Return Distributions: Normal, Lognormal, and Fat Tails

Risk models fail when their distribution assumption is wrong. Most assume normality; most markets are not normal. Knowing which distribution fits, and where it breaks, is the difference between reasonable sizing and a blow-up.

The Three Distributions

Normal: symmetric, bell-shaped, extends to negative infinity. Daily percentage returns are often roughly normal over short windows. The flaw: it allows returns below -100%, impossible for a long-only price series.

Lognormal: the natural log of price is normal, so price itself is lognormal. Returns are multiplicative, bounded below by -100%. Asset prices are better modeled as lognormal than normal, which is why Black-Scholes uses it.

Fat-tailed (leptokurtic): real returns have more extreme outcomes than either distribution predicts. The tails decay as a power law, not exponentially. kurtosis exceeds 3 (the normal value); large moves occur 5-50x more often than normality implies.

Which Fits Real Markets

None, fully. Daily equity returns look near-normal in the center but fat-tailed in the extremes. The practical pattern:

• Center of the distribution: approximately normal.
• Tails: fat, with kurtosis of 5-10 for equity indices, higher for single stocks and FX during events.
• Asymmetry: equity returns are negatively skewed; crashes are larger and faster than rallies.

Testing Your Market

Compute on at least 5 years of daily returns:

• Skewness: equity indices typically -0.3 to -1.0. Positive skew is rare and usually signals a different regime.
• Excess kurtosis: above 1 means fat tails are present; above 5 means severe.
• Jarque-Bera test: rejects normality for virtually every real financial series.

If your market shows excess kurtosis above 3, any risk model assuming normality understates tail risk by a factor of 3-10x.

Implications for Sizing

Normal-based volatility (standard deviation) understates tail risk. Adjust:

• Position sizing: cap single-trade risk at 0.5-1% rather than the 2% a normal model might permit. Fat tails make the worst case worse than sigma suggests.
• Stop placement: use ATR or recent realized range, not standard deviation bands. Sigma-based stops sit too close in fat-tailed markets and get picked off.
• Portfolio risk: scale Value-at-Risk estimates by the observed kurtosis. A 3-sigma normal event is a 99.9% rarity; in a fat-tailed market it occurs every few years.

Extreme Value Theory

For tail-specific modeling, fit a generalized Pareto distribution to returns beyond a high threshold (e.g., the worst 5%). EVT gives a more honest estimate of 1-in-100 and 1-in-500 day losses than fitting a normal to the whole series.

The Honest Use

You will not find a clean distribution that fits a real market. The useful knowledge is knowing where the normal approximation holds (the center, for short-horizon volatility) and where it fails (the tails, for risk). Size for the tails you actually have, not the tails a textbook assumes.