Return distributions: the setting that decides everything
A Monte Carlo simulation draws thousands of scenarios at random. The draw follows a statistical law (the “distribution”) that you choose yourself. And that choice decides everything: the same portfolio can show 95% success under a normal distribution and 78% under Student-t. Before you look at your success rate, check the distribution that produced it.
The normal (Gaussian) distribution
The best-known: the famous bell curve. Returns spread out symmetrically around the average. Most years deliver a “normal” return, and extremes (crashes or booms) are assumed to be very rare.
Concrete example
With an 8% average and 15% volatility, this model predicts that 68% of years land between -7% and +23%, and 95% between -22% and +38%. A -50% crash? Statistically near-impossible. Yet it happened in 2008 and again in March 2020. The model is wrong, the markets are not.
The log-normal distribution: a floor of reality
The log-normal fixes the main flaw of the normal: a return cannot drop below -100%. You cannot lose more than what you own. It is also asymmetric: losses are bounded, gains are not.
Why it is the industry standard
An asset at $100 that loses 50% is worth $50. To get back to $100, it needs +100%, not +50%. This multiplicative asymmetry governs markets: the log-normal accounts for it, the normal does not. That is why professional financial models (Black-Scholes, VaR, Markowitz) are built on it.
⚠️ “Fat tails”: the log-normal blind spot
Even the log-normal misses the truly extreme shocks. The 2008 crash (-38%), Black Monday 1987 (-22% in a single session), and the Covid drop of March 2020 (-34% in one month) are events the normal calls “impossible” and the log-normal calls “extremely rare.” The Student-t distribution adds fatter tails to reproduce these black swans.
Key Takeaways
- 1Log-normal is more realistic than normal because no return below -100% is possible.
- 2Fat tails (Student-t) reproduce the -40% crashes and +50% rallies the normal ignores.
- 3Brutal asymmetry: -50% demands +100% just to break even.
- 4For the final FIRE stress test, switch to Student-t before you decide.
Keep going
Frequently asked questions
It is the statistical shape of an asset's returns over a long period. Three common families: Normal (Gaussian), symmetric with thin tails; Log-normal, which forbids any return below -100% and tracks reality more closely; Student-t, with fat tails that model extreme crashes and rallies.
A return cannot drop below -100% (you cannot lose more than everything). The normal distribution still allows that impossibility. Log-normal imposes a natural lower bound and leaves gains unlimited, which sticks closer to market reality. It is the academic standard in finance.
Fat tails are the extreme events (-40% crashes, +50% rallies) that happen more often than the normal distribution predicts. The Student-t distribution captures this reality better (Black Monday 1987, 2008, March 2020). Ignoring fat tails systematically understates the real risk of a FIRE plan.
For the first iteration, log-normal: ~7% mean, ~15% volatility, the default for a 100% equity portfolio. For the final stress test before retiring, Student-t with 4 to 6 degrees of freedom, which captures fat tails. The result will be more pessimistic, and that is precisely the point of a stress test.