Guides
Why a rise and a fall of the same percent do not cancel out
A 50% gain followed by a 50% loss leaves you with 75%, not 100%.
Short answer
Because the second percentage is applied to a different base. Raising 100 by 50% gives 150, and 50% of 150 is 75, so a 50% fall leaves 75. To return to the start after a rise of p%, you need a fall of p / (100 + p) x 100 percent.
The example
| Step | Calculation | Value |
|---|---|---|
| Start | 100 | |
| Rise by 50% | 100 x 1.5 | 150 |
| Fall by 50% | 150 x 0.5 | 75 |
How much to fall to get back
| Rise | Fall needed to return |
|---|---|
| 10% | 9.09% |
| 25% | 20% |
| 50% | 33.3% |
| 100% | 50% |
Why it matters
This is why losing 50% of an investment needs a 100% gain to recover, and why two discounts of 20% and 10% do not make 30%. Always ask which value the percentage applies to. The reverse percentage calculator works this out.
Frequently asked questions
What is the formula for the percent needed to recover?
After a rise of p percent, the fall needed to return to the start is p / (100 + p) x 100. After a fall of p percent, the rise needed is p / (100 - p) x 100.