Why Simple Returns Can Be Approximately Normal
Summary
The document explains why a historical return histogram may compare simple returns with a normal distribution even though geometric Brownian motion implies normally distributed log returns and lognormally distributed prices. For small price changes, the logarithm of one plus the simple return is close to the simple return itself, making the two return measures approximately equal over short intervals.
Under the stated price process, the simple return over an interval is approximately a drift term plus a normally distributed shock, while the log return has an adjusted drift and the same shock. This offers a rationale for testing simple returns against a normal curve as an approximation. The argument depends on small changes and the assumed diffusion model; it does not establish that observed asset returns are normally distributed, especially when moves are large or market behavior departs from the model.
Key ideas
- For small price moves, simple and log returns are approximately equal.
- Under geometric Brownian motion, log returns are normally distributed while prices are lognormally distributed.
- The simple return over a short interval is approximately normally distributed under the model.
- The normality approximation weakens when price changes are large or the assumed model does not fit observed returns.
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# Why does Bloomberg's HRH test the simple returns for normality?
# Why does Bloomberg's HRH test the simple returns for normality?
On a Bloomberg terminal, it is possible to use the HRH (Historical Return Histogram) function on individual assets. It basically generates a histogram of the (simple) returns and overlays them with a theoretical normal distribution, indicating whether the distribution of the (daily, weekly, monthly,...) returns is approximately normally distributed.
With a geometric Brownian motion model, we would assume that the log return is normally distributed and the (simple) return is lognormally distributed. Hence my question: Why does it test for normality and not lognormality?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/19208
For small changes, the log-return $\ln \frac{S_{t_i}}{S_{t_{i-1}}}$ is close to the simple return $\frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}$: \begin{align*} \ln \frac{S_{t_i}}{S_{t_{i-1}}} &= \ln \Big(1+ \frac{S_{t_i}-S_{t_{i-1}}} {S_{t_{i-1}}} \Big)\\ &\approx \frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}}. \end{align*}
Note also that, assuming the SDE \begin{align*} \frac{dS_t}{S_t} = \mu dt + \sigma\, dW_t, \end{align*} then \begin{align*} \frac{S_{t_i}-S_{t_{i-1}}}{S_{t_{i-1}}} \approx \mu \Delta t_i + \sigma \Delta W_{t_i}, \end{align*} and \begin{align*} \ln \frac{S_{t_i}}{S_{t_{i-1}}} = \big(\mu -\frac{1}{2}\sigma^2\big) \Delta t_i + \sigma \Delta W_{t_i}, \end{align*} where $\Delta t_i=t_i-t_{i-1}$, and $\Delta W_{t_i} = W_{t_i}-W_{t_{i-1}}$ is normal.
That is, if the stock price is log-normally distributed, then the log-return is normally distributed, while the simple return is approximately normally distributed.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.