Fat Tails and Excess Kurtosis or why are Black Swans real

On December 9, 2009 Columbia University hosted a PRMIA sponsored discussion "Market Perspectives on TARP Programs". Afterward, at the networking reception, I overheard an industry professional begin to jump down the throat of a startled, young graduate student after he admitted he did not know what Kurtosis is. Students are used to professors who offer an explanation when a student does not understand a concept. This wide-eyed student clearly wanted to be rescued from this no-nonsense consultant. When I approached, the consultant asked me to explain Kurtosis. To diffuse matters, I casually responded, "What, who, me? I just had a breath mint". Much later I realized I was probably unprepared to give an elevator speech on the topic that most listeners could immediately understand. I have corrected this. Leaving out lots of technical bits and introducing a few sort of truths, here it is:

Often we hear of certain VaR estimates or system reliability projections or some other estimate immediately followed by the caveat, "assuming a normal distribution". Since most of us have no idea what this means, we accept the estimate and move on. But, what if the distribution is not normal? One way a Gaussian or bell shaped curve distribution can be abnormal is a property called excess Kurtosis. Gaussian distributions can be skewed and there are also lots of other kinds of distributions but they are not part of this discussion. Kurtosis is a measure of the peakedness or height to width of the curve. When a normal distribution is squeezed taller, it is said to have has excess Kurtosis. When a normal distribution is stretched wider, it is said to have negative excess Kurtosis. What causes this? One way, in financial markets, is leaps or gaps in prices, such as when a market opens after a weekend. Our problem with Kurtosis comes when we calculate the standard deviation, a value that helps us estimate the chances that various unusual outcomes will occur. When a normal distribution becomes abnormal due to excess Kurtosis, the tails of the curve are fatter than our standard deviation calculations estimate. The tails of a distribution represent the less likely outcomes. So, if a projection says something bad is likely to happen once in a hundred years, it might really be likely to occur once in twenty. This would be an extreme condition but it illustrates the point.

A more serious problem with VaR and certain other risk estimates than Kurtosis is something called heteroscedasticity which can lead to volatility clustering. In other words there are periods of high volatility followed by periods of lower volatility. Or, bad days are more likely to be followed by a worse day than a calm one. Volatility clustering is a more important cause of fat tails than Kurtosis. I will discuss this some other time.