6. The central limit theorem: why averaging tends toward the normal (and why that is not enough)

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

Roll a die. All six faces are equally likely: the chart of that distribution is flat, with no bell shape whatsoever.

Roll 30 dice, keep the average, and repeat the experiment thousands of times. The Gaussian bell appears out of nowhere, without any die having stopped being a die.

That is probably the single most important result in all of applied statistics. It is also one of the easiest to stretch beyond what it actually says.

What the theorem says exactly

The statement of the central limit theorem, put in words, is surprisingly simple.

Take many random variables that are independent of one another and sum or average them. The distribution of that sum or average moves closer and closer to a normal distribution as the number of variables you combine increases.

Two important qualifications to the statement:

• They need not follow a normal distribution, nor even resemble each other in shape.
• Each individual variable does need to have finite variance. That is, it must not be so extreme that no reasonable measure of dispersion can even be defined.

The thirty dice

A classic example helps the idea sink in.

Roll a six-sided die once. The outcome follows a uniform distribution: every face (1, 2, 3, 4, 5, 6) has exactly the same probability, and the chart of that distribution is flat.

Now roll 30 dice and compute the average of the 30 outcomes. That average no longer behaves like a uniform. It tends to cluster around 3.5, the mean of a single die.

Values such as 3.3 or 3.7 are far more frequent than extreme averages like 1.5 or 5.5.

And if you repeat the 'roll 30 dice and average' experiment thousands of times, the histogram of those averages looks remarkably like a Gaussian bell. Even though each individual die is perfectly uniform and nothing like a normal.

Applied to the markets

The application to financial markets is direct, thanks to the additive property of log returns you saw in module 4.

A monthly return is, approximately, the sum of about 21 daily log returns. An annual return is the sum of about 252.

According to the central limit theorem, you would expect that the more days you sum, the closer the distribution of the aggregate return gets to a normal shape. And that holds even if the individual daily return has fat tails and skew like those you saw in module 3.

And indeed, this is what is observed in practice up to a point. The distribution of an index's monthly or annual returns tends to show lower kurtosis (less tail 'fatness') than the distribution of that same index's daily returns. The aggregation process really does file down part of the irregularity.

What the theorem does NOT say

Here comes the part that is easy to misread, and so deserves careful emphasis.

The central limit theorem is about the distribution of the sum or the average of many variables, not about each individual variable on its own.

The fact that the annual return, the sum of 252 days, looks more normal does not imply in any way that any given day's return will stop having fat tails.

Tomorrow is still a single observation, not an average of hundreds of observations. And its individual distribution remains what it is: with heavier tails than a comparable normal.

The central limit theorem improves aggregate long-run behavior. It does not cure, and has no reason to cure, the behavior of each individual short-term event. And that is precisely where a good deal of the tail risk that matters to a trader lives.

Two conditions that markets satisfy only halfway

There are, in addition, two conditions of the theorem that financial markets satisfy only partially. That limits how far you can trust convergence to the normal, even over long horizons.

The first is independence. The theorem assumes the variables you sum are uncorrelated with one another.

But daily financial returns exhibit a well-documented phenomenon called volatility clustering: high-volatility days tend to cluster with other high-volatility days, and quiet days with other quiet days. It is a form of dependence between consecutive observations that the theorem, in its simplest version, does not accommodate.

The second is finite variance. If the true underlying distribution had extremely heavy tails, the very notion of variance could become unstable.

In that case, convergence toward the normal slows down, or simply fails to arrive as cleanly as the theorem promises in its idealized form. And it is not a purely theoretical case: these are tails heavier than what is usually observed in practice, yet present in some specific markets or assets.

The practical consequence, in two directions

The practical consequence of all this is twofold.

On one hand, the central limit theorem legitimately explains why assuming approximate normality over long horizons is a reasonable simplification and not a groundless fantasy. The annual return of a diversified portfolio would be the typical example. There is solid mathematics behind it.

On the other hand, that same theorem provides no alibi whatsoever for ignoring fat tails in decisions that depend on the short term:

• The size of a position for tomorrow.
• The margin needed to survive the next session.
• The level of a stop-loss.

In all those decisions, the relevant unit is the individual day, not the average of a whole year of days.

There is no contradiction

In short, the central limit theorem and the fat tails of module 3 do not contradict each other. They describe phenomena on different time scales and coexist perfectly well.

One explains why the aggregate tends to smooth out. The other reminds you that the individual, especially at the extremes, remains far wilder than the Gaussian bell would like to admit.