Why a 20% fall needs a 25% rise to recover
Equal-looking increases and decreases use different starting amounts. Recovering a loss requires working from the reduced value.
Percentage change →Follow both amounts
Start at 100. A 20% decrease removes 20, leaving 80. A subsequent 20% increase adds only 16 because it is calculated from 80, leaving 96. Adding the signed percentages, −20% and +20%, hides this change in the base. The sequence is multiplication by 0.8 followed by 1.2, whose product is 0.96.
Calculate the required recovery
To return from 80 to 100, enter 80 as the original and 100 as the new value. The increase is (100−80)/80 × 100% = 25%. More generally, a decline of d expressed as a decimal needs a rise of d/(1−d), provided 0 ≤ d < 1. A 50% decline therefore needs a 100% increase to recover.
Choose what recovery means
This comparison describes values only. It does not include withdrawals, additional purchases, fees, inflation or the time taken. Recovering the original price is not necessarily recovering purchasing power or the entire invested cost. A 100% loss leaves a zero base, for which percentage recovery is undefined; do not report it as merely another finite percentage. Use ROI or inflation tools for those different questions. The order of equal multiplicative changes does not rescue the lost value: 1.2 followed by 0.8 also ends at 0.96.
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