This presentation explores the left-tail of daily stock returns since 1950. It investigates whether the Cauchy distribution is an appropriate one to fit this tail as suggested by Benoit Mandelbrot. It also investigates the consequences of using the Normal distribution within a Value-at-Risk type model.

1. Black Swan(s) – the Fat Tail Issue - Guy Lion December 2009

8. An optical illusion The above frequency distribution curves in % of total daily stock returns do seem similar for all three distributions vs Actuals. The Cauchy tails go up because at this point the graph captures all returns beyond + or – 10%

9. Source of Radical Divergence The Cauchy distribution has fewer observations centered around the Mean. Those are redistributed in the tails. But, it is difficult to see proportions of <2%. As a result, this customary frequency distribution graph is somewhat misleading.

10. Overall Fit A complete picture The table allocates the 15,085 trading day stock returns since January 1, 1950 in various daily stock returns bins ranging from -24% or worse to + 24% or higher. As shown, the fit between the Actual data series and the Student’s t distribution is excellent. This is confirmed by the high Chi Square p value of 0.89 between the two data sets. The Normal distribution fit is not so good as it misses the 38 worse returns and the 39 best returns. However, surprisingly the Cauchy distribution fit is also poor as its tails are way too fat.

11. Assessing Risk Frequency The table at the top shows that the Cauchy distribution resulted in 134 days (out 15,085) having a negative monthly return of -16% or worse vs only 1 month in the actual data. Thus, the Cauchy distribution overstated this risk frequency by 134 times as shown in the second table. This table shows the # of months in each return bucket on a cumulative basis. Thus, in the Actual data there were 38 days with a negative return of – 4% or lower.

12. Risk Frequency narrative As shown, the Student’s t distribution captures the left tail risk at all levels almost perfectly. At every return cut off points, the number of days captured by this distribution is very close to the actual data (resulting in a multiple close to 1). Meanwhile, the Normal distribution completely misses out the entire left tail as the -4% return threshold is already over 4 standard deviation away (multiple of 0). For the Cauchy distribution it is the opposite problem. The tails are way too fat and it overstates the risk frequency at every cut off point by a factor ranging from 14 times to 134 times the actual risk frequency.

13. Risk Severity The table shows the left-tail consisting of the 38 worst daily returns out of 15,085 trading days since January 1 st , 1950. It also shows the corresponding 38 worst values for the Student’s t -, Normal - , and Cauchy – distributions. You can see how the Student’s t distribution matches the actual data very well. The Normal distribution misses out all 38 values as its very worst value (-3.7%) is still higher than the actual data’s 38 th worst value of 4.0%. On the other hand, the Cauchy distribution value are so much worst than the actual data as to be meaningless.

14. Risk Severity multiple This table divides the distribution return vs actual return. Doing so, shows that the Cauchy distribution overstates the worst return by a multiple of 107.6 times calculated as follows: -2202%/-20.5% = 107.6 times Meanwhile, the Normal distribution understates this return by 80%: -3.7%/-20.5% = 0.2 While the Normal distribution pretty much misses out this 38 worst observation left-tail risk and the Cauchy distribution overstates it by a factor of 14 to 121 times, the Student’s t distribution gets it just about right through the entire range (from very worst to the 38 th worst return).