Investors and managers are concerned with "fat tails". In part one of a two part article we look at where fat tails come from and how they can be managed.

The existence of fat tails in the financial markets results from the well-known fact that a Normal (or Gaussian) distribution doesn’t model returns in markets exactly. Despite strong statements in the press about quants in finance not having enough real world experience to know the limitations of their models(footnote) (see for example here, here, here, here, and just about every article about LTCM ever written) it turns out that those of us in the industry with a more mathematical approach aren’t surprised that a Normal model for returns doesn’t match markets perfectly. Although a Gaussian distribution is extremely good at modelling almost all of the returns of financial assets, it is less good at characterising the tails of the distribution. This is about the first thing you learn as a scientist when you arrive at a bank or hedge fund fresh faced, enthusiastic and desperate to apply the sexy new statistical techniques you learned in university.

The Normal distribution is completely characterised by two parameters.(footnote) These are the location parameter $\mu$ and the shape parameter $\sigma$. Using these two numbers, the probability density function at $x$ (roughly how likely it is that an event with value $x$ happens) is calculated from this equation:

$$ P(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

More informally, we refer to the location parameter as the

To model returns using a Normal distribution one has to estimate the standard deviation. One way to do this is to estimate the standard deviation of the history of the returns of the market using a standard maximum likelihood estimator. Then we assume that this standard deviation can be used in a Normal model for the returns of the market in the future. Since we didn't actually know what the sample standard deviation would be until the end of the data, this model does have a significant future peeking problem, but let's ignore that issue for the moment.

What is rather surprising is that the Normal distribution is pretty good. Here is the return distribution for the crude oil futures market since 1983 using daily close data.(footnote)

The crude oil futures market has an annualised volatility of 34.3% over its lifetime. We have overlaid a Normal distribution with a zero mean and a daily standard deviation of $\frac{0.343}{\sqrt{255}}$ on the graph. All the returns are scaled by this standard deviation and are z scores.

This is a reasonably good fit. Sure, there is more weight in the middle of the distribution, less in the shoulders and there are more of those nasty tail events, but it isn't too bad. Although there are a lot more plus four and minus four standard deviation events than the Normal distribution predicts (and way too many small events), it turns out that the Normal distribution is a good start.

A lot of really very smart mathematicians have made the rookie error of looking at this graph and saying “The Normal distribution obviously doesn’t work. How stupid those quantitative financial professionals are”. For an unusually insightful criticism of this problem see this letter in the FT. I wonder where that guy works now…

However, we can use a little bit of “real world insight”. Even to the average non-financial person it seems that financial markets have periods where not much is really happening interspersed with bursts of fear.(footnote) We could incorporate this in our model by estimating the volatility of the market over some recent period and assuming that the recent volatility is a better estimate of the volatility tomorrow.

A common way of doing this is to use an exponentially weighted estimator (EWMA) of the volatility, which places more weight on recent observations compared to older observations. These estimators are characterised by the length of time that it takes for weight of a point to decay to half its original weight. We are going to choose a 100 day exponentially weighted estimator, but across a wide range of parameters. The choice of weighting window isn't that important.

The figure supports our hypothesis that market volatility isn't constant.(footnote) So, we can use this observation to turn the crude oil series into something which is more “stationary”. We take each daily return and divide it by the volatility of the returns up to the day before (as estimated by the EWMA estimator) and then multiply by 10%. This should give us a 10% volatility series which (we hope) will be better described by a Normal distribution.

This is a much better fit to the distribution of real world returns. It's true that if you look really closely, there are still a few more large returns in the tails, but re-scaling the returns using recent volatility seems to be a considerable improvement on the stationary distribution approach.

We've made an improvement (which is always valuable), but there are still some extra extreme events. To understand the higher incidence of extreme events, we are going to have to introduce another statistical measure. Rather than just eyeball a distribution to work out if there are extreme events, we are going to use

Unfortunately, covering all that would make this piece excessively long and so we will cover this in “Does My Tail Look Fat In This: Part 2”, which will be coming out shortly.