Kurtosis measures how often extreme market events occur compared to a normal bell curve. Financial markets have high kurtosis — crashes and rallies happen far more frequently than standard models predict. Ignoring this in retirement simulations can overestimate your plan's safety by 5–10 percentage points.
Kurtosis measures the "tailedness" of a probability distribution — specifically, how much of the distribution's variance is driven by extreme outliers versus moderate deviations near the center. In retirement planning, kurtosis quantifies how often market crashes and rallies occur relative to what a normal (bell-curve) distribution would predict.
How It Works
Mathematically, kurtosis is the fourth standardized moment of a distribution — it takes each data point's deviation from the mean, raises it to the fourth power, and normalizes by the variance squared: Kurt[X] = E[(X - μ)⁴] / σ⁴.
A normal distribution has a kurtosis of exactly 3. To make comparisons easier, we often use excess kurtosis (kurtosis minus 3):
| Distribution | Excess Kurtosis | Tail Behavior |
|---|---|---|
| Normal (Gaussian) | 0 | Baseline — thin tails |
| Student-t (DOF=5) | 6 | Fat tails — extreme events ~4x more frequent |
| Student-t (DOF=4) | ∞ (undefined) | Very fat tails |
| Financial returns (empirical) | 3–10 | Significantly fatter than normal |
- Excess kurtosis > 0 (leptokurtic): fatter tails, more extreme events than a normal distribution
- Excess kurtosis = 0 (mesokurtic): normal distribution
- Excess kurtosis < 0 (platykurtic): thinner tails, fewer extremes
Interactive chart: kurtosis-comparison
Normal vs. fat-tailed distribution — notice the heavier tails
Coming soon
Why It Matters for Retirement Planning
Financial markets consistently exhibit positive excess kurtosis — typically in the range of 3 to 10 for monthly equity returns. This means:
- Market crashes like 2008 (-37%) and 2020 (-34%) are far more likely than a normal distribution predicts
- Extreme rallies also occur more frequently
- Traditional Monte Carlo simulators that assume normal returns underestimate tail risk, potentially giving retirees false confidence
A retirement plan that looks safe under normally distributed returns may have a meaningfully lower success rate when fat tails are accounted for. The difference can be 5–10 percentage points in success rate — the difference between "probably fine" and "significant risk of ruin."
Kurtosis in Retirement Lab
Retirement Lab captures excess kurtosis through the Student's t-distribution, where the degrees of freedom (DOF) parameter directly controls tail fatness:
- Higher DOF (e.g., 30): approaches a normal distribution, low excess kurtosis
- Lower DOF (e.g., 5): produces excess kurtosis of 6, matching empirical equity data
- DOF = 4: excess kurtosis becomes infinite — extreme events dominate
By calibrating DOF to match observed market behavior, the simulation produces a more realistic range of outcomes. Combined with skewness (via the Fernandez-Steel transformation), this captures both the frequency and asymmetry of extreme market events.
Frequently Asked Questions
- What is a normal kurtosis value for stock market returns?
- Financial returns typically show excess kurtosis of 3 to 10 for monthly data. This means extreme events (crashes and rallies) occur roughly 3 to 10 times more frequently than a normal distribution predicts. The exact value depends on the asset class and time period measured.
- Why does kurtosis matter for retirement planning?
- Higher kurtosis means more frequent extreme market events. A retirement plan that looks safe under normally distributed returns (kurtosis = 3) may have a meaningfully lower success rate — often 5 to 10 percentage points lower — when tested with realistic fat-tailed distributions that match actual market behavior.