Thursday, August 5, 2010

Wild Portfolio Outcomes

Most investment analysis is based on the assumption that returns follow a Gaussian or Normal distribution. However, examinations of available data show that returns don’t exactly follow this pattern. Benoit Mandelbrot of fractal fame suggested the Cauchy distribution as an alternative that may agree better with real-life investment data.

To illustrate the difference between these two theories, suppose that you invest money over a period of time, and based on historical data, you expect to have $1 million on a certain date. Suppose further that historical data suggests there is a one in ten chance that you'll actually have $750,000 or less. What is the chance that you'll actually end up with $250,000 or less?

The Gaussian distribution says that the odds of this bad outcome are less than one in a billion. However, the Cauchy distribution says that the odds of this bad outcome are just over 2%! This is an enormous difference.

The available evidence shows that real life investing is wilder than the Gaussian distribution, but not as wild as the Cauchy distribution. Beware any investing advice based too strongly on calculations using just one of these distributions.


  1. MJ, you're getting all university on us!

  2. lol @ Canadian Couch Potato.

    This is an interesting idea, and it's hard to wrap my head around. I suppose it's important because it points out that we often assume that things are a certain way because it's logical. Empirically, though, things don't necessarily follow our logic.

    Makes me think of Shrodinger's cat, for whom this joke was written:

    Shrodinger's cat walks into a bar... and doesn't.

    This joke will make physics folks smile, and confuse everyone else.

    I don't know a Cauchy distibution from a pull-out couch, but the post was clear enough for me to get a feel for the difference between the two distributions, and its importance in wealth forecasting.

  3. Just finished the misbehavior of markets here :)

    It's an interesting problem because the markets are riskier than the standard models say. Yet there is a risk premium for investing in equities if you can wait out the volatility. We just have to keep in mind that the great depression, tech bubble, black monday, etc. are real parts of the historical record, and aren't exceptions to be ignored and (hopefully) not repeated. These huge swings are what the market does.

  4. These aren't theories they are what the actual data shows. Plot the data yourself and you see the tails are fatter than is representative of a normal distribution. A normal distribution makes the math easier, for example it enables us to use the standard deviation as a risk measure, and hence its widespread us,
    This isn't all that complicated and you can get a good feel for the implications by reading a highly enjoyable and instructive book for anyone interested in investing:

    "Fooled By Randomness" by Nassim Taleb.

  5. @Canadian Couch Potato: I tried to make it sound less technical than it is. However, the embedded Gaussian distribution assumption was behind the assessing of the risk of mortgage-backed securities. It is also behind analyses that show you should use big leverage in investing. So, these assumptions can affect your life.

    @Gene: I think most students of statistics don't realize how deeply the assumption of a Gaussian distribution is embedded in the tools they think they understand. I tend to like the cat jokes -- they were a high point in some physics courses.

    @Potato: I agree that huge swings are the norm. I've always suspected that there is a tendency for markets to undo these large swings over time. It is as though there is some hidden underlying measure of value that pulls on the markets when they get too high or low.

    @DIY Investor: You're right that neither distribution exactly matches real life. Like any theory, these distributions are an approximation of the truth. The question is when is it safe to use these approximations? Unfortunately, this is a difficult question to answer. I don't find that Taleb's books add much beyond Mandelbrot's book. But Taleb may be easier for the general public to understand. As an added bonus, Taleb is in the running for the most arrogant person on earth.

  6. @MJ:

    Yes, I recall people talking about how mortgage default assumptions encouraged reckless lending. No regulator stepped in to say "If you jackasses start lending to people without the ability to pay, defaults are going to be a lot higher than your little model predicts."

    I think a problem with math-heavy models is they give a false sense of precision in complex situations that are hard to model. Plus, the situation complicates further when a large number of people invest based on the models.

  7. @Potato: I believe most models do account for the big events. The problem - as Nassim Taleb has explained very nicely in an obscure but interesting talk a couple of years ago - is that in September 1987 and many other cases the models based on all past data not only didn't show what was about to happen (since it was the first time) but couldn't even measure the possibility except by putting it in the same category as a meteor hitting the earth. That actually sounds like the obvious black swan argument but the explanation was really very good :) What model based on pre-2010 data would show you a 5-minute crash? Now that may be part of the models but if it never happens again they will probably break the models too (if you waited long enough).

    If that's the case no model will tell us what will actually happen next month, next year, or next decade. Numerical analysis may not help long-term investors as much as understanding the fundamental properties of things - how they produce investment returns, what guarantees you actually have, and how you can get out.

    I just requested The Misbehaviour of Markets from the library - maybe it sheds additional light on this but there are always likely to be things beyond the models that happen, no matter how good they are (if they aren't in the market then they'll be in your life and what you need the investments for).

  8. Gauss, Cauchy past history being repeated etc reminds me that the most important thing to examine are the assumptions and the consequences of those assumptions being wrong.

  9. @Canadian Investor: The same rule applies when doing math. A theorem has conditions that must be met before it applies. Problems occur when they get applied in the wrong situation. In finance, if we take option pricing as an example, the list of assumptions required for Black-Scholes option pricing to apply is laughable.