4. The log-normal distribution: why prices are not modeled as a normal
Level 2: Distributions and why markets are not "normal"
A +10% followed by a -10% does not bring you back to the starting point. It leaves you at 99.
That small loss of one euro is not a rounding artefact. It is the signal that prices do not add up: they multiply. And that changes the distribution you have to use to model them.
A different question
So far you have been talking about the distribution of returns. There is a different question, just as relevant: what shape does the distribution of prices themselves have, rather than of their changes?
The usual answer in finance is that prices are modeled as log-normal, not normal.
Understanding why is one of the most useful ideas in this level, because it connects directly with how gains and losses actually compound over time.
Two ways of measuring a return
The first step is to distinguish two ways of measuring a return.
The simple return is the one you already know. If a stock goes from 100 to 103 euros, the simple return is 3%.
The logarithmic return, or log return, is computed differently: by taking the natural logarithm of the ratio between the final price and the initial price.
For small moves its numerical value is very close to the simple return. In the example above, the log return would be roughly 2.96%, very close to the simple 3%.
The gap between the two grows as the move gets larger, and as the time horizon gets longer.
Log returns add up
The reason log returns are so useful is that they can be added directly over time, whereas simple returns cannot.
If an asset has a log return of 2% on Monday and 3% on Tuesday, its cumulative log return over those two days is, approximately, the sum: 5%.
With simple returns this does not work nearly as cleanly. It is the case from the opening: 100 rises to 110, and 110 minus 10% is 99, not 100. Simple returns combine multiplicatively, not additively.
This additive property of log returns is what makes it convenient to apply ideas such as the central limit theorem, from module 6, to the evolution of a price over many days.
From a normal in returns to a log-normal in price
Here is where the connection with the normal distribution appears.
Suppose you assume that an asset's daily log returns follow, approximately, a normal distribution. That is a reasonable approximation over short horizons, although you already know from module 3 that it has fatter tails than ideal.
Then the asset's future price — which is the initial price multiplied by the exponential of the sum of those log returns — necessarily follows a log-normal distribution.
The name is no accident. It is called log-normal precisely because it is the distribution whose logarithm follows a normal, even though the price itself does not.
Why price cannot be normal
Why does this distinction matter in practice? Because a normal distribution applied directly to price has a serious defect.
It is symmetric and extends, in theory, down to minus infinity. That means assigning a probability — small, but non-zero — to the stock price being negative.
And that makes no economic sense. A stock price has a very specific structure:
- It is bounded below at zero. At worst the company goes bankrupt and the value goes to zero, but not negative.
- It can rise with no clear upper limit.
The log-normal distribution respects exactly that structure. It is bounded at zero on the left, never produces negative values, and has a long right tail.
The reason for that tail is purely multiplicative: there is far more room to rise — from 100 to 200 is a doubling — than to fall, because from 100 you can only go down as far as 0.
What a log-normal looks like
A numerical example helps to picture it. Imagine a stock starting at 100 euros, with an average annual log return of, say, 8% and a considerable annual volatility.
Simulate thousands of possible one-year price paths under the log-normality assumption. The median final price might be around 108 euros. But the distribution will not be symmetric around that value:
- There will be a non-negligible probability of ending above 200 euros: a strong upward move, possible because multiplication has no ceiling.
- The possibility of ending below 50 euros exists, but it is far more compressed.
- Ending below zero is simply impossible by construction.
That upward skew, with a firm floor at zero, is the log-normal's distinctive signature next to the perfect symmetry of the normal.
Volatility drag
There is a further consequence of working with multiplicative returns, subtle but important.
The arithmetic mean of a series of simple returns is not what determines how much your capital actually grows over the long run. The geometric mean does, and it is always less than or equal to the arithmetic mean whenever there is variability.
This is known informally as volatility drag. Look at it with numbers: if an asset rises 50% one year and falls 50% the next, the arithmetic mean of those two returns is 0%.
But the actual capital has fallen. 100 rises to 150, then drops to 75: a net loss of 25%, not 0%.
The higher the volatility, the wider that gap between what the simple average suggests and what actually happens to compounded capital. One more reason to work with log returns when the time horizon is long.
In summary
When talking about daily returns or short-term relative moves, the normal approximation is a reasonable tool, with its fat tail limitations.
But when talking about the price level itself, especially over long horizons, the model consistent with the way prices actually compound over time is the log-normal, not the normal applied directly.