Formula, in Risk & Portfolio Construction
Kurtosis Formula
Kurtosis is the fourth standardized moment: the average of deviations from the mean raised to the fourth power, divided by the variance squared. It measures tail heaviness. A normal distribution has kurtosis 3, so excess kurtosis (kurtosis minus 3) above zero signals fatter tails and more frequent extreme moves than a normal model would predict.
4 variables. Published May 26, 2026. By AI Fin Hub Research.
Kurt = ( (1/n) x sum_i (R_i - mean_R)^4 ) / sigma^4
Excess kurtosis = Kurt - 3
where sigma = population standard deviation (divisor n) Variables
R_i
Period return
Individual return observation.
(R_i - mean_R)^4
Fourth-power deviation
Deviation from the mean raised to the fourth power. Because the exponent is even, all terms are positive and large deviations dominate enormously, so kurtosis is driven almost entirely by the tails.
sigma^4
Variance squared
The square of the variance, which standardizes the fourth moment so the measure is unitless and comparable across assets of different volatility.
Kurt - 3
Excess kurtosis
Kurtosis relative to the normal distribution's value of 3. Positive excess kurtosis (leptokurtic) means fatter tails; negative (platykurtic) means thinner tails. Asset returns are almost always leptokurtic.
Step by step
- 1
Compute the mean and population standard deviation of the returns.
Returns -10%, 0%, 0%, +10% have mean 0%.
- 2
Raise each deviation from the mean to the fourth power and average them.
Fourth powers: 0.0001, 0, 0, 0.0001; mean = 0.00005.
- 3
Square the variance to get sigma to the fourth power.
Variance = (0.01 + 0 + 0 + 0.01)/4 = 0.005; sigma^4 = 0.005^2 = 0.000025.
- 4
Divide the mean fourth-power deviation by sigma to the fourth, then subtract 3 for excess kurtosis.
0.00005 / 0.000025 = 2.0; excess = 2.0 - 3 = -1.0.
Worked example
Kurtosis of a symmetric four-return sample
Returns
-10%, 0%, 0%, +10%
Mean
0%
Deviations equal the returns. Fourth powers: (-0.10)^4 = 0.0001, 0, 0, (0.10)^4 = 0.0001. Sum = 0.0002; mean = 0.00005. Variance = (0.01 + 0 + 0 + 0.01)/4 = 0.02/4 = 0.005; sigma^4 = 0.005^2 = 0.000025. Kurtosis = 0.00005 / 0.000025 = 2.0. Excess kurtosis = 2.0 - 3 = -1.0.
Kurtosis of 2.0, or excess kurtosis of -1.0. This small, evenly spread sample is platykurtic (thinner-tailed than normal) because the values are bunched rather than tail-heavy. Real return series behave oppositely: daily equity returns commonly show kurtosis well above 3, meaning crashes and spikes happen far more often than a normal model assumes.
Common variations
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Sources
- The Variation of Certain Speculative Prices , Benoit Mandelbrot, Journal of Business (1963)
- Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals , Carlos M. Jarque and Anil K. Bera, Economics Letters (1980)
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