Why Log and Simple Returns Are Close for Small Price Moves
Summary
The document clarifies why log returns and simple returns can appear nearly identical in a price series. A log return is the logarithm of the ratio between consecutive prices, while a simple return is the price change divided by the previous price. Expanding the logarithm around a small price change shows that the log return is approximately the simple return, with the difference coming from higher-order terms.
This approximation is most useful when price changes over each observation interval are small, as is often the case at daily or shorter sampling frequencies. The two return definitions are not identical, however, and their difference can matter when moves are larger. The source explains the mathematical relationship but does not establish exactly how the cited software function labels or computes its output, so users should check the function’s documentation when that distinction matters.
Key ideas
- Simple return divides the price change by the previous price.
- Log return is the logarithm of the ratio of consecutive prices.
- For small price moves, a Taylor expansion shows that log returns approximate simple returns.
- The approximation becomes less reliable as the size of the price move increases.
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# Simple Compounding vs Continuous Compounding in return series
# Simple Compounding vs Continuous Compounding in return series
I'm creating a log price series in MATLAB. This is fairly easy to do using standard functions. Given a price series `prices`:
`r = diff(log(prices))` will give you the standard log return series calculated by
$r = \ln(P_t) - \ln(P_{t-1})$
However, I'd like to use the tick2ret function in the Financial Toolbox to keep my tooling consistent. Here's where the confusion starts:
I've always referred to `r` above as the "continually compounded return series". Perhaps this is incorrect. `tick2ret` calculates, by default, the "simple compounding return series". Curious, I made another return series:
`r2 = tick2ret(prices)`
and as it turns out `r` and `r2` are virtually the same aside from rounding. I expected the "simple compounded returns" to be calculated using
$r = \frac{P_t - P_{t-1}}{P_{t-1}}$
However, according to my calculation they are using the log return calculation.
Am I misunderstanding simple and compound returns? Can the log return series be calculated using simple compounding? I think I am getting lost in terminology.
Thank you!
## Answer by RRG (score 1, accepted)
https://quant.stackexchange.com/a/37096
Let $\Delta P = P_t - P_{t-1}$ and expand the continuously compounded return in a Taylor series $$ r = \log\left(\frac{P_t}{P_{t-1}}\right) = \log\left(\frac{P_{t-1}+\Delta P}{P_{t-1}}\right) = \log\left(1+\frac{\Delta P}{P_{t-1}}\right) \approx \frac{\Delta P}{P_{t-1}} - \mathcal{O}\left(\left(\frac{\Delta P}{P_{t-1}}\right)^2\right) = \frac{P_t - P_{t-1}}{P_{t-1}} - \mathcal{O}\left(\left(\frac{\Delta P}{P_{t-1}}\right)^2\right) $$ If $\Delta P/P_{t-1}$ is small, which is typically the case for daily or shorter time frames, the higher order terms can be neglected and the continuously compounded return is approximately equal to the simple return.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.