October 15, 2007

Kurtosis

The Black Swan is making me want to know as much as possible about statistics so I've been reading the statistics volume in the CFA1 reading material (not that I ever intend to take the CFA exam....actually I'm signed up for the December exam but will not take it for I am not and will not be ready....it's the accounting part that made me give up).
From what I understand, kurtosis is the statistical measure that tells us if a distribution is more or less peaked than a normal distribution. A leptokurtic distribution is more peaked than normal which I take to mean that it is fat-tailed. The normal distribution is a better approximation for U.S. equity returns for annual holding periods than for monthly returns which have very large excess kurtosis and are thus leptokurtic. My questions are: 1) Isn't the reason annual returns appear mesokurtic (normal-like) is that there isn't enough data to analyse? If we had 1,000 years of market data, wouldn't the distribution of annual returns turn out to be leptokurtic after all with many years of extreme returns? 2) Has anyone computed the kurtosis of daily, hourly, minutely, etc... returns? Does the distribution get more leptokurtic as we reduce the interval? If that's the case, it would explain why it is easier to make money trading off a 30-minute chart than off a daily chart, wouldn't it? More extreme/unexpected moves mean more trading opportunities. It would also mean that all the financial theories (which are based on the assumption that returns follow some sort of normal distribution) do not apply when we look at small intervals.

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