Reverse of the Law of Large Numbers
Paper: The Reverse of The Law of Large Numbers by Kieran Kelly, Przemyslaw Repetowicz, Seosamh macReamoinn
Abstract:
The Law of Large Numbers tells us that as the sample size (N) is increased, the sample mean converges on the population mean, provided that the latter exists. In this paper, we investigate the opposite effect: keeping the sample size fixed while increasing the number of outcomes (M) available to a discrete random variable. We establish sufficient conditions for the variance of the sample mean to increase monotonically with the number of outcomes, such that the sample mean “diverges” from the population mean, acting like an “reverse” to the law of large numbers. These results, we believe, are relevant to many situations which require sampling of statistics of certain finite discrete random variables.
Pop version: Damned lies, statistics and fair value
Scott Rickard, director of the complex and adaptive systems laboratory at University College Dublin, says: “Recent market bubbles defy explanation using the traditional efficient market hypothesis. The core idea of the ’reverse of the law of large numbers’ in Mr Kelly’s work provides the foundation for a new mathematical theory of markets that explains the continual misbehaviour of real markets. The exploitation of this new theory should lead to more accurate estimates of risk and better financial returns.”
June 9th, 2008 at 9:57 am
Cute idea. I have a different version of the opposite, which I call the “Law of Small Numbers.” It deals with the likelihood of financial firms to estimate the value of their assets/liabilities properly when they don’t have sufficient data (exposures/time) to do reasonable calculations. Errors tend to be made in the direction of over-estimating net worth, rather than the converse.
Back to RLLN — economic actors use foreshortening strategies to guide decision-making. Copying is a cheap strategy that works well in the beginning to middle of a move, helping create the conditions for an overshoot, and disaster at the turning points. This is just another reason for why momentum often works in the short run, but not in the intermediate-term.
June 9th, 2008 at 5:53 pm
Well the above is obvious if one assumes that asset price returns follow a scaleable power law regime rather than the incorrectly assumed Gaussian - under a scaleable/fractal regime the moments of the distribution do not tend to a stable value with increasing sample size, hence giving rise to ‘infinite’ volatility and other wacky outcomes
June 9th, 2008 at 6:32 pm
In the vein of making it simple, but not more so:
“The bigger they are…the longer it takes for them to fall.”
June 9th, 2008 at 6:41 pm
Apologies. I didn’t take my own medicine:
“The bigger they are…the LONGER they fall.”
June 10th, 2008 at 2:53 am
Very interesting paper. The pop version says they are trying to start up a hedge fund. I’m a bit skeptical on their ability to trade on a theory that markets misbehave. Think I read about this somewhere.
http://econpapers.repec.org/article/aeajecper/v_3A4_3Ay_3A1990_3Ai_3A2_3Ap_3A19-33.htm
June 10th, 2008 at 9:31 am
[...] The market implications of a reversal of the law of large numbers. (Alea) [...]
June 10th, 2008 at 5:01 pm
Boat Drinks - why are you a skeptic?
June 10th, 2008 at 5:49 pm
SP - Nothing personal about the idea itself. In fact, I like it. I’m a bit skeptical of it’s ability to find success as a hedge fund. I’m curious as to how far this particular type of market anomaly will allow for sustainable excess returns. I am sincerely curious as to how others think about this particular model.