Percentage Change
Overview
Relative movement between two values is measured against the starting point: Δ% = (new − old) ÷ old × 100. A price moving from 80 to 100 rises (100 − 80) ÷ 80, or 25 percent, while the same price falling from 100 back to 80 declines twenty percent. That asymmetry is the single most misread property of the formula: percentage gains and losses of equal magnitude do not cancel, and a portfolio down fifty percent requires a hundred percent gain to recover. The denominator is undefined when the old value is zero and the sign inverts confusingly when it is negative, which is why financial reporting substitutes absolute change or a note in those cases. Analysts comparing series that move in both directions often prefer the logarithmic return ln(new ÷ old), which is symmetric and additive across periods.
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| $old$ | Initial value | any | The reference value the change is measured from. |
| $new$ | Final value | same as initial | The value after the change. |
| $Δ%$ | Percentage change | % | Signed relative movement; negative denotes a decrease. |
Measuring Movement Against a Starting Point
Percentage change scales the absolute difference between two values by the earlier of the two:
$$\Delta\% = \frac{new - old}{old} \times 100$$
A price rising from 80 to 100 changes by $(100-80)/80 = 25\%$. The choice of denominator is what makes the measure relative: the same 20-unit move against a base of 200 would be only 10%.
Gains and Losses Do Not Cancel
Reversing the example exposes the most consequential property of the formula. Moving back from 100 to 80 is $(80-100)/100 = -20\%$, not −25%. Because the denominator changes with the starting point, a 50% loss requires a 100% gain to recover, and a 90% loss requires a 900% gain. Investment reporting relies on this asymmetry when distinguishing arithmetic from geometric mean returns.
Logarithmic Change
Analysts working with series that move in both directions frequently substitute the continuously compounded return $\ln(new/old)$. It is symmetric — a move up and back sums to zero — and additive across periods, so multi-period returns can be summed rather than chained. For small changes the two measures agree closely; a 5% simple change is a 4.88% log change.
Reading a Reported Figure
A percentage change is meaningless without its base. "Sales grew 300%" describes a quadrupling, whereas "sales are 300% of last year" describes a tripling. Careful writing states the base value alongside the rate.
Derivation & History
The expression is the discrete first difference of a series normalised by its own level, and it is the elementary case of the elasticity concept formalised by Alfred Marshall in Principles of Economics (1890). Rewriting it as $new/old - 1$ shows the relationship to the growth factor used in compound interest: a growth factor of 1.25 and a percentage change of 25% carry identical information. Multi-period compounding follows by multiplying growth factors, which is why an average of percentage changes overstates realised growth whenever the series is volatile.
Worked Examples
Quarterly revenue growth
- Difference: 1,437,500 − 1,250,000 = 187,500
- Divide by the earlier value: 187,500 ÷ 1,250,000 = 0.15
- Scale to a percentage: 0.15 × 100 = 15
Result: 15% increase
Recovering from a drawdown
- Fall: (50 − 100) ÷ 100 × 100 = −50%
- Required recovery: (100 − 50) ÷ 50 × 100 = +100%
Result: A 50% loss needs a 100% gain to break even
Edge Cases & Limitations
Zero starting value: Division by zero leaves the change undefined. Convention is to report the absolute change or mark the cell as not meaningful.
Sign changes: When a value crosses zero — a loss becoming a profit — the formula returns a figure whose sign misleads. Financial statements typically report the absolute movement instead.
Negative denominators: A move from −10 to −5 is an improvement but computes to −50%, inverting the intended reading.
Averaging percentage changes: The arithmetic mean of period changes exceeds the realised compound growth whenever the series varies. Use the geometric mean.
Real-World Applications
Consumer price inflation, gross domestic product growth and unemployment movements are all published as percentage changes on a prior period. Equity markets quote daily moves the same way, and retail analytics compares like-for-like sales against the corresponding period a year earlier to strip out seasonality.