## Applying the Gambler's Fallacy to Market Volatility

Newfound Research Follow |Nicholas Colas, chief market strategist at Convergex, published a piece this week called Tell ‘Em That It’s Human Nature.

The piece focuses on volatility within the S&P 500. To quote the summary:

*The surge in volatility over the past week enabled this year’s aggregate number of plus or minus 1% moves in the S&P 500 – currently 40 – to exceed last year’s total of 38. There were nineteen positive 1% or more days in 2014, and 19 negative days compared to 22 up days and 18 down days this year. We need 14 more 1% days in order to reach the annual average of 54 since 1958, representing only 16% of the next 86 trading days left in the year. *

*As we progress throughout the balance of 2015, we expect to encounter more of the volatility of the past week than the past few years. One percent or more days tend to pick up by the fifth or sixth year of bull markets such as the rallies of the 1980s, 1990s, and 2000s, and we are in our seventh. *

*Additionally, the VIX often hits its annual peak in October – for example last year – more so than any other month with a total of 5 since 1990. December may statistically register as the quietist month with 7 annual troughs over the past 25 years, but this year may prove different due to the uncertainty surrounding the Federal Reserve’s timing of an interest rate hike. *

So is the implication here that we should see more 1%-move days in the fall to make sure we “catch up” from 40 to the annual average of 54?

Emphatically *no.*

This is just a fundamental misunderstanding of statistics.

Believing that we should expect to see more volatility in the fall is actually an invocation of the *gambler’s fallacy*.

To quote Wikiepdia:

*The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature).*

To highlight why this type of thinking is wrong, consider this example.

We decide to flip a fair coin (i.e. just as likely to come up heads as it is tails) 100 times and count the number of times it comes up heads and the number of times it comes up tails. Before we start flipping, we expect (in the mathematical sense of the word) to see 50 heads and 50 tails.

After 50 flips, the coin lands on heads 32 times and on tails 18 times.

Should we expect to see more tails than heads in the next 50 flips to right this balance?

Of course not! Each coin flip is completely independent from the last. Over the next 50 flips we would still expect to see 25 heads and 25 tails regardless of what has happened in the past.

Let’s put this back in the context of 1%-move days.

Empirically, Colas’s analysis tells us that we expect to see 54 1%-move days every 252-trading days. That means we expect to see a 1%-move day on 21.42% of trading days. That’s similar to saying we have a coin that will land on heads 21.42% of the time and on tails 78.58% of the time.

If we’re going to flip that coin 252 times, we would expect to see 54 heads and 198 tails. Today we’re 165 flips in and have only seen 40 heads. Just like our prior coin example, however, we don’t expect the *next* flips to make up for past flips.

Now, that said, with about 90 trading days left in the year, naïve extrapolation of the probabilities tells us that we should expect to see about another 19 (90 x 21%) 1%-move days, which will actually put us *above* average.

How do we reconcile that fact?

This actually makes complete sense if we look at the stats through a different lens.

We’re currently 165 trading days into the year. Over 165-day trading day periods, going back to 10/06/1950, the S&P 500 has had, on average, 33.21 1%-move days.

In other words, in the last 165 trading days, we’ve actually seen about 21% more 1%-move days than average. At the end of the year, since the trailing 252-day period will contain this 165-day period, we can actually expect the stats to come in above average.

Not because we expect the fall to be any more volatile than usual, however – but because we’ve already lived through above average volatility times.

So remember: the next time someone tells you “because the past was low, the future must be high” or “because the past was high, the future must be low,” they’re probability invoking the gambler’s fallacy. Going back to the basics of just flipping a coin can be a great reminder that the events of the future need not make up the imbalances of the past.

Now, that said, historical evidence does point to volatility exhibiting “clustered” behavior. This implies that high volatility often follows high volatility and low volatility follows low volatility. We can actually see this in the chart provided by Colas where volatility – defined as the number of 1%-move days – goes through clustered high and low periods.

So seeing higher volatility in the fall may occur – but not because there some magnet causing reversion to the mean, but rather due to the clustered nature of volatility itself.

Finally, we think it is important to point out that this might all just be an interesting case of “magic number” analysis. What is the meaning behind 1% anyway? Consider that if we define a “large” move as 2% instead of 1%, the last 165 trading days has been below average.

**DISCLOSURE**:
*The views and opinions expressed in this article are those of the authors, and do not represent the views of equities.com. Readers should not consider statements made by the author as formal recommendations and should consult their financial advisor before making any investment decisions. To read our full disclosure, please go to: http://www.equities.com/disclaimer*

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