《投资学原理 Investments》课程教学资源（阅读资料）The adjustment of stock prices to new information

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INTERNATIONAL ECONOMIC REVIEW February, 1969 VoL. 10, No. 1 THE ADJUSTMENT OF STOCK PRICES TO NEW INFORMATION* BY EUGENE F. FAMA, LAWRENCE FISHER MICHAEL C. JENSEN AND RICHARD ROLL1 1. INTRODUCTION a THERE IS an impressive body of empirical evidence which indicates that cessive price changes in individual common stod eks are ve rly inc endent. 2 Recent papers by Mandelbrot [11] and Samuelson[16] show rigor ously that independence of successive price changes is consistent with an "efficient"market, i. e. a market that adjusts rapidly to new information It is important to note, however, that in the empirical work to date the sual procedure has been to infer market efficiency from the observed inde. pendence of successive price changes. There has been very little actual testing of the speed of adjustment of prices to specific kinds of new infor mation. The prime concern of this paper is to examine the process by which common stock prices adjust to the information (if any) that is implicit in a stock split PLITS, DIVIDENDS, AND NEW INFORMATION: A HYPOTHESIS More specifically this study will attempt to examine evidence on two related questions:(1)Is there normally some "unusual"behavior in the rates of return on a split security in the months surrounding the split? and (2) if splits are associated with "unusual"behavior of security returns, to what extent can this be accounted for by relationships between splits and changes Manuseript received May 31, 1966, revised October 3, 1966. to Professors Lorie, Merton H. Miller, and Harry V. Roberts for many helpful tar ments and criticisms The research reported here was supported by the Center for Research in Security Prices, Graduate School of Business, University of Chicago, and by funds made available to the Center by the National Science Foundation. s Cf Cootner [2]and the studies reprinted therein, Fama [3], Godfrey, Granger, and Morgenstern [8]and other empirical studies of the theory of random walks in specu lative prices a precise definition of"unusual " behavior of security returns will be provided

February, 1969 Vol, 10, No, 1 THE ADJUSTMENT OF STOCK PRICES TO NEW INFORMATION* BY EUGENEF. FAMA,LAWRENCEFISHER, MICHAELC. JENSENAND RICHARDROLL^ 1. INTRODUCTION THEREIS an impressive body of empirical evidence which indicates that successive price changes in individual common stocks are very nearly indeen dent.^ Recent papers by Mandelbrot [ll] and Samuelson [16] show rigorously that independence of successive price changes is colzsistent with an "efficient" market, i.e., a market that adjusts rapidly to new information. It is important to note, however, that in the empirical work to date the usual procedure has been to infer market efficiency from the observed independence of successive price changes. There has been very little actual testing of the speed of adjustment of prices to specijc kinds of new information. The prime concern of this paper is to examine the process by which common stock prices adjust to the information (if any) that is implicit in a stock split. 2. SPLITS, DIVIDENDS, AND NEW INFORMATION : A HYPOTHESIS More specifically, this study will attempt to examine evidence on two related questions: (1) Is there normally some "unusual" behavior in the rates of return on a split security in the months surrounding the split?3 and (2) if splits are associated with "unusual" behavior of security returns, to what extent can this be accounted for by relationships between splits and changes * Manuscript received May 31, 1966, revised October 3, 1966. 1 This study way suggested to us by Professor James H. Lorie. We are grateful to Professors Lorie, Merton H. Miller, and Harry V. Roberts for many helpful comments and criticisms. The research reported here was supported by the Center for Research in Security Prices, Graduate School of Business, University of Chicago, and by funds made available to the Center by the National Science Foundation. 2 Cf.Cootner [2] and the studies reprinted therein, Fama [3], Godfrey, Granger, and Morgenstern [8] and other empirical studies of the theory of random walks in speculative prices. a A precise definition of "unusual" behavior of security returns will be provided below

FAMA, FISHER. JENSEN AND ROLL in other more fundamental variables? In answer to the first question we shall show that stock splits are usually preceded by a period during which the rates of return(including dividends and capital appreciation) on the securities to be split are unusually high The period of high returns begins, however, long before any information (or even rumor) concerning a possible split is likely to reach the market. Thus we suggest that the high returns far in advance of the split arise from the fact that during the pre-split period these companies have experienced dra- matic increases in expected earnings and dividends in the empirical work reported below, however, we shall see that the highes verage monthly rates of eturn on split shares occur in the few months mmediately preceding the split. This might appear to suggest that the split itself provides some impetus for increased returns. We shall present evi- dence however, which suggests that such is not the case The evidence sup- ports the following reasoning: Although there has probably been a dramatic increase in earnings in the recent past, in the months immediately prior to the split (or its announcement)there may still be considerable uncertainty in the market concerning whether the earnings can be maintained at their new higher level. Investors will attempt to use any information available to luce this uncertainty, and a proposed split may be one source of such information In the past a large fraction of stock splits have been followed closely by dividend increases-and increases greater than those experienced at the same time by other securities in the market. In fact it is not unusual for the dividend change to be announced at the same time as the split. other studies(ef. Lintner [10] and Michaelsen [14]) have demonstrated that, once dividends have been increased, large firms show great reluctance to reduce them, except under the most extreme conditions. Directors have appeared to hedge against such dividend cuts by increasing dividends only when they are quite sure of their ability to maintain them in the future, i. e. only when they feel strongly that future earnings will be sufficient to maintain the dividends at their new higher rate. Thus dividend changes may be assumed to convey important information to the market concerning managements There is another question concerning stock splits which th ider. That is, given that splitting is not costless, and since the only apparent resul s to multiply the number of shares per shareholder without increasing the share- holder's claims to assets, why do firms split their shares? This question has been the subject of considerable discussion in the professional financial literature. (Cf. Bellemore and Blucher [1].) Suffice it to say that the arguments offered in favor of splitting usually turn out to be two-sided under closer examination -e.g. a split, by reducing the price of a round lot, will reduce transactions costs for some rela- tively small traders but increase costs for both large and very small traders (i.e for traders who will trade, exclusively, either round lots or odd lots both before and after the split). Thus the conclusions are never clear-cut. In this study we shall be concerned with identifying the factors which the market regards as important in a stock split and with determining how market prices adjust to these factors rather than with explaining why firms split their shares

2 FAMA, FISHER, JENSEN AND ROLL in other more fundamental variables?' In answer to the first question we shall show that stock splits are usually preceded by a period during which the rates of return (including dividends and capital appreciation) on the securities to be split are unusually high. The period of high returns begins, however, long before any information (or even rumor) concerning a possible split is likely to reach the market. Thus we suggest that the high returns far in advance of the split arise from the fact that during the pre-split period these companies have experienced dra rnatie increases in expected earnings and dividends. In the empirical work reported below, however, we shall see that the highest average monthly rates of return on split shares occur in the few months immediately preceding the split. This might appear to suggest that the split itself provides some impetus for increased returns. We shall present evidence, however, which suggests that such is not the case. The evidence supports the following reasoning: Although there has probably been a dramatic increase in earnings in the recent past, in the months immediately prior to the split (or its announcement) there may still be considerable uncertainty in the market concerning whether the earnings can be maintained at their new higher level. Investors will attempt to use any information available to reduce this uncertainty, and a proposed split may be one source of such information. In the past a large fraction of stock splits have been followed closely by dividend increases-and increases greater than those experienced at the same time by other securities in the market. In fact it is not unusual for the dividend change to be announced at the same time as the split. Other studies (cf. Lintner [lo] and Michaelsen [14]) have demonstrated that, once dividends have been increased, large firms show great reluctance to reduce them, except under the most extreme conditions. Directors have appeared to hedge against such dividend cuts by increasing dividends only when they are quite sure of their ability to maintain them in the future, i.e., only when they feel strongly that future earnings will be sufficient to maintain the dividends at their new higher rate. Thus dividend changes may be assumed to convey important information to the market concerning management's There is another question concerning stock splits which this study does not consider. That is, given that splitting is not costless, and since the only apparent result is to multiply the number of shares per shareholder without increasing the shareholder's claims to assets, why do firms split their shares? This question has been the subject of considerable discussion in the professional financial literature. (Cf. Bellemore and Blucher [I].) Suffice it to say that the arguments offered in favor of splitting usually turn out to be two-sided under closer examination -e.g., a split, by reducing the price of a round lot, will reduce transactions costs for some relatively small traders but increase costs for both large and very small traders (i.e., for traders who will trade, exclusively, either round lots or odd lots both before and after the split). Thus the conclusions are never clear-cut. In this study we shall be concerned with identifying the factors which the market regards as important in a stock split and with determining how market prices adjust to these factors rather than with explaining why firms split their shares

ADJUSTMENT OF STOCK PRICES assessment of the firm' s long- run earning and dividend paying potential We suggest, then, that unusually high returns on splitting shares in the months immediately preceding a split reflect the market,s anticipation of substantial increases in dividends which, in fact, usually occur. Indeed evidence presented below leads us to conclude that when the information effects of dividend changes are taken into account, the apparent price effects f the split will vanish. 5 3. SAMPLE AND METHODOLOGY a. The data. We define a"stock split" as an exchange of shares in which at least five shares are distributed for every four formerly outstanding. Thus this definition of splits includes all stock dividends of 25 per cent or greater We also decided arbitrarily, that in order to get reliable estimates of the parameters that will be used in the analysis, it is necessary to have at least twenty-four successive months of price-dividend data around the split date. Since the data cover only common stocks listed on the New York Stock Exchange, our rules require that to qualify for inclusion in the tests a split security must be listed on the Exchange for at least twelve months before and twelve months after the split. From January, 1927, through December 1959, 940 splits meeting these criteria occurred on the New York Stock Exchange. b. Adjusting security returns for general market conditions. Of course during this 33 year period, economic and hence general stock market condi- tions were far from static. Since we are interested in isolating whatever eatraordinary effects a split and its associated dividend history may have on returns, it is necessary to abstract from general market conditions in examining the returns on securities during months surrounding split dates We do this in the following way: Define Pit= price of the j-th stock at end of month Pit adjusted for capital changes in month t+1. For the method of adjustment see Fisher [5] Dit= cash dividends on the j-th security during month t(where the divi dend is taken as of the ex-dividend data rather than the payment date). Rit=(Pit+ D/Pi t-1= price relative of the j-th security for month t L t= the link relative of Fisher's"Combination Investment Performance Index"[6,(table A1)]. It will suffice here to note that L is a com- s It is important to note that our hypothesis concerns the information content of dividend changes. There is nothing in our evidence policy per se affects the value of a firm. Indeed, the first suggested by Miller and Modigliani in [15,(430)] = from information effects, in a perfect capital market dividend polic the total market value of a firm 8 The basic data were contained in the master file of monthly dividends, and capital changes, collected and maintained by the center for Resea Security Prices graduate School of Business, University of Chicago). At the time this study was conducted, the file covered the period January, 1926 to December, 1960. For a description of the data see Fisher and Lorie [7]

ADJUSTMENT OF STOCK PRICES 3 assessment of the firm's long-run earning and dividend paying potential. We suggest, then, that unusually high returns on splitting shares in the months immediately preceding a split reflect the market's anticipation of substantial increases in dividends which, in fact, usually occur. Indeed evidence presented below leads us to conclude that when the inforination effects of dividend changes are taken into account, the apparent price effects of the split will ~anish.~ 3. SAMPLE AND METHODOLOGY a. The data. We define a "stock split" as an exchange of shares in which at least five shares are distributed for every four formerly outstanding. Thus this definition of splits includes all stock dividends of 25 per cent or greater. We also decided, arbitrarily, that in order to get reliable estimates of the parameters that will be used in the analysis, it is necessary to have at least twenty-four successive months of price-dividend data around the split date. Since the data cover only common stocks listed on the New York Stock Exchange, our rules require that to qualify for inclusion in the tests a split security must be listed on the Exchange for at least twelve months before and twelve months after the split. From January, 1927, through December, 1959, 940 splits meeting these criteria occurred on the New York Stock E~change.~ b. Adjusting security returns for general market conditions. Of course, during this 33 year period, economic and hence general stock market conditions were far from static. Since we are interested in isolating whatever extraordinary effects a split and its associated dividend history may have on returns, it is necessary to abstract from general market conditions in examining the returns on securities during months surrounding split dates. We do this in the following way: Define Pjt= price of the j-th stock at end of month t. Pj', = Pjt adjusted for capital changes in month t + 1. For the method of adjustment see Fisher [5]. Djt = cash dividends on the j-th security during month t (where the dividend is taken as of the ex-dividend data rather than the payment date). Rjt = (Pjt + ~j~)/Pf,~-l = price relative of the j-th security for month t. Lt = the link relative of Fisher's "Combination Investment Performance Index" [6, (table Al)]. It will suffice here to note that Lt is a com- 5 It is important to note that our hypothesis concerns the information content of dividend changes. There is nothing in our evidence which suggests that dividend policy per se affects the value of a firm. Indeed, the information hypothesis was first suggested by Miller and Modigliani in [15, (430)], where they show that, aside from information effects, in a perfect capital market dividend policy will not affect the total market value of a firm. 6 The basic data were contained in the master file of monthly prices, dividends, and capital changes, collected and maintained by the Center for Research in Security Prices (Graduate School of Business, University of Chicago). At the time this study was conducted, the file covered the period January, 1926 to December, 1960. For a description of the data see Fisher and Lorie [7]

FAMA, FISHER, JENSEN AND ROLL plicated average of the Rit for all securities that were on the n..Se. at the end of months t and t-1. Lt is the measure of "general market conditions"used in this study. One form or another of the following simple model has often been sug gested as a way of expressing the relationship between the monthly rates f return provided by an individual security and general market conditions: a g.Bt=a;+月;log+wit, ajt is a random disturbance term. It is assumed that wje satisheecurity assumptions of the linear regression model. That is,(a)uit has zero ex pectation and variance independent of t;(b the ujt are serially independent and (c) the distribution of u, is independent of log L The natural logarithm of the security price relative is the rate of return with continuous compounding) for the month in question; similarly, the log of the market index relative is approximately the rate of return on a port lio which includes equal dollar amounts of all securities in the market. Thus (1) represents the monthly rate of return on an individual security as a linear function of the corresponding return for the market c. Tests of model speci fication. Using the available time series on R3 and Lt, least squares has been used to estimate ay and B, in (1)for each of the 622 securities in the sample of 940 splits. We shall see later that there is strong evidence that the expected values of the residuals from (1)are non-zero in months close to the split. For these months the assumptions of the regression model concerning the disturbance term in (1) are not valid Thus if these months were included in the sample, estimates of a and S rould be subject to specification error, which could be very serious. We have attempted to avoid this source of specification error by excluding from the estimating samples those months for which the expected values of the 7 To check that our results do not arise from any special properties of the index Lt, we have also performed all tests using Standard and Poor's Composite Price Index as the measure of market conditions; in all major respects the results agre completely with those reported below 8 Cf. Markowitz [13,(96-101) Sharpe [17, 18 and Fama The logarithmic form I is appea for two reasons. First, over the period covered by our ata the distribution of the monthly values of loge Le and loge rit are fairly sym metric, whereas the distributions of the relatives themselves are skewed right. Sym- metry is desirable since models involving symmetrically distributed variables present fewer estimation problems than models involving variables with skewed distributions. Second, we shall see below that when least squares is used to estimate a and a in ) the sample residuals conform well to the assumptions of the simple linear regres- sion model Thus, the logarithmic form of the model appears to be well specified from a sta- tistical point of view and has a natural economie interpretation (i.e. in terms of monthly rates of return with continuous compounding). Nevertheless, to check that our results do not depend critically on using logs, all tests have also been carried out using the simple regression of Rit on Lt. These results are in complete agree ment with those presented in the text

FAMA, FISHER, JENSEN AND ROLL plicated average of the Rjt for all securities that were on the N.Y.S.E. at the end of months t and t -1. Lt is the measure of "general market conditions" used in this st~dy.~ One form or another of the following simple model has often been suggested as a way of expressing the relationship between the monthly rates Of return provided by an individual security and general market condition^:^ (1) loge Rjt nj + $j loge Lt + ujt , where aj and Fj are parameters that can vary from security to security and ujt is a random disturbance term. It is assumed that ujt satisfies the usual assumptions of the linear regression model. That is, (a) ujt has zero expectation and variance independ.ent of t; (b) the ujt are serially independent; and (c) the distribution of uj is independent of log, L. The natural logarithm of the security price relative is the rate of return (with continuous compounding) for the month in question; similarly, the log of the market index relative is approximately the rate of return on a portfolio which includes equal dollar amounts of all securities in the market. Thus (1) represents the monthly rate of return on an individual security as a linear function of the corresponding return for the market. c. Tests of model specification. Using the available time series on Rjt and .Lt, least squares has been used to estimate nj and in (1)for each of the 622 securities in the sample of 940 splits. We shall see later that there is strong evidence that the expected values of the residuals from (1) are non-zero in months close to the split. For these months the assumptions of the regression model concerning the disturbance term in (1)are not valid. Thus if these months were included in the sample, estimates of n and f would be subject to specification error, which could be very serious. We have attempted to avoid this source of specification error by excluding from the estimating samples those months for which the expected values of the 7 To check that our results do not arise from any special properties of the index Lt, we have also performed all tests using Standard and Poor's Composite Price Index as the measure of market conditions; in all major respects the results agree completely with those reported below. 8 Cf. Markowitz 113, (96-101)], Sharpe [la, 181 and Fama 141. The logarithmic form of the model is appealing for two reasons. First, over the period covered by our data the distribution of the monthly values of log, Lt and log, Rjt are fairly symmetric, whereas the distributions of the relatives themselves are skewed right. Symmetry is desirable since models involving symmetrically distributed variables present fewer estimation problems than models involving variables with skewed distributions. Second, we shall see below that when least squares is used to estimate a and P in (I), the sample residuals conform well to the assumptions of the simple linear regression model. Thus, the logarithmic form of the model appears to be well specified from a statistical point of view and has a natural economic interpretation (i.e., in terms of monthly rates of return with continuous compounding). Nevertheless, to check that our results do not depend critically on using logs, all tests have also been carried out using the simple regression of Rjt on Lt. These results are in complete agreement with those presented in the text

ADJUSTMENT OF STOCK PRICES residuals are apparently non-zero. The exclusion procedure was as follows First, the parameters of (1)were estimated for each security using all avail able data. Then for each split the sample regression residuals were com- puted for a number of months preceding and following the split. When the number of positive residuals in any month differed substantially from the number of negative residuals, that month was excluded from subsequent alculations This criterion caused exclusion of fifteen months before the split for all securities and fifteen months after the split for splits followed by dividend decreases° Aside from these exclusions, however, the least squares estimates Bi for security 3 are based on all months during the 1926-60 period for price relatives are available for the security. For the 940 splits the smallest effective sample size is 14 monthly observations. In only 46 cases is the ample size less than 100 months, and for about 60 per cent of the splits more than 800 months of data are available. Thus in the vast ma jority of cases the samples used in estimating a and P in(1)are quite large Table 1 provides summary descriptions of the frequency distributions of the estimated values of ai, Bi, and ri, where ri is the correlation between monthly rates of return on security j(i.e, loge Rit)and the approximate monthly rates of return on the market portfolio (i.e., loge Lt). The table indicates that there are indeed fairly strong relationships between the market and monthly returns on individual securities; the mean value of the i, is 0.682 with an average absolute deviation of 0. 106 about the mean o TABLE 1 SUMMARY OF FREQUENCY DISTRIBUTIONS OF ESTIMATED COEFFICIENTS FOR THE DIFFERENT SPLIT SECURITIES Statistic Mean Median Mean absolute Standard reme 0.007-0.06, 0.04 Slightly left 08940.880 0.305-0.10*, 1.95 Slightly right 0. 132-0.04*, 0.91 Slightly left Only negative value in distribution. A Moreover, the estimates of equation (1) for the different securities conform irly well to the assumptions of the linear regression model. For example 9 admittedly the exclusion criterion is arbitrary. As a check, however the analysis of regression residuals discussed later in the paper has been carried out using the regression estimates in which no data are excluded The results were much the same as those reported in the text and certainly support the same conclusions 10 The sample average or mean absolute deviation of the random variable a is de fined as N where I is the sample mean of the as and N is the sample size

ADJUSTMENT OF STOCK PRICES 5 residuals are apparently non-zero. The exclusion procedure was as follows: First, the parameters of (1) were estimated for each security using all available data. Then for each split the sample regression residuals were computed for a number of months preceding and following the split. When the number of positive residuals in any month differed substantially from the number of negative residuals, that month was excluded from subsequent calcu'lations. This criterion caused exclusion of fifteen months before the split for a11 securities and fifteen months after the split for splits followed by dividend decreases9. Aside from these exclusions, however, the least squares estimates 8j and jj for security j are based on all months during the 1926-60 period for which price relatives are available for the security. For the 940 splits the smallest effective sample size is 14 monthly observations. In only 46 cases is the sample size less than 100 months, and for about 60 per cent of the splits more than 800 months of data are available. Thus in the vast majority of cases the samples used in estimating cu and P in (1)are quite large. Table f provides summary descriptions of the frequency distributions of the estimated values of aj, pi, and ri, where rj is the correlation between monthly rates of return on security j (i.e., log, Rjt) and the approximate monthly rates of return on the market portfolio (i.e., log, Lt). The table indicates that there are indeed fairly strong relationships between the market and monthly returns on individual securities; the mean value of the qi is 0.632 with an average absolute deviation of 0.106 about the mean.1° TABLE 1 SUMMARY OF FREQUENCY DISTRIBUTIONS OF ESTIMATED COEFFICIENTS FOR THE DIFFERENT SPLIT SECURITIES Statistic Mean PI'Iedian Mean absolute Standard I / deviation deviation Extreme Skewness values 1--1 a 0.000 0.001 0.004 0.007 -0.06, 0.04 Slightly left ,: 0.894 0.880 0.242 0.305 -0.10*, 1.95 Slightly right T 0.632 0.655 0.106 0.132 -0.04*, 0.91 Slightly left * Only negative value in distribution. Moreover, the estimates of equation (1)for the different securities conform fairly well to the assumptions of the linear regression model. For example, 9 Admittedly the exclusion criterion is arbitrary. As a check, however, the analysis of regression residuals discussed later in the paper has been carried out using the regression estimates in which no data are excluded. The results were much the same as those reported in the text and certainly support the same conclusions. 10 The sample average or mean absolute deviation of the random variable x is defined as where Z is the sample mean of the x's and N is the sample size

ADJUSTMENT OF STOCK PRICES Mandelbrot [12])argues that distributions of returns are well approximate by the non-Gaussian (i.e. infinite variance) members of the stable Paretian family. If the stable non-Gaussian distributions also provide a good descrip- tion of the residuals in (1), then, at first glance, the least squares regression model would seem inappropriate Wise [19] has shown, however, that although least square estimates "efficient, "for most members of the stable Paretian family they provide estimates which are unbiased and consistent. Thus, given our large samples least squares regression is not completely inappropriate. In deference to the stable Paretian model, however, in measuring variability we rely primarily on the mean absolute deviation rather than the variance or the standard deviation. The mean absolute deviation is used since, for long-tailed distri butions, its sampling behavior is less erratic than that of the variance or the standard deviation! In sum we find that regressions of security returns on market returns over time are a satisfactory method for abstracting from the effects of gen eral market conditions on the monthly rates of return on individual secu rities. We must point out, however, that although (1)stands up fairly well to the assumptions of the linear regression model, it is certainly a gros over-simplified model of price formation; general market conditions alone do not determine the returns on an individual security. In (1)the effects of these"omitted variables"are impounded into the disturbance term u. In particular, if a stock split is associated with abnormal behavior in returns during months surrounding the split date, this behavior should be reflected in the estimated regression residuals of the security for these months. The re mainder of our analysis will concentrate on examining the behavior of the estimated residuals of split securities in the months surrounding the splits 3. "EFFECTS OF SPLITS ON RETURNS: EMPIRICAL RESULTS In this study we do not attempt to determine the effects of splits for in- dividual companies. Rather we are concerned with whether the process of splitting is in general associated with specific types of return behavior. To abstract from the eccentricities of specific cases we can rely on the simple process of averaging; we shall therefore concentrate attention on the behavior of cross-sectional averages of estimated regression residuals in the months b a. Some additional definitions. The procedure is as follows: For a given olit, define month 0 as the month in which the effective date of a split occurs. (Thus month 0 is not the same chronological date for all securities, and indeed some securities have been split more than once and hence have more than one month 0). 12 Month 1 is then defined as the month immediately Essentially, this is due to the fact that in computing the variance of a sample large deviations are weighted more heavily than in computing the mean absolute deviation, For empirical evidence concerning the reliability of the mean absolute deviation relative to the variance or standard deviation see Fama [3,(94-8) split more than once

ADJUSTMENT OF STOCK PRICES 7 Mandelbrot [12]) argues that distributions of returns are well approximated by the non-Gaussian (i.e., infinite variance) members of the stable Paretian family. If the stable non-Gaussian distributions also provide a good description of the residuals in (I), then, at first glance, the least squares regression model would seem inappropriate. Wise [I91 has shown, however, that although least square estimates are not "efficient," for most members of the stable Paretian family they provide estimates which are unbiased and consistent. Thus, given our large samples, least squares regression is not completely inappropriate. In deference to the stable Paretian model, however, in measuring variability we rely primarily on the mean absolute deviation rather than the variance or the standard deviation. The mean absolute deviation is used since, for long-tailed distributions, its sampling behavior is less erratic than that of the variance or the standard deviation1'. In sum we find that regressions of security returns on market returns over time are a satisfactory method for abstracting from the effects of genera,l market conditions on the monthly rates of return on individual securities. We must point out, however, that although (1)stands up fair!y well to the assumptions of the linear regression model, it is certainly a grossly over-simplified model of price formation; general market conditions alone do not determine the returns on an individual security. In (1)the effects of these "omitted variables" are impounded into the disturbance term u. In particular, if a stock split is associated with abnormal behavior in returns during months surrounding the split date, this behavior should be reflected in the estimated regression residuals of the security for these months. The remainder of our analysis will concentrate on examining the behavior of the estimated residuals of split securities in the months surrounding the splits. 3. "EFFECTS" OF SPLITS ON RETURNS: EMPIRICAL RESULTS In this study we do not attempt to determine the effects of splits for individual companies. Rather we are concerned with whether the process of splitting is in general associated with specific types of return behavior. To abstract from the eccentricities of specific cases we can rely on the simple process of averaging; we shall therefore concentrate attention on the behavior of cross-sectional averages of estimated regression residuals in the months surrounding split dates. a. Some additional definitions. The procedure is as follows: For a given split, define month 0 as the inonth in which the effective date of a split occurs. (Thus month 0 is not the same chronological date for all securities, and indeed some securities have been split more than once and hence have more than one month 0).12 Month 1is then defined as the month immediately " Essentially, this is due to the fact that in computing the variance of a sample, large deviations are weighted more heavily than in computing the mean absolute deviation. For empirical evidence concerning the reliability of the mean absolute deviation relative to the variance or standard deviation see Fama [3, (94-8)]. '2 About a third of the securities in the master file split. About a third of these split more than once

FAMA, FISHER, JENSEN AND ROLL following the split month, while month-1 is the month preceding, etc. Now define the average residual for month m(where m is always measured rela- tive to the split month)as there Aim is the sample regression residual for security j in month m and nm is the number of splits for which data are available in month m. 3 Our principal tests will involve examining the behavior of um for m in the in- terval-29 s m s 30, i.e., for the sixty months surrounding the split month. We shall also be interested in examining the cumulative effects of abnormal return behavior in months surrounding the split month. Thus we define the cumulative average residual Um as The average residual um can be interpreted as the average deviation (in month m relative to the split month) of the returns of split stocks from their normal relationships with the market. Similarly, the cumulative average residual Um can be interpreted as the cumulative deviation(from month-29 to month it shows the cumulative effects of the wanderings of the re- turns of split stocks from their normal relationships to market movements Since the hypothesis about the effects of splits on returns expounded in Section 2 centers on the dividend behavior of split shares, in some of the tests to follow we examine separately splits that are associated with increased dividends and splits that are associated with decreased dividends. In addi- tion, in order to abstract from general changes in dividends across the market, "increased and "decreased"dividends will be measured relative to the average dividends paid by all securities on the New york Stock Exchange during the relevant time periods. The dividends are classified as follows: Define the dividend change ratio as total dividends (per equivalent unsplit share) paid in the twelve months after the split, divided by total dividends paid during the twelve months before the split. 4 Dividend"increases"are then defined as cases where the dividend change ratio of the split stock is greater than the ratio for the Exchange as a whole while dividend"decreases include cases of relative dividend decline. s We then define u+, u- and U+ 13 Since we do not consider splits of c that were not on the New York Stock Exchange for at least a year be a year after a split, nm will be 940 for -11 s m s 12. For other months, a dividend day the security trades ex-dividend on the excha 15 When dividend"increase"and "decrease"are defined relative to the market it turns out that dividends were never unchanged. " That is, the dividend change ratios of split securities are never identical to the corresponding ratios for the ex (Continued on neat page

8 FAMA, FISHER, JENSEN AND ROLL following the split month, while month -1 is the month preceding, etc. Now define the average residual for month m (where m is always measured relative to the split month) as where iij, is the sample regression residual for security j in month m and n,, is the number of splits for which data are available in month m.13 Our principal tests will involve examining the behavior of u, for m in the interval -29 5 m 5 30, i.e., for the sixty months surrounding the split month. We shall also be interested in examining the cumulative effects of abnormal return behavior in months surrounding the split month. Thus we define the cumulative average residual U, as The average residual u, can be interpreted as the average deviation (in month m relative to the split month) of the returns of split stocks from their normal relationships with the market. Similarly, the cumulative average residual Urn can be interpreted as the cumulative deviation (from month -29 to month m); it shows the cumulative effects of the wanderings of the returns of split stocks from their normal relationships to market movements. Since the hypothesis about the effects of splits on returns expounded in Section 2 centers on the dividend behavior of split shares, in some of the tests to follow we examine separately splits that are associated with increased dividends and splits that are associated with decreased dividends. In addition, in order to abstract from general changes in dividends across the market, "increased" and "decreased" dividends will be measured relative to the average dividends paid by all securities on the New York Stock Exchange during the relevant time periods. The dividends are classified as follows: Define the dividend change ratio as total dividends (per equivalent unsplit share) paid in the twelve months after the split, divided by total dividends paid during the twelve months before the split.I4 Dividend "increases" are then defined as cases where the dividend change ratio of the split stock is greater than the ratio for the Exchange as a whole, while dividend "decreases" include cases of relative dividend decline.15 We then define ud, u; and Uk, '3 Since we do not consider splits of companies that were not on the New York Stock Exchange for at least a year before and a year after a split, rzm will be 940 for -11 5 m 5 12. For other months, however, nm. < 940. '4 A dividend is considered "paid" on the first day the security trades ex-dividend on the Exchange. '5 When dividend "increase" and "decrease" are defined relative to the market, it turns out that dividends were never "unchanged." That is, the dividend change ratios of split securities are never identical to the corresponding ratios for the Exchange as a whole. (Continued on next page)

ADJUSTMENT OF STOCK PRICES Um as the average and cumulative average residuals for splits followed by "increased"(+) and"decreased"() dividends These definitions of "increased"and"decreased"dividends provide a simple and convenient way of abstracting from general market dividend changes in classifying year-to-year dividend changes for individual securities. The def initions have the following drawback, however. For quarterly dividends an increase in its dividend rate at any time during the nine months before or twelve months after the split can place its stock in the dividend "increased"class. Thus the actual increase need not have oc- curred in the year after the split. The same fuzziness, of course also arises in classifying dividend"decreases. " We shall see later, however that this fuzziness fortunately does not obscure the differences between the aggregate behavior patterns of the two grou b. Empirical results. The most important empirical results of this study are summarized in Tables 2 and 3 and Figures 2 and 3. Table 2 presents the average residuals, cumulative average residuals, and the sample size for each f the two dividend classifications ("increased, and"decreased")and for the total of all splits for each of the sixty months surrounding the split. Figure phs of ge residuals for the total sample of splits and Figure 3 presents these graphs for each of the two dividend classifications. Table 3 shows the number of splits each year along with the end of June level of the stock price index. Several of our earlier statements can now be substantiated. First, Figure Ba, 3a and 3b show that the average residuals(um) in the twenty-nine months prior to the split are uniformly positive for all splits and for both classes of dividend behavior. This can hardly be attributed entirely to the splitting process. In a random sample of fifty-two splits from our data the median time between the announcement date and the effective date of the split was 44.5 days. Similarly, in a random sample of one hundred splits that occurred etween 1/1/1946 and 1/1/1957 Jaffe [9 found that the median time between announcement date and effective date was sixty-nine days. For both samples in only about 10 per cent of the cases is the time between announcement date and effective date greater than four months. Thus it seems safe to say that the split cannot account for the behavior of the regression residuals as far as two and one-half years in advance of the split date Rather we sug gest the obvious-a sharp improvement, relative to the market, in the earn- ings prospects of the company sometime during the years immediately pre- ceding a split Thus we conclude that companies tend to split their shares during "ab- normally"good times-that is during periods of time when the prices of their shares have increased much more than would be implied by the normal In the remainder of the paper we shall always use“ Increase”and“ decrease”as defined in the text. That is, signs of dividend changes for individual securities are measured relative to changes in the dividends for all N Y S.E. common stocks

ADJUSTMENT OF STOCK PRICES 9 U; as the average and cumulative average residuals for splits followed by "increased" (+) and "decreased" (-) dividends. These definitions of "increased" and "decreased" dividends provide a simple and convenient way of abstracting from general market dividend changes in classifying year-to-year dividend changes for individual securities. The definitions have the following drawback, however. For a company paying quarterly dividends an increase in its dividend rate at any time during the nine months before or twelve months after the split can place its stock in the dividend "increased" class. Thus the actual increase need not have occurred in the year after the split. The same fuzziness, of course, also arises in classifying dividend "decreases." We shall see later, however, that this fuzziness fortunately does not obscure the differences between the aggregate behavior patterns of the two groups. b. Empirical Results. The most important empirical results of this study are summarized in Tables 2 and 3 and Figures 2 and 3. Table 2 presents the average residuals, cumulative average residuals, and the sample size for each of the two dividend classifications ("increased," and "decreased") and for the total of all splits for each of the sixty months surrounding the split. Figure 2 presents graphs of the average and cumulative average residuals for the total sample of splits and Figure 3 presents these graphs for each of the two dividend classifications. Table 3 shows the number of splits each year along with the end of June level of the stock price index. Several of our earlier statements can now be substantiated. First, Figures 2a, 3a and 3b show that the average residuals (u,,) in the twenty-nine months prior to the split are uniformly positive for all splits and for both classes of dividend behavior. This can hardly be attributed entirely to the splitting process. In a random sample of fifty-two splits from our data the median time between the announcement date and the effective date of the split was 44.5 days. Similarly, in a random sample of one hundred splits that occurred between 1/1/1946 and 1/1/1957 Jaffe [9] found that the median time between announcement date and effective date was sixty-nine days. For both samples in only about 10 per cent of the cases is the time between announcement date and effective date greater than four months. Thus it seems safe to say that the split cannot account for the behavior of the regression residuals as far as two and one-half years in advance of the split date. Rather we suggest the obvious-a sharp improvement, relative to the market, in the earnings prospects of the company sometime during the years immediately preceding a split. Thus we conclude that companies tend to split their shares during "abnormally" good times-that is during periods of time when the prices of their shares have increased much more than would be implied by the normal In the remainder of the paper we shall always use "increase" and "decrease" as defined in the text. That is, signs of dividend changes for individual securities are measured relative to changes in the dividends for all N.Y.S.E. common stocks

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