## Is it rude to ask “How fat is your tail?”

In fact, not only is it not rude, the question is central to understanding risk models.  The Coefficient of Riskiness(COR) allows us for the first time to talk about this critical question.

You see, “normal” sized tails have a COR of three. If everything were normal, then risk models wouldn’t be all that important. We could just measure volatility and multiply it by 3 to get the 1 in 1000 result. If you instead want the 1 in 200 result, you would multiply the 1 in 1000 result by 83%.

Amazing maths fact – 3 is always the answer.

But everything is not normal. Everything does not have a COR of 3. So how fat are your tails?

RISKVIEWS looked at an equity index model. That model was carefully calibrated to match up with very long term index returns (using Robert Shiller’s database). The fat tailed result there has a COR of 3.5. With that model the 2008 S&P 500 total return loss of 37% is a 1 in 100 loss.

So if we take that COR of 3.5 and apply it to the experience of 1971 to 2013 that happens to be handy, the mean return is 12% and the volatility is about 18%. Using the simple COR approach, we estimate the 1 in 1000 loss as 50% (3.5 times the volatility subtracted from the average). To get the 1/200 loss, we can take 83% of that and we get a 42% loss.

RISKVIEWS suggests that the COR can be an important part of Model Validation.

Looking at the results above for the stock index model, the question becomes why is 3.5 then the correct COR for the index? We know that in 2008, the stock market actually dropped 50% from high point to low point within a 12 month period that was not a calendar year. If we go back to Shiller’s database, which actually tracks the index values monthly (with extensions estimated for 50 years before the actual index was first defined), we find that there are approximately 1500 12 month periods. RISKVIEWS recognizes that these are not independent observations, but to answer this particular question, these actually are the right data points. And looking at that data, a 50% drop in a 12 month period is around the 1000th worst 12 month period. So a model with a 3.5 COR is pretty close to an exact fit with the historical record. And what if you have an opinion about the future riskiness of the stock market? You can vary the volatility assumptions if you think that the current market with high speed trading and globally instantaneously interlinked markets will be more volatile than the past 130 years that Schiller’s data covers. You can also adjust the future mean. You might at least want to replace the historic geometric mean of 10.6% for the arithmetic mean quoted above of 12% since we are not really taking about holding stocks for just one year. And you can have an opinion about the Riskiness of stocks in the future. A COR of 3.5 means that the tail at the 1 in 1000 point is 3.5 / 3 or 116.6% of the normal tails. That is hardly an obese tail.

The equity index model that we started with here has a 1 in 100 loss value of 37%. That was the 2008 calendar total return for the S&P 500. If we want to know what we would get with tails that are twice as fat, with the concept of COR, we can look at a COR of 4.0 instead of 3.5. That would put the 1 in 1000 loss at 9% worse or 59%. That would make the 1 in 200 loss 7% worse or 49%.

Those answers are not exact. But they are reasonable estimates that could be used in a validation process.

Non-technical management can look at the COR for each model can participate in a discussion of the reasonability of the fat in the tails for each and every risk.

RISKVIEWS believes that the COR can provide a basis for that discussion. It can be like the Richter scale for earthquakes or the Saffir-Simpson scale for hurricanes. Even though people in general do not know the science underlying either scale, they do believe that they understand what the scale means in terms of severity of experience. With exposure, the COR can take that place for risk models.