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03-Feb-2006
January Barometers, Super Bowls and Part-Whole Fallacies, by Dr. Victor Niederhoffer and Dr. Alex Castaldo

This is the time of year when the tendency to use the beginning of a period to predict the end of the period runs amok. For one, there are the proposition bets on the Super Bowl. The team that scores first in the Super Bowl is 27 of 38 to win the game. Thus, many of the sports gambling books will offer a dollar bet that equates to 71% (before their vig) that the team that scores first will win. Similarly, the team that scores first in soccer is 80% to at least draw the game. And if you're a confirmed gambler, you can get similar proposition bets on the team that scores first in a basketball game.

Gamblers know that such statistics are purely random phenomena due to the part-whole fallacy. Similar statistics would apply to the team that scores the last point, or any point in these games. As Doc has shown in a widely referenced article the part-whole fallacy occurs because:

The correlation between x and x + y is  Var(x) / SQRT[ Var(x) (Var(x)+Var(y)) ]

Regrettably, stock market people aren't as sophisticated as gamblers. And they often refer to the January effect, where if the first month of the year is up, the whole of the year is likely to be up as if it's a non-random phenomenon. However, as Doc has shown, and as I showed 40 years before him in my correction of a similar error that a future Nobelist made in his research showing that first-quarter earnings were not predictive of the full year's earnings, such correlations are guaranteed to exist by randomness. Indeed, the correlation between January, and the whole year is 30% by randomness.

Let's look at some actual results to put this in perspective, considering first what happens in the next 11 months after January is up, and comparing that to what happens in the next 11 months when any month is up.

```January as Predictor
(S&P 500 Point Changes, 1980-2005)
Avg Pt Chg
SP500 Cash Index
Obs.     Next 11 mos       Stdev.

January up                    17         56.9            105.1
Any non-January month Up     168         48.6            103.0

```

Thus, after an up January, the average move the next 11 months is 57 points. But after any other month is up, the average move the next 11 months is up 49 points. The difference of 8 points between them, less than 8% of the standard deviation of the point move, is completely consistent with randomness.

```January as Predictor
(S&P 500 Percentage Changes, 1980-2005)

Avg % Chg
SP500 Cash Index
Obs.     Next 11 mos       Stdev.

January up                    17         12.2%            12.6
Any non-January month up     168         10.2%            14.0
Any non-January month down   107          9.9%            18.2
```

Looking at the expected change in percentage terms, and after a any month is down, the results are again consistent with randomness. But they show clearly that after any month is up, there is a very high expectation of some 10 to 12% for the next 11 months. Regrettably, the expected move for the next 11 months after any month is down is 10%, quite consistent with the 10% to 12% average following the up months, so there is no evidence that moves following up months, January up months, down months, or for that matter any other months, are any different in expectation from the others.

However, comparing the standard deviation of 18% for the moves following the down months to the 14% standard deviation for the moves following the up months (using a variance ratio test), we notice a slight tendency, perhaps a 10-to-1 shot by chance alone for the variability to be greater following down months than following up months.

We turn now to another aspect of the January effect. Do the stocks that tend to go up the most in January continue to go up to an inordinate extent in the next 11 months of the year. The results will tend to vary with the sample chosen. Mr. Dude Pomada has shown in a survival-adjusted study for the last 10 years that there is a small tendency, about 10 in 100 to be consistent with randomness for the stocks in the S&P 500 that are in the highest 10% of performers to continue to outperform during the next 11 months, with their expected gain being about 3 percentage points a year better than the average. We note that the 10 best performing S &P 500 companies are Broadcom, Allegheny Tech, Adv Micro Device, Ciena, Engelhard, JDS, Schlumberger, Halliburton, Applied Micro Circuits, and Archer Daniels. However, there was much variability in the results between years, with down market periods like 2000-2002 showing completely opposite results to the main finding of continuity in the study. Thus, the results are highly likely to be spurious as well as useless, and we can pass them safely through the minister.

Inspired by all this beginning-of-period, end-of-period correlation melange, I completed a hand study of whether superior performance in the first month for individual companies of the Dow might be predictive for the rest of the year. I will report the results tomorrow morning as I am preparing for a trip to Vegas with daughter Victoria.

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