Message Font: Serif | Sans-Serif
No. of Recommendations: 3
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).

Print the post  


What was Your Dumbest Investment?
Share it with us -- and learn from others' stories of flubs.
When Life Gives You Lemons
We all have had hardships and made poor decisions. The important thing is how we respond and grow. Read the story of a Fool who started from nothing, and looks to gain everything.
Community Home
Speak Your Mind, Start Your Blog, Rate Your Stocks

Community Team Fools - who are those TMF's?
Contact Us
Contact Customer Service and other Fool departments here.
Work for Fools?
Winner of the Washingtonian great places to work, and Glassdoor #1 Company to Work For 2015! Have access to all of TMF's online and email products for FREE, and be paid for your contributions to TMF! Click the link and start your Fool career.