THE JOURNAL OF BUSINESS given interval. For example, the number deviations than would be expected under in column(5)opposite Allied Chemical the Gaussian hypothesis. In columns(4) implies that about 1. 21 per cent less total through( 8) the overwhelming prepon- frequency is within 2.5 standard devia- derance of negative numbers indicates tions of the mean than would be expected that there is a general deficiency of rela- under the Gaussian hypothesis. This tive frequency within any interval 2 to 5 means there is about twice as much fre- standard deviations from the mean and quency beyond 2.5 standard deviations thus a general excess of relative fre CABLE 2 COMPARISON OF EMPIRICAL FREQUENCY DISTRIBUTIONS WITH UNIT NORMAL INTERVALS S8s③334#885 >50S Allied Chemical inaencan tobacco arrester 008293 份8:三湖 3822 0:1-:别4=:3=:09 averages 0.08370.06360.0183-0.0064-0.0120-0,0086-0.002979-0.00112-0.001132 than would be expected if the distribu- quency beyond these points. In column tion were normal (9)twenty-two out of thirty of the nur The most striking feature of the tables bers are positive, pointing to a general is the presence of some degree of lepto- excess of relative frequency greater than kurtosis for every stock. In every case five standard deviations from the mean the empirical distributions are more At first glance it may seem that the peaked in the center and have longer absolute size of the deviations from nor- tails than the normal distribution. The mality reported in Table 2 is not im- pattern is best illustrated in Table 2. In portant. For example, the last line of the columns(1),( 2), and (3)all the numbers table tells us that the excess of relative are positive, implying that in the empiri- frequency beyond five standard devia cal distributions there are more observa- tions from the mean is, on the average tions within 0.5, 1.0, and 1.5 standard about 0. 12 per cent. This is misleading
BEHAVIOR OF STOCK-MARKET PRICES however, since under the Gaussian hy- illuminating form. In sampling from a pothesis the total predicted relative fre- normal distribution the probability that quency beyond five standard deviations an observation will be more than two is 0.00006 per cent. Thus the actual standard deviations from the mean ecess frequency is 2,000 times larger 0.04550. In a sample of size N the expect than the total expected frequency. ed number of observations more than Figure 1 provides a better insight into two standard deviations from the he nature of the departures from nor- is N X004550. Similarly, the expected mality in the empirical distributions. The numbers greater than three, for dashed curve represents the unit normal five standard deviations from the mean density function, whereas the solid curve are, respectively, N X00027, N x presents the general shape of the em- 0.000063, andn X0.0000006 Following pirical distributions. A consistent depar- this procedure Table 3 shows for each ture from normality is the excess of ob servations within one-half standard de- viation of the mean On the average there is 8.4 per cent too much relative fre- quency in this interval. The curves of the empirical density functions are above the curve for the normal distribution. Before -3 -2 23 1.0 standard deviation from the mean, however, the empirical curves cut down through the normal curve from above Standardized variabl ation, in twenty-four out of thirty cases the excess is not as great as that within stock the expected and actual numbers of one-half standard deviation. Thus the observations greater than 2, 3, 4, and 5 empirical relative frequency between 0.5 standard deviations from their means and 1.0 standard deviations must be less The results are consistent and impres- than would be expected under the Gauss- sive. Beyond three standard deviations ian hypothesi there should only be, on the average Somewhere between 1.5 and 2.0 stand- three to four observations per security rd deviations from the mean the em- The actual numbers range from six to pirical curves again cross through the twenty-three. Even for the sample sizes normal curve this time from below. This under consideration the expected number is indicated by the fact that in the em- of observations more than four standard pirical distributions there is a consistent deviations from the mean is only about defie of relative frequency within 0.10 per security. In fact for all stocks 2.0, 2.5.3.0. 4.0, and 5.0 standard devia- but one there is at least one observation tions,implying that there is too much greater than four standard deviations relative frequency beyond these inter- from the mean, with one stock having as vals. This is, of course, what is meant by many as nine observations in this range In simpler terms, if the population of The results in Tables 1 and 2 can be price changes is strictly normal, on the cast into a different and perhaps more average for any given stock we would
THE JOURNAL OF BUSINESS expect an observation greater than 4 These results can be put into the form standard deviations from the mean about of a significance test. Tippet [44] in 1925 once every fifty years. In fact observa- calculated the distribution of the largest tions this extreme are observed about value in samples of size 3-1, 000 from a four times in every five-year period. Sim- normal population. In Table 4 his results ilarly, under the Gaussian hypothesis for for N=1, 000 have been used to find any given stock an observation more the approximate significance levels of the than five standard deviations from the most extreme positive and negative first mean should be observed about once differences of log price for each stock every 7, 000 years. In fact such observa- The significance levels are only appro tions seem to occur about once every mate because the actual sample sizes are three to four years greater than 1,000. The effect of this TABLE 3 AN ALYSIS OF EXTREME TAIL AREAS IN TERMS OF NUMBER OF OBSERVATIONS RATHER THAN RELATIVE FREQUENCIES INTERVA >2S >4S >5S Expected Actual Expected Actual Expected Actual ExpectedActual 553.3160 4|0.000 19 American Tobacco 25527 568122 08 nternational Nickel 455855755746255556567 3.3 100142 1,44665.8863.9 20445 3,211 778 Tota 1,78742,05810584482.511200.023345 Total sample size
BEHAVIOR OF STOCK-MARKET PRICES to overestimate the significance level, tion P of all samples, the most extreme since in samples of 1, 300 an extreme value of a given tail would be smaller in value greater than a given size is more absolute value than the extreme value probable than in samples of 1,000. In actually observed most cases, however, the error intro- As would be expected from previous duced in this way will affect at most the discussions, the significance let third decimal place and hence is negligi- Table 4 are very high, implying that the ble in the present context. observed extreme values are much more Columns(1)and (4)in Table 4 show extreme than would be predicted by the the most extreme negative and positive Gaussian hypothesis changes in log price for each stock. Col- D. NORMAL PROBABILITY GRAPHS measured in units of standard deviations Another sensitive tool for examining from their means. Columns(3)and(6)departures from normality is probability show the significance levels of the ex- graphing. If u is a gaussian random vari- treme values. The significance levels able with mean u and variance o?, the should be interpreted as follows: in sam- standardized variable ples of 1, 000 observations from a norma population on the average in a propor (2) TABLE 4 SIGNIFICANCE TESTS FOR EXTREME VALUES Smallest Standardized Largest Standardized Stock Allied Chemical gk|=818:8数 merican Can 07238 ectric 二 Procter gamble 0,99998 Standard Oil (n.J. 10039.013 ted Aircraft Westinghous Woolworth 0.06744 5.8900.99999008961 7.743 1.0000
THE JOURNAL OF BUSINESS ill be unit normal. Since g is just a are determined by the two most extreme linear transformation of 4, the graph of values of u and g. The origin of each s against u is just a straight line graph is the point(umin, %min), where The relationship between s and u can umin and amin are the minimum values of be used to detect departures from nor- u and s for the particular stock. The last mality in the distribution of u. If ui, i- point in the upper right-hand corner of 1,..., N are N sample values of the var- each graph is(umax, max). Thus if the iable u arranged in ascending order, then Gaussian hypothesis is valid, the plot of a particular u is an estimate of the f s against u should for each security ap- fractile of the distribution of u, where proximately trace a 45 straight line from the value of f is given by 20 Several comments concerning the (3)graphs can be made immediately. First probability graphing is just an Now the exact value of z for the f of examining an empirical frequency dis- fractile of the unit normal distribution tribution and there is a direct relation eed not be estimated from the sample ship between the frequency distributions data. It can be found easily either in examined earlier and the normal proba- any standard table or(much more rap- bility graphs. When the tails of empirical dly by computer. If u is a Gaussian frequency distributions are longer than random variable, then a graph of the those of the normal distribution, the ample values of u against the values of slopes in the extreme tail areas of the a derived from the theoretical unit nor- normal probability graphs should be mal cumulative distribution function lower than those in the central parts of (c.d. f ) should be a straight line. There the graphs, and this is in fact the case may, of course, be some departures from That is, the graphs in general take the nearity due to sampling error. If the de- shape of an elongated S with the curva- partures from linearity are extreme, how- ture at the top and bottom varying ver, the Gaussian hypothesis for the directly with the excess of relative fre- distribution of u should be questioned. quency in the tails of the empirical dis- The procedure described above is called tribution normal probability graphing. A normal Second, this tendency for the extreme probability graph has been constructed tails to show lower slopes than the main r each of the stocks used in this report, portions of the graphs will be accentu with u equal, of course, to the daily first ated by the fact that the central bells of diffe of log price. The graph distributions ar found in Figure 2 higher than those of a normal distribu The scales of the graphs in Figure 2 tion. In this situation the central por- tions of the normal probability graphs ao This particular convention for estimating f is should be steeper than would be the case y one of many conventions are 2 The reader should note it is not mates of the fractiles, and always visible in the graphs because it falls at the of this report, it makes very little difference which point of intersection of the two axes. It is probab 里J. Gumbel [20, p. 15] or Gunnar Bl0m,p、 produced by the cathode ray tube of the Universit of Chicago's I B M