Kurtosis in Return Distributions

At each step in defining a distribution, you may have noticed that the exponent used on each deviation from the mean has been increasing. For mean, you leave each observation at the first power, sum, and divide by _n_. For variance, it's the squared deviations from the mean that are important. For skewness, it's the cubed deviations. The next step, and the last one, is __kurtosis__, which is a measurement of deviations from the mean taken to the fourth power. What that means is that this measurement is sensitive to particularly large deviations from the mean. What do you think kurtosis would be if there were more large deviations from the mean than normal?

No, kurtosis is never negative, actually. Since the deviations are taken to the fourth power, each of these adjusted deviations must be positive.

What do you think would affect the kurtosis measurement more: large positive deviations from the mean or large negative deviations from the mean?

Incorrect. Remember that any deviation from the mean is taken to the fourth power.

So think about this: If you had a distribution with just six observations, -2, 0, 0, 1, 2, and 5, which of them would have the largest effect on the kurtosis measurement?

Incorrect. This isn't one of the most extreme values, so it couldn't have the greatest impact.

Incorrect. Try calculating the mean first.

Yes! The observation of 5 is the most extreme, since the mean is $$\displaystyle \frac{-2+0+0+1+2+5}{6} = 1$$, and 5 is the largest observation away from 1 in the listing.

The normal distribution is __mesokurtic__. This means that it has no excess kurtosis, but rather the "normal" level of kurtosis, which happens to be 3.  A distribution with excess kurtosis (kurtosis greater than 3) is called __leptokurtic__, and is shown above. A distribution with kurtosis of less than 3 is called __platykurtic__, but this kind is rarer.

Suppose a distribution of portfolio returns had a mean of 6.5%, and an analyst was using this portfolio in a forecasting model that assumed that returns were normally distributed when the returns were actually leptokurtic. What would you say about this analyst's model?

Incorrect. The number of returns above the mean would be the same since a leptokurtic distribution and a normal distribution are both still symmetric.

To summarize this discussion: [[summary]]

Incorrect. Consider that the analyst is relying on a normal distribution, which does not have the "fat tails" of the actual distribution.

Exactly! Since the deviations are taken to the fourth power, each of these adjusted deviations must be positive.

It works a lot like variance: variance is never negative because deviations are squared in that calculation.

Absolutely! This poor analyst is planning on a thinner tail than what will be observed. That's the danger in ignoring kurtosis.

You're quite right! Either one is taken to the fourth power, so the direction of the deviation doesn't matter.

You can see why a leptokurtic distribution is known as having "fat tails." The additional extreme values that fatten up the tails weigh heavily in the kurtosis calculation, leading to excess kurtosis.

Large and positive

Large and negative

Large and either positive or negative

It wouldn't matter

Large positive deviations

Large negative deviations

-2

0

5

The model will predict more negative values than what will actually be observed

The model will predict fewer negative values than what will actually be observed

The model will predict more returns in excess of 6.5% than what will actually be observed

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