Degrees of freedom (DOF) is the parameter that controls how fat the tails are in a Student's t-distribution. Lower DOF = fatter tails = more frequent extreme market events. At DOF=5, extreme events occur roughly 4x more often than a normal distribution predicts.
Degrees of freedom (DOF) is the key parameter of the Student's t-distribution that determines tail thickness. It provides a single, interpretable dial for controlling how frequently extreme events occur in a Monte Carlo simulation.
How It Works
The relationship between DOF and distribution behavior:
| DOF | Excess Kurtosis | Practical Meaning |
|---|---|---|
| 4 | ∞ | Extremely fat tails — DOF must exceed 4 for finite kurtosis |
| 5 | 6.0 | Close match to empirical equity return data |
| 8 | 3.0 | Moderate fat tails |
| 15 | 0.5 | Mild fat tails |
| 30+ | ~0 | Nearly indistinguishable from normal distribution |
Critical constraints:
- DOF > 2 required for finite variance
- DOF > 4 required for finite kurtosis
- Retirement Lab uses DOF values between 4 and 10
As DOF increases toward infinity, the t-distribution converges exactly to the normal distribution. This makes DOF a smooth control — you can dial from "realistic fat tails" to "textbook bell curve" with a single parameter.
Why It Matters for Retirement Planning
DOF directly affects success rate estimates. Lower DOF generates more extreme returns in the simulation, which — combined with sequence-of-returns risk — typically reduces the percentage of scenarios where the portfolio survives.
The difference is meaningful: the same retirement plan might show a 92% success rate at DOF=30 (near-normal) and an 84% success rate at DOF=5 (fat tails). The latter is a more honest estimate of real-world risk, which is why DOF calibration matters for retirement security.
Frequently Asked Questions
- What happens if degrees of freedom is too low?
- Very low DOF values (below 3) produce distributions where the variance is infinite or undefined, making the simulation results unstable. DOF must exceed 2 for finite variance. Retirement Lab uses DOF values of 4-10, which produce meaningful fat tails while keeping the distribution mathematically well-behaved.
- How do I choose the right degrees of freedom?
- Empirical research on equity markets suggests DOF of 4-6 for monthly returns. Retirement Lab defaults to DOF=5. Higher values (8-10) are more conservative and produce milder fat tails. You can adjust this in the advanced settings to see how tail thickness affects your plan's success rate.