#### Subject 10. Kurtosis in Return Distributions

We've just discussed skewness, which refers to the deviation from a normal distribution due to asymmetry. A distribution may deviate from the normal distribution by being more or less peaked than a normal distribution. Kurtosis is based on the size of a distribution's tails. It is the statistical measure that tells us when a distribution is more or less peaked than a normal distribution.

• Distributions with small tails (that is, less peaked than normal) are called "platykurtic." If a return distribution has more returns with large deviations from the mean, it is platykurtic.
• Distributions with relatively large tails (that is, more peaked than normal) are called "leptokurtic." If a return distribution has more returns clustered closely around the mean, it is leptokurtic.
• A distribution with the same kurtosis as the normal distribution is called "mesokurtic." Kurtosis is critical in a risk management setting. Most research on the distribution of securities returns has shown that returns are not normal. Actual securities returns tend to exhibit both skewness and kurtosis (sounds like fungus!). Skewness and kurtosis are critical for risk management because if securities returns are modeled after a normal distribution, predictions from those models will take into consideration the potential for extremely large negative outcomes. In fact, most risk managers put very little emphasis on the mean and standard deviations of a distribution and focus more on the distribution of returns in the tails of the distribution, since this is where the risk is.

To calculate skewness (when N is large): where μ is the mean and σ is the standard deviation.

The normal distribution has a skew of 0, since it is a symmetric distribution.
If skewness is positive, the average magnitude of positive deviations is larger than the average magnitude of negative deviations.

To calculate excess kurtosis (when N is large): The kurtosis equals excess kurtosis + 3.

For all normal distributions, kurtosis is equal to 3; excess kurtosis is equal to 0.

A leptokurtic distribution has an excess kurtosis greater than 0, and a platykurtic distribution has an excess kurtosis less than 0.