Kurtosis
Kurtosis is a statistical measure of the shape of a probability distribution's tails relative to a normal distribution, with high kurtosis indicating fatter tails and a higher likelihood of extreme return outcomes in either direction.
Kurtosis is the fourth statistical moment of a distribution, measuring how heavy or light the tails are compared to a normal distribution. A normal distribution has a kurtosis of 3. To make comparisons easier, analysts often use excess kurtosis, which subtracts 3 from the raw value so that a normal distribution scores zero.
A distribution with positive excess kurtosis (leptokurtic) has fatter tails and a sharper, more concentrated peak than normal. Most individual stock and index return series exhibit positive excess kurtosis, meaning extreme returns — both up and down — occur more often than a bell curve would predict. A distribution with negative excess kurtosis (platykurtic) has thinner tails and a flatter peak; returns cluster more closely around the mean with fewer extreme outcomes.
For risk management, high kurtosis is a warning signal that a portfolio or strategy has more exposure to extreme outcomes than a volatility measure alone would reveal. Two strategies might have identical standard deviations but very different kurtosis profiles. The high-kurtosis strategy might look calm for extended periods but periodically experience violent drawdowns; this is the classic signature of short-volatility or carry-trading strategies that earn steady income until a crisis unwinds the trade.
Option pricing models that assume log-normal returns (equivalent to normally distributed log returns with kurtosis of 3) will underprice options with extreme strikes because they do not account for the fat tails. The implied volatility smile — where out-of-the-money options trade at higher implied volatility than at-the-money options — is the market's empirical correction for the kurtosis that theoretical models ignore.
Kurtosis is most meaningfully evaluated alongside skewness. A portfolio with both negative skewness and high positive kurtosis is particularly dangerous: not only are the worst outcomes worse than the distribution average, but they also occur more frequently than normal. Understanding both moments together gives a more complete picture of downside risk than variance or standard deviation alone.