Comparison, in Risk & Portfolio Construction
Arithmetic vs Logarithmic Returns
Both express the change in value of a position, and both are correct, but they have mirror-image additivity properties that decide where each belongs. The arithmetic return is the percentage change. The log return is the natural log of the price ratio. Arithmetic returns sum across a portfolio at a point in time, weighted by allocation; log returns sum across time for a single asset, because compounding becomes addition under the logarithm. Picking the wrong one introduces subtle errors that grow with horizon and magnitude. This matrix lays out when to reach for each.
6 criteria. Published May 26, 2026. By AI Fin Hub Research.
The percentage change in value: end minus start, divided by start. The intuitive, directly interpretable definition used in most reporting.
Pros
- Aggregates across assets: a portfolio return is the weighted sum of its holdings' arithmetic returns
- Directly interpretable as the actual percentage gain or loss a position experienced
- Matches how investors and clients think and how performance is usually reported
- No transformation needed, so it is the natural unit for cross-sectional comparison at a date
Cons
- Does not add across time: multi-period returns must be chained multiplicatively, not summed
- Asymmetric around zero in a way that distorts averages, since a minus 50 percent then plus 50 percent is not flat
- Less convenient for statistical modeling that assumes additive, normally distributed increments
- Volatility does not scale by the simple square-root-of-time rule as cleanly as for log returns
Portfolio aggregation across positions, performance reporting, and any cross-sectional comparison at a single point in time
The natural log of the price ratio between two dates. Continuously compounded returns that add over time and are symmetric around zero.
Pros
- Adds across time: the multi-period log return is the sum of single-period log returns
- Symmetric around zero, so a gain and an equal-magnitude loss in log terms cancel exactly
- Convenient for modeling, since many processes assume additive, roughly normal increments
- Volatility scales naturally with the square root of time under common assumptions
Cons
- Does not aggregate across assets: the log return of a portfolio is not the weighted sum of component log returns
- Less directly interpretable, since a log return is not the actual percentage gained
- Requires conversion back to arithmetic terms for reporting to non-technical audiences
- Undefined for non-positive prices, which is rarely an issue for equities but matters for some series
Time-series modeling, compounding over horizons, volatility scaling, and any analysis that wants additive increments
Decision table
See the tradeoffs side by side
| Criterion | Arithmetic (Simple) Returns | Logarithmic Returns |
|---|---|---|
| Adds across time | No, chain multiplicatively | Yes |
| Adds across assets | Yes, weighted sum | No |
| Symmetry around zero | Asymmetric | Symmetric |
| Interpretability | Direct percentage | Needs conversion |
| Modeling convenience | Lower | Higher, additive increments |
| Volatility time-scaling | Less clean | Square root of time |
Verdict
There is no universally right choice; there is a right choice per task, set by which additivity you need. When you are aggregating across positions in a portfolio at a single date, use arithmetic returns, because they sum with the portfolio weights and a portfolio's simple return really is the weighted average of its holdings' simple returns. When you are working across time, compounding over periods, modeling a return series, or scaling volatility, use log returns, because they turn compounding into addition and play nicely with the statistical assumptions most models make. For small daily returns the two are numerically close, so the distinction is academic; over long horizons or large moves it is material, and mixing them, for example summing arithmetic returns over time, is a quiet but real error.
Run the numbers next
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Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.
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Paste daily returns; get Sharpe, Sortino, Calmar, and Omega side-by-side with a recommendation on which ratio fits your distribution.
Questions people ask next
Why don't arithmetic returns add over time?
Returns compound multiplicatively: a 10 percent gain followed by a 10 percent loss leaves you at 0.99 of the start, not 1.0, so the simple returns do not sum to zero. To get a multi-period simple return you multiply the gross returns and subtract one. Log returns sidestep this because the logarithm turns the product of gross returns into a sum, which is exactly why they are the natural unit for anything that aggregates across time.
When are the two effectively interchangeable?
For small returns the log return closely approximates the arithmetic return, because the natural log of one plus a small number is nearly that number. So for daily equity returns the difference is usually in the third decimal place and rarely changes a conclusion. The gap widens as returns get large or as you compound over many periods, which is when the choice of definition starts to matter materially.
Which should I use to compute a Sharpe ratio?
Either can work, but be consistent and aware of the convention. Many practitioners compute log returns, take their mean and standard deviation, and annualize using the square-root-of-time scaling that log returns support cleanly. Whichever you pick, use the same definition for the numerator and denominator and for the risk-free rate, and state which convention you used, since mixing arithmetic and log returns in one calculation produces a number that does not mean what it appears to.
Sources
- Returns: Arithmetic and Logarithmic , Campbell, Lo, MacKinlay, The Econometrics of Financial Markets (1997)
- Geometric or Arithmetic Mean: A Reconsideration , Jacquier, Kane, Marcus, Financial Analysts Journal (2003)
Keep the topic connected
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- Learn Sharpe Ratio
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- Learn Monte Carlo Simulation
Monte Carlo simulation in trading: when it's the right tool, when it's overkill, and the seed-discipline gotcha that ruins most published examples.
- Learn Drawdown
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