How To Calculate Percentage Change

The formula has three parts and only one of them causes trouble. The numerator is the subtraction — how much changed in absolute terms. The multiplication by 100 is formatting, turning a ratio back into a percentage. The denominator is the interesting piece: it is always the value you started from, sometimes called the base or the reference value, and getting it wrong is how percentages in ordinary reporting go astray.

That choice of denominator is not arbitrary. A change has to be measured relative to the situation it changed from, because that is the only meaningful baseline for judging size. A $5,000 raise matters enormously on a $40,000 salary and barely registers on a $400,000 one; the arithmetic captures that automatically, provided the original — not the larger, not the newer — goes in the denominator.

The immediate consequence is the asymmetry that trips up nearly everyone: increases and decreases of the same absolute size are different percentages. From 80 to 100 is +25%. From 100 back to 80 is -20%. Nothing is inconsistent — the two moves act on different bases, and the base for the second leg is the endpoint of the first. The general rule is sharp: to undo a rise of x%, you need a fall of x/(1+x)%.

This becomes dramatic at the extremes, and it explains a category of misleading headlines. A portfolio that loses 90% of its value must gain 900% merely to break even, because what remains is one tenth of the original. A 50% drop requires a 100% rise. There is no fixed percentage gain that restores an arbitrary loss, and any statement implying otherwise has quietly switched denominators part way through.

Another consequence is that equal-and-opposite sequential changes always leave you worse off. Rising 50% and then falling 50% on a starting value of 100 gives 75, not 100, because the two moves multiply: 1.5 x 0.5 = 0.75. In general an equal rise and fall of x leaves you down x squared as a fraction. At ±20% the loss is 4%; at ±50% it is 25%. Smaller swings hurt less, quadratically so.

Averages across percentage changes carry the same trap. Reporting that a portfolio gained 50% in year one and lost 50% in year two averages to zero percent, but the actual position is down twenty-five percent. The correct average for compounded series is geometric, not arithmetic — which is precisely why growth rates are summarised with compound annual figures rather than simple averages of annual returns.

Percentage points deserve separate handling because they are a different unit entirely. They measure the arithmetic difference between two rates, whereas percentage change measures movement relative to one of them. An unemployment rate moving from 5.0% to 5.5% has risen half a percentage point; relative to where it began, that is a ten percent increase. Both descriptions are correct, and mixing them is a common way to make a small movement sound enormous.

Finally, there are genuine edge cases with no clean answer. When the original value is zero, the denominator is zero and the percentage change is undefined rather than infinite — going from zero users to fifty users is better described with the absolute figure and a growth multiple than with a percentage. When the original value is negative, the formula mechanically produces a sign that reads backwards: moving from -20 to +10 computes as -150% despite plainly being an improvement. In both cases the absolute change and the narrative are more honest than the ratio.