# What is the distribution of percentage return in general?

In finance, we often assume that the log-returns $\ln(1+R(t))$ follow a normal distribution.

Since $\ln(1+R(t)) \approx R(t)$ when $R(t)$ is small, \begin{equation*} dS/S \sim \text{Normal}. \end{equation*}

However, I have seen sometimes people assuming that \begin{equation*} \Delta S/S \sim \text{Normal}, \end{equation*}

so I wonder if the result holds in general (e.g. for percentage returns over a long time period, my understanding is that percentage return will follow a Normal distribution only when its value is small, i.e. for $dS/S$ ). In particular, what conclusion can we draw about the distribution of $\Delta S/S$ if we assume that log-returns $d\ln(s)$ follow a Normal distribution?

What is the mapping between log return $r_l$ and arithmetic return $R_A$? It is $r_l=\ln(1+R_A)$ and $R_A=e^{r_l}-1$.
If $r_l$ has the normal distribution then $e^{r_l}$ has the lognormal distribution (by definition) and $e^{r_l}-1=R_A$ has the "lognormal distribution shifted to the left by 1". I don't think there is a name for this distribution, which has support on $-1\le R_A \le\infty$.
• Thank you for your answer. Is it correct that the assumption: $\Delta S/S∼Normal$ is valid only when $\Delta t$ or $\Delta S/S$ is small enough? In other conditions, $\Delta S/S$ will just follow a shifted lognormal distribution. Feb 5 '17 at 10:25