How to Use This Tool
Enter a starting value and a list of percentage changes. Each one is applied to the result of the last, which is what actually happens and is not what adding them up gives you.
Why percentages multiply
A 20% rise means multiplying by 1.20. Two of them is 1.20 × 1.20 = 1.44, so the total is 44%, not 40. The extra 4 points are the growth on the growth — the second rise applies to the amount the first one added as well as the original.
The same reasoning explains the more surprising case. Up 20% then down 20% is 1.20 × 0.80 = 0.96, leaving you 4% down. The fall is a fifth of 120, which is 24, while the rise was only a fifth of 100, which is 20. Equal percentages taken from unequal bases are unequal amounts.
The recovery asymmetry
This is where it stops being an arithmetic curiosity. To undo a loss you need a proportionally larger gain, and the gap widens fast:
- Lose 10% → need 11.1% to get back
- Lose 25% → need 33.3%
- Lose 50% → need 100%
- Lose 90% → need 900%
The reason is the same one throughout: after a loss you are compounding from a smaller base, so each percentage point of recovery is worth fewer units than each point of the original loss cost. This is why avoiding large drawdowns matters more than capturing large gains, and why "it fell 50% but then rose 50%" describes ending 25% down.
The per-period average
The last panel shows the single steady rate that would produce the same final result over the same number of periods — the geometric mean, or in a financial context the compound annual growth rate. It is not the average of the percentages, and the difference matters.
Returns of +50% then −50% average to 0% arithmetically, and the geometric mean is −13.4%, which is what actually happened to the money. Whenever the numbers vary, the arithmetic mean of percentage changes overstates performance, and it overstates it more the more volatile the series is.
Where this shows up
- Stacked discounts. "20% off, then a further 10% off" is 28% off, not 30, because the second discount applies to the already reduced price.
- Compound interest. The entire mechanism, and the reason a small rate over a long period beats intuition.
- Inflation over several years. Adding annual rates understates the cumulative effect.
- Growth targets. "10% a month" is 214% a year, not 120%.
