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Read the amount lost
A 20% loss on starting wealth 100 removes 20 and leaves 80. Recovery requires adding the same amount of 20, but the gain is now measured relative to the smaller starting amount 80.
Use the recovery denominator
The required gain is 20/80 = 25%. Applying a 20% gain instead produces 96. The unequal percentages reflect different bases rather than contradictory arithmetic.
| Operation | Base | Change | Endpoint |
|---|---|---|---|
| 20% loss | 100 | −20 | 80 |
| 20% gain | 80 | +16 | 96 |
| 25% gain | 80 | +20 | 100 |
Write the factor identity
For loss fraction L below 1, recovery rate is 1/(1−L)−1 = L/(1−L). As the loss approaches a complete loss, the required positive-base recovery becomes arbitrarily large. At a zero base, ordinary percentage recovery is undefined.
Separate recovery arithmetic from a forecast
The calculated percentage identifies the gain needed to reach the old wealth level. It does not say the investment will deliver that gain or how long recovery will take. New owner contributions are a different source of capital.