Kurtosis and Excess Kurtosis

What is kurtosis?

Kurtosis is the fourth standardized moment of a distribution: the average of standardized deviations from the mean raised to the fourth power. Large deviations dominate that average, so it measures tail weight on both sides of the mean at once. A normal distribution has raw kurtosis 3; excess kurtosis subtracts 3, so the normal benchmark has excess kurtosis 0.

How is kurtosis calculated?

For a population with positive variance and finite fourth moment, raw kurtosis is E[(R − mu)⁴] / sigma⁴.

Excess kurtosis subtracts 3. The normal distribution therefore has raw kurtosis 3 and excess kurtosis zero. Other distributions can share that fourth moment.

Sample kurtosis conventions

For the unadjusted sample coefficient, average squared deviations with divisor n to obtain m2 and fourth-power deviations with divisor n to obtain m4. Raw sample kurtosis is m4 / m2².

Sample-adjusted estimators apply an additional formula. Keep the convention with the reported number. Standard deviation measures dispersion; kurtosis measures a different standardized moment.

Kurtosis example: 200 returns with two outliers

Use 198 zero returns, one −12% return and one +12% return. Mean return is zero. Divisor-n variance is (144 + 144) / 200 = 1.44 in squared percentage points, giving standard deviation 1.2 percentage points.

The fourth moment is (20,736 + 20,736) / 200 = 207.36. Divide by 1.44² to obtain raw kurtosis 100 and excess kurtosis 97.

Each ±12% outcome is ten standard deviations from the mean. Together they carry 1% of the empirical probability.

Kurtosis and tail probabilities at a cutoff

Beyond three standard deviations, the example has probability 1%. A normal distribution has 0.27%, giving an expected 6.8 such observations over 2,520 observations with that same normal marginal distribution.

Beyond two standard deviations, the example still has probability 1%, while the normal distribution has 4.55%. Higher kurtosis does not order tail probabilities at every possible cutoff.

This finite-support example also separates a large fourth moment from an asymptotic heavy-tail definition. Calculate the tail probability relevant to the position directly.

How a variance mixture creates excess kurtosis

For a zero-mean mixture of normal regimes, let v be the regime variance. Overall variance is E[v] and the fourth moment is 3 × E[v²]. Raw kurtosis is 3 × E[v²] / E[v]².

Different regime variances raise that value above 3. This construction supplies a concrete explanation of how variance changes alter unconditional return shape.

Kurtosis, skewness and option payoffs

Skewness uses signed cubes and changes sign when returns are reversed. Kurtosis uses fourth powers and stays unchanged. Neither statistic alone specifies the complete return distribution.

Compare option payoffs under a fat-tailed distribution and include the premium; the Options Profit Calculator shows which price outcomes drive each structure’s payoff across price, date and volatility, so you see what a tail move does to the position.

Limitations of kurtosis

• Calculate the tail at the cutoff you trade. The example has excess kurtosis 97 yet 1% beyond two standard deviations, against 4.55% for the matched normal distribution.
• Read skewness for direction. Kurtosis treats large gains and large losses alike, and skewness describes the asymmetry.
• Check which observations drive the sample value, and name the estimator. In the example, two returns out of 200 produce all of the fourth moment, for an excess kurtosis of 97, and unadjusted and sample-adjusted estimators print different numbers on the same data.
• Test normality on the full distribution. Matching the normal value of 3 does not make a distribution normal, because other distributions share that fourth moment.
• Match the return horizon and data definition before you compare two readings.

Bottom line

Kurtosis summarizes how heavily extreme observations weigh in a distribution, which makes it a quick check for departures from a normal model. Use it as a flag, then calculate the actual tail probabilities and payoff scenarios that matter for the position.