Why Normal Log Returns Do Not Imply Normal Simple Returns

Summary

The discussion separates price levels, log prices, log returns, and simple returns, which are different quantities with different distributions. If a price ratio has a normally distributed logarithm, exponentiating that log ratio produces a lognormal gross return; subtracting one gives a shifted lognormal simple return, which is generally not normal. The response illustrates this transformation algebraically and addresses the apparent mismatch between normal probability plots.

A second reply emphasizes that the distribution depends on how returns or price dynamics are defined, distinguishing simple returns, differences of log prices, and an additive-noise price equation. These answers clarify a distributional relationship but do not establish that a particular empirical series follows a lognormal model. The question’s wording may also conflate normality of log price levels with normality of log returns; normal log prices alone do not guarantee normally distributed returns. Model choice therefore requires specifying the process and checking its assumptions against the data.

Key ideas

• A normal logarithm of a price ratio implies a lognormal gross simple return.
• Subtracting one from a lognormal gross return yields a shifted lognormal simple return, not generally a normal one.
• Simple returns and log returns are distinct transformations and should not be conflated.
• The assumed price process and return definition determine the implied distribution.
• Normality plots alone do not establish that a chosen model fits the data.

Tags

# Price is Log-normal distributed, yet the return is non-normal


# Price is Log-normal distributed, yet the return is non-normal












I have a price series. The natural logarithm of the price shows good normality. As shown in the standardized normal probability plot below:

However, by viewing the standardized normal probability plot, the returns (or say, change of the price), do not show good normality.

My questions are why the price is log-normal, yet the return can be non-normal? And given the situation, what can be the feasible model to describe the price process?

Thank you in advance.

## Answer by user9403 (score 2)

https://quant.stackexchange.com/a/37256

If I'm understanding you correctly, the log returns are normal, but the simple returns are not. While I'm surprised your plots are that different, simple returns will not be normal even if the log returns are normal; they will instead be (shifted) log-normal. If $S_t =S_0 e^{y\sqrt{t}}$ where $y$ is Gaussian, then the simple return is $\frac{S_{t+1}-S_{t}}{S_{t}}=\frac{S_{t+1}}{S_t}-1=e^{y}-1$, which is also log-normal (though shifted by 1).

## Answer by Dave Harris (score 0)

https://quant.stackexchange.com/a/37257

There is an extensive paper on this. It is at

> Harris, D.E.(2017) The Distribution of Returns. Journal of Mathematical Finance , 7, 769-804.

It will provide you a methodology to calculate the distribution that should be present in section two of the paper. It does depend upon whether you calculate returns as $$r=\frac{p_{t+1}}{p_t},$$ or $$r=\log(p_{t+1})-\log(p_t)$$ or $$p_{t+1}=rp_t+\epsilon_{t+1}.$$ Each one generates a different answer.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.