Kurtosis

Two distributions can have the same average and standard deviation but very different shapes. Kurtosis captures differences in the tails, showing whether extreme outcomes are more or less likely than under a normal distribution.

A normal distribution has a kurtosis of 3. Analysts often use excess kurtosis, which subtracts 3, so a normal distribution has excess kurtosis of zero. Distributions with positive excess kurtosis have fat tails, meaning extreme values are more common than a normal distribution would suggest. Those with negative excess kurtosis have thin tails.

For example, daily stock market returns typically show significant positive excess kurtosis. Large daily moves, both up and down, occur more often than a normal distribution would predict.

High kurtosis matters for risk management, because models assuming normal distributions can seriously underestimate the probability of large losses. This is why risk managers use stress tests and measures such as expected shortfall alongside standard deviation and Value at Risk.