Distributions — normal, and why returns aren't

The normal distribution

The normal (Gaussian) distribution has the famous bell-shaped density:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

It has two parameters:

• $\mu$ — the mean (center)
• $\sigma$ — the standard deviation (width)

Useful facts to memorize:

• ~68% of mass within 1σ of mean
• ~95% within 2σ
• ~99.7% within 3σ
• ~99.99% within 4σ

The normal is the default assumption in much of classical statistics for one reason: the Central Limit Theorem. The average of many independent random variables tends to be normally distributed regardless of the underlying distribution. This is why means of samples behave nicely, even when individual observations don't.

Why people assume returns are normal

If you have a few minutes and a copy of pandas, plot the histogram of SPY daily log-returns. It looks bell-shaped. The mean is near zero. The standard deviation is around 1% per day. Easy story: returns are normal.

This assumption is built into a lot of classical finance:

• Mean-variance portfolio optimization (Markowitz, 1952)
• Black-Scholes option pricing (1973) — assumes log-returns are normal
• Value-at-Risk computed from normal quantiles
• Most introductory textbook problem sets

Why returns aren't normal

Look more carefully and the story breaks:

1. Tails are fatter than normal. Days more extreme than 3σ happen much more often than the normal predicts. Days like Black Monday 1987 (−20%) or COVID's March 2020 (−12% in a day) shouldn't happen in a normal universe. They do happen. Often.
2. Volatility clusters. A high-volatility day is more likely to be followed by another high-volatility day. The normal model assumes returns are independent over time; they aren't (a phenomenon called heteroscedasticity).
3. Asymmetric. Down moves tend to be sharper than up moves of the same probability. This is called negative skewness.

These are not minor refinements. They are the difference between strategies that work and strategies that blow up. Most blow-ups happen because someone assumed normality and the market produced a fat-tail event. LTCM (1998), the Quant Quake (August 2007), countless others.

Better models — fat-tailed distributions

The classic alternative is the Student t-distribution, which has heavier tails. A t-distribution with 4 degrees of freedom has tail probabilities orders of magnitude larger than the normal at the same standard deviation.

Other useful models:

• Mixture of normals (a few "calm" normal distributions plus one "panic" wider one)
• Stable distributions (Mandelbrot's preference — they have infinite variance in the heaviest cases)
• GARCH and EGARCH (model the time-varying volatility explicitly)

We won't fit any of these yet. The point right now is: if you model returns as normal, you will systematically underestimate extreme risk.

Practical guidance

Even though returns aren't normal, the normal distribution shows up usefully:

• Sample averages of returns (rolling means) are approximately normal by CLT, even though individual returns aren't.
• Squared returns (a rough proxy for variance) are not normal — they're highly skewed.
• Many derived statistics (Sharpe estimates, regression coefficients) are approximately normal due to CLT.

A practical rule: assume normality when computing summary statistics about averages over many days; do not assume normality when sizing positions around tail risk.

Try it

How fat are the tails of a return series? Count how much of the data sits beyond k standard deviations from the mean:

Predict before the next lesson

Tomorrow we'll formalize expected value and variance — the first two moments of a distribution. Predict:

• Why do you think variance (the squared deviation from the mean) is the standard measure of "spread," instead of, say, mean absolute deviation?
• For two random variables $X$ and $Y$, is $E[XY] = E[X] \cdot E[Y]$ always? When might it not be?