SV · Foundations
Log returns
Also called: logarithmic returns, continuously compounded returns
A log return is the natural log of the price ratio between two dates; log returns add across time, which makes them the default unit in quant work.
Definition
The log return between two prices is the natural logarithm of their ratio. It is the continuously compounded rate of return that would take the first price to the second.
The simple (arithmetic) return is the percentage change. The two are close for small moves and diverge for large ones.
Intuition
Log returns are additive over time: the log return over a week is the sum of the daily log returns. Simple returns compound multiplicatively instead. Additivity makes statistics easier: means, variances and the square-root-of-time scaling of volatility all apply directly to sums.
Log returns are also symmetric in a useful way. A rise from 100 to 200 and a fall back to 100 have log returns of +0.693 and −0.693, which cancel, while the simple returns of +100% and −50% do not.
How it's calculated
rt = ln(Pt / Pt−1) Rt = Pt / Pt−1 − 1 = ert − 1
rt→t+n = rt+1 + rt+2 + … + rt+n
For small moves r ≈ R; a Taylor expansion gives r ≈ R − R²/2. Over a period, the mean log return is approximately the mean simple return minus half the variance, which is why the compounded growth rate of a volatile asset is lower than its average simple return.
Prices should be adjusted for dividends and splits before computing returns, or the series will show false jumps.
Example: a price that rises from 100 to 110 and falls back to 100 has simple returns of +10% and −9.09%, which average to about +0.45% even though the price ended where it started. The log returns are +0.0953 and −0.0953, which sum to exactly zero.
How to read it
Use log returns for time aggregation, volatility estimation and most statistical models. Use simple returns when aggregating across assets in a portfolio at one point in time, because portfolio simple returns are weighted sums of asset simple returns while log returns are not.
For reporting, convert back: a cumulative log return of 0.10 is a simple return of about 10.5%.
Common mistakes
- Averaging simple returns over time and reporting the result as a growth rate.
- Combining log returns across assets with portfolio weights as if they were simple returns.
- Computing returns from unadjusted prices across splits or dividends.
- Assuming log returns are normally distributed; they are closer than prices but still fat-tailed.
Go deeper
Saqrid Institute lesson: Prices, Returns, and Why We Use Log-Returns. Lesson text is open to read without an account.