Kurtosis and skewness in investment terms

[sanjose04]

Hi, David.

I have a question regarding kurtosis and skewness.
In investment terms skewness is supposed to mean “bias toward positive or negative return”.
Kurtosis captures the tendency of the price of this investment to jump either direction. In FRM, I’ve encountered EVT, and its objective is to capture severe events. These severe events come with fat-tail, and fat-tail is higher kurtosis. My question is then, what is the difference between skewness and kurtosis?

If a distribution exhibits skewness, doesn’t that also imply that it has a tendency to move toward either directions, which eventually leads to higher kurtosis than normal distribution? For example, let’s say a statician is conducting a research to find out the average income of people in a town, and there is one very rich guy. The average of the people’s income excluding him is 100$, and if he is included the average jumps up to one million$. The skewness is definitely positive, and doesn’t this also exhibit high kurtosis, since he’s a really big outlier and an extreme event?

 

[David Harper CFA FRM]

Hi Donald,

In your example, yes, adding "Bill Gates" will postive skew and introduce a fat/heavy tail; but if you could add, to the left hand side of your distribution, a "negative Bill gates", then you'd get back to a symmetric distribution (skew = 0) but with fat tails (negative and positve Bill Gates in the tails, so to speak). If you match "fatness" on both sides (so that mean = median), you can have fat but skew=0. To your point, they often do tend to associate with each other; e.g., in Saturday's webinar, we looked at the lognormal distribution for asset prices (which is implied by a normal distribution for log returns). This lognormal is positively skewed and heavy tailed (kurtosis > 3) and they graphically "feel" associated (to me).

But the skew is a function of behavior about the mean (a 3rd moment function). I personally find it easiest to associate positive skew (non symmetry) with: mean > median. If the mean > greater than median, the weighted average is being "pulled higher" than the 50th percentile median. This is largely a function of behavior around the body/middle of the distribution.

But kurtosis is technically independent of mean and variance (i.e., a fourth moment indifferent to the 1st and 2nd). I realize it is often called "peakedness" but since, for example, the student's t distribution is not peaked and yet leptokurtotic (heavy tailed), I find peakedness to be confusing. Same with "fat tails" because sometimes long tails have kurtosis but appear skinny. Therefore, I find mose precise to be: excess kurtosis = heavy tails. Or, simply, the probability of extreme loss exceed P[normal]. I've been told we shouldn't get too religous about skew/kurtosis; that they are incomplete per se.

So I view skew as "tilt" that speaks to symmetry around the mean (mean >? median) and kurtosis as a tail metric. You might find the normal mixture distribution (a distribution that adds n normals together), because combinations of normals can produce any combination of skew/kurtosis. I hope that helps with the distinction...David

 

[sanjose04]

Thank you for your thorough explanation. I've got the point clear.

Sincerely,

Donald.