Price Movements in Speculative Markets: Trends or Random Walks by SIDNEY S. ALEXANDER, Professor of Industrial management There is a remarkable contradiction between the con behavior of speculative prices held by professional stock market analysts on the one hand and by academic statisticians and econo mists on the other. The profes sional analysts operate in the belief that there exist certain trend generating facts, knowable today, that will guide a speculator to profit if only he can read them cor rectly. These facts are believed to generate trends rather than ins tantaneous jumps because most of those trading in speculative markets have imperfect knowledge of these facts, and the future trend of prices will result from a gradual spread of awareness of these facts throughout the market. Those who gain mastery of the critical information earlier than others will, accordingly, have an opportunity to profit from that early knowledge The two main schools of profes sional analysts, the fundamen talists"and the "technicians, ' agree on this basic assumption. They differ only in the methods used to gain knowledge before others in the market, The fundamentalist seeks this early knowledge from study of the external factors that lie behind the price changes. In a commodity market he tries to estimate the future balance of supp In the stock market he general business conditions and the profit prospects for various i dustries, and for the individual firms within those industries, with The 'technician''operates on the same basic assumption, that facts existing at one time will govern the pi time, but he operates in a different manner. He leaves to others the study of the fundamental facts in the reliance that as thos others act on their knowledge there will be a detectable effect on te of the stock The technician, accordingly, studies price movements of the immediate past for telltale indica he Both schools of analysts thus assume the existence of trends hich represent the gradual recognition by the market of emergent factual situations trends which, if they exist, must depend for 4 The data gathering and processing. underlying this research were sup Po2 ch Fund of the School of Industrial man etts Institute of technol 7
their existence on a lagged response of the market prices to the derlying factors governing those prices. It might, at first blush, seem possible that the trends arise not from a lagged response of the market price to the fundamental circumstances, but rather from a trend in those underlying circumstances themselves. Thus although a stock's price might at all times represent a given mul tiple of its earnings, its earnings might be subject to a long run trend, If, however, there are really trends in earnings, so that an increase in earnings this year implies a higher probability of increase next year than do stable or declining earnings, the stock rice right ld reflect these prospects by a higher p and by a higher ratio of price to current earnings. Consequentl if there is no lagged response there should be no trend in prices By a trend in this connection we mean a positive serial correlation of successive changes or, more generally, a probability of future price change dependent on present The professional analysts would certainly not subscribe to th notion that the best picture of the future movements of prices can be gained by tossing a coin or a set of coins. Yet that is just what academic students of speculative markets say is the best way. The academic students of speculative marl the very existence of trends in speculative prices, claiming tha where trends seem to be observable, they are merely interpreta after the fact, of dom walk. A pri k if at any time the change to be expected can be represented by the result of tossing a coin, not necessarily a 50-50 coin, however. In par ticular, a random walk would imply that the next move of the speculative price is independent of all past moves or ever This probabilistic view of speculative prices is consistent with the theoretical bent of economists who like to talk about perfect markets. If one were to start out with the assumption that a stock commodity speculation is afair game' with equal expectation of gain or loss or, more accurately, with an expectation of zero in, one would be well on the way to picturing the behavior of speculative prices as a random walk. But in fact, this picture of a speculative price movement is as much based on empirical findings as on theoretical predispositions. In a pioneer work Bachelier,I a student of the great French mathematician Poincare derived, in his doctoral thesis in 1900, a theory that speculative prices follow random walks, largely from the as sumption of zero expectation of gain. He then compared the statistical dis ibution served distributions of price changes of certain government securities (rentes)on the Paris Bourse, and he found a close cor espondence between the observed distribution and that to be ex pected from his theory. M. L. Bachelier, Theorie de la Speculation, Gauthier-Villars, Paris 1900
The most impressive recent findings confirming the random walk hypothesis are those of Kendall. He calculated the firs twenty-nine lagged serial correlations of the first differences of twenty-two time series representing speculative prices. Nineteen ese were indexes of Britishindustrial share prices on a weekl basis.(See Table 1). Two of the remaining three were cash wheat t Chicago, one weekly and one monthly, and the last was the spot cotton price at New York, monthly. Essentially, Kendall was ask ing with respect to each weekly series: How good is the bes estimate we can make of next week's price change if we know this week's change and the changes of the past twenty-nine weeks and correspondingly for the monthly series Contrary to the generalimpression among traders and analysts hat stock and commodity prices follow trends, Kendall found, with two or three exceptions, that knowledge of past price changes yields substantially no information about future price change More specifically, he found that each period's price change was not significantly correlated with the preceding period's price hange nor with the price change of any earlier period, at least as p to twenty-nine periods. Essentially, timate of the next period's price change could have been drawn at random from a specified distribution with results as satisfactory as the best formula that could be fitted to past data. In the case of hat distribution was studied in detail and it turned out to be very close to a normal distribution There was one notable exception, however, to this pattern of random behavior of price changes, That was the monthly series on cotton prices in the United States since 1816 with, of course,a few interruptions for such events as the Civil War. For this seri there did appear to be some predictability, and Kendall felt im pelled to draw the moral that it is dangerous to generalize eve from fairly extensive sets of data, For, from the behavior of wheat prices and the stock prices, one might have concluded that speculative markets do not generate autocorrelated price changes and here was cotton providing a notable excepti Alas, Kendall drew the wrong moral, The appropriate one is that if you find a single exception, look for an error. An error e was, for the cotton price series was different from the others nvestigated by Kendall. Almost all the others were series of ob servations of the price at a specified time- say, the closing price on Friday of each week. Each observation of the cotton series was an average of four or five weekly observations of the corresponding month. It turns out that even if the original data - the Friday osing prices - were a random walk, with successive first differ ences uncorrelated, the first differences of the monthly average of The Analysis of Economic Time Series- Part I: Prices. I Journal of the R ety( Series A), vol Pp.I1-25
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four or five of these weekly observations would exhibit first-order serial correlations of about the magnitude Kendall found for cotton. So Kendall's exception vanishes, and we are left with the conclusion that at least for the series he investigated the serial correlations were not significantly different from zero. 4 But the question immediately arises whether a week is not an inappropriate period of observation, The market analysts might Pro ng smooth underlying movement on which is typically superimposed lot of short-term fluctuation. With weekly observations the short term fluctuations might very easily swamp the underlying trends In particular, the give and take of the market leads to a phenome on,recognized by all analysts, of reactions, usually called tech nical presumably associated with profit hese reactions are, of course, negatively correlated with the main price swings. That's what makes them reactions. Kendall's correla tions, close to zero, could possibly be a consequence of the com bination of the negative contributions of the reactions and the positive contributions of the trends The path of a speculative price might, accordingly, be repre sented by a sum of two components, a smooth underlying trend or cycle changing direction only infrequently, and a much shorter cycle of action and reaction,. Under this hypothesis the first-order serial correlations of daily price changes might be negative, the first order correlation of weekly changes might be close to zero, phile the first order serial correlations of monthly or bimonthly changes might be significantly larger than zero We can test this possibility by studying the first order serial correlations of Kendall's data using successively longer intervals of differen , serial correlation of one week changes, then of two week, four eek, eight week, and sixteen week changes, the influence of the reactions should become smaller and smaller and the trend effect if there is one, should become do relations, roughly calculated, 5 are given in Table 1 his point was independently The latter r discove ered by the author and by Holbrook ver he pleasure of first publishing it in Note on the Correlation of First Differences of Averages in a Randor Chain,"Econometrica, Vol. 28, No. 4, October 1960, pp.916-918. Another possible exception may be noted for Kendall's series 3, Invest ment Trusts, whose first five serial correlations were.301.0.356, 0.158 0. 164 and 0.066. This series will be mentioned again below. oughly, because they were computed, not from the original data, but rom the serial correlations published by Kendall. Since successive serial correlations are based on fewer observations because of the necessity of sacrificing end terms, a certain #lend term error is introduced by this procedure
While an occasional high value correlation occurs in Table 1 for intervals greater than one week, it must be remembered that for the given total time period under study, the number of observa tions drops in proportion to the length of the differencing interval Since the variance of the correlation coefficient is inversely pro portional to the number of observations, it is directly proportional to the length of the differencing interval. An occasional high com puted value of the correlation becomes increasingly probable as the differencing period is lengthened, even if the true correlation s zero It must be concluded that the data of Table l do not give any substantial support to the hypothesis that, as differences are taken er longer and longer intervals, the firat order serial correlations of the first differences generally increase Once again, Series 3, investment trusts, is an exception. It seems to have a particularly high level of serial correlation o two-week period of differencing. Other occasional higher valt mong the various series, for changes at sixteen week intervals in particular, have to be seriously discounted although they sugges intriguing possibilities for further study. These higher correlations for sixteen week changes (for Series 4, 8, 13, 15, and possibly 17, 18, and 19) proceed fr rather curious relationship that holds for 18 of the 19 series. The first order correlations on an eight-week basis tend to be algebraically maller than the second order correlations (see Table 1).The implication of the existence of an eight week half cycle may be an interesting suggestion, although it could hardly be said to be es tablished by the data One further attempt was made, in spite of Kendall's findings that the serial correlations were not significantly different from zero, to see if some nugget of systematic trend behavior might still be found in his data. It is possible that while the lagged autocorre from zero when taken one at a time, they might jointly form a pat- tern that is significant. a simple test of this possibility was at tempted. A trend was fitted to the first differences of each stock price series by a Spencer 2l-term moving average. Then the ratio of the variance of the moving average to the variance of the first diff The variance ratios given in Table 2 are to be interpreted as follows. If each first difference lay exactly on the moving average See E, T. Whitaker and G. Robinson, The Calculus Observations 4th ed. )London 194, P. 290, for the formula used. Actually necessary to fit the trends to the series themselves, but the variance of the ving average, expressed in units of the variance of the first difference could be computed by applying the smoothing coefficients of the Spencer formula directly to the lagged serial correlations
TABLE 2 RATIOS OF THE VARIANCE OF SMOOTHED FIRST DIF FERENCES TO THE VARIANCE OF UNSMOOTHED FIRST DIFFERENCES KENDALL'S STOCK PRICE INDEXES (Smoothing Performed by 21 Term Spencer Moving Average) Industry Group Ratios Banks and Discount Companies 0.16 Insurance Companies Investment Trusts 162 Cotton 210 7 Electric Light and Power 206 Iron and Steel Total Industrial Productive 212 Home Rails 138 Shipping 151 14 Stores and Catering 212 219 Breweries and Distilleries 17 Miscellaneous 221 T 218 19 Industrials (All Classes Combined) trend line, that is, if the trend line were a perfect fit, the variance ratio would be unity. If, on the other hand, all the serial corre lations of order greater than zero were identically zero, the ex pected values for a random walk, the variance ratio would be 0.143 the sum of the squares of the coefficients in the smoothing formula sible for Except for Series 3, the trend variance is not a much large proportion of the original variance of the first differences than would be expected in the case of a random walk. It must be con cluded that, with this exception, if trends exist in the first differ ences, they are very weak. All in all, Kendall's data do seem to confirm the random walk hypothesis. Further work by Osborne strengthens the random walk hypothesis from a different point of view. While Kendall 7 M. F. M. Osborne, Brownian Motion in the Stock Market, Opera tions Research, Vol, 7, No. 2, March-April 1959, pp. 145-173. See also comment and reply in Operations Research, Vol. 7, No. 6, November December 1959, pp. 806-811 13
worked with serial correlations for each series separately, Osborne worked with ensembles of price changes, Roughly stated, he found that the changes in the logarithms of stock prices over any period a given market, principally the New York Stock Exchange, con stituted an ensemble which appeared to be approximately normally distributed with a standard deviation proportional to the square root f the length of the period. This proportionality of the standard deviation of price differences to the aquare root of the differencing period is a characteristic of a random walk and had been pointed out much earlier by Bachelier. In Bachelier's case, however, the differences were arithmetic, while in Osbornes they were loga It must be noted that Osborne's measurements do not concern trends in the prices of stocks but merely the statistical distribution f the changes in the logarithms, which, as Osborne pointed out orrespond quite closely to percentage changes. That they do not correspond exactly to percentage changes has an important bearing on one of Osbornes principal findings, as we shall see Osborne also supplied a theoretical mechaniam that could ex e observed pattern of pric a random walk in the logarithm of prices with each step being a constant logarithmic value, depending on the time length of the step The basic step is a transaction of which there might be ten or hundred a day. The compounding of such steps in familiar prob bility sequences would, over any period of time, yield a normal distribution of changes in the logarithms of price, with standard deviation proportional to the square root of the period over which made One peculiar result of Osborne's proposed mechanism merit further study. Bachelier, the pioneer in regarding speculative price behavior as a random walk, derived the theoretical proper ties of the distribution of changes in the prices of rentes on the assumption of a fair game, "I that is a zero expectation of gain a price change in either direction of a given amount was equally probable in Bachelier's model. Osborne made a somewhat different assumption with a radically different result; he assumed that a change in either direction of a given amount in the logarithm of price was equally likely, no longer a fair game Thus, under Bachelier's as sumption, given an initial invest ment value, say $100, it would be equally probable, at the end of time T to be worth $100+k or $100-k. Exactly how large k would be for any stated probability would depend on the fundamental con stant of the distribution and the square root of the length of time T. But whatever the value of k, so long as the probability of a gain of k is equal to the probability of a loss of k for all k within the pe mitted range, the expected value in any future period remains $1
Under Osborne's assumption, however, there will be an ex- pectation of gain. Suppose, using logarithms to base 10 and starting from $100, a gain over some particular period of time say five years, is equally likely to be a gain of logarithm l or loss of logarithm -l. These would correspond to an equal probability that the price at the end of five years would be either $10 or sl The mathematical expectation in this case would be $505, or an ex ected gain of $405. This example illustrates the familiar dif ference between the arithmetic and the geometric mean. Over the long run, then, it makes a great deal of difference whether there is an expectation of zero arithmetic gain or zero logarithmic gain. In the latter case there will be a tendency for an investment value to grow, independent of any growth in the economy other than the growth implicit in the existence of a random walk in the loga How clearly established an empirical finding is the logarithmi rather than the arithmetic step in the random walk Osborne was led to the logarithmic form, while Bachelier was not, because Bachelier considered only a single type of security at a time hereas Osborne considered an ensemble of prices, usually all th common stock prices on a particular exchange. Osborne assumed without much explicit consideration, that it was appropriate to try to fit one distribution of expected change to all common stocks whether priced at $100 or $10 or whatever. Very little empirical investigation is required to show that the relative frequency of price rises of $10 in one month is much smaller among stock selling at $10 than among stocks selling at $100. On the other hand it is quite reasonable to expect that the relative frequency of a $ price rise in a month among $10 stocks would be about equal to the relative frequency of a $10 price rise in a month among $100 stocks irical tests seem to If then we have to choose a single distribution that will fit stocks of all different prices, and if our only choice were between equal probabilities of dollar amount changes and equal probabilities of proportional changes, we are necessarily led to choose the latter The assumption of equal probabilities of given changes in the loga rithm of price falls in the latter class But there are other possible models which yield equal probabil ities of changes of given proportions. One is of particular interest to us in that it certainly fits the data as well as the logarithmic model and does not imply a built-in growth of values as does the logarithmic. It postulates equal probability of given percentage changes, almost the same as equal probability of given logarithmic changes, but not quite. On the tiny difference hinges the existence or nonexistence of the remarkable property of speculation being a game biased in favor of winning. In both the logarithmic form and the percentage form of the hypothesis it is equally probable that a $100 stock rises by $10 in
a month or that a $10 stock rises by $1. Under both schemes it is equally probable that a $100 stock declines by $10 in a month or a s10 stock by $1. But under the percentage form it is equally prob able that a $100 stock goes to $101 or to $99 in a given hereas in the logarithmic form it is equally probable that a $100 stock goes to $101 or $99.01 in a given time. This difference of one cent in a dollar change from $100 spells the difference between zero expectation of gain and positive expectation. If, then, the percentage hypothesis is adopted instead of the logarithmic hypothesis, the expectation of gain disappears. The difference between the distributions generated by the two hypotheses would, over most time periods of practical interest, be so fine that any test delicate enough to distinguish between them is likely to throw them both out In fact, both hypotheses would generate normal distributions of e changes in the logarithms of prices, differing only in their means and standard deviations. In testing various models of this sort we generally infer the mean and the standard deviation from the data and assume that the mean was influenced by general e nomic conditions separate from the random walk. Under these ircumstances we can say whether the observed distribution is or is not close to normal, but we cannot say whether it is closer to the percentage hypothesis or to the logarithmic, To discriminate be tween these hypotheses we would need an independent measure of the random step, or of the standard deviation, Bachelier actually derived such an independent measure from the price of options, but Osborne me rely showed that the distribution resembled a normal distribution and the standard deviation increased with the square root of the differencing period. To whatever extent his findings support the logarithmic hypothesis, they also support the percentage hypothesis But Osborne did not rigorously test the normality of the distri bution. A rigorous test, for example the application of the chi square test to some of the data used by Osborne, would lead us strongly to dismiss the hypothesis of normality. (See Table 3).It yields a chi-square of over 60 for 8 degrees of freedom, although almost all of the discrepancy between actual and expected frequen- cies arises from the extreme classes of increases or decreases reater than 10 per cent. It may be presumed that special factor ated to produce far more large price changes than are charac teristic of a normal distribution. This sort of situation(leptokurto sis)is frequently encountered in economic statistics and would ertainly overshadow any attempt to test fine points such as the difference between a logarithmic and a percentage scheme It should be noted that Osborne remarked that the tails of the observed stribution did not appear to correspond to those of the normal distribution
