Kurtosis

What Is Kurtosis?

Kurtosis is a measure of the shape of a probability distribution, specifically how much of its total variation comes from rare, extreme observations rather than typical ones. It does not measure width, which is the job of standard deviation, or lopsidedness, which is skewness. Instead, kurtosis focuses on the tails, the far left and far right ends of the distribution where unusually large gains or losses sit.

A normal distribution, the familiar bell curve, has a kurtosis of 3 under the standard formula. Because of this, many analysts report "excess kurtosis," which subtracts 3 so that a normal distribution scores zero. Positive excess kurtosis means fatter tails than a normal curve, while negative excess kurtosis means thinner tails.

How It Is Calculated

The basic formula compares the fourth power of deviations from the mean to the square of the variance:

Kurtosis = E[(X − μ)⁴] / σ⁴

In practice, analysts compute this from a sample of observations. Software packages often report both the raw value and the excess version, so it is worth checking which one a given source uses.

Types of Distributions

• Leptokurtic: excess kurtosis above zero, with a sharp peak and fat tails.
• Mesokurtic: excess kurtosis near zero, similar to a normal distribution.
• Platykurtic: excess kurtosis below zero, with a flatter peak and thinner tails.

Why It Matters to Investors

Kurtosis helps show whether an average-based view of risk tells the full story. Many financial return series, such as daily stock returns, have shown positive excess kurtosis, meaning large moves happen more often than a normal distribution would predict. An investor who assumes returns are normal may underestimate the chance of severe losses. Measures like volatility alone can miss this, which is why risk analysts often examine tail risk alongside it.

Kurtosis is descriptive, not predictive. A high reading shows that past returns had fat tails, but it does not indicate when the next extreme move will occur. Estimates are also sensitive to outliers and need reasonably large samples to be reliable.