How Normal Log Returns Map to Arithmetic Returns

Summary

The document clarifies the distribution of an arithmetic percentage return when log returns are assumed normal. It gives the exact mapping: the log return is the natural logarithm of one plus the arithmetic return, so the arithmetic return is the exponential of the log return minus one. Exponentiating a normally distributed variable produces a lognormal variable; subtracting one shifts its support and distribution accordingly.

Therefore, normal log returns imply arithmetic returns with a shifted lognormal distribution, not a normal distribution in general. The approximation between log and arithmetic returns is useful for small returns, but it does not hold as an exact identity, particularly over longer horizons or for larger moves. The document presents this distributional transformation without estimating parameters, testing market data, or addressing how returns may depart from the assumed normal log-return model.

Key ideas

• The arithmetic return equals the exponential of the log return minus one.
• Normally distributed log returns imply shifted lognormal arithmetic returns.
• Arithmetic and log returns are approximately similar only when returns are small.
• The result follows from the assumed log-return distribution and does not establish that real returns are normal in logs.

Tags

# What is the distribution of percentage return in general?


# 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?

## Answer by Alex C (score 1, accepted)

https://quant.stackexchange.com/a/32249

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 $-1\le R_A \le\infty$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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