Thursday, June 11, 2009

Valuing Extreme Events

Much of economic theory is built on the assumption that financial returns follow the well-known Bell curve. However, there is much evidence to support the idea that the Bell curve understates the likelihood of extreme events.

Some theorists believe that financial returns follow a Cauchy distribution, which is superficially similar to the Bell curve:

However, the Cauchy distribution is narrower in the middle and higher at the sides. This means that mild events are a little less likely and extreme events are more likely.

To see just how different these curves are for extreme events we need to look at the same curves on a log plot where each horizontal gridline represents a factor of 10:

A 5-standard deviation event on the Bell curve is very unlikely, but the same event on this Cauchy distribution is about 6000 times more likely.

So, if a financial institution was selling insurance against unlikely events based on the Bell curve, it might charge a million dollars, but really have a 6-billion dollar exposure if the events actually follow a Cauchy distribution.

It’s not hard to see the potential danger of using the wrong math to value extreme events. Of course, this danger is mostly a problem for taxpayers who bail out financial institutions.


  1. Cool! I like geeking out with different distributions, though I've really only ever worked with Guassian and Poisson, so now I can add another to the list!

    I've been meaning to ask you: what do you use to make your graphs? They look too pretty/smooth to be Excel...

  2. Potato: I've been using Excel. As long as you use enough points the graphs come out fairly smooth. It took me a while to figure out how to get them out of Excel without butchering them. I've been pasting the charts into photoshop and saving them as png files.