Aug 21

Skewness and kurtosis

by David Harper, CFA, FRM, CIPM

FRM |

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Learning objectives

  • Define, calculate and interpret skewness and kurtosis

Third and fourth standardized moments

The k-th moment about the mean (mu) is given by:

moment

In this way, the difference of each data point from the mean is raised to a power (k=1, k=2, k=3, and k=4). There are the four moments of the distribution:

  • If k=1, refers to the first moment about zero: the mean.
  • If k=2, refers to the second moment about the mean: the variance.
  • If k=3, refers to the standardized third moment about the mean: skewness
  • If k=4, refers to the standardized fourth moment about the mean: peakedness.

Skew is the third moment divided by the cube of the standard deviation (the third standardized moment):

skew

Kurtosis is the fourth moment divided by the standard deviation to the fourth power (the fourth standardized moment)

kurtosis

Skewness refers to whether a distribution is symmetrical. A nonsymmetrical distribution is either positively or negatively skewed. The measure of “relative skewness” is given by the equation below, where zero indicates symmetry (no skewness), a negative value indicates skew left and a positive indicates skew right.

  • Relative skewness > 0 indicates positive skewness (a longer right tail) and relative skewness < 0 indicates negative skewness (a longer left tail).

Kurtosis measures the degree of “peakedness” of the distribution. A value of three (3) indicates normal peakedness.

  • Kurtosis greater than three (>3), which is the same thing as saying “excess kurtosis > 0,” indicates high peaks and fat tails (leptokurtic). Kurtosis less than three (<3), which is the same thing as saying “excess kurtosis < 0,” indicates lower peaks.

Tips for FRM candidates

  • A normal distribution has relative skewness of zero and kurtosis of three (or the same idea put another way: excess kurtosis of zero).
  • Note that technically skew and kurtosis are not, respectively, equal to the third and fourth moments; rather they are functions of the third and fourth moments.
  • Leptokurtic distributions (kurtosis > 3, excess kurtosis > 0) (fat tails) are often observed in asset returns.

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