Fat Tails

. July 21, 2011
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This post is an appendix to my original article about how to beat the market.

Appendix: Fat Tails

Since drift in the random-walk theory has to be constant by definition, and since all stock movement other than drift is random, the most common form of the random-walk model can actually say, "On average, this stock is xx% likely to move this far once every yyy years."

Let's look at some examples of real market behavior.

  • Crude oil dropped 8.8% in one day on May 5, 2011. According to the traditional random-walk model, a one-day move this large only has a 0.08% chance of happening if we sit and watch crude oil for an entire century. (Reuters)

  • "The 20th century saw 48 days in which the Dow Jones Industrial Average swung more than 7 per cent. “Normal” statistical modelling predicts such swings should happen once every 300,000 years." (Financial Times)

  • "If the market followed a normal distribution, we would expect to see days greater or less than 5 standard deviations (or roughly +/- 4.7% by my calculations) about once every 6,922 years…we’ve seen FIVE so far this month [October 2008]." (MarketSci)

  • "The 1987 market plunge represents a change equaling 22 standard deviations. The odds of such a 22 standard deviation event occurring are 1050." (SharpeInvesting)

    In other words, if the random-walk model is true, price movements the size of the 1987 crash should happen once every 300,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 years. That's a three with forty-seven zeroes after it.
The large, sudden price movements above are called fat tails. An economist who read this article pointed out that there are forms of the random-walk model that are consistent with fat tails. Fat tails can serve as evidence against the form of investment management called Modern Portfolio Theory without refuting more sophisticated random-walk models.

Most money managers use Modern Portfolio Theory and variance-at-risk models to manage money, which don't manage the risk of fat tails properly. They have good reasons for doing this: reasons that are mostly good for them, less so for you.

The random-walk model assumes that beating the market is impossible to do consistently over time. People who try to beat the market sometimes mistakenly trade so much that they pay more in commissions than they make in the market. If you assume you can't beat the market, you won't over-trade. Your manager is trying to help you think long-term.

But if you in fact can beat the market without over-trading, using the random-walk model is counterproductive.

Another reason: If the model says it's impossible to beat the market, your manager won't be blamed if he doesn't don't beat the market.

Modern Portfolio Theory has come to dominate business schools so much that graduates don't know how to manage money any other way. The math is easy, so the model is easy to teach and simple to apply. Even managers who think they can beat the market use it because it's what they know how to do.

Old habits die hard. Science is no exception.

The hedge fund Long-Term Capital Management used similar math to manage money. It worked for a few years, but the fund failed spectacularly in 1998. The failure was so big that the Federal Reserve had to organize a bailout.

Instead of learning from this, investment banks continued to use similar methods to manage mortgage-backed securities, leading to the crash of 2008.

Modern Portfolio Theory is more than just inaccurate. It led investors into the jaws of financial death.

For more information about fat tails and the problems with traditional financial models, read The Misbehavior of Markets by Benoit Mandelbrot.