A recent article by Bloomberg “*Brexit Spurred a Bunch Of VaR-y Rare Market Gyrations**”* *spurred* our own risk analysts to run a related study using BISAM’s Cognity fat-tailed models.

The article refers to Friday’s Brexit markets as “Sigma-tized,” noting, “Statistical theory holds that 68 percent of all observations in a normal distribution lie within one standard deviation of the average of that sample. Events that lie on the fringes of this distribution are defined by a number of sigmas, which denote the increasing improbability of this outcome being realized.”

The piece goes on to demonstrate that the Friday market moves for several key market variables were 18 to four-times sigma events, meaning *practically impossible events* according to the typical VaR models. The BISAM team took the same market variables and calculated the probability of such events according to our Cognity risk engine at market close on Thursday, June 23, the day prior to the EU Referendum vote. We then translated that probability into a return magnitude that the Normal VaR model would assign and measured its sigma-multiple^{[1]}.

**The Results**

** *****Note: in the above table, 2.14+31 = 21378598016546600000000000000000.00!**

**To interpret the table, let’s use the Eurostoxx 600 Banks index as an example:**

The observed return was negative 14.46%. According to the normal model such a return has practically zero probability and can happen once every **968,210,050 years**. It is a seven-sigma event (6.8)^{[2]}. However, according to the Cognity Fat-Tailed models, such an event has very small, but not at all zero probability (0.014%), and can happen on average every **28 years:** this makes the event much less improbable.

If we transfer the probability assigned by the Fat-Tailed model back into the “Normal world” we will find out that such probability corresponds to four-sigma events (3.6), but that the drop with equivalent probability is twice as low at 7.69%. Thus, if we want to characterize the severity of such events (i.e. unlikeliness) as multiples of sigma, meaning we measure the “**Sigma-tization”**, then by virtue of a fat-tailed model we make such a 14.46% loss to appear two times smaller in multiples of sigma space (3.6 sigma event versus the original 6.8 times sigma with the normal approach).

The above chart shows the Eurostoxx 600 Banks index Post-Brexit return and the “tail” fit of the Cognity Fat-tailed model in blue, with the corresponding probability of such a return. The red line is the “Normal world” tail, where we have also indicated the loss magnitude that would have the same probability according to the Normal model.

The following charts for the rest of the variable are self-explanatory, but drop us a line at [email protected] if you’d like to discuss.

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^{}^{[1] }According to Bloomberg it is a 10-sigma event since they use longer time-period to estimate the sigma^{}

^{[2] }Our Normal sigma-multiples are slightly different from the Bloomberg numbers since we decided to use a unified 2250 days observation period for the sigma estimates across all variables.