Kurtosis: Definition, Types, and Practical Use

What is kurtosis?

Kurtosis is a statistical measure that describes the weight of a distribution’s tails relative to its center. In practical terms, it indicates how frequently extreme values (outliers) occur compared with a normal distribution. High kurtosis means fatter tails (more extreme events); low kurtosis means thinner tails (fewer extreme events).

Key points

• Kurtosis measures “tailedness,” not “peakedness.” A distribution can be sharply peaked yet have low kurtosis, or relatively flat and have high kurtosis.
• In finance, kurtosis is used to assess tail risk — the likelihood of large positive or negative returns.

Types of kurtosis

Kurtosis is typically compared to the normal distribution. Two common conventions exist: one where the normal distribution’s kurtosis = 3, and another using excess kurtosis where the normal = 0. Using the 3-based convention:

• Mesokurtic: kurtosis ≈ 3 (excess kurtosis ≈ 0). Tails are similar to a normal distribution.
• Leptokurtic: kurtosis > 3 (excess kurtosis > 0). Tails are “fat”; more frequent extreme values. Often interpreted as higher tail risk.
• Platykurtic: kurtosis < 3 (excess kurtosis < 0). Tails are thin; fewer extreme values and generally more stable behavior.

How it’s calculated (brief)

Common forms:
* Spreadsheet: use the built-in function — e.g., KURT(range) in Excel/Google Sheets.
– Example: for sample data in A1:A10, =KURT(A1:A10) returns the sample’s excess kurtosis (negative value indicates platykurtic).
* Moment-based formula (excess kurtosis):
k = m4 / (m2^2) − 3
where m2 is the second moment (variance) and m4 is the fourth moment about the mean.

Notes:
* Different formulas and corrections exist for small samples; exercise caution with small n or heavily skewed data.
* “Excess kurtosis” = kurtosis − 3, which sets normal distribution to zero.

Kurtosis vs. skewness

• Skewness measures asymmetry (direction and degree of tilt). Positive skew = long right tail; negative skew = long left tail.
• Kurtosis measures tail weight regardless of symmetry. A distribution can be symmetric (zero skewness) but have high kurtosis (fat symmetric tails).

Why kurtosis matters in finance

• Tail risk: High kurtosis implies more frequent extreme returns, increasing the chance of large losses or gains beyond what normal models predict.
• Portfolio construction: Investors or managers with low risk tolerance may prefer assets with lower kurtosis; those seeking occasional large gains might tolerate higher kurtosis.
• Risk management and stress testing: Kurtosis helps identify when models that assume normal returns understate extreme-event probabilities.

How it complements other metrics

• Alpha: measures excess return relative to a benchmark — not about tail shape.
• Beta: measures sensitivity to market movements — volatility relative to market, not tail weight.
• R-squared: explains how much of a fund’s movement is captured by a benchmark — model fit, not distribution shape.
• Sharpe ratio: compares return to volatility — a performance metric that still assumes return distribution properties; kurtosis highlights departure from normality that can affect Sharpe interpretation.

Practical cautions

• Sensitive to sample size and outliers — small samples can give misleading kurtosis estimates.
• Use together with other diagnostics (skewness, histograms, quantile measures, stress scenarios).
• For heavy-tailed data, consider using robust risk measures (e.g., value at risk (VaR) with fat-tailed models, conditional VaR).

Short FAQs

Q: Is high kurtosis good or bad?
A: Neither inherently; it depends on investor objectives and risk tolerance. High kurtosis signals higher probability of extreme outcomes (good or bad).

Q: What is excess kurtosis?
A: Excess kurtosis = measured kurtosis − 3. It sets the normal distribution’s excess kurtosis to 0 and is commonly reported by statistical software.

Q: Should I rely solely on kurtosis to assess risk?
A: No. Kurtosis is a useful indicator of tail risk but should be combined with other statistics, visual inspection, and scenario analysis.

Conclusion

Kurtosis is a compact way to quantify how prone a dataset — or an investment’s returns — is to extreme outcomes. Understanding whether returns have fat tails (leptokurtic) or thin tails (platykurtic) helps inform risk management, portfolio selection, and model choice, but it should not be used in isolation.