BEHAVIOR OF STOCK-MARKET PRICE as well as the main body of the data. normal or Gaussian distribution. when a The distributions referred to are mem- is in the intervalo a 2, the extreme bers of a special class which mandelbrot tails of the stable Paretian distributions has labeled stable Paretian. The mathe- are higher than those of the normal dis matical properties of these distributions tribution, and the total probability in are discussed in detail in the appendix to the extreme tails is larger the smaller the this paper. At this point we shall merely value of a. The most important conse- introduce some of their more important quence of this is that the variance exists descriptive properties (i.e., is finite) only in the extrem Parameters of stable Paretian distri- a= 2. The mean, however, exists as long butions -Stable Paretian distributions as a >1.2 have four parameters:(1)alocation pa- Mandelbrots hypothesis states that rameter which we shall call 8, (2)a scale for distributions of price changes in spec parameter henceforth called y, (3)an ulative series, a is in the interval 1 <a< index of skewness, B, and(4)a measure 2, so that the distributions have means of the height of the extreme tail areas of but their variances are infinite. The the distribution which we shall call the Gaussian hypothesis, on the other hand characteristic states that a is exactly equal to 2. Thus When the characteristic exponent a is both hypotheses assume that the distri- greater than 1, the location parameters bution is stable Paretian. The disagree- bution. The scale parameter r can be any of the characteristic exponent e value the expectation or mean of the distri ment between them concerns th positive real number, but B, the index of skewness, can only take values in the in- lions.Two important properties of sta terval -1<B< 1. when 8=0 the dis- ble Paretian distributions are(1)stabil- tribution is symmetric. When B>0 the ity or invariance under addition, and (2) distribution is skewed right (i.e, has a the fact that these distributions are the long tail to the right), and the degree of only possible limiting distributions for value of p. imi is larger the larger the sums of independent,identically distrib- tribution is skewed left, and the degree By definition, a stable Paretian distri of left skewness is larger the smaller the bution is any distribution that is stable value of B r invariant under addition. That The characteristic exponent a of a the distribution of sums of independent stable Paretian distribution determines identically distributed, stable Paretian the height of, or total probability con- variables is itself stable Paretian and, tained in, the extreme tails of the distri- except for origin and scale, has the same bution, and can take any value in the form as the distribution of the individual interval< a< 2. When a= 2, the rel- summands. Most simply, stability means evant stable Paretian distribution is the that the values of the parameters a and u The derivation of most of the important B remain constant under additio erties of stable Paretian distributions is due to p The property of stability is responsible Levy [29. A rigorous and compact mathematical 1 For a proof of these statements see Gnedenko t of the theory can be found in B. v nd Kolmogorov[171, pp. 179-83 comprehensive mathematical treatment can be 1a A more rigorous definition of stability is given found in Mandelbrot[37] in the appendix
THE JOURNAL OF BUSINESS for much of the appeal of stable Paretian the underlying distributions of price distributions as descriptions of empirical changes from transaction to transaction distributions of price changes. The price are allowed to have infinite variances. In hange of a stock for any time interval this sense, then, Mandelbrot's version of can be regarded as the sum of the changes the theory of random walks can be re from transaction to transaction during garded as a broadening rather than a e interval. If transactions are fairly contradiction of the earlier Bachelier- uniformly spread over time and if the Osborne model changes between transactions are inde Conclusion.-Mandelbrot's hypothesis pendent, identically distributed, stable that the distribution of price changes is Paretian variables, then daily, stable Paretian with characteristic expo- and monthly changes will follow stable nent a 2 has far reaching implications Paretian distributions of exactly the For exampl DIe. if the variances of distribu- same form, except for origin and scale. tions of price changes behave as if they For example, if the distribution of daily are infinite, many common statistical changes is stable Paretian with location tools which are based on the assumption parameter 8 and scale paremeter y, the of a finite variance either will not work distribution of weekly (or five-day) or may give very misleading answers changes will also be stable Paretian with Getting along without these familiar location parameter 58 and scale parame- tools is not going to be easy, and before ter 5y. It would be very convenient if parting with then y host impressiry must be sure that changes were independent of the differ- At the moment, the most impressive encing interval for which the changes single piece of evidence is a direct test rere computed of the infinite variance hypothesis for It can be shown that stability or in- the case of cotton prices. Mandelbrot[37, der addition leads to a most Fig. 2 and pp. 404-7] computed the sam important corollary property of stable ple second moments of the first differ- Paretian distributions; they are the only ences of the logs of cotton prices for ossible limiting distributions for sums increasing sample sizes of from 1 to 1, 300 of independent, identically distributed, observations. He found that the sample random variables. 14 It is well known that moment does not settle down to any if such variables have finite variance, the limiting value but rather continues to limiting distribution for their sum will be vary in absolutely erratic fashion, pre- the normal distribution. If the basic vari- cisely as would be expected under his ables have infinite variance, however, hypothesis. 5 and if their sums follow a limiting dis- As for the special but important case tribution, the limiting distribution must be stable Paretian with0 a<2. The second moment of a random variable x is just E(a ) The variance is just the second moment In light of this discussion minus the square of the mean. Since the mean is Mandelbrot's hypothesis can actually assumed to be a constant, tests of the moment are also tests of the sample variance. be viewed as a generalization of the In an earlier privately circulated version of[37] central-limit theorem arguments of Mandelbrot tested his hypothesis on various other Bachelier and osborne to the case where series of specu 14 For a proof see Gnedenko and Kolmogorov neither as extensive nor as conclusive as the tests [17],pp.162-03. on cotton prices
BEHAVIOR OF STOCK-MARKET PRICES of common-stock prices, no published the actual departures from normality are evidence for or against Mandelbrots the- sufficient to reject the Gaussian hypothe ory has yet been presented. One of our sis. The only goal will be to see if the main goals here will be to attempt to test departures are usually in the direction Mandelbrots hypothesis for the case of predicted by the Mandelbrot hypothes B. THE DATA C. THINGS TO COME The data that will be used throughout Except for the concluding section, the this paper consist of daily prices for each remainder of this paper will be concerned of the thirty stocks of the Dow-Jones with reporting the results of extensive Industrial Average. 16 The time periods tests of the random walk model of stock vary from stock to stock but usually run price behavior. Sections III and IV will from about the end of 1957 to September examine evidence on the shape of the 26, 1962. The final date is the same for distribution of price changes. Section III all stocks, but the initial date varies from will be concerned with common statisti- January, 1956 to April, 1958. Thus there cal tools such as frequency distributions are thirty samples with about 1, 200- and normal probability graphs, while 1,700 observations per sample Section IV will develop more direct tests The actual tests are not performed on of Mandelbrot's hypothesis that the the daily prices themselves but on the characteristic exponent a for these dis- first differences of their natural loga tributions is less than 2 Section V of the rithms. The variable of interest is paper tests the independence assumption of the random-walk model. Finally, Sec Pu+1-loge tion VI will contain a summary of pre- where pei is the price of the security at vious results, and a discussion of the im- the end of day t+ 1, and Pr is the price plications of these results from various at the end of day t points of vie There are three main reasons for using hanges in log price rather than simple III. A FIRST LOOK A' E price changes. First, the change in log PIRICAL DISTRIBUTIONS price is the yield, with continuous com A. INTRODUCTION pounding, from holding the security for A> In this section a few simple techniques that day. 7 Second, Moore [41, pp. 13-151 be used to examine distributions of has shown that the variability of simple daily stock-price changes for individual pnce changes for a given stock is an in- securities. If Mandelbrot's hypothesis stock. His work indicates that taking creasing function of the price level of the that the distributions are stable paretian with characteristic exponents less than 2 16 The data were very generously correct, the most important feature of Professor Harry B Ernst of Tufts Unive the distributions should be the length of The proof of this statement goes as follows their tails. That is, the extreme tail areas p:+1= exp(obe pt 力t+1 than would be expected if the distribu tions were normal. In this section no Pi+1= pie attempt will be made to decide whether PI exp(loge Pu+1-loge Pr)
THE JOURNAL OF BUSINESS logarithms seems to neutralize most of behavior of these blue-chips stocks differs this price level effect. Third, for changes consistently from that of other stocks in less than t 15 per cent the change in the market, the empirical results to be log price is very close to the percentage presented below will be strictly appli price change, and for many purposes it is cable only to the shares of large impor- convenient to look at the data in terms tant companies of percentage price changes. 8 One must admit, however that the In working with daily changes in log sample of stocks is conservative from the rice, two special situations must be point of view of the Mandelbrot hypoth- noted. They are stock splits and ex-divi- esis, since blue chips are probably more dend days. Stock splits are handled as stable than other securities. There is follows: if a stock splits two for one on reason to expect that if such a sample day t, its actual closing price on day t is conforms well to the Mandelbrot hypoth doubled, and the difference between the esis, a random sample would fit even logarithm of this doubled price and the better. t- 1 is the first difference for day t first difference for day t+ 1 is the differ- One very simple way of analyzing the ence between the logarithm of the closing distribution of changes in log price is to price on day t+ 1 and the logarithm of construct frequency distributions for the the actual closing price on day t, the day individual stocks. That is, for each stock of the split. These adjustments reflect the fact that the process of splitting a within given standard deviations of the stock involves no change either in the mean change can be computed and com asset value of the firm or in the wealth pared with what would be expected if the of the individual shareholder distributions were exactly normal. This On ex-dividend days, however, other is done in Tables 1 and 2 In Table 1 the things equal, the value of an individual proportions of observations within 0.5 share should fall by about the amount 1.0,1.5, 2.0, 2.5, 3.0, 4.0, and 5.0 stand of the dividend. To adjust for this the ard deviations of the mean change, as first difference between an ex-dividend well as the proportion greater than 5 ay and the preceding day is computed standard deviations from the mean, are computed for each stock. In the first line of the body of the table the proportions for the unit normal distribution are where d is the dividend per share. 9 One final note concerning the data is Table 2 gives a comparison of the unit in order. The Dow-Jones Industrials are normal and the empirical distributions not a random sample of stocks from the 1 I recognize that because of tax effects and other Ne ew York Stock Exchange. The compo- considerations, the value of a share may not be ex anies are among the largest pected to fall by the full amount of the dividend and most important in their fields. If the adjustment should be, the price changes on ex-divi- dend days were discarded in an earlier version of the Since, for our purposes, the variable of intere per. Since the results reported in the earlie will always be the change in log price, the rear on differ very little from those to be presented ould note that henceforth when the words"price below, it seems that adding back the full amount change "appear in the text, we are actually referring of the dividend produces no important distortions to the change in log price. in the empirical results
BEHAVIOR OF STOCK-MARKET PRICES Each entry in this table was computed umn(1)opposite Allied Chemical im- by taking the corresponding entry in plies that the empirical distribution con Table 1 and subtracting from it the entry tains about 7.6 per cent more of the for the unit normal distribution in Table total frequency within one-half standard 1. For example, the entry in column(1) deviation of the mean than would be Table 2 for Allied Chemical was found expected if the distribution were normal ibtracting the entry in column(1) The number in column(9)implies that Table 1 for the unit normal, 0.3830, from in the empirical distribution about 0.16 the entry in column(1) Table 1 for per cent more of the total frequency is Allied Chemical. 0.4595 greater than five standard deviations a positive number in Table 2 should from the mean than would be expected be interpreted as an excess of relative under the normal or Gaussian hypothe frequency in the empirical distribution sis over what would be expected for the Similarly, a negative number in the iven interval if the distribution were table should be interpreted as a normal. For example, the entry in col- ciency of relative frequency within the TABLE 1 FREQUENCY DISTRIBUTIONS INTERVAls 1.5S20s2.5s3.0S 50S Alcoa American Can 60 American Tobacco Kodak 0.9983840.9983 1010.9958510.9979253.0020 4638.74 rnational harvester 710.9951731.000000 octer gamble 9378 58530.9986178,0013822 Standard Oil (Calif 697.951 United Aircraft Averages 70:20810908195098096890.99830162