Percent Change and Its Traps — Does a 50% Drop Then a 50% Rise Break Even?
The percent change formula, why successive changes multiply, why a 50% fall followed by a 50% rise does not break even, and how to handle zero or negative starting values.
'Sales rose 20%', 'the stock fell 30%' — percent change is the most widely used percentage in news and reports. The formula is simple, but when changes follow one another or the starting value is small, the result can defy intuition.
Basic formula
Percent change (%) = (new value − old value) ÷ old value × 100
- Price of 1 kg of coffee beans $28.00 → $32.20: 4.20 ÷ 28.00 × 100 = 15% rise
- Electricity bill $86.00 → $73.10: −12.90 ÷ 86.00 × 100 = 15% fall
A 50% drop followed by a 50% rise
A stock worth $1,000,000 that falls 50% is worth $500,000. If it then rises 50%, it gains 50% of $500,000, which is $250,000, ending at $750,000. It is still 25% short of breaking even.
The reason is that the second change is based on the already changed value. Successive changes are not added; they must be multiplied.
Final multiplier = (1 + r₁) × (1 + r₂) × …
- −50% then +50%: 0.5 × 1.5 = 0.75 → −25% overall
- +10% then −10%: 1.1 × 0.9 = 0.99 → −1% overall
- +10% three years in a row: 1.1³ = 1.331 → +33.1% overall (not 30%)
How much must it rise to recover?
To recover from an x% fall, the value must rise by x ÷ (100 − x) × 100.
- 10% fall → 11.1% rise needed
- 20% fall → 25% rise needed
- 50% fall → 100% rise needed
- 80% fall → 400% rise needed
The bigger the loss, the faster the required recovery grows. That is the mathematical reason behind the investing advice 'avoid big losses'.
Average percent change is not the arithmetic mean
If sales went +20% in the first year and −20% the next, is the average 0%? Since 1.2 × 0.8 = 0.96, sales fell 4% over the two years. The average growth rate over several periods is found with the geometric mean.
Average annual change = (final multiplier)^(1/number of periods) − 1
In the example above, 0.96^(1/2) − 1 ≈ −2.02%, an average decline of about 2% per year.
When the starting value is zero or negative
- If the starting value is 0, you cannot divide by zero, so percent change is undefined. For something that started at zero, such as new revenue, label it 'new' or describe it with the amount of change.
- If the starting value is negative (e.g. an operating loss of −$50 million → −$25 million), a percentage is confusing. This calculator uses the absolute value in the denominator and shows the narrowing loss as +50%, but in a report it is clearer to write the amount, such as 'the deficit narrowed by $25 million'.
The small-base illusion
If subscribers grow from 2 to 6, that is a 200% increase. The number looks dramatic, but the actual change is 4 people. When reading a percent change, check the size of the base and the absolute change as well.
Try it in the calculator
Enter the old and new values in the third row of the Percent tab to see the percent change and the difference together. The fourth row gives 'the value increased by x%', and the fifth row gives 'the value before the change'.