Our previous posting on kurtosis seems to have generated considerable interest. Who knew that â€˜fat tails’ could be so popular?
One reader who contacted us was William Shadwick, founder of Omega Analysis, a quantitative research firm based on London. Bill is known to many as the developer of the Omega Ratio. And along with co-author Ana Cascon, he recently won the Investment Management Consultants Association’s Journalism Award for his paper on new methods of calculating tail risk (the paper first appeared in the Journal of Investment Consulting‘s Spring 2006 edition). Unfortunately, the paper is only available to JOIC subscribers. But Bill has allowed us to host Omega Analysis’ Primer on tail risk analysis here and also spent some time on the phone with me last week to further explain his ideas.
The document introduces a new statistic Shadwick calls the C-S Character of a distribution. He says this measure is vastly superior to kurtosis in dealing with the sort of data sets available from hedge fund managers.
Kurtosis, he argues, is a “wildly volatile measure” that would require thousands of data points to become truly accurate. Unfortunately, hedge fund investors don’t have the luxury of thousands – or even hundreds – of data points. And even if they did, says Shadwick, the world (and the fund) would have evolved significantly over the period of time being analyzed – making analysis difficult at best. In essence, says Shadwick, kurtosis cries wolf all the time.
According to his firm’s paper, the C-S Character (or “Tail Risk Index”) provides accurate results with as few as 18 data points (e.g. monthly returns) by identifying what Shadwick calls the tail risk “fingerprint”. While the mathematics behind the index can be mind-numbing, Omega Analysis’ paper boils it down to the following:
“The first C-S Character is defined as the ratio of the standard deviation of a distribution to its ‘om’…The ‘om’ is equal to half the mean absolute deviation.”
A normal distribution always has a C-S Character of 2.51 (ed: don’t ask why). Anything higher than this signifies higher than normal tail-risk while anything lower signifies a tail-risk that is lower than that of a normal distribution.
You’d be excused for thinking this sounds a lot like kurtosis. But as Shadwick explains, “Kurtosis gives off so many signals, you just don’t know which ones to trust.”
The following graphical illustration taken from his Primer makes the point:
This is where things get interesting. If the C-S Character is less susceptible to statistical noise than kurtosis, can it discern between situations of real tail risk and situations where kurtosis might just experience short-term spikes? Yes it can, says Shadwick. He turns to everyone’s favorite risk management case study, Amaranth, to prove point:
While Shadwick says that Amaranth’s kurtosis would have also spiked in mid-2006, we wouldn’t know whether to trust this signal. However, he argues, the C-S Character leaves little doubt that the fund’s risk profile had significantly changed by April 2006 (or even as early as September 2005).
Shadwick concludes with a blunt warning for those in the hedge fund replication business:
“For distributions with fatter tails than normal, estimates of kurtosis from small samples such as this are so volatile as to be completely useless. This is a very serious objection to the creation of ‘synthetic hedge funds’ based on matching mean, variance, skew and kurtosis”.