Simple and log returns
Why do a gain and a loss of similar percentage size not cancel out?
In plain words
A simple return is the percentage change in price from one point to the next. A log return is the natural logarithm of the ratio of the two prices, which adds up exactly across several periods. Simple returns compound by multiplication instead.
Why it matters
Simple returns do not add. A large down day and a large up day of similar size look like they cancel, but because a loss shrinks the base the next gain is calculated on, the combination is negative. Averaging simple returns across days or weeks therefore misstates what actually happened to a position.
In a real week: ETH, week of 6 October 2025
In the week of 6 October 2025, ETH's worst day, 10 October, fell 12.33%, and its best day, 12 October, rose 10.82%. Because the fall came first, the later gain was applied to a base already reduced by it, so the two did not offset. Across the full week ETH opened at 4514.32 and closed at 4152.29, a simple return of -8.02%.
A common mistake
Summing percentage changes across periods. Two days that look like they offset can still leave the market lower, because each day's return applies to a different price base.