More on the law of small numbers

Behavioral finance will help improve our understanding of how markets operate and how investors make decisions. Unfortunately, it will often generate competing hypotheses for different price behavior in asset markets that need to be reconciled. A recent paper attempts to reconcile the law of small numbers (LSN) with the disposition effect and trending prices to within a single model. See "The Law of Small Numbers in Financial Markets: Theory and Evidence". 

The law of small numbers states that there is an incorrect belief that small samples will represent the properties of the actual population. This mistake may suggest that some managers will have a hot hand,  and it will also suggest that if there is a sequence of the same result there is a higher likelihood of a reversal to bring the sample closer to the population, the gambler's fallacy. Note that the law of small numbers can suggest either trends or reversals. If the investor knows the data generating process, it will assume mean reversion. If the process is not known, it is assumed that the few data positions can be extrapolated.  

The disposition effect states that investors will sell assets that have increased in value while holding asset that decrease in value. The disposition effect means selling your winners because you believe there will be a reversal from winner to loser and holding your losers because you believe that losers will soon turn into winners. 

Note that when you have a certain view based on the law of small numbers it may explain or reinforce the disposition effect. The behavioral error on the small sample leads to a behavioral error associated the disposition of assets, yet the law of small numbers can also tell us something about return extrapolation, trends. 

Investors will sell assets that have recently gone up because there is expectation that they will reverse, price changes will revert to the long-term population; however, they will buy assets that have gone up for a while, the hot hand. The overall result is that trends and reversals can be generated because of the law of small numbers. Additionally, the disposition effect will be more pronounced for shorter horizons than longer horizons. If you fall for the law of small numbers, you will have a stronger disposition effect and a doubling down in buying.  

The LSN investors assume a mental model that a risky asset has a quality term along with noise, and they can make some inference about this term over a small sample of past prices. If the noise is negatively autocorrelated, then you will get the gambler's fallacy and expected price reversal. If the quality term is positively correlated than you get the hot hand. This type of model can be applied to describe the characteristics of investors, negative beliefs about short-term performance and extrapolative beliefs based on longer-term price behavior.

These are very interesting traits for individual investors and may show-up with aggregate pricing behavior. You may not trade upon this information, but you can ensure that you do not fall prey to the law of small numbers or the disposition effect.  

Can this help with trend-following? It may not tell you when you can make money, but it provides another explanation for why trending in prices may occur. LSN investors can drive aggregate price behavior.