A 20% increase followed by a 20% decrease does not return you to the original. That single fact catches almost everyone the first time they encounter it, and it underlies a surprising number of real-world surprises: from investment recovery math to discount stacking to inflation math.
Key Takeaways
- Percentage change formula: (New − Old) / Old × 100. The old value is always in the denominator.
- A percentage increase and decrease of the same size do not cancel out.
- To recover from a percentage loss, you need a larger percentage gain.
- Percentage decreases are bounded by 100% (you can't lose more than everything); increases have no upper bound.
- Stacked percentage changes multiply, they don't add.
The Two Formulas
Percentage Increase = (New − Old) / Old × 100
Percentage Decrease = (Old − New) / Old × 100
These are the same formula with the sign swapped. The general form is:
Percentage Change = (New − Old) / Old × 100
Positive result = increase. Negative result = decrease.
Examples:
- $80 rises to $100. Change = (100 − 80) / 80 × 100 = +25%
- $100 falls to $80. Change = (80 − 100) / 100 × 100 = −20%
Same dollar difference, different percentage. This is the source of the asymmetry: the denominator changes.
Why a 20% Drop Isn't Undone by a 20% Gain
Walk through the math:
- Start at $100
- Apply −20%: $100 × 0.80 = $80
- Apply +20%: $80 × 1.20 = $96
You are $4 short. The second 20% is calculated against $80, not $100. The recovery percentage required:
Recovery Gain Required = Loss / (1 − Loss)
For a 20% loss: Recovery = 0.20 / (1 − 0.20) = 0.20 / 0.80 = 25%
For a 50% loss: Recovery = 0.50 / 0.50 = 100%
A full doubling. This is why crashes are so painful in investing: recovering from a 50% loss requires the asset to double from the new low.
| Loss | Gain Required to Recover |
|---|---|
| 5% | 5.3% |
| 10% | 11.1% |
| 20% | 25% |
| 33% | 49.3% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |
Stacked Percentage Changes
Sequential percentage changes multiply. The combined effect of applying rates r1, r2, ..., rn in sequence:
Final = Original × (1 + r1) × (1 + r2) × ... × (1 + rn)
Use negative values for decreases.
Example: A portfolio starts at $100,000, gains 10%, loses 8%, gains 15%, loses 5%.
Final = 100,000 × 1.10 × 0.92 × 1.15 × 0.95 = $110,665
Total change: +10.67%. Arithmetic sum of the percentages would have been +12%, which is wrong. The compounded sequence is what counts.
Worked Examples
Salary negotiations. A $60,000 salary rises 5% one year, then 3% the next. Final salary = 60,000 × 1.05 × 1.03 = $64,890. Total raise: 8.15%, not 8%.
Price hikes. A subscription priced at $40 increases 10%, then another 5%. New price: 40 × 1.10 × 1.05 = $46.20. Effective increase: 15.5%, not 15%.
Sales recovery. Quarterly sales drop 15%, then recover 18%. Net change: 0.85 × 1.18 = 1.003. About 0.3% above starting point. Almost flat despite the apparent strong recovery.
Currency. A currency drops 20% against the dollar, then recovers 20%. Recovery is only to 96% of original, still 4% down.
Percentage Change vs Percentage Points
These two terms are constantly conflated and they describe different things.
Percentage points = absolute difference between two percentages. Percentage change = relative difference.
Interest rates move from 5% to 6%:
- +1 percentage point
- +20% change
Unemployment rises from 4% to 6%:
- +2 percentage points
- +50% change
A news headline that says "rates jumped 50%" might mean a 50-point relative move on a small base: alarming-sounding but small in absolute terms. Always check which one is meant.
When the Asymmetry Matters Most
Investing. A 50% drawdown requires 100% recovery. This is the math behind why diversification and risk management compound: avoiding the deep losses means avoiding the much larger gains required to recover.
Inflation. A 5% inflation rate followed by 5% deflation does not return prices to the original. The same goods cost 99.75% of the starting price.
Body weight. Losing 15% of body weight and then regaining 15% (in pounds) does not return you to the original weight. The percentage applied to the lower base is smaller in absolute terms.
Engagement metrics. A 30% drop in website traffic followed by a 30% gain leaves traffic at 91% of the original. Recovery looks healthier than it is.
Common Mistakes
Adding percentage changes. Sequential changes multiply, they don't add.
Using the new value as the denominator. Always divide by the original value.
Mixing percentage points and percentage change. They are not interchangeable.
Assuming symmetry. A 30% loss is not undone by a 30% gain. It is undone by a 42.9% gain.
Quoting "X% larger" loosely. "200% larger than $50" can mean $100 (200% of) or $150 ($50 + 200%). Phrasing matters.
Ignoring base size when reporting changes. A 200% increase from 2 cases to 6 cases is mathematically true and practically misleading. Small bases produce dramatic percentages.
Practical Scenarios
Scenario 1: The 10/10 trade. A stock rises 10%, then falls 10%. Net: 0.99, a 1% loss. Two trades of equal magnitude cost you 1%.
Scenario 2: Salary versus inflation. Your salary rises 4%; inflation is 5%. Real change: (1.04 / 1.05) − 1 = −0.95%. Slight loss in purchasing power despite a nominal raise.
Scenario 3: Marketing channel performance. Channel A: conversion rate up 25% (from 0.8% to 1.0%). Channel B: conversion rate up 5% (from 4.0% to 4.2%). Channel A looks more impressive but added 0.2 percentage points; Channel B added 0.2 percentage points. Same absolute gain.
Scenario 4: Tax bracket changes. A tax bracket moves from 22% to 24%. That is a 2 percentage point increase and a 9.1% relative increase. Both numbers are correct; they describe different things.
FAQ
What is the formula for percentage increase? (New − Old) / Old × 100. The result is positive when the new value is larger than the old.
What is the formula for percentage decrease? (Old − New) / Old × 100, or equivalently (New − Old) / Old × 100 with a negative result.
Why don't a percentage increase and decrease of the same size cancel out? Because they are calculated on different bases. The decrease is applied to the original value; the increase is then applied to the smaller post-decrease value.
How much gain do I need to recover from a loss? Gain Required = Loss / (1 − Loss). A 20% loss requires 25% gain; a 50% loss requires 100% gain.
Can a percentage decrease be more than 100%? No. A 100% decrease is total; the value reaches zero. Greater than 100% would imply going negative, which can happen in some contexts (e.g., earnings, where a loss is possible) but is unusual.
Can a percentage increase be more than 100%? Yes. A 200% increase means tripling. A 500% increase means 6x the original. There is no upper bound.
How do stacked percentage changes work? They multiply. Apply each change as (1 + rate), with negative rate for decreases, and multiply all factors together.
Related Tools
The Percentage Change Calculator handles single and multi-step changes. The Percentage Calculator covers the three core forms (part, whole, percentage). For sale-price math specifically, see the Discount Calculator.
Related Articles
Final Thoughts
The asymmetry between percentage increases and decreases is the math behind several of life's most expensive surprises. Investment recoveries take longer than they should; sequential price hikes compound; inflation outruns nominal raises. The fix is mechanical: never add percentage changes, always work from the correct base, and remember that the same number measured forward and measured backward usually means different things.