Log returns
See why +10% then −10% doesn't get you back, and why log returns simply add up.
Up and back down is not a round trip
To get from $110.00 back to $100, the price only needs to fall 9.09%.
| Step | Simple return | Log return |
|---|---|---|
| $100 → $110.00 | +10.00% | +9.53% |
| $110.00 → $100 | -9.09% | -9.53% |
| Round trip adds to | +0.91% | 0.00% |
A 10% gain followed by a 10% loss leaves you at $99.00, not $100. Simple returns are lopsided: up 10.00% but back down only 9.09%. Log returns are symmetric: +9.53% up and -9.53% back, adding to exactly zero.
Log returns add up, simple returns don't
Adding up this year's monthly simple returns gives +28.58%, but $100 actually became $128.30, a +28.30% return. The gap of 0.28% comes from compounding. The log returns add to +24.92%, and e raised to that gives the real total back.
Model log returns and prices stay positive
Both panels use the same random shocks over three years. Added as fixed dollar amounts, 22% of prices fall below zero, which no real price can do. Treated as normally distributed log returns, the price is the exponential of the process, so it never reaches zero.
Try this
- Set the move to 10%: $100 goes to $110, then back down to $99. Recovering only takes a 9.09% fall.
- Try a 50% move. How far from $100 do you end up?
- Draw a new year of returns a few times. The log sum always matches ln(end ÷ start); the simple sum never quite does.
- Raise the volatility in the last part and count how many dollar-step prices go negative.
The math
- log return in period t
- price at the end of period t
- natural logarithm
- prices are the exponential, so always positive
