《投资学原理 Investments》课程教学资源（阅读资料）the behavior of stock-market prices（fama,1965）

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THE BEHAVIOR OF STOCK-MARKET PRICES* EUGENE F FAMA I. INTRODUCTION havior will tend to recur in the future JoR many years the following ques- Thus, if through careful analysis of price tion has been a source of continuing charts one develops an understanding controversy in both academic and of these patterns, this can be used to business circles: To what extent can the predict the future behavior of prices an past history of a common stocks price in this way increase expected gains be used to make meaningful predictions By contrast the theory of random concerning the future price of the stock? walks says that the future path of the price level of a security is no more pre- Answers to this question have been po, dictable than the path of a series of vided on the one hand by the varie chartist theories and on the other hand cumulated random numbers. In statisti by the theory of random walks al terms the theory says that successive Although there are many different price changes are independent, identical- chartist theories, they all make the same ly distributed random variables. Most basic assumption. That is, they all as- simply this implies that the series of price sume that the past behavior of a securi- changes has no memory, that is, the ty's price is rich in information concern- past cannot be used to predict the future ng its future behavior. History repeats in any meaningful way itself in that"patterns"of past price be-,. The purpose of this paper will be to This study has profited from the criticisms, underlying the random-walk model and ggestions, and technical assistance of many dif- then to test the model's empirical validi ratitude to professors William Alberts, lawrence ty. The main conclusion will be that the Fisher, Robert Graves, James Lorie, Merton Miller, data seem to present consistent and ate School of Business, University of Chicago. I wish strong support for the model. This im- pecially to thank Professors Miller and Roberts for plies, of course, that chart reading roviding not only continuous intellectual stimula- though perhaps an interesting pastime, ading the v is of no real value to the stock market in Many of the ideas in this t of the vestor. This is an extreme statement and work of Benoit Mandelbrot of the IBM Watson Re- the chart reader is certainly free to take ritten work of Dr Mandelbrot many exception We suggest, however, that nce the empirical evidence produced by Work on this paper was supported in part by this and other studies in support of the funds from a grant by the Ford Foundation to Graduate School of Business of the University of random-walk model is now so volumi Chicago, and in part by funds granted to the Center nous, the counterarguments of the chart time was provided by the 7094 Computation Center force if they are not equally well supp s for Research in Security, Prices of the School by the reader will be completely lacking University of Chica ed by empirical work f Assistant professor of finance, Graduate School I The Dow Theory of course, is the best known of Business, University of Chicage example of a chartist theory

BEHAVIOR OF STOCK-MARKET PRICES II. THEORY OF RANDOM WALKS one is trying to solve. For example, some IN STOCK PRICES one who is doing statistical work in the The theory of random walks in stock stock market may wish to decide whether actually involves two separate dependence in the series of successive hypotheses: (1)successive price changes price changes is sufficient to account for are independent, and (2)the price some particular property of the distribu- changes conform to some probability bon of price changes. If the actual de- distribution. We shall now examine each pendence in the series is not sufficient to of these hypotheses in detail account for the property in question, the the independence hypothesi quate description of reality. In statistical termsindependence means By contrast the stock market trader that the probability distribution for the has a much more practical criterion for price change during time period t is inde- judging what constitutes important de his purposes the random walk model is during previous time periods. That is, valid as long as knowledge of the past knowledge of the sequence of price changes behavior of the series of price changes leading up to time period t is of no help in assessing the probability distribution cannot be used to increase expected gains for the price change during time period More specifically, the independence as- Now in fact we can probably never reality as long as the actual degree of hope to find a time series that is charac- dependence in the series of price changes terize ed by perfect independence. Thus, is not sufficient to allow the past history strictly speaking, the random walk the- of the series to be used to predict the ory cannot be a completely accurate de- future in a way which makes expected scription of reality. For practical pur- profits greater than they would be under poses, however, we may be willing to a naive buy-and-hold model accept the independence assumption of Dependence that is important from the model as long as the dependence in the trader 's point of view need not be im the series of successive price changes rtant from a statistical point of view, not above some "minimum acceptable, and conversely dependence which is im- level portant for statistical purposes need not What constitutes a" minimum accept. be important for investment purposes. able"level of dependence depends, of For exanple, we may know that on alter course, on the particular problem that nate days the price of a security always increases by e and then decreases by e From a statistical point of view knowl Pr(!=xJ*L1,2t-g,.)=Pr(ac=x), edge of this dependence would be impor tant information since it tells us quite a where the term on the right of the equality sig the unconditional probability that the bit about the shape of the distribution during time t will take the value a, of price changes. For trading purposes, term on the left is the conditional pr the price change will take the value however, as long as e is very small, this on the knowledge that previous price perfect, negative, statistical dependence the values xt-1,at-2, etc. is unimportant. Any profits the trader

THE JOURNAL OF BUSINESS may hope to make from it would be with the random-walk hypothesis. In washed away in transactions costs order to justify this statement, however, In Section V of this paper we shall be it will be necessary now to discuss more concerned with testing independence fully the process of price determination from the point of view of both the statis- in an intrinsic-value-random-walk mar tician and the trader at this point, how- ket ever, the next logical step in the develop- Assume that at any point in time ment of a theory of random walks in there exists, at least implicitly, an intrin stock prices is to consider market situa- sic value for each security. The intrinsic tions and mechanisms that are consistent value of a given security depends on the with independence in successive price earnings prospects of the company which hanges. The procedure will be to con- in turn are related to economic and po- sider first the simplest situations and litical factors some of which are peculiar then to successively introduce complica- to this company and some of which affect other companies as well. s We stress. however, that actual mar 2. MARKET SITUATIONS CONSISTENT ket prices need not correspond to intrin- sic values. In a world of uncertaintSas55 Independence of successive price trinsic values are not known exac changes for a given security may simply Thus there can always be disagreemer reflect a price mechanism which is totally among individuals, and in this way ad litical events. That is, stock prices may Henceforth uncertainty e 3 unrelated to real-world economic and po- tual prices and intrinsic values of randomly generated noise, where by concerning intrinsic values will emer be just the accumulation of many bits noise in this case we mean psychological the market under the general heading of“ noise”in and other factors peculiar to different In addition. intrinsic values can them- the types selves change across time as a result of of bets"they are willing to place on either new information or trend. New in different companies Even random walk theorists, however, the success of a current research and de would find such a view of the market un- velopment project, a change in manage- appealing. Although some people may be ment, a tariff imposed on the industry's primarily motivated by whim, there are product by a foreign country, an increase many individuals and institutions that in industrial production or any other seem to base their actions in the market actual or anticipated change in a factor painstaking) of economic and politica/ which is likely to affect the companys circumstances. That is, there are many 3 We can think of intrinsic values in either of two ways. First, perhaps they just cur ities have conventions for e ting the worth of a security "intrinsic values"which depend on eco- by relating it to various factors which affect nomic and political factors that affect in- values may actually represe dividual companies lve from The existence of intrinsic values for some dynamic general equ vlew one individual securities is not inconsistent takes

BEHAVIOR OF STOCK-MARKET PRICES On the other hand, an anticipated however, the assumptions upon which long-term trend in the intrinsic value of the Bachelier-Osborne model is built are a given security can arise in the following rather extreme. There is no strong reason way. Suppose we have two unlevered to expect that each individuals estimates companies which are identical in all re- of intrinsic values will be independent spects except dividend policy. That is, of the estimates made by others (i.e. both companies have the same current noise may be generated in a dependent and anticipated investment opportuni- fashion). For example, certain individ ties, but they finance these opportunities uals or institutions may be opinion lead- in different ways. In particular, one com- ers in the market. That is, their actions pany pays out all of its current earnings may induce people to change their opin as dividends and finances new invest- ions concerning the prospects of a given ment by issuing new common shares. company. In addition there is no strong The other company, however, finances reason to expect successive bits of new new investment out of current earnings information to be generated independ- and pays dividends only when there is ently across time. For example, goo money left over. Since shares in the two news may tend to be followed more often a ompanies are subject to the same degree by good news than by bad news, and bad risk, we would expect their expected news may tend to be followed more often ates of returns to be the same. This will by bad news than by good news. Thus be the case, however, only if the shares there may be dependence in either the payout have a higher expected rate of generating new information, and price increase than do the shares of the may in turn lead to dependence in suc- igh-payout company. In this case the cessive price changes trend in the price level is just part of the Even in a situation where there are expected return to equity. Such a trend dependencies in either the information is not inconsistent with the random-walk or the noise generating process, however, it is still possible that there are offsetting pendence assumption of the random walk produce independence in price changes model was proposed first, in a rather for individual common stocks. For ex- much later but more explo] and then ample, let us assume that there are many vague fashion, by Bachelier itly by Os- sophisticated traders in the stock market borne[42]. The argument runs as follows: and that sophistication can take two If successive bits of new information forms: (1)some traders may be much arise independently across time, and if better at predicting the appearance of noise or uncertainty concerning intrinsic new information and estimating its ef values does not tend to follow any con- fects on intrinsic values than others sistent pattern, then successive price while (2)some may be much better at changes in a common stock will be inde- doing statistical analyses of price be pendent havior. Thus these two types of sophis- As with many other simple models, ticated traders can be roughly thought 4 A trend in the price level, of course, corresponds to a non-zero mean in the distribution of price 5 A lengthy and rigorous justification for these statements is given by Miller and Modigliani [40]

THE JOURNAL OF BUSINESS and superior chart readers. We further The effectiveness of their activities in assume that, although there are some- erasing dependencies in the series of price times discrepancies between actual prices changes can, however, be reinforced by and intrinsic values, sophisticated trad- another neutralizing mechanism. As long ers in general feel that actual prices usu- as there are important dependencies ally tend to move toward intrinsic val- the series of successive price changes, op- portunities for trading profits are avail Suppose now that the noise generating able to any astute chartist. For example process in the stock market is dependent. once they understand the nature of the More specifically assume that when one dependencies in the series of successive person comes into the market who thinks price changes, sophisticated chartists will the current price of a security is above be able to identify statistically situations or below its intrinsic value, he tends where the price is beginning to run up to attract other people of like feelings above the intrinsic value. Since they ex- and he causes some others to change pect that the price will eventually move their opinions unjustifiably. In itself this back toward its intrinsic value, they will type of dependence in the noise generat- sell. Even though they are vague about ing process would tend to produce"bub- intrinsic values, as long as they have bles""in the price series, that is, periods sufficient resources their actions will tend of time during which the accumulation to erase dependencies and to make actual of the same type of noise causes the price prices closer to intrinsic values. level to run well above or below the in- Over time the intrinsic value of a trinsic value common stock will change as a result of If there are many sophisticated traders new information, that is, actual or an in the market, however, they may cause ticipated changes in any variable that these"bubbles"to burst before they affects the prospects of the company. If have a chance to really get under way. there are dependencies in the process For example, if there are many sophist- generating new information, this in it- cated traders who are extremely good at self will tend to create dependence in estimating intrinsic values, they will be successive price changes of the security. able to recognize situations where the If there are many sophisticated traders ice of a common stock is beginning to in the market, however, they should run up above its intrinsic value. Since eventually learn that it is profitable for they expect the price to move eventually them to attempt to interpret both the back toward its intrinsic value, they have price effects of current new information an incentive to sell this security or to and of the future information implied by sell it short. If there are enough of these the dependence in the information gen- sophisticated traders, they may tend to erating process. In this way the actions prevent these "bubbles ' from ever oc- of these traders will tend to make price curring. Thus their actions will neutral- changes independent ize the dependence in the noise-generat- Moreover, successive price changes ing process, and successive price changes may be independent even if there is usu will be independent ally consistent vagueness or uncertainty n fact of course. in a world of uncer tainty even sophisticated traders cannot erating process is itself relevant information which always estimate intrinsic values exactly. the astute trader should consider

BEHAVIOR OF STOCK-MARKET PRICES surrounding new information. For exam- In sum, this discussion is sufficient to pIe, if uncertainty concerning the im- show that the stock market may conform portance of new information consistently to the independence assumption of the causes the market to underestimate the random walk model even though the effects of new information on intrinsic processes generating noise and new in- values, astute traders should eventually formation are themselves dependent. We learn that it is profitable to take this into turn now to a brief discussion of some account when new information appears of the implications of independence in the future. That is, by examining the history of prices subsequent to the infux 3. IMPLICATIONS OF INDEPENDENCE of new information it will become clear In the previous section we saw that that profits can be made simply by buy- one of the forces which helps to produce ing(or selling short if the information is independence of successive price changes pessimistic)after new information comes may be the existence of sophisticated into the market since on the average ac- traders, where sophistication may mean tual prices do not initially move all the either(1) that the trader has a special way to their new intrinsic values. If talent in detecting dependencies in series many traders attempt to capitalize on of prices changes for individual securi- this opportunity, their activities will ties, or( 2)that the trader has a special tend to erase any consistent lags in the talent for predicting the appearance of adjustment of actual prices to changes new information and evaluating its ef in intrinsic values fects on intrinsic values. The first kind plies, of of trader corresponds to a superior chart course, that, if there are many astute reader, while the second corresponds to traders in the market, on the average a superior intrinsic value analyst the full effects of new information on in- Now although the activities of the trinsic values will be reflected nearly in- chart reader may help to produce inde- stantaneously in actual prices. In fact, pendence of successive price changes however, because there is vagueness or once independence is established chart surrounding new informa- reading is no longer a pr In a series of independent has two implications. First, actual prices the past history of the series cannot be will initially overadjust to the new in- used to increase expected profits trinsic values as often as they will under- Such dogmatic statements cannot be djust. Second, the lag in the complete applied to superior intrinsic-value analy djustment of actual prices to successive sis, however. In a dvnamic econom new intrinsic values will itself be an in- there will always be new information dependent random variable, sometimes which causes intrinsic values to change preceding the new information which is over time. As a result, people who can he basis of the change (i.e., when the consistently predict the appearance of information is anticipated by the market new information and evaluate its effects before it actually appears)and some- on intrinsic values will usually make ase successive price changes in individ- have this talent. The fact that the ac mot times following. It is clear that in this larger profits than can people who do ual securities will be independent random ities of these superior analysts help to variables make successive price changes independ-

THE JOURNAL OF BUSINESS ent does not imply that their expect Unfortunately, by this criterion this profits cannot be greater than those of author does not qualify as a superior the investor who follows some naive buy- analyst. There is some consolation, how and-hold policy er. since. as we shall see later, other It must be emphasized, however, that more market-tested institutions do not the comparative advantage of the supe- seem to qualify eithe rior analyst over his less talented com- Finally, let us now briefly formulate a petitors lies in his ability to predict rational investment policy for the aver consistently the appearance of newe in- age investor in a situation where stock formation and evaluate its impact on prices follow random walks and at every intrinsic values. If there are enough su- point in time actual prices represent good perior analysts, their existence will be estimates of intrinsic values. In such a sufficient to insure that actual market situation the primary concern of the prices are, on the basis of all available average investor should be portfolio anal- information, best estimates of intrinsic ysis. This is really three separate prob- values. In this way, of course, the supe- lems. First, the investor must decide rior analysts make intrinsic value analy- what sort of tradeoff between risk and sis a useless tool for both the average expected return he is willing to accept analyst and the average investor. Then he must attempt to classify securi- to three ties according to riskiness, and finally he obvious question: (1) How many superior must also determine how securities from analysts are necessary to insure inde- different risk classes combine to form analysts? and(3)What is a rational in- risk and return, arious combinations of rtfolios with va vestment policy for an average investor In essence in a random-walk market faced with a random-walk stock market? the security analysis problem of the aver- It is impossible to give a firm answer age investor is greatly simplified If actu to the first question, since the effective al prices at any point in time are good ness of the superior analysts probably estimates of intrinsic values, he need not depends more on the extent of their re- be concerned with whether individual sources than on their number. Perhaps a securities are over-or under-priced. If he single, well-informed and well-endowed decides that his portfolio requires an specialist in each security is sufficient. additional security from a given risk It is, of course, also very difficult to class, he can choose that security ran- identify ea anle those people that qualify domly from within the class. On the aver as superior analysts. E post, however, age any security so chosen will have there is a simple criterion. a superior about the same effect on the expected re analyst is one whose gains over many turn and riskiness of his portfolio periods of time are consistently greater B. THE DISTRIBUTION OF PRICE CHANGES than those of the market. Consistently is the crucial word here, since for any 1. INTRODUCTION given short period of time, even if there The theory of random walks in stock are no superior analysts, in a world of prices is based on two hypotheses random walks some people will do much (1)successive price changes in an indi better than the market and some will do 1E complete formulation of the port much worse folio ana blem see Markowitz [39

BEHAVIOR OF STOCK-MARKET PRICES vidual security are independent, and changes occur quite frequently, it may (2)the price changes conform to some be safe to infer that the economic struc- probability distribution Of the two hy- ture that is the source of the price changes potheses independence is the most impor- is itself subject to frequent and sudden tant. Either successive price changes are shifts over time. That is, if the distribu independent (or at least for all practical tion of price changes has a high degree of purposes independent)or they are not; dispersion, it is probably safe to infer and if they are not, the theory is not that, to a large extent, this is due to the valid. All the hypothesis concerning the variability in the process generating new distribution says, however, is that the information price changes conform to some probabili- Finally, the form of the distribution of ty distribution. In the general theory of price changes is important information random walks the form or shape of the to anyone who wishes to do empirical distribution need not be specified. Thus work in this area. The power of a statis- any distribution is consistent with the tical tool is usually closely related to the theory as long as it correctly character- type of data to which it is applied. Ir izes the process generating the price fact we shall see in subsequent sections changes. 8 that for some probability distributions From the point of view of the investor, important concepts like the mean and however, specification of the shape of the variance are not meaningful distribution of price changes is extremely helpful. In general, the form of the dis- BACHELIER-OSBORNE MODEL tribution is a major factor in determining The first complete development of a the riskiness of investment in common theory of random walks in security prices stocks. For example, although two differ- is due to Bachelier [6], whose original ent possible distributions for the price work first appeared around the turn of changes may have the same mean or ex- the century. Unfortunately his work did pected price change, the probability of not receive much attention from econo- very large changes may be much greater mists, and in fact his model was inde for one than for the other. pendently derived by Osborne [42 ]over The form of the distribution of price fifty years later. The Bachelier-Osborne hanges is also important from an aca- model begins by assuming that price demic point of view since it provides de- changes from transaction to transaction scriptive information concerning the na- in an individual security are independ ture of the process generating price ent, identically distributed random vari- changes. For example if very large price ables. It further assumes that transac- 8 Of course the theory does imply that the tions are fairly uniformly spread across time, and that the distribution of ionarity can be independence holds, however, sta- changes from transaction to transaction if independence holds in a strict fashion, then for the has finite variance. If the number of tion to reality even though the parameters of the probability distribution of the ry large, then price changes across these differencing intervals will be sums For statistical purposes stationarity implies of many independent variables. Under mply that the parameters of the distribution should be fixed at least for the time period covered by the these conditions the central-limit thec rem leads us to expect that the daily

THE JOURNAL OF BUSINESS weekly, and monthly price changes will that, in the past, academic research has each have normal or Gaussian distribu- too readily neglected the implications of tions. Moreover, the variances of the dis the leptokurtosis usually observed in tributions will be proportional to the re- empirical distributions of price changes spective time intervals. For example, if The presence, in general, of leptokur o' is the variance of the distribution of tosis in the empirical distributions seems the daily changes, then the variance for indisputable. In addition to the results the distribution of the weekly changes of Kendall [26] and Moore [41] cited should be approximately 502. above, Alexander [1] has noted that os Although Osborne attempted to give borne's cross-s il data do not really an empirical justification for his theory, support the normality hypothesis; there most of his data were cross-sectional and are too many changes greater than t 10 could not provide an adequate test. per cent. Cootner [10] has developed Moore and Kendall, however, have pro- whole theory in order to explain the lor vided empirical evidence in support of tails of the empirical distributions. Final the Gaussian hypothesis. Moore [41, pp. ly, Mandelbrot [37, Fig. 1] cites other 116-23] graphed the weekly first differ examples to document empirical lepto- ences of log price of eight NYSE common kurtosis stocks on normal probability paper. Al- The classic approach to this problem hough the extreme sections of his graphs has been to assume that the extreme seem to have too many large price values are generated by a different mech changes, Moore still felt the evidence anism than the majority of the observa was strong enough to support the hy- tions. Consequently one tries a posteriori pothesis of approximate normality to find“ causal” explanations for the Similarly Kendall [26] observed that large observations and thus to rational weekly price changes in British common ize their exclusion from any tests carried stocks seem to be approximately nor- out on the body of the data. 10 Unlike the mally distributed. Like Moore, however, statistician, however, the investor cannot he finds that most of the distributions of ignore the possibility of large price price changes are leptokurtic; that changes before committing his funds, and there are too many values near the mean once he has made his decision to invest, and too many out in the extreme tails. he must consider their effects on his In one of his series some of the extreme wealth. observations were so large that he felt Mandelbrot feels that if the outliers compelled to drop them from his subse- are numerous, excludin quent statistical tests away much of the significance from any tests carried out on the 3. MANDELBROT AND THE GENERALIZED CENTRALLIMIT THEOREM the data. This exclusion process is all the more subject to criticism since probabil- The Gaussian hypothesis was not seri- ity distributions are available which ac- ously questioned until recently when the curately represent the large observations work of Benoit mandelbrot first began to 1o When extreme values are excluded from th appear. 9 Mandelbrot's main assertion is sample, the procedure is often called "trimming Another technique which involves reducing the size His main work in this area is 3 erences of extreme observations rather than excluding them to his other works are for J. w and in the bibliography. ukey[45]

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