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Simple vs log returns
Log returns add across time, simple returns add across a portfolio.
The short answer
A simple return is the change in price over the starting price, P₁ / P₀ − 1, and a log return is ln(P₁ / P₀). Log returns add up across time, so use them for models and multi-day sums; simple returns average across holdings, so use them for a portfolio. For a 1% daily move the two differ by only 0.00005, but the gap grows with big moves and long stretches.
Simple returns
Use it when
- You need a portfolio's return.
- You report results to people, who read a return as a change in money.
- You compare assets over one and the same period.
- What it costs
- Easy to read, but chaining them means multiplying growth factors: (1 + r₁)(1 + r₂) − 1. Adding them over time gives the wrong answer.
- Watch out
- Their average over time flatters growth: +50% then −50% averages zero, yet turns 100 into 75.
Log returns
Use it when
- You add returns over many days.
- You build statistical models, which quants usually fit on log returns.
- You want a rise and the fall back to the start to cancel exactly.
- What it costs
- One log per price, and a conversion back at the end for anyone reading the number: e raised to the log return, minus 1.
- Watch out
- They don't average across a portfolio: its log return is not the weighted average of its holdings' log returns. Convert to simple returns first.
At a glance
| Question | Simple returns | Log returns |
|---|---|---|
| Formula | P₁ / P₀ − 1 | ln(P₁ / P₀) |
| Over several days | Multiply: (1 + r₁)(1 + r₂) − 1 | Add: ℓ₁ + ℓ₂ + ℓ₃ |
| Across a portfolio | Weighted average of the holdings | Doesn't average; convert first |
| Up 10%, then down 10% | +10% and −10%, which look like zero but lose 1% | +0.0953 and −0.1054, which add to −0.0101 |
| Range | From −100% upwards, with no ceiling | Any number; a total loss is minus infinity |
| For small daily moves | Almost equal to the log return | The simple return minus about half its square |
How to choose
- For one asset over time, such as multi-day sums, volatility or a model, compute log returns.
- For a portfolio over one period, use simple returns weighted by the holdings.
- For a number someone will read, convert to a simple return at the end.
- Whichever you use, compute it from prices adjusted for splits and dividends, so a 2-for-1 split doesn't look like a crash.
When you need both
Learn it properly
Questions
Why do quants use log returns?
They add across time, so a month's log return is the sum of its daily ones, which keeps models and multi-day maths simple. They are also symmetric: a rise and the fall back to the starting price cancel exactly.
When are simple and log returns about the same?
For small moves. The log return is about the simple return minus half its square, so at 1% they differ by 0.00005, and for most daily stock returns it hardly matters. Over a big move or a long stretch, the gap is large.
How do I convert between them?
Log from simple: ln(1 + r). Simple from log: e raised to the log return, minus 1. Log returns that add to 0.03 over five days mean a simple return of about 3.05%.
What is volatility drag?
The gap between the average return and the rate your money actually compounds at, about half the variance. A fund that makes +50% then −50% averages zero but turns 100 into 75, and of two funds with the same average return, the wilder one ends with less.