Fat Tails: How Often Price Moves Beyond Three Sigma

If daily returns were normally distributed, a day beyond three standard deviations should occur roughly once a year. We measured the actual frequency across 518 daily bars on twelve pairs — it turned out to be more than four times higher.

Fat tails in the distribution: what exactly we measured

For each pair we took daily returns on closes, computed the mean and standard deviation, then the share of days deviating from the mean by more than two, three and four deviations. A normal distribution predicts 4.6% for two sigma, 0.27% for three and 0.006% for four — those are the numbers we compare against.

The measurement

• Share of days outside ±2σ: 5.22% (a normal distribution predicts 4.6%)
• Share of days outside ±3σ: 1.16% (predicted 0.27%) — a median of about 6 days per pair
• Days beyond ±4σ across all pairs: 18
• Largest observed deviation: 8.7σ (AUDUSD)

Measured across 518 daily bars for each of 12 currency pairs. The sample is public, the methodology is in how we run backtests, and the calculation is reproduced by a script in the repository.

A factor of four is not a detail

At two sigma the actual frequency is close to theory: 5.22% against 4.6%. At three the discrepancy is fundamental: 1.16% against 0.27%, more than fourfold. Beyond four sigma, 18 days accumulated across all pairs, though a normal distribution predicts almost none on a sample this size. The largest observed deviation is 8.7 sigma on AUDUSD, and the probability of that under the normal law is indistinguishable from zero.

What this changes in risk calculation

Any formula that assumes normality understates extreme events. That covers maximum drawdown estimates, required margin, and every conclusion of the form 'this happens once a century'. The practical takeaway is not to swap one distribution for another but to stop building decisions on the assumption that rare events are genuinely rare.

How it looks in practice

Three consequences. A stop set as a multiple of ordinary volatility gets taken out by slippage more often than the model implies. An account load that looks safe under a normal distribution leaves too little room for a tail day. And position size is best computed from a bad-day scenario rather than a median one.

Limits of the measurement

The sample is 518 bars, about two years. For tails that is short by definition: rare events are rare, and their frequency on a short sample is itself estimated with wide error. The direction of the conclusion is robust — the divergence from the normal law is too large to be noise — but the specific percentages should be read as orders of magnitude. More on why this matters in our piece on sample size.

This material is for educational purposes and is not individual investment advice.