Divergent Moments in Financial Data - Blowups Happen

In an earlier post I described noted that hedge fund trading had a propensity to blow up but provided no justification for that assertion. This post is about the reasons why I believe this is so.

Risk management is contigent on understanding the second central moment of the distribution of portfolio returns i.e. the variance. The variance is a statistic, meaning a random number derived from observable data, and as such it has a sampling error which represents the uncertainty of the measured value of the statistic about the true population value. The sampling error of the variance is discussed in The Econometrics of Financial Markets and is shown there to be proportional to the kurtosis of the distribution.

Everybody knows that financial data is leptokurtotic, but I'm not sure how many people appreciate that knowledge of the kurtosis is critical to the entire practice of risk management -- and not just as far as it affects long-tail events. In one of my favourite books Fractals and Scaling In Finance, by Benoit Mandelbrot, there is an extensive discussion of the possibility that moments of the distributions of financial data may not exist (meaning that their population expected values do not converge). The sample value of a moment is always finite, for it just a function of the data sample, so an interesting question is how do you measure a moment in such a way that you can accept the hypothesis that the population value is divergent?

Mandelbrot suggests looking at the cumulative sample moment, i.e. to compute the sample moment for every sub-period starting with the beginning of the data we have and ending on each successive date. If the population moment is finite then this will converge on the true value, following the Law of Large Numbers. If the population moment is divergent then the cumulative sample moment will not show signs of convergence.

The chart below shows the cumulative kurtosis and cumulative standard deviation for the daily change in the yield of three month treasury bills (the data is from the Federal Reserve Bank of St. Louis available here).

This chart shows the first two cumulative even moments from what is perhaps the principal risk factor for the entire U.S. economy, if not the world. It is annotated with the terms of the Chairman of the Board of Governors of the Federal Reserve System who was responsible for interest rate management during their terms.

I think one could argue that it is suggestive that the kurtosis is not converging to any finite value. In fact, it would appear that the entire term of Alan Greenspan, and that of Ben Bernanke until the recent financial crisis forced an abrupt policy change directed at curtailing interest rate risk, contributed to steady divergence in the kurtosis of the distribution of changes of interest rates. I also include the cumulative standard deviation, which steadily declines during the Greenspan period.

The lack of clear convergence is worrying because it requires us to consider the possibility that the kurtosis is undefined, which implies that the variance is not reliable measurable, and (to be deliberately provocative) that our risk management infrastruture is a self-deluding fantasy -- i.e. that blowups will happen and not "once in a hundred years."