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specific predictions relating to fractional moves in prices

We come now to the specific predictions relating to fractional moves in prices. There are two facts from which our predictions are derived:

1. Limit orders tend to cluster more strongly at 8/8 than 4/8, and less strongly elsewhere. One expects, for example, to find continuations less likely after any of the four events ending at fractional price 8/8 than at events ending at 2/8.

2. Since the limit orders act as a barrier to continued price movement, the specialist and his floor trading competitors have a special incentive to sell for their own accounts one eighth below 8/8 and 4/8 after Events RR and FR, and to purchase one eighth above 8/8 and 4/8 after Events RF and FF. The incentive should be strongest at prices 1/8 away from the fractional price 8/8. For example, we predict that the relative frequency of continuation should be less after occurrences of Event FF terminating at fractional price 1/8 than after an occurrence of FF ending at price 5/8.

Forty predictions derived from these items are set forth in Table VI. The pre ­dicted inequality between chances of continuation at two different fractional prices are given in columns 1 and 4. The numbers under the four events in columns 2, 3, 5, and 6 refer to the actual difference between chances of continua ­tion as calculated from Figure 2 and Figure 3. For example, the first predicted inequality in Table VI is that, when an event terminates at 0/8, the chances of a continuation arc less than when an event terminates at 4/8. That is

Pe(0/8) - Pe(4/8) < 0.

The corresponding number under Event RR in column 2 shows that for the specific Event RR, the difference

Р .(0/8) - РЇ.(4/8) = - 0.05.

All these occasions in which the facts were not in agreement with theory are underlined.

Assuming somewhat heroically that the chances of a correct prediction were 1/2 and independent of the success of any other prediction, we find that the binomial probability of 29 or more successes (the observed number in Table VI) out of 40 trials is a mere 0.003. In addition, examination of the table will disclose that the observed size of the difference between proportions of continu ­ations was greater for the correct predictions than for incorrect predictions.

Note that the chance of a continuation after Event RR terminating at price 4/8 appears to be out of line with the chance of a continuation after an occur ­rence of this event at any other fractional price. An explanation was offered by an odd lot broker on the New York Stock Exchange. He suggested that since limit orders cluster at fractional price 4/8 and 8/8, single transactions involving large volume would probably be traded at these levels. But the transactions of 1000 or more shares are printed out in full on the tape. That is, 175 T 59 1/2 as against T 58 5/8 for small orders. And "everybody knows that tape readers will rush in on the same sid& of the market as the large orders, thus continuing the move. If this is true, continuations in price ought to be more likely when the preceding transaction was of 1000 or more shares.

Some interesting properties of price movement were masked by our tech ­nique. We considered movements of 1/8 or more only after the occurrence of two consecutive non-zero changes of 1/8. Therefore, the total number of rises and falls at the even and odd eights listed in Table V does not provide a reason ­able estimate of the probability that a stock transaction took place at an even eighth. In fact, the last different price from the terminal price of all the events in our sample at even (odd) eighths must have occurred at odd (even) eighths. Thus the distribution of terminal fractional prices of events RR, RF, FR, and FF is biased to make the numbers of transactions at even and odd eighths nearly equal. A separate investigation, reported elsewhere [ll], gives more complete information which bears on the relative frequency of odd and even eighths. An examination of all transactions on the NYSE in 1964 showed that 58.5% of all transactions on the NYSE during 1964 fell on an even eighth. The symmetric 95% confidence limits were at 55.9% and 61.1%. This preference for even eighths is largely a consequence oi the tendency for 62% of all trans ­actions at the same price as the previous price to occur at even eighths. This, in turn, is a consequence of the heavy concentration of limit orders at even eighths (six to one in the typical specialist's book of Table IV).