Kurtosis (Excess)

Measures fat tails — how often extreme events occur vs. normal distribution.

Excess kurtosis measures tail risk: how much more likely extreme events are compared to a normal distribution. Positive excess kurtosis (leptokurtic) means fatter tails — more frequent outliers. Most financial returns have excess kurtosis of 3-10, meaning 3-sigma+ events happen far more than the normal distribution predicts. Example: A normal distribution says a 4-sigma event occurs once in 15,000 days (~60 years). With excess kurtosis of 5, it's once every 1,000 days (~4 years). This is why VaR underestimates risk — it assumes normality. The 2008 crisis was a supposed '25-sigma event' under normality, but given realistic kurtosis, it was more like 4-sigma. Key: Higher kurtosis = don't trust VaR, use stress tests.

Formula

\text{Kurt} = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{r_i - \mu}{\sigma} \right)^4 - 3

Related Terms

Skewness

Measures asymmetry — are big losses or big gains more likely?

Monte Carlo Simulation

Generating thousands of possible future scenarios through random sampling.

Value at Risk (VaR)

The maximum expected loss at a given confidence level — but doesn't tell you how bad the tail is.