The Student's t-distribution is a bell curve with heavier tails than the normal distribution, controlled by a degrees-of-freedom (DOF) parameter. It's the standard model for capturing the fat-tailed behavior of real financial markets in Monte Carlo simulations.
The Student's t-distribution resembles the familiar normal (Gaussian) bell curve but with heavier tails - meaning extreme values occur more frequently. The thickness of the tails is controlled by a single parameter: degrees of freedom (DOF). Lower DOF produces fatter tails; as DOF approaches infinity, the distribution converges to the normal. Retirement Lab offers two DOF settings: DOF 3 for extreme fat tails and DOF 5 for moderate fat tails that closely match observed equity market behavior.
How It Works
The key relationship between DOF and tail behavior:
| DOF | Excess Kurtosis | Behavior |
|---|---|---|
| 4 | ∞ (undefined) | Extremely fat tails |
| 5 | 6.0 | Fat tails - matches equity markets |
| 8 | 3.0 | Moderate tails |
| 30 | 0.2 | Nearly normal |
| ∞ | 0 | Exactly normal |
Retirement Lab generates t-distributed returns using the Marsaglia-Tsang gamma sampling method - an O(1) algorithm that efficiently produces Student-t random variables from a seeded pseudorandom number generator.
Why It Matters for Retirement Planning
The Student-t distribution is the workhorse model for fat-tailed retirement simulations. Combined with the Fernandez-Steel transformation for skewness, it captures two critical properties of real markets:
- Excess kurtosis: extreme events occur more often than a bell curve predicts
- Negative skewness: large losses are more likely than large gains of equal magnitude
This combination produces harsher worst-case scenarios and usually a lower success rate than normal distribution assumptions - a useful stress test for any retirement plan.
How Retirement Lab Addresses This
Retirement Lab uses the Student's t-distribution as its Pro fat-tail model, offering DOF 3 (extreme fat tails) and DOF 5 (moderate fat tails) - both deliberately cautious. Enable fat tails in your simulation settings to see how heavier tails change your success rate and worst-case outcomes.
Frequently Asked Questions
- What degrees of freedom should I use for stock market returns?
- Lower degrees of freedom mean heavier tails. Retirement Lab offers DOF=5 (moderate, excess kurtosis of 6) and DOF=3 (extreme); both are deliberately cautious compared with yearly U.S. stock returns. Higher values (DOF=10+) approach a normal distribution.
- How is the Student-t distribution different from the normal distribution?
- Both are symmetric and bell-shaped, but the Student-t has heavier tails. At DOF=5, extreme events (3+ standard deviations) occur roughly 4 times more frequently than under a normal distribution. As DOF increases toward infinity, the t-distribution converges to the normal distribution.