I was looking at the definition of kurtosis on the Corporate Finance Institute website.
“Kurtosis identifies whether the tails of a given distribution contain extreme values.”
“Kurtosis is a statistical measure that defines how heavily the tails of a distribution differ from the tails of a normal distribution.”
“In finance, a leptokurtic distribution shows that the investment returns may be prone to extreme values on either side. Therefore, an investment whose returns follow a leptokurtic distribution is considered to be risky.”
“In the finance context, the platykurtic distribution of the investment returns is desirable for investors because there is a small probability that the investment would experience extreme returns.”
I posted the following question on Twitter and tagged finance professor Rick Nason, author of Rethinking Risk Management.
“The graphs on this page confuse me. It looks like the narrow leptokurtic graph has few outliers. It looks like the wide platykurtic graph has more outliers. Opposite of definition?”
He replied, “It is at the extreme tails that you want to look. There you will see the black lines on top, showing that the probability of an extreme event (Black Swan) is higher. BTW – hard to draw these graphs from equations. Graphs from real life data tell better story.”
Ah so. I should be looking at the magnitude of the outliers outside the blue box (the far edge of the tails).
Basically I was mixing up more/less-extreme cases (magnitude) with more/fewer extreme cases (frequency):
- more extreme events = greater frequency
- more-extreme events = greater severity.
As the CFI website states: “In finance, kurtosis is used as a measure of financial risk. A large kurtosis is associated with a high level of risk for an investment because it indicates that there are high probabilities of extremely large and extremely small returns. On the other hand, a small kurtosis signals a moderate level of risk because the probabilities of extreme returns are relatively low.”
“An excess kurtosis is a metric that compares the kurtosis of a distribution against the kurtosis of a normal distribution. The kurtosis of a normal distribution equals 3. Therefore, the excess kurtosis is found using the formula: Excess Kurtosis = Kurtosis – 3”
- mesokurtic – the data follows a normal distribution. excess kurtosis = zero.
- leptokurtic – more-extreme outliers… more risky. positive excess kurtosis.
- platykurtic – less-extreme outliers… less risky. negative excess kurtosis. Flat tail like a platypus.
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Good clarification. While it is good that people have gotten away from the incorrect “peakedness” interpretation, many people still don’t have it right. They think that higher kurtosis means “more outliers,” but this is not true. If you take a bunch of returns (say 100) ranging from -2.0 to 2.0 that are roughly normal, then the kurtosis is about 3.0. If you then add a single return at 10.0, the kurtosis jumps to around 25. But if you add 10 returns, all at 10.0, the kurtosis becomes much smaller, around 8.0. The point is that when adding a “bunch of outliers,” they are no longer so much outliers. An outlier is a *rare*, *extreme* value. If you add a bunch of them, they are no longer so rare or extreme.
So kurtosis does not measure the number of outliers, it measures their extremity.
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