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]