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When discussing probability distribution, I always read something such as excess kurtosis, positive kurtosis, positive skewed and negative skewed. What exactly do these concepts indicate? In practical applications, such as market return or something else, what can these characteristics tell us?

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Excess kurtosis $=$ kurtosis $-$ 3, since the normal distribution has kurtosis $=$ 3 (that is what the "excess" refers to). Also, kurtosis is always positive, so any reference to signs suggests they are saying that a distribution has more kurtosis than the normal. Skew indicates how asymmetrical the distribution is, with more skew indicating that one of the tails "stretches" out from the mode farther than the other does.

Practically: High kurtosis indicates a high propensity of a distribution to give you "outliers", in the sense that you will tend to get a lot of rather closely spaced outcomes, followed by a few, rare, way-out-there values. In the markets, this type of distribution can lull you into a sense of complacency, with well localized values most of the time, only to ruin your day with a crazy loss. For skewed distributions, a right-skew to a financial product indicates that its positive returns tend to be higher than its losses, for a simple example, which all other things being equal, is good.

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  • $\begingroup$ Broadly this is okay (+1), but I want to chase two points: 1) "Skew indicates how asymmetrical the distribution is" - notionally the concept of skewness does relate to asymmetry, but the third-moment skewness can be zero yet the distribution isn't symmetric, so third moment skewness isn't just "asymmetry". ... 2) on kurtosis, see the discussion here and here ... $\endgroup$
    – Glen_b
    Commented Oct 23, 2013 at 1:25
  • $\begingroup$ (ctd)... and more especially in this paper (pdf link) or the classic by Balanda & MacGillivray -- your discussion is good (in that it emphasises both peakedness and heavy tails), but it would be easy to miss that both matter. What is critical with the standardized 4th moment is that measure of kurtosis decreases the more concentrated the distribution is around $\mu \pm \sigma$. $\endgroup$
    – Glen_b
    Commented Oct 23, 2013 at 1:26
  • $\begingroup$ @Glen_b: Thanks for the additions - great point on skewness..there is a trade-off between long vs fat tails that complicates matters, but I was trying to keep it relatively simple - a zero skew means that the asymmetries "balance out" either by being absent or finding different ways of creating the same conditional expectation (above/below) the mean. $\endgroup$
    – user31668
    Commented Oct 23, 2013 at 11:56
  • $\begingroup$ (cont'd) For kurtosis - I thought that "heavy-tailed" distributions had more of the data within one $\sigma$ of the mean, due to the heavy effect of outliers on $\sigma$. Wouldn't the forth moment be much more influenced by the density far from the mean than closer to the mean, in the sense that no matter how concentrated the probability is within 1 sd, you can adjust the tails to counterbalance it with only minor changes? Either way, I agree that kurtosis has a controversial interpretation (see Taleb "The Black Swan") and peakedness and tails are part of it, but asymmetry affects it too. $\endgroup$
    – user31668
    Commented Oct 23, 2013 at 12:00
  • $\begingroup$ Exactly right - concentration in the $\mu\pm \sigma$ range is nearly irrelevant to kurtosis. You can have a constant probability in that range while kurtosis tends to infinity, and you can have probability increasing to 1.0 in that range while kurtosis tends to its lower bound. On the other hand, higher kurtosis is mathematically related to greater tailweight, where tailweight is measured by $E\{Z^4 I(|Z| \ge 1)\}$, and where $Z = (X-\mu)/\sigma$. $\endgroup$ Commented Jul 11, 2018 at 21:53

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