# A Belief in the Law of Small Numbers & The Gambler's Fallacy

We often remind our readers that when dealing with random variables, like returns in the market, it is crucial that you are able to speak the language of statistics, which we call Riskese in Step 8 of The 12-Step Recovery Program for Active Investors.  In the world of random variables, there is no “black” or “white.” The only certainty that exists is that there is no certainty; only levels of confidence.

One of the key elements of making statistical inferences is the concept of representativeness in terms of a sample. Representativeness can be defined as the degree to which a sample is similar to or representative of the characteristics of the entire population. In terms of investing, is the last 3 years representative of the population of 89 years of market returns? It is an important concept to grasp because most people make daily decisions based on experiences or “samples” that are not representative of the population as a whole. How many times have we observed a single instance or maybe even a few instances and then jump to a conclusion as to what it means for the future?

Making conclusions based on small sample sizes that we think truly reflect the overall population can lead to misinformed policy decisions or even money management decisions with potential devastating consequences. According to psychologists Daniel Kahneman and Amos Tversky in their paper titled "Belief in the Law of Small Numbers", it happens all of the time by ordinary people and professional scientists alike. Take the example of flipping a fair coin:

“When subjects are instructed to generate a random sequence of hypothetical tosses of a fair coin, for example, they produce sequences where the proportion of heads in any short segment stays far closer to 0.50 than the laws of chance would predict (Tune, 1964). Thus, each segment of the response sequence is highly representative of the “fairness” of the coin… Subjects act as if every segment of the random sequence must reflect the true proportion: if the sequence has strayed from the population proportion, a corrective bias in the other direction is expected. This has been called the gambler’s fallacy.”[1]

Let’s put this in a context that may be more palpable. You are sitting at a roulette table and betting whether or not the ball will land on a red color or black color.  In American roulette (38 slots), there are 18 red slots, 18 black slots, and 2 green slots, which allows the casino to have a slight edge over the gambler (a 47% chance that the ball will land in either black or red). Let’s say you start betting on “red” and 4 times in a row the ball lands on “black.” The gambler’s fallacy states that since you believe that each spin of the wheel represents the true proportion of the population that sooner rather than later the color “red” is more likely to pop up than not. In reality, the odds haven’t changed and, more importantly, there is the potential to experience many more “blacks” in a row just by chance. It is highly likely that you will lose all of your money before possibly seeing a favorable result.

In the context of investing, the same principle applies when investors believe that they are likely to see a “reversion to the mean” or a correction of a "trend" after seeing a sequence of either favorable or unfavorable returns, which can also be referred to as Market Timing. For example, last year was a tremendous year in terms of returns, especially for investors at Index Fund Advisors. IFA Index Portfolio 100 delivered an annualized return that was 5% above its historical average. Is it now more likely that we will experience a less than average return this year so that there is a "reversion to the mean?" The answer is no because, with 5 million buyers and 5 million sellers every day, every price is a fair price, just like the fair coin. Tomorrow's price depends on random news that we do not know today. We could experience another year just like last year. There is also the chance that we could experience a market like we did from 1975 to 1989 where we had 15 straight years of positive returns ranging between 4% and 49%. The point is that 1-year, 3-years, 5-years, or 10-years' worth of data provides us no reliable insight about what the subsequent decade will hold or what the annualized return has been over the last 89 years since 1928. Small samples of returns are not reliable in terms of their representativeness of the entire population (89 years).

Imagine if we solely relied on returns from 1975 to 1989 to plan our financial future? The annualized return of IFA Index Portfolio 100 was 24.12%. The return over the subsequent 15-year period (1990 to 2004) ended up being less than half of what we experienced during the prior 15-year period (11.74%). It is safe to say that our savings assumptions or withdrawal assumptions would be highly impacted by this result. Again, 1975 to 1989 was not representative of what we could expect from the market over any 15-year span. This is where rolling period returns using large data sets can give helpful insight in terms of setting financial planning expectations. Based on 890 individual 15-year (180 months) monthly rolling returns for IFA Index Portfolio 100, the median outcome was 13.03% with a range between -0.18% (12/28-11/43) and 23.99% (10/74-9/89).

See IFA Index Portfolio page for the full interactive chart

Relying on small samples, like 1, 3, 5, or 10 year of returns, to draw conclusions about what to expect in the future opens investors up to the error that their sample is not representative of the population. We prefer to have at least 50 years of data. We have often heard people state that things are different this time and therefore the long term data is less useful. However, returns are based on changes in price over some period of time, not on events, and what remains constant over time is the ability of willing buyers and seller to determine a fair price for a security given the current events. Please remember that the job of a free market is to set prices based on current uncertainties, so that investors are positioned to earn a risk-appropriate expected return. In other words, prices move inversely proportional to uncertainty so that the expected return remains essentially constant over time

This is why IFA bases its decisions off of the most robust returns data in the world, which is sourced from the Center of Research in Security Prices (CRSP) at the University of Chicago Booth School of Business. Drawing on over 1,000 individual monthly returns for many different indexes, we can observe the entire population of quality stock market data and make far better decisions about the expected probability distributions of future returns. This means that we have understanding and acceptance of the Law of Large Numbers, not a "belief" in the law of small numbers, otherwise know as the Gambler's Fallacy.

[1] Kahneman, Daniel & Amos Tversky. “Belief in the Law of Small Numbers.” Psychological Bulletin, 1971. The American Psychological Association.