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# Black Swan. The Fat Tail Issue

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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.

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• The concept of Kurtosis as a function of degrees of freedom in a Student-T has really got me obsessed and so I check my statistics/probability text-book from over 30 years ago as well as Googling the Internet, and it turns out that the Kurtosis(ie. fat tailedness) of a Student-T is 3(df-2)/(df-4) for df > 4 and the excess Kurtosis of a Student-T is Student-T Kurtosis - 3 = 6(df-4). In the limit as df goes to infinity this is the same as the Normal distribution. So then I recheck what is the Kurtosis of a Cauchy-Lorentz and it is undefined, same as the Kurtosis of any Student-T with df less than or equal to 4. So then a Monte Carlo simulation using Crystal Ball Software on annualized returns of the stock market that yields 3.4 degrees of freedom, does indeed seem to give credence to Mandelbrot/Taleb premise that the annualized return of the stock market is indeed NOT a Normal Distribution.

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• So if we wrote a function y=f(df)=(1/df)^100, then for df=1 we get 1 and for df=2 we get 7.88*10^-31 and for df=3 we get 1.94*10^-48 and for df=100 we get 0. This would model the property you are stating which is that for every increase of df by 1 we get an enormous reduction in kurtosis(ie. the fat tail phenomena). Or for every decrease in df by 1, we get an enormous increase in kurtosis. And in this way the kurtosis of df=3 if far closer to df=100 than it is to df=1. Would this be the power law relationship that Taleb to a greater extent and Mandlebrot to a lesser extent speak of???

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• cmm7mmc, actually it is not. The Student t with 3.4 degrees of freedom is oddly enough a heck of a lot closer to the Normal distribution than the Cauchy distribution. That's actually the main subject of this presentation. I document on every other slide how the difference between the Normal and the Student t distribution with 3.4 DFs is not that different. Meanwhile, the Cauchy distribution is way out there with tails that are absurdly fat vs the two other distributions and the actual data. The degrees of freedom are not a linear metric. There is a gigantic difference between 1 vs 3.4 DFs that is far greater than the difference between 3.4 DFs vs an infinite # of DFs. As you stated earlier once you go beyond 100 DFs, you essentially get the Normal distribution.

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• To be fair to Mandlebrot/Taleb, I have to say that a Student-T with degrees of freedom at 3.4 as determined by Gaetan is far closer to a Cauchy distribution(ie. Student-T with one degree of freedom) than it is to the Normal dIstribution(ie. Student-T with infinite degrees of freedom). Actually, my 30 year old statistics/probability textbook says with 100 degrees of freedom or higher there are NO tables given to estimate Student-T percentiles, since the Student-T and Normal are so similar, one only needs to refer to the Normal tables to estimate percentiles for the Student-T, when degrees of freedom are greater than or equal to 100. Again, 3.4 is far closer to 1 than it is to 100.

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• Interesting! I started thinking about this after looking at some of talebs stuff. I think the larger issue of quantitative financial models being 'brittle' or missing sources of uncertainty deserves attention

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### Black Swan. The Fat Tail Issue

1. 1. Black Swan(s) – the Fat Tail Issue - Guy Lion December 2009
2. 2. The Black Swan Paradigm <ul><li>Taleb stated in “The Black Swan” that stock prices are more volatile than the Normal distribution entails. Thus, he relegated to “Mediocristan” all models of modern finance that rely on the Normal distribution. He referred to the intractable volatility of the real world as “Extremistan.” </li></ul>
3. 3. Benoit Mandelbrot <ul><li>Taleb’s argument is based on Benoit Mandelbrot’s work. The latter, a French mathematician, uncovered that stock price returns are more dispersed than the Normal distribution. He suggested the Cauchy distribution fits the tails of stock returns much better. </li></ul>
4. 4. The Cauchy Distribution <ul><li>T he Cauchy distribution is not well known. Even university classes on statistical distributions may not cover it. And, quantitative software don’t pre-program it. Fortunately, the construction of its cumulative distribution function is not too difficult as it relies on two simple parameters: </li></ul><ul><li>1) The Median; </li></ul><ul><li>2) The difference between the 75 th and 25 th percentile divided by 2 (called Gamma). </li></ul><ul><li>The above are the rough equivalent of the Mean and the Standard Deviation for the Normal distribution. </li></ul>
5. 5. Cauchy Cumulative Distribution formula <ul><li>x o = Median </li></ul><ul><li>= (75 th percentile – 25 th percentile)/2 </li></ul><ul><li>Excel includes the arctan formula (it calls it ATAN). </li></ul>
6. 6. The Cauchy Conundrum <ul><li>Even though Mandelbrot uncovered the benefit of the Cauchy distribution (fatter tail) near the onset of the development of modern finance that relied on the Normal distribution, the Cauchy distribution has remained mysterious. Why is that? </li></ul>
7. 7. Distribution Analysis <ul><li>To investigate the fat tail benefit of the Cauchy distribution, I looked at daily S&P 500 returns since 1950 (15,085 trading days) and compared the overall fit and the left tail fit of the Cauchy distribution vs the Normal distribution and the Student’s t distribution (best fit per Crystal Ball software). </li></ul>
8. 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. 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. 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. 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. 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. 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. 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).
15. 15. How often would a VAR model blow up? <ul><li>Let’s say you use a Value-at-risk model as a risk management tool. You use a Normal distribution because it is a lot more transparent than any other distributions. And after all, it works 99.75% of the time [(15,085 – 38)/15,085]. How often would such a VAR model have blown up since 1950? And, when… </li></ul><ul><li>As shown on the table, this VAR model would have blown up in 14 different years a total of 38 times. But, it would have blown up 15 times in 2008 alone! </li></ul>
16. 16. 99.75% of the time is not good enough! <ul><li>As reviewed a VAR model relying on the Normal distribution that works 99.75% of the time would have still blown up in 14 out of the past 59 years (or in 24% of the years). And, it would have also blown up 15 times in 2008 alone. </li></ul><ul><li>The blown-up proofing remedy: use the Student’s t distribution. </li></ul>