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|Subject: Re: fat tails - thin ice||Date: 4/24/2002 5:08 PM|
|Author: TWA40||Number: 205 of 297|
Maybe the problem isn't so much fat tails as it is that one tail is fatter than the other. But if that were the case we wouldn't have a normal distribution and financial markets wouldn't be random walks.
I suspect that this is intended as a tongue-in-cheek jab at "A Random Walk Down Wall Street", but just in case it's not, I'll clarify that something can be normally distributed, but not random. Or random, but not normally distributed. They're different concepts.
Random implies a selection process whereby every individual in the population is equally likely to experience a particular fate (i.e. being selected for a portfolio, going up by 2%, whatever). Roll a fair die and the result is random, but the frequency distribution isn't normal, it's uniform: 1,2, 3,4,5,6 could each occur with a probability of 0.167.
Normal means that the data approximate a normal probability density function, which wouldn't translate very well in ASCII format. But the formula includes constants like Pi and e, as well as 2 inputs derived from the sample population: the mean, and the standard deviation from the mean. In part, statisticians like to assume that data are normal because it makes everything so easy to work with--an entire frequency distribution can be generated using only 2 inputs.
You know you're working with data that are approximately normal if they have a classic bell-curve shape that's fairly symmetrical about the middle. The "middle" can be defined by the mean, median, or mode, and in a normal distribution they should all be roughly the same. In addition, the mean minus 2 standard deviations should exclude about 2.5% of all observations and the mean plus 2 SD should also exclude about 2.5%.
Skewness is the problem you mention where one tail is fatter than the other. Most real world data sets are skewed to varying degrees. The average American household has net assets of $150,000, but Buffett and Gates have north of $30 billion. That's positive skew. Risk-arb returns cluster around 4-8% annualized, but the deals that blow up result in -40 to -80%. That's negative skew (both these are hypothetical examples). A lot of times skewness can be reduced with an appropriate transformation, but usually it's just a work-around, rather than an actual correction.
But it's almost impossible to have enough data to accurately understand what's happening out in the tails. Two sigma events only occur about 1 time out of 19, so to observe 200 of them we need to sample 3,700 independent events. To witness that many 3-sigma events, we need to sample 45,000 independent events. For 4-sigma, we'd need 1.5 million events, and for 5-sigma, we'd need 135 million. There simply aren't enough independent data, on anything, to understand what's happening way out in the tails. That word "independent" is critically important too. Stock market data are notoriously autocorrelated, and so the return from Company A isn't independent of Company B, at least in the short run. And in the really long run, we don't get very many periods of non-overlapping data. And even if enough data are accumulated, there'd be no guarantee that they would mean anything going forward ("it's different this time" sometimes actually applies).
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