The normal (Gaussian) distribution is the classic bell curve used in most financial models. While mathematically convenient, it dangerously underestimates extreme market events — crashes happen far more often than it predicts. For realistic retirement planning, fat-tail distributions like the Student-t are essential.
The normal distribution — also called the Gaussian or bell-curve distribution — is the most widely used probability distribution in finance and statistics. It is fully described by two parameters: the mean (center) and standard deviation (spread). Most traditional Monte Carlo simulators assume investment returns follow a normal distribution.
How It Works
The normal distribution has several convenient properties:
- Symmetric: gains and losses of equal magnitude are equally likely
- Thin tails: extreme events are vanishingly rare (99.7% of values fall within 3 standard deviations)
- Additive: the sum of normal variables is also normal, simplifying portfolio math
These properties made it the foundation of Modern Portfolio Theory and the standard assumption in financial modeling for decades.
Interactive chart: normal-vs-empirical
Normal distribution vs. actual S&P 500 monthly return distribution
Coming soon
Why It Matters for Retirement Planning
The normal distribution's fatal flaw for retirement planning is its thin tails. Real financial markets exhibit fat tails — extreme events occur far more frequently than the bell curve predicts:
- A 4-sigma daily move (like a single-day drop of ~6%) should occur once every 126 years under normal assumptions. In practice, it happens every few years.
- The 2008 crisis, 2020 COVID crash, and 1987 Black Monday are all events that the normal distribution assigns near-zero probability.
For retirees, this matters because sequence-of-returns risk makes portfolios asymmetrically sensitive to crashes. Underestimating crash frequency means overestimating plan safety.
Retirement Lab uses the normal distribution as its free-tier baseline model, while the pro tier upgrades to the Student's t-distribution with Fernandez-Steel skewness — capturing both the frequency (kurtosis) and asymmetry (skewness) of real-world extreme events.
Frequently Asked Questions
- Why do most financial models use the normal distribution?
- The normal distribution is mathematically convenient — it is fully defined by just two parameters (mean and standard deviation), has well-known properties, and simplifies portfolio optimization (Modern Portfolio Theory is built on it). However, this convenience comes at the cost of accuracy: it underestimates extreme events, which matters enormously for retirement planning.
- How wrong is the normal distribution for stock market returns?
- Under a normal distribution, a daily market drop of 10%+ should occur roughly once every 10,000 years. In reality, it happens every few decades. Monthly equity returns show excess kurtosis of 3-10, meaning extreme events occur 3 to 10 times more frequently than the normal distribution predicts.