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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?

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  • $\begingroup$ Note that if $e^{x}$ is log normal, then, by definition, $x$ is normal. $\endgroup$ – mark leeds Dec 21 '18 at 13:19
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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.

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