2. The normal distribution: the Gaussian bell curve

Level 2: Distributions and why markets are not "normal"

Under the Gaussian bell curve, a five or six standard deviation move is so unlikely that it should essentially never happen over the entire life of a market.

That is a strong prediction. And it is exactly the point where the theory starts to collide with what has actually happened in financial markets.

Before breaking it, you have to understand it properly.

Two numbers and that is it

The normal distribution, also called Gaussian, is the symmetric bell-shaped curve you have probably already seen on some chart.

Its most important mathematical feature is that it is completely determined by just two parameters:

• The mean, which fixes where the center of the bell sits.
• The standard deviation, which fixes how wide or narrow it is.

If you are told that a variable follows a normal distribution with a mean of 0.05% and a standard deviation of 1.1%, you already know the entire curve. No further data is needed to draw it, to compute any probability from it, or to compare it with another normal.

Why that is rarer than it looks

This is a strong and, at the same time, rather unintuitive property.

For the vast majority of distributions out there, knowing the mean and the standard deviation is NOT enough to reconstruct the full curve. Infinitely many different shapes — with more or less skew, with heavier or lighter tails — could share exactly those two numbers.

The normal is the exception in which those two numbers suffice. That is why it is so convenient to work with: it reduces the whole problem of describing a distribution to estimating two quantities.

The 68-95-99.7 rule

A practical rule associated with the normal, known as the empirical rule or the 68-95-99.7 rule, says that:

• Roughly 68% of observations fall within one standard deviation of the mean.
• 95% fall within two standard deviations.
• 99.7% fall within three.

Applied to the earlier example, with a mean of 0.05% and a standard deviation of 1.1%:

• 68% of days would have a return between -1.05% and 1.15%.
• 95% between -2.15% and 2.25%.
• 99.7% between -3.25% and 3.35%.

Outside that last range, that is more than three standard deviations from the mean, the normal leaves barely 0.3% of days: some 3 or 4 days out of every 1,250.

Why finance leans on it

Why does finance lean so heavily on this particular distribution? For three mutually reinforcing reasons.

The first is mathematical tractability. Central formulas in the field become simple and manageable once returns are assumed to be normal: from the Black-Scholes option pricing model to the classical portfolio variance formulas or the parametric calculation of Value at Risk. Without that assumption, many of them no longer have a closed-form solution and you have to fall back on numerical simulation.

The second is historical. Before cheap computers existed, working with normal tables was the only practical way to do these calculations by hand.

The third is that there is a kernel of truth behind it. The central limit theorem, which you will see in module 6, explains why summing or averaging many sources of randomness tends to produce shapes close to the normal, even when the individual pieces are not normal at all.

The z-score: comparing falls across different assets

A useful tool that follows directly from the normal is the z-score, or standardized score. It measures how many standard deviations away from the mean a particular value sits.

Take two stocks:

• Stock X has a mean of 0% and a standard deviation of 1.5%. A day with a -3% return sits two standard deviations below the mean: z = -2.
• Stock Y has a mean of 0.2% and a standard deviation of 2.5%. A day with a -4% return sits at (-4 - 0.2) / 2.5 = -1.68 standard deviations, that is, z = -1.68.

Notice what happens. Although -4% is a larger nominal fall than -3%, relative to each stock's own typical behavior X's decline (z = -2) was more extreme than Y's (z = -1.68).

The z-score lets you compare events across assets with very different volatilities on a single scale.

The tails that fall off a cliff

There is one feature of the normal worth keeping in mind from now on, because it is the seed of the next module.

The curve extends, in theory, from minus infinity to plus infinity. But the probability of values far from the mean falls off extremely fast — exponentially — the further out you go.

This means that, under strict normality, a five or six standard deviation move is considered so astronomically unlikely that it 'should essentially never happen' over the entire life of a market.

Where this shows up in practice

In trading practice, normality is assumed either implicitly or explicitly in many places:

• Risk models.
• Position sizing rules, of the 'never risk more than two standard deviations of your capital on a trade' kind.
• Derivative pricing models.

The reason, again, is that it makes the arithmetic manageable.

The problem is one you will develop in the next module. When the real distribution of returns is not as well behaved as the normal, those risk assumptions turn out to be systematically optimistic: they underestimate both how often and how violently extreme moves can occur.