# Why should we use log returns? Log normality

According to this link, there are some reasons we have to use log returns.

But I can not understand the first reason provided in the link:

First, log-normality: if we assume that prices are distributed log normally (which, in practice, may or may not be true for any given price series), then $$\log(1 + r_i)$$ is conveniently normally distributed, because:

$$\tag{1} 1 + r_i = {p_i \over p_j} = e^{\log\left({p_i \over p_j}\right)}$$

I can't understand how equation $$(1)$$ is related to the normal distribution.

Anyone can help?

• Note that if $e^{x}$ is log normal, then, by definition, $x$ is normal. Dec 21 '18 at 13:19
• See here for a general explanation of the use of logarithms in finance and statistics (including log returns). May 16 at 20:16

## 1 Answer

Saying that prices are lognormally distributed here means that $$p_i, p_j$$ are assumed to be lognormally distributed. Then it is easy to verify that $$\frac{p_i}{p_j}$$ is also lognormal. Hence, by the definition of the lognormal distribution, $$\log \left( \frac{p_i}{p_j} \right)= \log \left( 1 + r_i \right)$$ is normally distributed.