On to Kurtosis

Kurtosis describes the shape of a random variable’s probability density function. Consider the three curves shown in the chart below (click the chart to expand it into a pop up).

The chart illustrates the notion of kurtosis. The red curve has the highest kurtosis. It is the most peaked at the center and has wider tails. The blue curve is in the middle. The green curve has the least kurtosis. It is flatter at the center and has the narrowest tails.

Incidentally, one can’t tell from looking at the chart which curve has the highest or lowest standard deviation*. The red line is more peaked at the center, which might lead one to believe that it has a lower standard deviation. It has fatter tails, which might lead one to believe that it has a higher standard deviation. If the effect of the peakedness exactly offsets that of the fat tails (or vice versa in the case of the green line), each curve will have the same standard deviation.

A normal random variable (the blue curve) has a kurtosis of 3 irrespective of its mean or standard deviation. If a random variable’s kurtosis is greater than 3 (the red curve), it is said to be leptokurtic. If its kurtosis is less than 3 (the green curve), it is said to be platykurtic. Leptokurtosis is associated with densities that are simultaneously “peaked” and have “fat tails.” Platykurtosis is associated with densities that are simultaneously less peaked and have thinner tails.

What does this mean to a hedge fund investor? Well it means that a fund with high kurtosis (i.e. 3+ and therefore, leptokutic) has exhibited more frequent behavior at the extremes (the "tails") than a fund with a lower kurtosis.

If you want to see a relatively accessible discussion of kurtosis and (from my prior blog entry) skew I suggest you look at this working paper by Harry M Katt entitled "Integrating Hedge Funds into the Traditional Portfolio".

*"Standard deviation" is a very traditional portfolio risk measure which will be discussed in a future blog.