Distributions — normal, and why returns aren't

The normal distribution

The normal (Gaussian) distribution has the famous bell-shaped density — the curve whose height shows how thickly probability is packed near each value:

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

In words: this gives the height of the bell curve at any point $x$. The new symbols, piece by piece: $\exp(\cdot)$ means "e (a constant, about 2.718) raised to the power of whatever is inside the parentheses." $\pi$ is pi, about 3.14159. $\sqrt{2\pi}$ is the square root of two pi. The inner fraction $(x-\mu)^2 / 2\sigma^2$ measures how far $x$ sits from the center $\mu$, scaled by the spread $\sigma$. The farther out $x$ is, the more negative the exponent, and the lower the curve. You will never compute this by hand — libraries do it. What matters is the shape: tallest at $\mu$, falling off symmetrically on both sides.

It has two parameters:

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

Useful facts to memorize ("mass" here means probability — the share of area under the curve):

• ~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 (CLT). Take many independent random variables — independent meaning knowing one tells you nothing about the others (formal definition in A1-05) — and average them. That average tends to be normally distributed, no matter what shape the underlying distribution has. 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. (Log return — a return computed with logarithms; nearly the same as the percent change for small daily moves.) 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) — choosing portfolio weights by trading off average return against variance
• Black-Scholes option pricing (1973) — assumes log-returns are normal
• Value-at-Risk — a "how much could I lose on a bad day" number — computed from normal quantiles (quantiles — cut points of a distribution, like percentiles)
• Most introductory textbook problem sets

Why returns aren't normal

Look more carefully and the story breaks:

1. Tails are fatter than normal. The "tails" are the far left and far right ends of the distribution — the extreme outcomes. 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. Volatility — how big the moves are, regardless of direction. 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. (The technical name for this non-constant spread is heteroscedasticity.)
3. Asymmetric. Down moves tend to be sharper than up moves of the same probability. This is called negative skewness — skewness being the lopsidedness of a distribution.

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 (Long-Term Capital Management, a hedge fund that collapsed in 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. It has a tuning knob called "degrees of freedom" — lower values mean fatter tails. A t-distribution with 4 degrees of freedom has tail probabilities orders of magnitude larger — larger by factors of 10 — 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 (models that let volatility itself change over time, instead of assuming one fixed σ)

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 — averages over a sliding window of recent days) 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 are approximately normal due to CLT. Examples: Sharpe estimates (Sharpe ratio — return per unit of risk) and regression coefficients — the fitted slopes in a regression.

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. In the code, |x - mean| — the vertical bars mean absolute value — is the distance from the mean, ignoring sign:

Predict before the next lesson

Tomorrow we'll formalize expected value and variance — the first two "moments" of a distribution (a moment is a summary number computed from 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 — the average distance from the mean, ignoring sign?
• For two random variables $X$ and $Y$, is $E[XY] = E[X] \cdot E[Y]$ always? In words: is the average of "X times Y" always equal to the average of X times the average of Y? When might it not be?