporating risk in the analysis and abstracting from general market movements. The market model, which is closely related to the capital asset pricing model first formulated by Sharpe [32], Lintner [23] and Mossin [28], was utilized. The general method utilized in this paper is quite simple. The Wall Street Journal was examined on a daily basis in selected months to determine the dates on which EPS numbers first became generally available to the public. At this point certain preliminary screening criteria (to be discussed presently) were applied to define and limit the sample. Once a sample which met the preliminary requirements was obtained, The Standard & Poor's Stock Guide was examined to determine the quarterly EPS numbers of the chosen companies. Comparison of the estimated expected annual EPS number and the actual annual EPS number of each firm in an earnings forecast model yielded the direction and degree of forecast error (equation 3). At this point certain secondary screening criteria were applied, and the sample was further limited. Companies with both negative and positive forecast errors were included in the sample. In this paper a firm with a positive forecast error will be defined as a firm whose actual reported EPS number is greater than its estimated expected EPS number, estimated on the basis of the earnings models in this paper. A negative forecast error will be defined as a firm whose actual EPS is less than estimated EBS. Once the companies had been selected, one year of daily price (return) data was obtained from the Standard & Poor's ISL Daily Stock Price Tapes, beginning approximately four months before and ending eight months after the actual annual EPS number was published in The Wall Street Journal. Part of this year was utilized for an analysis of the reaction to the arrival of the EPS information (residual analysis), and part was utilized to estimate market para meters (discussed presently) necessary for the residual analysis. Some General Considerations The Standard and Poor's ISL Daily Stock Price Tapes were available in the University of Florida Systems Library from the second quarter of 1962 through the second quarter of 1972. The tapes contain one calendar quarter of daily prices, dividends, and stock split and dividend data for all companies on the New York and American Stock Exchanges. A program was written to retrieve the daily price data from the tapes and to calculate the daily holding period return (HPR) for each security in the sample in any particular quarter. The holding period return is defined as Pjt Pjtl + D jft 3,ti j,t HPR = (4) Pj,t1 where Pj,t = price of security j in time t Dj,t = dividend of security j in time t. The t in this case corresponds to an increment of one day. The HPR may be thought of as the rate of return accruing to the holder of security j from holding this security from time period tl to time period t, in this case one day. One of the problems of using daily price data is the difficulty of keeping track of the individual dates on which the annual EPS numbers of different securities are published. This problem was held to within manageable limits by including in the sample only those securities whose annual EPS numbers appeared in The Wall Street Journal between the dates of February 15 and March 20 of any particular year. These dates were chosen because most industrial firms have fiscal years ending on December 31, and a few months elapse before the results of a year's operations are summarized and made available. Thus, a majority of annual EPS numbers are made available between the two abovementioned dates. The above procedure may possibly limit the generality of the sample somewhat, although there is no evidence that companies with December 31 fiscal years are systematically different from companies with different fiscal years. Somewhat more than half of all industrial companies in the United States have fiscal years ending on December 31. Some companies furnish preliminary EPS numbers to The Wall Street Journal and then later report the final numbers. Other companies have only the final numbers published. This study did not distinguish between prelim inary and final numbers. Each security's publication date was crosschecked in The Wall Street Journal Index to insure that no previous annual EPS information had been reported either as a preliminary number or as news such as reported at a meeting of security analysts. Thus, all firms in the study had the first mention of the EPS number published between February 15 and March 20. A residual analysis (discussed presently) was performed beginning twenty days before the publication of the EPS information and was continued until sixty days after this date. Since the annual EPS number was always published in the first quarter of the year after the year which it summarized, the above period of time always fell in the first two quarters. These two quarters were the middle two of the year of HPR's which were collected for each firm. The two outside quarters of this year were used to estimate the parameters of the market model. These two quarters corresponded to the fourth quarter of the year which the annual number summarized and the third quarter of the following year. In no case was the residual analysis performed during a time period which was also used to estimate the market parameters, preventing a bias in the estimation of those coefficients. Earnings Models Two models were used to estimate the expectation of annual EPS. The first model, a naive model, estimated that the market predicts this year's annual EPS number would be the same as last year's EPS number. This model shall be called the AF model, which stands for annual forecast. The second model estimated that the market predicted that the current year's annual EPS number would equal the sum of the first three quarters of actual EPS (previously published) and the fourth quarter results of the previous year multiplied by the ratio of the first three quarters of the current year to the first three quarters of the previous year. This model will be called the QF model, which stands for quarterly forecast. The AF model was used as a preliminary screening device. The Wall Street Journal reports the previous year's results and compares these to the current year's results. The preliminary screen consisted of requiring at least a 20 percent difference between the current year's annual EPS and the previous year's annual EPS. Those companies which made it through this initial screen were then subjected to the QF model. These models may be stated more precisely and rigorously in mathematical notation. Assume that a company's actual annual EPS number was published in The Wall Street Journal. The year of operations which it summarizes contains four quarters, the first three quarters having been previously published and the fourth quarter results being implied in the annual number. Similarly, the previous year's annual EPS number summarized four quarters of results. The subscript k is used to denote the kth quarter relative to the first quarter of the previous year; the quarter just prior to the annual report (fourth quarter) is eight. Similarly, the first three quarters of the same year would be subscripted five, six, and seven, respectively; and the four quarters of the previous year would be subscripted one, two, three, and four, respectively. Thus, the actual annual results for the previous year for company j, which is nothing more than the sum of four quarters of operations for that year, may be denoted as 4 E EPSj,k k=l and the immediate past year's results may be denoted as 8 EPS SEPS j,k. k=5 The AF model assumed that the market predicts that this year's annual EPS will be the same as the previous year's EPS. Denoting the AF model estimate of this number as EA, then 4 EA = EPSj,k (5) k=l The annual forecast error (AFE) may now be defined as 8 E EPSjk k=5 AFEj= 1 (6) EA This annual forecast error is always expressed as a fraction. It can easily be seen that the AFE is nothing more than the actual current year's EPS number divided by the estimate of its market expectation (EA) in the AF model, less one. For instance, if a company had earned $1.00 the previous year and $1.25 in the current year, then the AFE would be ($1.25/$1.00 1) = .25 or 25 percent. Now, the preliminary screen may be denoted mathe matically as: IAFEI 2: 20%. There are several things to note about the AF model. First, it is not a very sophisticated model since it does not adjust for the expected annual growth in EPS. Since, however, few companies have growth rates in excess of 20 percent, the possibility of incorrectly classifying a company is small. Suppose, for example, that in the previous year a company's annual EPS were $1.00, and that in the current year the company earned $1.20. The company would be included in the sample as its earnings increased by 20 percent. If the market projected that the company's earnings were going to increase by 15 percent (to $1.15), then the actual earnings of $1.20 would be viewed as an unexpected earnings increase by the market, and the price of the security would be likely to rise. Thus, the model "correctly" classifies companies as long as the market expects less than a 20 percent growth in annual EPS. A second problem with the AF model is that it ignores interim reports of EPS. Thus, even though all companies in the sample have at least a 20 percent change in EPS, no assurance is given that there will be some price adjustment subsequent to the arrival of the actual number. Indeed, some people in the market may use only an AF model or an AF model with growth; i.e., they may ignore interim reports, but there is no assurance of this. A second model, which takes into account quarterly reports, is needed. This quarterly model was the second screen on EPS. The 20 percent AFE screen was utilized as the pre liminary method for identifying those companies which were most likely to have a price adjustment. The second or QF (quarterly forecast) model was applied to all companies which survived this preliminary screen. Mathematically the estimate of the expected EPS using historical quarterly EPS information and denoted as EQ is defined as 7 Z EPSjk 7 k=5 EQ = EPSj,k +3 (EPSj,4). (7) k=5 E EPSj,k k=l Verbally, this model predicts that this year's annual EPS number will be equal to the sum of the actual first three quarters of EPS plus an estimate of the current year's fourth quarter, which is estimated to be the previous year's fourth quarter results multiplied by the ratio of the current year's three quarter results to the previous year's three quarter results. Another way of stating this is that this year's fourth quarter results are expected to be the same as last year's but adjusted for the proportion of change in the first three quarters of results to the previous year's first three quarters of results. The quarterly forecast error (QFE) for firm j, always expressed as a fraction, is defined as 8 E EPSj,k k=5 QFEj = 1. (8) EQ The calculated QFE1 may be of either sign; i.e., a company with an AFE of +20 percent will not necessarily have a positive QFE. It may have been that the first three quarters were exceptionally good for the firm and that the fourth quarter failed to live up to expectations. Companies which had a conflict in the predicted direction of forecast error between the AFE and QFE models were excluded from the sample. This procedure was designed so that the direction of forecast error could be predicted with some reliability, constituting the second screen on earnings. For instance, suppose that a company had earned $1.00 per share the previous year and that each of the four quarters of operations that year had provided $.25. Suppose also that in the current year the company had earned $2.25, $.50 a quarter for the first three quarters and then $.75 for the fourth quarter. These increased quarterly results for the fourth quarter were better than would have been predicted on the basis of the first three quarters of results, and the model showed this; i.e., the company would have been included in the sample. This firm's AFE was $2.25/$1.00 1 or 125 percent and its QFE was ($2.25/[$1.50 + ($1.50/$.75)$.25]) 1 = ([2.25/2.00]1) = 12.5 percent. Thus, both the AFE and QFE models predicted a positive forecast error. The above firm would not have been included in the sample if it had earned the same $.50 a share in each of the first three quarters but only earned $.25 in the fourth quarter even though the AFE is a positive 75 percent ($1.75/$1.00 1). The EQ model would have made the same estimate, but since the fourth quarter was less than expected, the QFE would be ($1.75/$2.00 1) = 8.75 percent. The above procedure was designed so that the direction of forecast error could be predicted with some reliability. Thus, we are relatively confident that any adjustment in price which occurs as a result of the new EPS information will be directly related to the direction of forecast error. Residual Analysis The method of residual analysis is fairly standard and straightforward. It relies on the fact that under certain conditions the return of a security may be dis aggregated into a component which is related to the return on capital assets in general and a component which is related only to the specific security. This proposition may be stated mathematically in what has come to be called the market model. The model states that the returns of a security are a linear function of a general market factor such that Rjt = j + jRm,t + uj,t (9) where E(uj,t) = 0 cov(Rm,t,uj,t) = 0 and Rj,t = return on security i in time t, including dividends and capital gains. Rm,t = return on the general market factor in time t. uj,t = the individualistic factor representing the part of security i's return which is independent of Rm,t. aj ,j = intercept and slope, respectively, of the linear relationship between the securities return and the return on the general mark factor. The Rj,t and Rm,t correspond to the holding period return (eq. 4) of the individual securities and the market index, respectively. The return generating mechanism of (9), under some crucial assumptions, is consistent with the capital asset pricing model. These assumptions and the capital asset pricing model will be discussed in the next section. The a and a obtain their economic interpretations from the capital asset pricing model. The B is a measure of the systematic or nondiversifiable risk of the security. The a is the average value, over time, of the systematic or individualistic portion of a security's return. It is assumed that the systematic portion of a security's return is captured by the dj and 8 Rm,t. The individualistic component represents the classes of events which have impact only on security j. In an efficient market this uj,t will fluctuate randomly as random informal tion about the security is instantaneously reflected in the market price (and return) of the security. It is this interpretation of uj,t which has allowed researchers to use (9) to test market efficiency. The method, outlined below, estimates the market parameters of (9) for a large sample of securities and then tests whether, in fact, the uj,t fluctuates randomly. The general method is quite simple. First, the market parameters of (9) are estimated during a time period when one believes that the assumptions of the model are met; most importantly E(uj,t) = 0 and cov(uj,t,Rm,t) = 0. (To the extent that this is not true, the estimated intercept will be biased.) In the present case, (9 ) was estimated using the two outside quarters of the one year of holding periodreturns (HPR'seq. 4) which were collect ed for each security. Using the same quarterly designa tions as used in the earnings forecast models, these four quarters of HPR's would correspond to quarters 8, 9, 10, and 11 with 8 and 11 being used to estimate the market parameters. Notice that quarter 8 occurs during the fourth fiscal quarter of the companies in the sample. The middle two quarters of the year (9 and 10) contain the possible adjustment period of the security price to the new EPS information. These two quarters were not used in estimating (9). The S&P 425 Industrial Index was used as a proxy for the market index, Rm,t. The next general step in the method is to use the estimated parameters (9) with the returns to the individual securities and the market portfolio to estimate u, uj,t = Rj,t (&j + jRm,t) (10) The ujt in any particular t represents that portion of the security's return which reflects events which are company specific after abstracting from risk and general market movements. In this study these uj,t were estimated for each security in the sample for eightyone separate days. These eightyone days began twenty days prior to the publication of the annual EPS number in The Wall Street Journal and continued for sixty days after that date. Since there are typically five trading days in each week, this period corresponds roughly to one month before and continues for three months after the publication date. Only three months of residuals were collected after publication because any residual after that time cannot be attributed solely to the arrival of the annual number as the first quarter results of the new fiscal year were likely to be reported soon after the end of three months. The next step in the analysis is to cross sectionally average the estimated residuals, u t's. This averaging is performed relative to the time period in which the expected disequilibrating information becomes publicly available; e.g., in this study all t are measured relative to the publication of the annual EPS number. For instance, on day 0, the publication date, the firms in the sample would have all of their estimated residuals (Uj,0's) averaged together regardless of the chronological dates of publication. This is called an average residual, ut n l ujt j=1 "t = (11) n where there are n securities in the sample. In this study the sample was split into those firms with positive and those firms with negative forecast errors. There were eightyone separate average residuals (Ut's) calculated for each of these subsamples. Note that this u may be thought of as an average rate of return which would have been earned on a portfolio of n securities had they been purchased and held for the time period t (in the present case, one day) after abstracting risk and general market movements. If the market is efficient, the u in any t will be very close to zero as the price movement of the securities from which it is generated fluctuates randomly. A u which is signifi cantly different from zero in a statistical sense is an indication of market inefficiency with respect to the information being examined. If more than one time period is being examined, then these out's may be accumulated (added) to form a cumulative average residual (hereafter CAR) T CART = t (12) t=0 where there are T time periods to be examined. The CAR shows the cumulative effects of the wandering of the return of the securities of interest around the market line. A CAR in this study was calculated from day zero to day sixty after the publication of the earnings announce ment and from day 20 to day +10 relative to the publica tion. Each was calculated for both negative and positive forecast errors. The u's may be accumulated in such a way that the resulting index, called an Abnormal Performance Index (API), may be interpreted as the rate of return accruing to the holder of an equally weighted portfolio of all the n securities in the sample, had the holder purchased them at time period t relative to the arrival of the information of interest and held them until day T. T+N APIT+N = T (l+t) (13) t=T+l where the API is compounded from period T+1 to T+N. It can be seen that the CAR and API correspond to the wellknown arithmetic and geometric rates of return. The differences between calculated values of arithmetic and geometric rates of return are trivial at low rates of return (less than 15 percent). In this study, where the typical u was less than onehalf of one percent, the API and CAR gave essentially similar results. The cumulative difference between the two was never greater than .2 per cent after sixty days. An operational definition of market efficiency with respect to publicly available information has not been satisfactorily formulated in the literature. In some cases the market is deemed efficient if it "rapidly" adjusts to new information. The term "rapidly" has never been specified in terms of time periods. As Ball [1] says, "The Efficient Market Hypothesis is limited in operational content until the speed of adjustment is specified precisely." In other cases, the market is deemed efficient if it was not possible to earn an abnormal rate of return by trading on the information as soon as it became publicly available. Similarly, some studies have compared the abnormal rate of return to transactions costs. Regardless of the criterion of efficiency deemed appropriate, the u's and CAR may be used to test efficiency. Since the u's may be expected to fluctuate randomly about the market line in an efficient market, a test of market efficiency is whether or not this average residual (u) is significantly different from zero. That is, if it were possible to earn an abnormal rate of return by trading on the basis of some information available in time period t, then the market had not fully reflected that information. If the market is inefficient, the average residual may also be used to test the speed of adjustment by examin ing the number of u's (time periods) after the arrival of the new information it takes for the average residual to become insignificantly different from zero. If the disequilibrium lasts more than one time period, then the CAR may be used to measure the rate of return from trading on the new information when it became publicly available and holding the securities for as long as the disequilibrium persisted. The ending value of the CAR may be compared to percentage transactions cost as a further test of efficiency. The CAR is also an appropriate device with which to examine the market adjustment process. An increase in the CAR implies an increase in the prices of the securities in the sample, and vice versa. A CAR which is wandering randomly suggests no systematic market adjustment to the information of interest. Thus, the pattern of market adjustment may be inferred from examination of the CAR. 37 The Capital Asset Pricing Model, the Parameter Estimation Problem, and the Normality Assumption The Capital Asset Pricing Model The most important assumptions of the Capital Asset Pricing Model (CAPM) are that (1) investors are risk averters, (2) returns are normally distributed, (3) a riskless rate of interest at which investors can borrow and lend as much as they want is in existence, and (4) expecta tions are homogeneous. If we define Rf as the riskfree rate and the other variables as before, then the CAPM suggests that the expected value of the return on any security j is: E(Rj) = Rf + (j(E[Rm] Rf) (14) where 8 = cov Rj,Rm/q2(Rm) (15) The above model suggests that the return generating process is the market model (9) if aj = (l8j)Rf. Written in terms of excess returns, the model may be tested by add ing an intercept to (14) and running the following regression using ex post rates of return Rj,t Rf,t = "j + 8j(Rm,t Rf,t) + uj,t (16) The prime is to distinguish this intercept from aj in (9). If the CAPM is correct, then the implication is that the a in (16) will equal zero. The in (16) is merely the systematic portion of a security's return expressed in units of market risk. If we estimate (9) ex post instead of (16); then, if we believe that the assumptions of the CAPM are correct, the estimated a in (9) impounds the riskfree rate and the systematic risk of the security as well as any systematic return a' that may accrue to the security due to an incorrectly specified CAPM or violation of the CAPM assumptions. The estimated intercept in (9) will be aj = Rf(lj) + a (17) Note that the riskfree rate in (16) is time subscripted. This means that variance in a estimated in (.9.) may in part be due to shifts in the riskfree rate although this rate, which is thought to be closely approximated by the treasury bill rate, has historically been fairly stable relative to stock market stability. Researchers have found that (14), which is called the onefactor model, is not well specified empirically (see Fama and MacBeth [16] and Blume and Friend [7 ]). The reason for this is that investors may not be able to borrow and lend as much as they prefer at the riskfree rate. To overcome this difficulty, researchers have formulated what is called a twofactor model (see Fama and MacBeth [16]). The twofactor model relies on the assump tion of the existence of a risky portfolio, called the zero beta portfolio, which is uncorrelated with the market portfolio and which, when used in lieu of the riskless rate, yields similar conclusions as to the linear relation ship between market risk and return (see Black [ 4]). The point of this is that if we replace Rf,t with Rzt (the zero beta return) then we arrive at similar conclusions as to the estimated intercept in (9). In this case the a impounds the zero beta rate of return instead of Rf and some of the variance in & may be due to changes in Rz,t. The Parameter Estimation Problem The typical procedure for testing market efficiency is to estimate (9) by pooling (using) ex post rates of return from time periods before (preperiod) and after (postperiod) the arrival of the information of interest. These pre and posttime periods are typically quite long; five to ten years of monthly returns are not unusual (see for example Ball and Brown [ 3] and Fama et al. [15]). It is assumed that by pooling these pre and posttime periods to estimate the parameters in (9), accurate long term estimates of these parameters are obtained. Most studies make the assumption that the parameters do not change between the pre and posttime periods. If, in fact, the parameters do shift, the calculated average residuals and CAR may not give a true indication of the adjustment process. The above problem may not be serious in the present case as only one year of return data is examined and it is unlikely that there will be a marked shift in the market estimate of the risk of the firm during this short time period. This short time period does, however, give rise to another problem. Since only about three months of daily observations are used in each of the pre and post time periods to estimate the parameters in (9), shortterm shifts in the riskfree rate (or the zero beta rate) or the systematic return to the security, see (17), may bias the estimate of the market parameters. The problems of parameter stability and bias in the estimated parameters between the pre and posttime periods are serious in this study. As a means of over coming it, the parameters were estimated not only by pool ing the two time periods but by estimating them in each period individually. The average residuals and CAR were also calculated using these parameters estimated in the pre, post, and pooledtime periods. Comparison of these three series of CAR's, should they be similar, would lead to the conclusion that parameter stability and biased parameters were not a problem. Significant differences in the CAR's would require the explicit consideration of the degree of bias present. The Normality Assumption One problem with using stock price rates of return is that observed return distributions have "fat tails" as compared to normal or Gaussian distributions. It has been found that monthly rates of return probably conform better to a nonnormal, symmetric, stable distribution than to the normal (see Mandelbrot [26] and Fama [13]). Since regression analysis and the planned statistical test of the average residual require the assumption of normality, there are some potential problems with using normal statis tical techniques on stock returns, as members of the stable class of distributions do not converge to normality as the classical central limit theorem suggests. Researchers have found that regression analysis and standard errors of means are robust with respect to violations of the normality assumption for monthly rates of return as long as the dispersion of the parameted stable return distribution is defined (see Officer [31] and Fama and MacBeth [16]). The evidence for daily stock price rates of return also suggests that the violation of the normality assumption is not serious. In a recent study, Blattberg and Gonedes [ 6] find that "for daily rates of return, the student model has greater descriptive validity than does the symmetric stable model," and "the student model has fat tails as does the stable model, but converges to normality for large sum sizes." Thus, the observed "fat tails" of stock return distributions is not a problem for statistical inference in this case because, as Fama observes, "as long as one is not concerned with precise estimates of probability levels (always somewhat of a meaningless activity) interpretation of t statistics in the usual way does not lead to serious errors [16]." Statistical Tests Each of the eightyone average residuals was tested to determine if its average value was significantly different from zero. The specific test performed was to divide the average residual by its calculated standard error. The resulting statistic, called a t statistic (not to be confused with t as in time period) is well known. A t statistic greater than two (approximately) indicates that the mean is significantly different from zero at the .05 level of confidence (approximately). A statistically significant average residual in the time period after the receipt of the EPS information implies that the market was inefficient; i.e., it had not completely adjusted for or fully reflected the information. It was discovered that during certain time periods relative to the announcement date there was no observed statistical significance in the average residuals but that the CAR appeared to trend strongly. This trend was due (apparently) to noise in the average residuals after a certain length of time which inflated their standard errors and thus mitigated against statistical significance. This trend in the CAR resulted from many of the average residuals being positive (or negative) but not statistically significant where market efficiency would imply that the average residuals fluctuate randomly. Thus, it became necessary to develop a statistical test of this observed trend in the CAR, as the existence of such a statistically significant trend implies market inefficiency. One way to statistically test for the existence of such a trend in the CAR is to regress it against some time variable, i.e., fit a linear trend line through it. CARt = a + b t + e (18) where the t variable is nothing more than the integer value of the day t relative to the announcement day. If the slope coefficient of the time variable is significantly different from zero, then one can infer that a trend in the CAR exists, whereas market efficiency would imply that the CAR should wander randomly. The above test is not appropriate if the error term of (18) is autoregressive (serially correlated). The least squares estimates in such a case are unbiased and consistent but are not efficient, which means that the estimated standard error of the estimators (a and b) will be biased. If the disturbances are autoregressive and we use (18), the calculated acceptance region for the estima tors will be narrower than they should be for the specified level of significance. This would inflate the calculated t statistics and bias them toward inferring significance. In fact, the above regression does have an auto correlated error term. The very procedure of accumulating (adding) the average residuals into the cumulative average forces the level of the cumulative average at one point in time to be dependent on its level in previous time periods. The nature of this autocorrelation would seem to be well approximated by a first order autoregressive scheme, which imposes an autocorrelation which dies out exponentially. Adding the average residuals over time would create an autocorrelation function which also asymptotically approaches zero, so that the first order autoregressive process should provide an adequate approximation. An accurate estimate of the standard error of the estimator of the slope coefficient of (18) may be obtained by estimating the serial correlation coefficient in the data and using it to fit a first order autoregressive scheme to the same data. CARt PCARt1 = a(lp) + b(tptl) + (etPetl) (19) where pis the estimated serial correlation coefficient. Again, one must believe that a first order autoregressive scheme best describes the data (and it seems very reason able) for (19) to be appropriate (for a discussion of this, see Cochrane and Orcutt [10]). The above procedure is a test of whether or not the adjustment process is instantaneous. It can also be used to test the abnormal rate of return which could have been earned. Since the CAR is expressed in terms of the rate of return which would accrue to the holder of an equally weighted portfolio of the securities in the average residual, the slope coefficient of (19) may be interpreted as an excess rate of return per day. An estimate of the cumulative rate of return is to compound this slope coefficient times the number of days over which the CAR was calculated (T in [121). A confidence interval around this average cumulative rate of return may be closely approximately by 2 S CART (12)L (20) where CART = a + (T), where a and 8 are the estimated coefficients in (19). S = standard error of the regression. 1The confidence interval is only approximate because the above procedure ignores the fact that a and b in (19) are estimates and are not known with certainty. The factor ignored is of order 1/n, so that for large n its contribu tion will be small. In general terms, if we estimate Y = a + bX + e and we want a confidence interval around a predicted Y, then E(Y) + E(a + b [X0]) Var (Y) = 2(1 + +X) n = ) EX. The above procedure (20) is equivalent to ignoring the two terms on the right in the Var (Y). But, since n and Xi are fairly large in this study (in fact, never less than sixty) the error in the confidence interval is very small. 46 This confidence interval was used to test the reliability of the abnormal rate of return indicated by the average cumulative rate of return. CHAPTER III RESULTS Description of the Sample The Earnings Digest section of The Wall Street Journal was examined between the dates of February 15 and March 20 in the years 1963 to 1971. These nine years corresponded to the availability of the ISL Daily Stock Price Tapes in the University of Florida Systems Library. The initial step in drawing the sample was to examine the current year's reported annual EPS numbers to determine those firms which had at least a 20 percent change from the previous year. The previous year's number is also reported in the Earnings Digest. The next step was to examine Standard & Poor's Stock Guide to determine third quarter results for the current and previous years. At this point all Overthe Counter securities were discarded as the Stock Guide includes the exchange listing of all firms. Annual and quarterly earnings were used to calculate the Quarterly Forecast Error (QFE) (equation 8) for each security. Those securities whose QFE conflicted in direction with their Annual Forecast Error (AFE) (equation 6) were discarded. At this point The Wall Street Journal Index 47 was utilized to crosscheck the publication dates of annual EPS numbers and to insure that no previous publication of the number had taken place in The Wall Street Journal. The final sample consisted of 158 firms, 113 with positive forecast errors and 45 with negative forecast errors. Table 1 presents a breakdown of these firms across security exchanges, and it also presents the average AFE and QFE for both positive and negative forecast errors. Of the 158 firms, 105 were listed on the New York Stock Exchange and 53 on the American Exchange. No attempt was made to manage the proportion of the two different exchanges in the sample. TABLE 1 SUMMARY OF THE NUMBER OF FIRMS WITH NEGATIVE AND POSITIVE FORECAST ERRORS, OF EXCHANGE LISTINGS, SAND AVERAGE OF PERCENTAGE FORECAST ERRORS Forecast # NYSE # AMEX Average Average Error # Firms Firms Firms AFE QFE Positive 113 74 39 68.6 21.5 Negative 45 31 14 44 21.6 All Firms 158 105 53 36 9.3 The range of the positive AFE was 20 percent to 480 percent, and similarly, the range of the positive QFE was from 0 to 170 percent. The median QFE was 10 percent. Thus there was a positive skew to the distribution of Quarterly Forecast Error. The range of the negative AFE was 20 percent to 93 percent, and the range of the negative QFE was 0 to 89 percent. Since no firms were examined with negative earnings, the AFE is bounded in a negative direction by 100 percent while the positive AFE is not bounded. This accounts for the difference between the average positive and negative AFE's. The positive and negative QFE's were of a similar magnitude. Table 2 is a summary of the distribution of the firms whose numbers were examined among the nine years in the sample. The firms were fairly evenly distributed among the nine years, but there was a slight tendency for the negative forecast errors to occur more frequently in the later years and the positive forecast errors to occur more frequently in the earlier years. An Analysis of Regression Results The daily holding period returns (HPR'sequation 4) of all 158 securities were regressed against the daily HPR's of the S&P 425 Industrial Index. These regressions were run three times for each security in the sample; once using HPR's from the quarter just prior to the earnings announcement quarter (preperiod or period 8), once for the second quarter after the earnings announcement quarter (postperiod or period 11), and once by pooling the HPR's from both time periods. This procedure yielded three sets TABLE 2 SUMMARY OF THE DISTRIBUTION OF FIRMS AMONG DIFFERENT TIME PERIODS FOR THE TOTAL SAMPLE AND FOR POSITIVE AND NEGATIVE FORECAST ERRORS Positive Negative Year Total Error Error 1963 29 24 5 1964 20 17 3 1965 13 10 3 1966 31 27 4 1967 9 5 4 1968 9 7 2 1969 14 10 4 1970 16 6 10 1971 17 7 10 158 113 45 of parameters for each security. Table 3 is a summary of the three sets of regression results for the total sample and for the sample divided into positive and negative forecast errors. The market index, on average, explained about 11.7 percent of the variance of the individual security returns. This level of explained variance (R2) is roughly comparable to the results reported by Kaplan and Roll [21] using a weekly differencing interval. The range of R2's was from less than 1 percent for several firms to 62 percent for one firm (Boeing). The most striking feature of the results presented in Table 3 is the differences between the pre and post periods. There was a systematic tendency for the post period to have a higher proportion of the variance of the individual security returns explained by the market index. The R2 is almost 70 percent higher in the post than in the preperiod for the whole sample. The average estimated intercept is more than three times as large in the pre than in the postperiod for those firms with positive forecast errors (.149 percent versus .04 percent). The average intercept is negative in the preperiod and positive in the postperiod for negative forecast error firms. There is also a significant shift (.05 level) in the estimated market risk of the firms in the sample. The average beta increased from .897 in the preperiod to 1.076 in the post period. TABLE 3 AVERAGES OF THE ESTIMATED MARKET PARAMETERS AND EXPLAINED VARIANCES IN THE PRE, POST, AND POOLED TIME PERIODS FOR ALL 158 FIRMS AND FOR FIRMS WITH POSITIVE AND NEGATIVE FORECAST ERRORS Estimation ^ A 2 Period R Total Sample (158 Firms) Pre .001013 .897 .086 Post .000442 1.076 .146 Pooled .000823 .98 .118 Positive Forecast Errors (113 Firms). Pre .00149 .949 .097 Post .0004 1.08 .152 Pooled .00113 1.04 .130 Negative Forecast Errors (45 Firms) Pre .000194 .776 .056 Post .000544 1.063 .131 Pooled .000054 .817 .085 It is apparent from the above results that the estimates of the market parameters between the pre and postperiods are quite different. The problem arises, therefore, of attempting to determine the cause of this estimation problem and of attempting to determine which set of market parameters is the most appropriate to use in performing the residual analysis to follow. Briefly, it was determined that the market parameters estimated in the postperiod were more appropriate, as the parameters estimated in the preperiod were biased due to the way the sample was drawn. The remaining pages of this section will be used to present the analysis which led to this deter mination. The estimated residual in each time period for each security is calculated by (10): uj,t = Rj,t (j + 8jRm,t), (10) and the average residual is calculated by n j=1 Vt = (11) n If the average residual is calculated for a sample of securities, it will be related to the estimates of their market parameters, i.e., substituting (10) into (11) n n n z Rj,t E &j E RjRm,t j=l j=l j=l t = ( + ), (21) n n n or t Rt ( + R t) (22) Thus, the average residual in any time period is nothing more than the average return of the securities in the sample for that time period less the average systematic return of those securities and less the average systematic risk of the securities times the average return of the market portfolio in the time period. Note that in (22) the time period is measured relative to the announcement date of the new EPS information and thus the Rt and Rm t represent many different chronological dates. Equation (22) clearly demonstrates the purpose of using the market model (9) in a residual analysis to test market efficiency. If the market is efficient, then subtracting the average systematic return (T + f ERt) of a sample of securities from the average return of those securities in some time period (trelative to some informa tion availability) Rt, should yield approximately a zero return. This is because the Rt represents many different chronological dates and thus should be a random number if the securities in the sample move randomly relative to the announcement date. If the market is not efficient with respect to the securities in the sample, then subtracting the systematic return from the average calculated return in a time period when the market is systematically adjusting for some new information will not yield a zero average return. The above argument assumes that an unbiased estimate of the systematic portion of each security's return is obtained, as well as unbiased estimates of the market risk parameters. The estimated 8 in the preperiod is slightly smaller than in the postperiod. These differences are not likely to have a substantial impact upon the estimated residuals because the average Rm,t is likely to be quite small on a daily basis.1 In a famous study, Fisher and Lorie [17] estimated that the average longterm annual rate of return of all securities listed on the New York Stock Exchange was about 9 percent. This means that, on average, the value of Rmt on any particular day is likely to be on the order of .09/250 = .036 percent if there are 250 trading days in a year. The difference between the two estimated B's is small (1.076 .897 = .179), and thus the differences between calculated average residuals as a result of different estimates of B is likely to be very small (.179 x .00036 = .0000644). iTime does not mean that the individual Rmt's in different time periods cannot be quite large. It is the 8iRm.t term which is used to adjust individual returns for systematic risk; i.e., if Rm t is 1 percent in some time period, then a security wiTHEa a of .5 would have ' percent subtracted from its return in the residual analysis, and a security with a f of 2 would have 2 percent subtracted from its return (eq. 10). Contrary to the 8j, the differences in the esti mates of the acj's in the pre and postperiods could cause a significant difference in the estimated average residuals and one does not know which set of estimated parameters is appropriate. Thus, the problem arises of determining whether or not the observed instability in the estimated average intercepts of (9) between the pre and postperiods was caused by some CAPM related instability (perhaps instability in the zero beta rate of return) or by some sample specific instability. If the former is the case, there can be no confidence that the estimated intercept parameter is appropriate for the period of estimation of the average residual. If the latter is the case and if it can be established that either the pre or postperiod estimated parameters are an accurate estimate of the extant market relationship, then estimation of the average residuals is appropriate. The average &j in the preperiod is .001013 (more than onetenth of 1 percent) which is equivalent to an annual return of 25 percent (.001013 x 250). The equivalent annual return in the postperiod is 11 percent (.000442 x 250). Thus the difference between the average d's estimated in the pre and postperiods is on the order of 14 percent on an annual basis and near onetenth of 1 percent in each period (.081 percent). Thus the observed differences in the V's can significantly affect the average residual. The remainder of this section will identify potential sources of instability and bias in the s's and will analyze the most probable source of bias in this study. Recall from (17) that if one believes that the assumptions of the capital asset pricing model are met in the real world, then the estimated aj impounds the riskless rate Rf, the systematic risk of the security Bj, and the systematic return of the security aj (if it exists). If one believes that the assumption of riskless borrowing and lending is not tenable, then Rf in (17) can be replaced by the zero beta rate of return Rz. There is a potential source of bias in the estima tion of aj. If the regression model assumption E(ujt) = 0 is not met, then the estimated aj will be biased; i.e., if E(uj,t) = uj 0 over some period of estimation, then this specification error will cause the estimated intercept to be biased by an amount equal to the uj. Adding this j to (17) we find aj = Rf(laj) + aj + j. (23) A market disequilibrium in the estimation period will cause E(uj,t) ; 0 and by the above argument this trend in the residual can bias j. Equation (23) identifies those sources of variance which are the most likely to affect the estimate of aj. Note that if all of the assumptions of the CAPM are met, the !a's (and the aj's) will equal zero. If the two factor (zero beta model) CAPM is correct, then the Rf will be replaced by Rz (the zero beta return), a will be equal to zero, and the aj will equal zero if there is no trend in the residual. The CAPM in both forms assumes that the market is efficient; i.e., there is no 3 4 0. Equation (23) may be used in an analysis of the [ resulting from estimating (9) in different time periods. The problem is to identify the most probable cause of the different c and to determine which set of parameters is appropriate for the residual analysis. Note that an average systematic risk of about one for the securities in the sample means that the riskless rate (23) is likely to contribute little to the &. The average treasury bill rate (thought to be a reasonable surrogate for Rf) was about 4 percent during the period. The treasury bill rate also had a slight uptrend during the nineyear period, and since, chronologically, the pre period is always before the postperiod this uptrend, if it influenced the 6, should have influenced the postperiod minutely. In fact, the postperiod & was systematically lower than the preperiod. A similar line of reasoning suggests that the zero beta rate of return (with the average $ close to one), Rz, probably did not cause the differences between the pre and postperiod &'s, although the argument is less forceful than in the case of Rf since the zero beta rate has been typically estimated to be of a greater magnitude and variance than the riskless rate (see Fama and MacBeth [16]). Fama and MacBeth and Black, Jensen, and Scholes [5 ] estimate the monthly zero beta rate over many different time periods, and in no time period do they find a level of Rz which implies an annual return of greater than about 15 percent. Thus it seems very unlikely that the estimated annual return of 25 percent in the preperiod could have been caused by an exceptionally high zero beta return. However the possibility should be kept in mind if a more likely explanation cannot be found. A fairly simple analysis demonstrated that the most likely explanation of the high T in the preperiod was a systematic tendency in the error term of (9) to have an expectation different from zero. This misspecification of (9) caused the estimated intercept to be biased; i.e., in (23) the u was different from zero. It was found that (9) was not subject to this same misspecification when it was estimated in the postperiod. The preperiod corresponds to the fourth calendar quarter of a year in which each security in the sample had at least a 20 percent change in EPS from the previous year. Table 1 shows that, on average, EPS was about 69 percent greater than the previous year for the securities with positive forecast errors and about 44 percent lower for securities with negative forecast errors. The quarterly forecast error was about 21 percent (plus and minus) for both positive and negative forecast errors. This means that it is very likely that the firms in the sample had third quarter earnings reports which were different from those expected by the market. Since AFE was substantially greater than QFE, on average, the increase in AFE was likely to be caused by increased third quarter earnings as well as increased fourth quarter earnings. If the market is inefficient, then it may be adjusting for the third quarter EPS report during the fourth quarter: that is, the quarter used to estimate the preperiod market parameters. To test whether or not there was any statis tical association between the direction and degree of Annual Forecast Error and the level of a', the following regression was run for the whole sample: ai,pre = a + b(AFEj) + e, (24) where aj,pre = the estimated intercept of the market model of security j in the pretime period. AFEj = the percentage difference between this year's annual EPS number and the previous year's annual EPS number. The result of the regression was j,pre = .0013 + .00000795 (AFE) 125) (4.06) (3.415) The relationship was as significant when the regression was run for the pooledtime period estimate of &j. There was no significant relationship found when the regression was run for the postperiod &j's. Thus, the high level of a in the preperiod and in the instability between periods appears to have been caused by a misspecification of the market model (9) during the preperiod.2 Note that the postperiod is not subject to the same misspecification. Although the manner in which the sample was drawn forced a high level of EPS in the pre period, there is nothing inherent in the procedure which leads to a systematic price disequilibrium in the post period. Researchers have found that high levels of EPS in one period are typically not followed by high levels of EPS in subsequent periods; i.e., EPS numbers seem to vary randomly. Since the postperiod corresponds to the quarter in which stock prices would be adjusting for the second 2Note that the same results could have been obtained had the market been systematically anticipating fourth quarter EPS; i.e., it was very efficient. However, the residual analysis which is to be presented in the next section suggests that this is unlikely. quarter EPS number, there is no reason to believe that this number will be related to the previous year's annual results. For empirical evidence in this regard, see Lintner and Glauber [24] and especially Brealey [8]. Comparison of the & in the preperiod between the sample of positive and negative forecast errors is revealing. Those securities with positive forecast errors had an average estimated intercept of .00149, which is an equivalent annual return of 37 percent (.00149 x 250). If the market were adjusting for increased third quarter EPS during the fourth quarter (preperiod), then a u greater than zero is expected and the estimate of [ would impound the . For firms with negative forecast errors the u would be negative during the fourth quarter, and indeed the estimated & is negative during that period (Table 3). This is the most convincing evidence that the different levels of & were not caused by shifting levels of the zero beta return. Since there is a significant overlap in the time period of estimation for negative and positive fore cast errors (Table 2), it is highly unlikely that the zero beta return biased the a downward for negative forecast error firms and during the same time period biased the i's upward for positive forecast error firms. The reason for the differences in the average level of explained variance (R2) between the pre and post time periods may now be rationalized. During the preperiod there is reason to believe that some of the total variance of return was caused by the adjustment process u and, thus less of the total variance was explained by the market index. The above analysis strongly suggests that the market parameters estimated in the preperiod are systemati cally biased away from the normal relationship and thus should not be used in the residual analysis. Examination of Table 3 reveals that the parameters estimated by pooling the pre and posttime periods may also be severely biased. It was therefore determined that only the post period estimated market parameters were appropriate for the residual analysis. That analysis will be presented in the next section. Residual Analysis for Positive Forecast Error The market parameters of (9) estimated in the post time period were used in (10) to estimate residual for each firm starting twenty days before and running through sixty days after the publication of the EPS number. These estimated residuals were averaged crosssectionally (11) relative to the publication date and accumulated (12) to form a cumulative average residual (CAR). Table 4 shows the average residual (11) for all firms with positive forecast errors for the publication date of EPS and ten days thereafter. The corresponding t statistic for each of these TABLE 4 THE AVERAGE RESIDUAL, t STATISTIC, AND CUMULATIVE AVERAGE RESIDUAL FOR POSITIVE FORECAST ERRORS ON SELECTED DAYS AFTER THE EARNINGS ANNOUNCEMENT Day t of u CAR 0 .01097 3.89 .01097 1 .00517 2.22 .01614 2 .00375 1.53 .01989 3 .00204 1.02 .02193 4 .00312 1.22 .02505 5 .00125 .54 .02381 6 .00033 .10 .02347 7 .00218 1.16 .02128 8 .00360 1.58 .01769 9 .00212 1.01 .01555 10 .00244 1.28 .01312 15 .0153 20 .0172 25 .0260 30 .0307 35 .0327 40 .0463 45 .0514 50 .0491 55 .0495 60 .0577 average residuals is also presented, as is the CAR for these eleven days. After day 10 the CAR is shown every five days until day 60. The t statistic for the average residual is calculated by dividing the average residual by its standard error (estimated standard deviation divided by the square root of the sample size). A t statistic greater than two implies an average residual which is statistically different from zero at approximately the .05 level of confidence. The Cumulative Average Residual for firms with positive forecast errors is plotted in Figure 1. The first day plotted (day 0) corresponds to the day the actual EPS number was published in The Wall Street Journal. The CAR is plotted for sixty days after publication, which corresponds to about three calendar months. The plot of the CAR corresponds exactly to those numbers presented in Table 4. The remainder of this section will analyze and interpret Table 4 and Figure 1. This analysis will include the realized abnormal return, the adjustment process, and the speed of adjustment. The Abnormal Return The average residuals were statistically different from zero (.05 level) on the date of publication of the EPS number (day 0) and on the following day (day 1) as indicated by the t statistic of each individual average W44 a 0 5 0 4e .*4 *r ma u 0 to 4 I , r 4 04 .40 a) * * * * * * * * 909 egS LS *69 oe, *LS *r6 ozs * OSt 6V *LV Sto *cO *es *c 6C 0ZC *9 **Z OE 061 *LI 53T SC *St 6 OL 99 *I Co *I *I * * * residual in Table 4. The cumulative return for these two days was about 1.6 percent. Current transactions costs are on the order of 1 percent on both the buy and sell sides for large roundlot transactions. Since the CAR did not reach the necessary 2 percent on the two days in which the average residuals were statistically signifi cant, it was not possible to have earned an abnormal return. Examination of Figure 1 reveals that, while the individual average residuals were not statistically signi ficant after day 1, the CAR appeared to trend strongly upward from about day 12 to about day 45. The CAR main tained its relatively high level until day 60; its ending value was 5.77 percent. The cumulative return of 5.77 percent is 3.77 percent greater than the average 2 percent transactions costs necessary to buy and sell securities in large round lots. (This assumes that the securities were purchased at the closing price on the days prior to day 0.) This abnormal return may be an indication of market inefficiency, but since the individual average residuals which comprise the CAR are not generally statistically significant (.05 level), the 3.77 percent abnormal return may have been due merely to chance occurrence. To test for the existence of a statistically significant trend, the CAR in Figure 1 was regressed against the integer value of the day relative to the publication day; i.e., (19) was run. The results of that regression are summarized in Table 5.3 The t statistic of the slope coefficient indicates that the slope is statistically significant at the .01 level. This regression line is plotted against the CAR in Figure 2. TABLE 5 SUMMARY OF REGRESSION TESTING THE STATISTICAL SIGNIFICANCE OF THE CUMULATIVE AVERAGE RESIDUAL FOR POSITIVE FORECAST ERRORS AFTER ADJUSTING FOR SERIAL CORRELATION Days Rho a b S R2 (t) (t) 060 .85 .0129 .00069 .00286 .962 (2.38) (5.08) SBecause the CAR is scaled in percentage units and the independent variable in (19) is an integer representing one day, the slope coefficient in Table 5 may be inter preted as an average rate of return per day, and the cumulative return on any particular day may be predicted from (19). This predicted cumulative average return is of interest because the regression results may be used to 3Examination of the R2 term indicates that the first order autoregressive scheme was appropriate. The regression coefficient and the serial correlation coefficient together explain 96 percent of the variance of the CAR. a 4) go 0 0 I4J ' $ r4 0 4 $4 4J 0 Ape r14 4a 'o . $4 0 0 0 4 Ua +) 0 * * * * * * * 0*39 0 5s o*as 3"LS 0199 0*9 3039 o*os 060S 0*6V 0 *3f 0*1 00*L 0* ;f 0* 9 o*aE 0*9C 0*0? 0"6 OCL O*tE 0*6I 0029 01?t 0.o 0 g* 0*0v 0O6Z 0*0t 0;61 0o91 V sa 0* 0*61 O*1 009 06*1 OOL O*Zg 0"0 * * * generate a confidence interval around any particular day's predicted cumulative average return (20). The upper and lower 95 percent confidence limits for the regression line plotted in Figure 2 are plotted in Figure 3. Note that these confidence limits are only approximate and that the true lines can be expected to bow slightly at the ends. As indicated in Chapter 2, the error in the confidence limits is likely to be small.4 The most interesting part of the regression line and its confidence limits is the predicted cumulative average rate of return on day 60, the ending day in the analysis. This predicted cumulative average rate of return may be interpreted as the average rate of return accruing to the holder of an equally weighted portfolio of all of the securities in the sample if held from day 0 to day 60. Using the numbers in Table 5, the predicted cumulative average rate of return on day 60 is 5.43 percent. The 95 percent confidence limits around this predicted point (20) are approximately 4.35 percent to 6.51 percent. The lower confidence limit of 4.35 percent is an indication that the purchaser of an equally weighted portfolio of the securities in the sample would have earned a minimum of 2.35 percent on average, after transactions 4The error is on the order of 1/n times the absolute value of the confidence limit. Since n is 60, the error is on the order of 1.6 percent of the confidence limit. 0 O5 0 1 0M 10 4 14 ,414 Ed > J $4 1 00 *40 09: O0 4) 0 A 0 O roc E4 0 04 A 494 * * * DOLS O*I9 0*99 O*9S O 9S 0"3; 0* 3 eos; O'OC 0 *I 0.L9 0*;t 0*LI *"9* 0* I O Et 0'*0 00*s 0*9c O0 se DO*C o* cc 0621 0' L 0"91 0" T 03 1 0"5a 0*391 0*0t 0"3I 0091 0"1l O*2 O*lI 002 001 * * * costs, with a 95 percent level of confidence. This return was in excess of the normal market return. Since this return accrued over a three calendar month period, if one assumes such investment opportunities are available year round, then this 2.35 percent is equivalent to an annual excess return of 9.4 percent (2.35 x 4). If one assumes that the average rate of return of 5.43 percent can be earned each three months, then the equivalent annual excess return is 13.7 percent ([5.432] x 4), after allowing for transactions costs and normal market returns. The above results and analysis are an indication that market inefficiencies existed with respect to the securities in the sample in the time periods in which they were examined. The existence of a statistically significant (.01 level) trend in the CAR indicates that the market did not instantaneously adjust to the new information, and the 95 percent confidence interval around the predicted average CAR indicates that an excess return could have been earned had one acted upon the publicly available EPS information as soon as it became available. The above abnormal return could have been earned by using a fairly simple investment strategy or decision rule, although the actual profit from such a strategy may not have been as large as it would seem. The strategy would require examining the earnings reports in The Wall Street Journal on a daily basis and immediately purchasing those securities which met the necessary criteria implied in (6) and (8). To the extent that one is not able to immediately purchase such securities, the minimum 2.35 percent abnormal return may be decreased. For instance, if the approximately 1 percent average increase on day 0 occurs early in the day, then an investor who was unable to purchase early on this day would have the average 2.35 percent abnormal return decreased to 1.35 percent. This might be a problem for some investors who only receive The Wall Street Journal after a day's delay. If the individual investor could not afford to buy in large round lots, then the abnormal return could be further pared in transactions costs which can be greater than 4 percent for oddlot purchases and sales. Another danger is that the individual investor may be insufficiently wealthy to properly diversify himself, and thus he may be subjected to the gambler's ruin. The point of this discussion is that while this study may have demonstrated a fairly strong degree of market inefficiency, the individual investor should be very cautious in trying to implement such an investment strategy unless he tests such a strategy over a long period of time without actually investing. The Adjustment Process There appear to be three distinct stages in the adjustment process of stock prices to unexpected increases in EPS. These three stages are illustrated in Figure 4; (lines are drawn as illustrations and with no pretense of accuracy). The first two stages took place during the first ten to twelve days after publication of the actual EPS number in The Wall Street Journal. The first stage consisted of a rather smooth and uninterrupted runup in price for about five market days (one week) after publica tion. The greatest percentage move (about 1 percent, see Table 4) occurred on the publication day. This move was statistically significant at the .05 level as was the move on the following day. The average cumulative price increase was 24 percent which was reached after five market trading days. After the rather strong spurt of price increases, there was a five to seven day period of rather smooth and persistent price decreases. None of these price decreases during the second stage were statistically significant (.05 level) although several tended toward significance. The average cumulative price decrease was about 1i percent. This reduced the initial price runup to about 1D percent. After the initial two stages in the adjustment process, the market appeared to enter a third stage lasting * * 0 s *v4 V 4U 0 '.4 0 4( 0 Co V0 U 0 uk me 0 04 (4J 4) 0 n4 io a ,w 0m S00 Uh 0 0 0l)> 5414 0 0 *r4 0 (4 I0 * * * * * * * * * * * * * * * * * 0 009 0 o Sg 0 *~S 0 otL 0 *9S 0 9L 0 SS 0 *09 0 o S 0 "IS 0 t03 0 *0; 0 6*1 0 *It O *; 0 9* 0 *S6 0 *I 0 *r1 0 *61 0 0o 0 SC 0 *L 0 it o 9C 0 *Ce 0 *cc 0 *ZE 0 *6; 0 OC 0 6Z 0 *8 0 "LZ 0 "9 0 *; 0 foZ 0 Ut 0 o 2 0 12 0 *0 0 06t 0 *at 0 *Lt 0 91 0 SI 0 & vl 0 *OE 0 *21 0 e1 0 01 0 *6 0 "9 0 *; 0 09 0 S 0 0 0 0 0 &  0 00 o L o 3 o i o 2 o t o 2 o gjI o 'a o *L o o '1 from about day 15 after publication of EPS to about day 45. This stage consisted of a persistent drift upward in the cumulative average residual, while only a few of the average residuals which comprised the CAR were statistically significant. Several of the average residuals between days 15 and 30 were negative, causing the increase in the CAR to be quite jerky. Beginning at about day 30 the CAR began to trend upward more smoothly than during days 15 to 30. This trend lasted until about day 45 after the announcement. After about day 45 the CAR appeared to vary more randomly than during the first fortyfive days. It is possible that the market had completed its adjustment by day 45 and was in equilibrium, although it is not clear that this was so. It is possible that the market began to anticipate first quarter EPS during this period (days 45 to 60). The preannouncement movement in the CAR will be examined and discussed in a subsequent section. From the above discussion it appears as though there is a threestage adjustment process involved in the market reaction to earnings forecast errors. A rationaliza tion of this situation may be offered involving the mecha nism whereby information is disseminated to and interpreted by investors. Specifically, the initial two stages of the adjustment may be a market reaction to incomplete earnings information in The Wall Street Journal. That is, the price may initially increase as a result of publication of the EPS number, but the market may not have sufficient detail to sustain the initial price runup. Publication of the annual EPS number in The Wall Street Journal is accompanied by only a bare minimum of the fundamental conditions surrounding the EPS number, usually only total sales, total costs, earnings and EPS numbers.5 The initial price reaction may cause technical interest which may cause the market to overreact in the first stage and as a result to decrease in the second stage. Thus, it appears as though the initial runup in price and subsequent decrease is a possible overreaction to incomplete information. The third stage may result from a lag in the availability of the annual report to investors and analysts.6 After about day 15, details of the year's operations become available in the annual report. This 5A supplementary cause of the initial two stages may be the actions of a group of investors who trade on the basis of annual EPS numbers as compared to the previous annual EPS number and not on the basis of improved fourth quarter results. 6A small study conducted at the University of Florida revealed that over a onemonth period in March 1974, there was an average of about a threeweek lag between publication of the annual EPS number in The Wall Street Journal and the arrival of the annual report of the company at the University of Florida libraries. more complete information may cause the cumulative average residual to begin to drift upward. The detail in the annual report may cause shareholders to reevaluate the longterm investment worth of the firm in question and to, perhaps, rebalance portfolios. At the same time it is to be expected that analysts will reevaluate the firms on the basis of the annual report and possibly issue buy or sell recommendations. The price changes in the third stage are permanent in nature reflecting the reevaluation of the longterm investment worth of the firms (price changes in the first stage were partially eliminated). The fact that all of the securities in the sample have at least a 20 percent change in annual EPS may tend to exaggerate the initial adjustment process. If there are indeed some traders who buy and sell solely on the basis of annual EPS numbers and ignore interim earnings, then this study would tend to overemphasize the action of such traders when compared to the "normal" adjustment mechanism. It may very well be that the initial adjust ment stage is much less pronounced for securities with less than a 20 percent change in annual earnings. The Speed of Adjustmeat If the above analysis is correct, then the adjust ment process of the stock market to published annual EPS information apparently takes about fortyfive days or about 21 calendar months after the publication of the annual EPS number. This is entirely in keeping with the Ball and Brown [3 ] results, as they also found about a twomonth adjustment process. If one concludes that the market had completed the adjustment process by the end of the fortyfifth day after the earnings announcement, then it is apparent that the market was out of equilibrium for about 75 percent of those sixty days. In this case we call the market out of equilibrium if it is moving in other than a random fashion. It appears from the statis tical analysis that this is the case. It would seem as though the twomonth adjustment process would contradict the Jones and Litzenberger results that the adjustment took place over a sixmonth period. This, however, is not true. Since this twomonth adjustment resulted in a permanent price change, it is to be expected that this price change would have persisted on average until the end of the six months. It is interesting to note that the conclusions for market efficiency would be quite different had the third stage of the threestage adjustment not taken place. The fact that the market took about ten days to adjust for the earnings announcement and that it eliminated most of the price change associated with the announcement would have been strong evidence of a fairly efficient market. The fact that the third trend existed and, what is more important, that the price increase appeared to persist is evidence that the market has reevaluated its fundamental estimate of the worth of the securities in the sample and that this reevaluation and adjustment takes a considerable length of time. It is not entirely clear that the adjustment process has been completed by the end of the fortyfifth day. The findings with respect to the anticipation of the annual EPS would suggest that a similar process might occur for quarterly EPS. If the first quarter earnings reports for most companies in the sample are published about three months (one quarter) after the annual report, then it may well be that there is some adjustment occurring during the last fifteen days also. If this is so, then it may be that the securities with highly unpredictable components of their EPS may never reach equilibrium in the sense that the market has completely adjusted for all available infor mation. Of course, it may very well be that the adjustment process for securities whose EPS are fairly predictable is indeed rapid because the arrival of the annual report could be considered to be no news to the market and thus little if any adjustment would be expected. The conclusions seem inescapable, however, that the capital markets were not very efficient with respect to the securities in this sample if one uses the speed of adjustment as the criterion for efficiency. Residual Analysis for Negative Forecast Error Table 6 lists the average residual, t statistic, and CAR for the fortyfive firms in the sample with negative forecast errors. Each is listed for the eleven days subsequent to the publication of the actual EPS number. In addition, the CAR is presented every five days from day 15 to day 60. Figure 5 is a plot of the CAR on a daily basis for days 0 to 60. Examination of Figure 5 reveals that the CAR generally drifted downward after the announcement date as would be expected for firms with unexpected decreases in EPS. As with positive forecast errors, the first two average residuals are statistically significant at the .05 level. Those two days correspond to the EPS publication date and the following day. The total price move on the first two days was a negative 2.8 percent. This is .8 percent greater than average transactions cost. Thus, it appears profitable to sell short the securities in the sample, as an excess rate of return of .8 percent could have been earned. The initial price move was reversed after only two days, and the CAR appeared to have a slight upward drift from about days 2 to 10 after publication. TABLE 6 THE AVERAGE RESIDUAL, t STATISTIC, AND CUMULATIVE AVERAGE RESIDUAL FOR NEGATIVE FORECAST ERRORS ON SELECTED DAYS AFTER THE EARNINGS ANNOUNCEMENT Day u t of u CAR .01740 .01164 .00108 .00077 .00457 .00147 .00346 .00142 .00539 .00288 .00306 2.79 2.15 .23 .20 .92  .41 .50  .32 1.46 .96 1.16 .01740 .02857 .02749 .02672 .02215 .02362 .02016 .02157 .01618 .01331 .01637 .02525 .03705 .03492 .04761 .04983 .06539 .08736 .08012 .08254 .08827 '44 O c c "q 0 4r 34 .9 90 44 9at vo > 14 >jj 01 * so 14 0 ) 0 94 0)4 44 ou'J 10 rtA * * * * * * "39 *SS 0o9 *LS *99 *r3 * LI *OS *st o00 0 *GE "3 *?c * L * I *11 0S *O *E *0? oI OLr .91 *z oz oI 6O 031 *I 0 t 00 01 Beginning at about day 10 or 11 the CAR started to drift downward again. The ending value of the CAR was 8.8 percent after sixty days, but this level had been essentially attained by the end of fortyfive days. Between days 10 and 60 few of the individual average residuals were statistically signi ficant (.05 level), yet the CAR trended downward fairly strongly. As a test of the statistical significance of that trend, the CAR was regressed against the integer value of the day relative to the announcement day; i.e., equation (19) was run. The results of that regression are presented in Table 7. Table 7 reveals that the slope coefficient is statistically significant at the .01 level. This is an indication of a statistically significant trend in the CAR and this indicates that the market did not instantaneously adjust to the new EPS information. TABLE 7 SUMMARY OF REGRESSION TESTING THE STATISTICAL SIGNIFICANCE OF THE CUMULATIVE AVERAGE RESIDUAL FOR NEGATIVE FORECAST ERRORS AFTER ADJUSTING FOR SERIAL CORRELATION Days Rho a b S R2 (t) (t) 060 .77 .0064 .00144 .0039 .976 (1.28) (10.95) Using the estimated coefficients in Table 7, the estimated cumulative average return after sixty days was a negative 9.28 percent. The 95 percent confidence interval around this average return is negative 8.06 percent to negative 10.5 percent. Assuming average transactions costs of 2 percent, the minimum expected return for short sales is 6.06 percent with a 95 percent level of confidence. This translates to a 24.3 percent excess return on an annual basis if one assumes that such investments are available year round. It appears as though the rate of return from selling short securities with negative forecast errors is considerably higher than the rate of return from long purchases of securities with positive forecast errors, at least for the securities in this sample. Recall that the Quarterly Forecast Error for both negative and positive earnings changes was an almost identical 21 percent (Table 1). One reason that the ending value of the CAR was greater for the negative series may be the finding that there is generally a greater percentage decline in price relative to a percentage change in earnings for unfulfilled earnings expectations than there is for earnings greater than expectations (see Neiderhoffer and Regan [30]). Another reason may be that the market actually projected some longterm growth in quarterly earnings, and thus the actual Quarterly Forecast Error was actually greater than indicated by the negative QFE. The similarities in the adjustment process of the stock market to negative and positive forecast error are illustrated in Figure 6, which is a plot of the CAR's for both negative and positive QFE's. The pattern of adjust ment for the two series looks remarkably similar, almost like mirror images of each other; both exhibit statistically significant price changes on the day of the earnings announcement and the following day. These price changes are directly related to the direction of forecast error. Both series exhibit a second stage just after the first in which the prices of the securities move inversely to the direction of forecast error. This situation appears to be a correction for a possible overreaction to the earnings announcement. Both series exhibit a price trend which is directly related to the direction of forecast error which begins 10 to 12 days after publication of the EPS number and lasts until about day 45. From days 45 to 60 both series appear more or less horizontal and thus appear to change in a random fashion, although it is not altogether clear that this is so. The fact that both series begin to trend in a direct relation to the change in EPS between days 10 and 15 seems to indicate that some common occurrence causes this trend, perhaps the arrival of the 4. 4. 4. 4. 4. 4. 4. O 0 I M 4%0 CS 0 4) 443 ow 00 *r4 Si id 414 ^rl 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 S 0 * 0 + 0 + 0 0 0 0 t4 0 0 + 0 + 0 +* 0 0 + 0 + 0 + 0 0 0 0 + 0 + 0 + 0 0 0 0 0 0 + 0 0 0 0 + 0 0 * * * * * * 4. * * * * * * * * o09 *SS *s; *9S Ls S *ess * *5s *C3 S *09 *6g 6tp * Lt * 3 o *I *03 *Re *1C OLZ *o9 *O; e*Z *cc *oZ IZ 61 *9I *LI *91 *Sl o*I *II *3I "6 o9 "* *t S0 ol actual annual report in the hands of investors. Although the two series exhibit some remarkable similarities, there are also some distinct differences. The adjustment process for the first ten days appears to be slightly different. Both the negative and positive series peak in absolute value at about 2 percent, yet it takes the positive series about five days to reach this level, while the negative series achieves a 2.8 percent in two large and significant jumps. (Recall from Table 1 that the average QFE for the two series were almost identical [21.5 percent versus 21.6 percent] and that the average AFE's were greater for the positive series [68.6 percent versus 44 percent].) One possible explanation for this is that the two significant moves for the positive series trigger certain technical indicators such as relative strength tests and cause investor interest on the basis of this information. This technical interest could cause the price to keep increasing for the extra few days. Such a mechanism is not available for negative price moves, so that the entire adjustment appears to take place in just two days. The second stage of the adjustment process seems to be more smooth and less random for the positive than the negative series. Notice that the average residual tended toward statistical significance between days 5 and 10 for the positive forecast error series, but that the t statistics are near zero between days 2 and 10 for the negative series (see Tables 4 and 6). The indication is that whatever causes the second stage of the adjustment process, it is not as strong for the negative series as it is for the positive series. PreAnnouncement Movements in the CAR Examination of the average residuals and CAR prior to the actual announcement of EPS is of interest because it gives an indication of how the market anticipates the earnings announcement. The market may use many different sources of information in the anticipation process. Economywide movements and earnings of firms in the same industry may allow investors to form estimates of the futureEPS of a firm and, perhaps, cause them to purchase or sell securities on the basis of such information. Move ments in the CAR as a result of investors acting on such information would be an indication of an efficient market. However, movement in the CAR caused by investors taking advantage of other sources of information could be an indication of market inefficiencies. Such sources of information include news leakages and insiders taking advantage of monopolistic sources of information. Unfortunately, it is impossible to tell if movements in the CAR prior to the EPS announcement are caused by information of this sort or information about economywide or industrywide movements. Examination of the CAR prior to the earnings announcements is, nonetheless, still of interest. Figure 7 is a plot of the CAR for both positive and negative forecast errors beginning twenty days prior to and ending ten after the earnings announcement. Tables 8 and 9 show the average residual, t statistic of the average residual, and the cumulative average residual. The CAR's in both cases started accumulating at day 20 or about one month before the announcement. The direction of movement in the CAR's prior to the announcement date is directly related to the direction of forecast error. The CAR's begin a rather smooth trend in anticipation of the announcement approximately fifteen days (three weeks) before the actual announcement. Both positive and negative series reach a cumulative absolute value of 3.9 percent on the day before the announcement. The average residual is significantly different from zero (.05 level) on the day before the announcement for the positive series and on the second day prior to the announcement for the negative series. The patterns of adjustment for the positive and negative series are again remarkably similar. The indication is that the anticipation and adjustment mechanism 0 11 + uO 0 04.1 0 P . 4 I 0 4 m 0 ul c? 04o 0m *0w u.44> rt4 I 0 co $ 0 0444 1.4 C^ *D Cr s *05 3 *L 09 aS *0 *1 0l; * * c * 6 O6 *1 II 2E 3l *'l Lt l 61 *0 a OZ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S0 *0 0 *0 *04 *0 S0 0* * * * * * TABLE 8 PREANNOUNCEMENT AVERAGE RESIDUAL, t STATISTIC, AND CUMULATIVE AVERAGE RESIDUALS FOR POSITIVE FORECAST ERRORS Day u t of CAR 20 .00008 .04 .00008 19 .00091 .03 .00099 18 .00298 1.29 .00396 17 .00079 .42 .00495 16 .00033 .13 .00442 15 .00385 1.80 .00827 14 .00097 .45 .00730 13 .00261 1.11 .00991 12 .00252 1.25 .01243 11 .00077 .41 .01165 10 .00058 .30 .01108  9 .00341 1.62 .01448  8 .00374 1.74 .01822  7 .00077 .37 .01745  6 .00521 1.64 .02266  5 .00137 .52 .02404  4 .00488 2.11 .02892  3 .00252 1.10 .03143  2 .00302 1.15 .03445  1 .00504 2.03 .03950 0 .01097 3.89 .05047 1 .00517 2.22 .05564 2 .00375 1.52 .05940 3 .00204 1.02 .06143 4 .00312 1.22 .06455 5 .00125 .54 .06330 6 .00033 .10 .06300 7 .00218 1.16 .0688 8 .00359 1.58 .05720 9 .00212 1.01 .05510 10 .00244 1.28 .05262 TABLE 9 PREANNOUNCEMENT AVERAGE RESIDUAL, t STATISTIC, AND CUMULATIVE AVERAGE RESIDUALS FOR NEGATIVE FORECAST ERRORS Day t of W CAR 20 19 18 17 16 15 14 13 12 11 10  9  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 9 10 .00324 .00309 .00032 .00171 .00066 .01255 .00285 .00356 .00253 .00317 .00398 .00203 .00018 .00602 .00059 .00598 .00029 .00633 .00776 .00244 .01740 .00112 .00108 .00077 .00457 .00147 .00346 .00142 .00539 .00288 .00306  .90 1.26  .08  .45 .18 3.71 .95  .89  .03  .08  .96 .62 .05 1.67  .17 1.15 .09 1.69 2.45 .53 2.79 2.15 .23 .20 .92  .41 .50  .32 1.46 .96 1.16 .00324 .00015 .00047 .00218 .00152 .01407 .01123 .01478 .01731 .01414 .01812 .01609 .01591 .02193 .02252 .02850 .02820 .03454 .04230 .03986 .05727 .06843 .06735 .06658 .06201 .06348 .06002 .06144 .05605 .05317 .05623 