Daily Speculations 

The Chairman

10/18/2005
Momentum Versus Randomness, from Victor Niederhoffer
The question often arise as to whether the stocks are random. To test this, I propose to use the average absolute deviation. Let's take the week from 8/5/2005 to 8/12/2005, where the consecutive prices of the S&P futures were 1236, 1234, 1241, 1240, 1245, 1238. The absolute changes are 2, 7, 1, 5 and 7, summing to 22. Thus, the average absolute deviation is 4. The absolute deviation is always computed around the mean change, but in this case, the mean was 0.
It turns out the average absolute daily move during the last 9 years, from yearend 1995 to 10/15/2005, was 9.6. Indeed the actual average absolute moves for the market are as follows:
Number of Days Average Absolute Move Lower 2.5% Bound 1 9.6 2 13.7 13.9 3 16.5 17.3 5 21.0 22.1 10 28.0 30.7 20 39.7 42.2
Note that the 5day average absolute change of 21.0 is just 2.18 times as great as the average absolute change for the 1day. It turns out that if we repeatedly take random samples with replacement from the distribution of actual daily price changes, only 2.5% of the time will the actual change be less than 22.1. Thus, the 21.0 average absolute change for the 5day interval is nonrandomly less than would be expected by change. Similarly the 2day, 3day , 5day, 10day and 20day changes are less than would occur by chance in 97.5% of all samples.
Another way of putting this is that empirically in real life markets, the 2day absolute change turns out to be 1.43 times greater than the 1day absolute change, the 3day 1.70 times, the 5day 2.18 times, and the 10day 2.90 times, and so on. The random ratios from simulation would be as low as 1.45, 1.79, 2.30, 3.20, and so on, by chance alone just 2.5 times out of 100. Thus, I conclude there is a inordinate tendency to reversal in the daily moves for all intervals up to 20 in daily prices. This conclusion was confirmed using the variance ratio, with virtually identical results.
Thanks to Tom Downing for his excellent artful simulation.