Normal distribution and financial returns, the big mistake

The distribution of returns on a financial instrument tells us a lot about the risk we could take by investing money in it. In fact, it allows us to be able to evaluate how many times in the past a certain loss has occurred, and therefore to make decisions about our ability to be able to bear it.

If the fluctuations in yields that have occurred in the past were indicative of the future, and if we had a large enough history to observe, no further reasoning would be required. The empirical distribution, that is, the one that has occurred throughout the history of the financial instrument, would be sufficient to give us all the information we need about the investment risk.

Unfortunately, this is not always the case. Very often we have historians “limited” to a few hundred observations, or even less, and therefore it is not certain that in the past a certain loss has occurred a sufficient number of times to have statistically significant information. This is all the more relevant the greater the magnitude of the loss we are considering, that is, the rarer it is, the more we move towards the left tail.

Moreover, it is not even said that the dynamics of prices will remain the same over time, that is, it is not said that something has not changed in the market and / or in the financial instrument itself. In fact, events can occur that change the risk profile of the financial instrument, and therefore not necessarily, despite having a broad history, what has been seen in the past is indicative for the future.

For both of these reasons, it is important to try to trace the empirical distribution of returns to a model, that is to a form, which has a priori validity, beyond the empirical observations that have been recorded. In short, it is a question of moving from the simple observation of data to the use of reason, identifying, if possible, assumptions that may have general validity

over time and then using methodologies capable of reading the past through said assumptions.

By far, the most used model to describe the distribution of returns is the normal statistic, the famous Gaussian “bell”. There are many theoretical reasons that could lead to this choice. In all scientific fields the normal distribution plays an important role, as it is shown (the central limit theorem) that by adding, from a statistical point of view, a very high number of distributions, one tends to a normal distribution. Many quantities are thus regulated by this statistic: the measurement of the length of a table, the distribution of shots around a target, the strokes of bad luck in gambling …

There are so many areas in which this distribution is reflected in nature that it has been called “normal”, that is, found as a rule. It is therefore not surprising that this distribution immediately found use in the financial field, in a somewhat uncritical way, without being questioned … thus causing one of the most sensational measurement errors in history.

In the next article, we will see the implications for investors of this mistake and how Egonon looked for tools to remedy it.

by Riccardo Donati,

Scientific Committee Egonon