Fat-Tailed Probability Distributions
January 18th, 2010 5:11 pm Category: Operations Research, by: John Hughes
Given the recent negative turn of events in the economy as a whole, and on Wall Street in particular, it is critically important for decision makers to always be aware of those ‘Black Swan’ occurrences that might be lurking in the future ready to derail the best-laid plans and forecasts. Such incidents are extreme and/or catastrophic in nature. They are the outliers that we think we can safely ignore. These are the events that when we talk about them in hindsight we begin by saying “Who would have thought that …” Well, although we’ve all heard of Murphy’s Law “if it can go wrong, it will”; as recent events have shown us, Murphy was an optimist!
Although the use of the term ‘black swan‘ probably originated in the 17th century, it has more recently come into vogue as a result of the book “The Black Swan: The Impact of the Highly Improbable” by Nassim Nicholas Taleb. Since Black Swan events are so very rare, we think we can forget about them when making plans and forecasts. The problem is that when they occur, they can wreak complete devastation on an organization or business. They cannot be ignored; they need to be properly evaluated and anticipated so that the organization will survive the event, and hopefully rebound and prosper in the future.
An example of a Black Swan Event would be the destruction and devastation resulting from Hurricane Katrina in New Orleans. Another obvious one would be the attacks of 9/11 and the collapse of the World Trade Towers. And let’s not forget the Tylenol tampering scare. At the time, society was shocked in its naivety that anybody would ever do such a thing. And these examples are not meant to imply that a Black Swan need be something negative. There can be tectonic and unexpected events, such as the sudden collapse of the U.S.S.R and the fall of the Berlin Wall, that are generally considered as being of a positive nature.
Although various methods are available to analyze random or unpredictable processes, the analytical modeling techniques that are typically used today are frequently insufficient to meeting the challenge of the Black Swan. In order to provide insight into the risks associated with some situation or decision, modelers have at their disposal tools ranging from “merely” solving a closed-form algebraic equation of a probability distribution, to the use of a full blown Monte Carlo simulation program. But fundamental to many analyses is the Normal Probability Density Function, the familiar ‘bell-shaped’ curve where the two tails of the graph (at the far left and right) taper down to equal 0. The problem is that since the late 1800s, researchers have recognized that this curve, with its ‘near-0’ tails does not accurately model Black Swan events.
Such extreme circumstances demand the use of a group of probability functions that are being called “fat-tailed” or stable-Paretian distributions. These, functions, based on the work of Vilfredo Pareto in the late 19th century, give higher probabilities of occurrence to events in the tails of the curve. A specific example of one is the Cauchy Distribution. Whereas the Normal curve approaches 0 at plus or minus 3.5 standard deviations, a Cauchy (depending on its parameters) is still not close to 0 at plus or minus 5 standard deviations.
Gen. Carl Strock of the Army Corps of Engineers, addressed a press conference shortly after Hurricane Katrina regarding the New Orleans levee system. He said, “… when the project was designed … we figured we had a 200 or 300 year level of protection. That means that the event we were protecting from might be exceeded every 200 or 300 years. That is a 0.05% likelihood. So we had an assurance that 99.5% of this would be okay. We unfortunately have had that 0.5% activity here.” The General’s analysis was based on a Normal distribution. If however a fat-tailed distribution had been used, that 300 years would have been much less, perhaps in the range of 60 to 80 years, and perhaps remedial actions would have been taken to avert the disaster that nearly destroyed the city of New Orleans.
The most generalized version of the equation of a fat-tailed probability function is below.
The parameter is what determines the thickness of the two tails, what is called the kurtosis of the function. Generally, as ￼ decreases, tail thickness increases. In fact, the standard Normal Distribution is merely a special case of this equation where the parameters have certain specific values.
The problem with these fat-tailed distributions is that, depending on the specific values chosen for the parameters (which determine the exact shape of the graph), they may not be solvable algebraically. With the Normal Distribution, it’s possible to mathematically solve the equation and state that the probability of a certain event is some specific value. However, fat-tailed distributions do not lend themselves to this kind of a closed-form analysis. To be able to estimate the chances of specific events, numerical methods such as Monte Carlo simulations or binomial decision trees are required.
So the conclusion is that in analyzing any decision or plan that involves random processes, it is critical to realize and anticipate both the worst and best case scenarios. The business person or decision maker should discuss with the modeler/analyst what might really happen under a wide range of possible scenarios. Realizing the drawbacks of the Normal Distribution, together they need to decide on whether or not a probability function with a fat-tail (embodying this kind of “the sky’s the limit” thinking) is appropriate. And if a fat-tail distribution is required, the modeler/analyst must decide on the best numerical technique to address the needs of the decision maker.