Kurtosis is the standardized fourth central moment of a random variable. It indicates the likelihood of observing extreme positive or negative events relative to the mean. Random variables with high levels of kurtosis are often described as having distributions that are fat-tailed.
Mathematically, for a random variable X with mean μ and standard deviation σ, kurtosis is defined as:
The kurtosis of a normal distribution is always equal to three. Because of this we often talk of excess kurtosis which is simply kurtosis minus three. Distributions with positive excess kurtosis are said to be leptokurtotic. Distribution with negative excess kurtosis are said to be platykurtotic.
All other things being equal, investors tend to prefer investments with lower kurtosis. The returns of many securities display excess kurtosis. Certain hedge fund strategies can display high levels of kurtosis. Hedge fund managers can use risk management to reduce kurtosis.