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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 year-end 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 5-day average absolute change of 21.0 is just 2.18 times as great as the average absolute change for the 1-day. 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 5-day interval is non-randomly less than would be expected by change. Similarly the 2-day, 3-day , 5-day, 10-day and 20-day 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 2-day absolute change turns out to be 1.43 times greater than the 1-day absolute change, the 3-day 1.70 times, the 5-day 2.18 times, and the 10-day 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.

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