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.
“Extreme events”, “nonlinear dynamics”, “power laws”, “flash crashes”, “fractal processes”… a lot of academic, journalistic and practitioner ink has been spilled about “fat tail events” in financial markets. As is so often the case, a great deal of confusion has been generated by the casual use of subtle statistical concepts. Although this article isn’t attempting to be a statistical primer, it might help to illustrate and illuminate some of these subtleties and to manage the risk of fat tail events.
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(1) (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 characterizing 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.
Quantitative Modelling of Markets
Unless we want to throw away all quantitative techniques and decide on investment strategies and risk allocation by using chicken entrails, astrology or tarot cards, we are going to have to use some mathematical techniques to model the prices or returns of financial assets. When one compares the real returns of financial assets to the distributions shown on this page it seems like the Normal distribution is a good place to start.
The Normal distribution is completely characterized by two parameters.(2) These are the location parameter and the shape parameter . Using these two numbers, the probability density function at (roughly(3) how likely it is that an event with value happens) is calculated from this equation:
More informally, we refer to the location parameter as the mean and the shape parameter as the standard deviation and, in the world of finance, standard deviation is known as volatility.(4) In finance, it is returns which are modeled, not prices(5) and it is those that we expect to be Normally distributed. So, how closely can we model returns using a Normal distribution?(6)
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.(7)
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 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.