Percentage Points Are Not Percent

Two true statements about one move

The confusion is not that one description is right and the other wrong. It is that both are correct and they measure different things. A percentage point is a difference on the scale itself: 5% minus 4% is one percentage point, in the same way that five metres minus four metres is one metre. A percent change is a ratio: the increase of one, divided by the starting value of four, is 0.25, so the rate rose by twenty-five percent. The first answers "how far did it move on the scale", the second answers "how much bigger is it than it was". A rate can move a lot on one measure and barely at all on the other, and nothing about the words percent and percentage point tells a reader which one is meant.

The table above makes the divergence visible by running seven pairs. Notice that the point column and the percent column move independently: a one-point rise is worth a hundred percent starting from 1%, fifty percent starting from 2% and twenty-five percent starting from 4%, while a five-point rise is worth twenty-five percent from 20% and only ten percent from 50%. The relative reading is inversely proportional to the base, which means anyone reporting a change from a small base can produce an impressively large percent almost at will. The reverse trick works too: report a change from a large base in relative terms and it sounds trivially small. Neither is a lie. Both are choices about which true number to print.

Where the difference changes the decision

Take an interest rate. A thirty-year loan of $300,000 at 4% costs $1,432.25 a month; at 5% it costs $1,610.46. That is one percentage point on the rate, 12.4% more on the monthly payment, and $64,158.83 more in total interest — a rise of 29.8% in the amount you hand to the lender over the life of the loan. Three legitimate percentages, one underlying move, and each of them supports a different sentence. If a broker tells you the rate rose "only one point" and a commentator says borrowing costs "jumped thirty percent", they are describing the same event and neither is lying to you.

The same split shows up in tax and in conversion. Raise an income tax rate from 20% to 22% and the rate moved two points, the tax bill rose 10%, and take-home pay fell from 80% to 78% of gross — a drop of only 2.5% relative. On a 50,000 income that is 10,000 of tax becoming 11,000 and 40,000 net becoming 39,000. Three headlines, all defensible, all pointing in different directions. On the commercial side the effect works in your favour: lifting a conversion rate from 2.0% to 2.5% is half a percentage point, which sounds negligible, but it is a 25% relative gain — on 100,000 sessions, 2,000 orders become 2,500, and at an average order value of 60 the revenue goes from 120,000 to 150,000. Half a point is worth 30,000. The rule for reading and for writing is the same: always say which measure you are using, and if a source does not, compute the other one before you act on it.

Up then down by the same percent does not return to the start

Multiply, do not add. A rise of x is a multiplication by (1 + x) and a fall of x is a multiplication by (1 − x), so doing both in either order multiplies by (1 + x)(1 − x). Expand it and the middle terms cancel: the product is 1 − x². It is always less than 1 for any non-zero x, so you always end up below where you started, and the shortfall is exactly the square of the move. Rise 10% then fall 10% and you keep 0.99, a loss of 1%. Rise 20% then fall 20% and you keep 0.96, a loss of 4%. Rise 50% then fall 50% and you keep 0.75 — a quarter of the value has vanished from a pair of moves that look like they cancel. The order does not matter, because multiplication commutes, which is another way of saying the loss is structural and not a matter of sequencing.

The practical corollary is the recovery rule, and it is the one people get wrong with money on the table. To undo a fall of d you need a rise of d ÷ (1 − d), which is always larger. After a 20% fall you need 25% to get back. After a 25% fall you need 33.33%. After a 50% fall you need 100% — a doubling — to return to where you were. This is why portfolio drawdowns are quoted so carefully, why a discount and the markup that reverses it are never the same number, and why averaging percentage changes across periods is meaningless unless you multiply the factors and take the root. Add percentages only when they share a base; otherwise multiply.

Percentages of different bases cannot be added

A percentage is meaningless without the number it is a percentage of, and two percentages computed against different bases are not commensurable. If one department cuts costs by 10% and another by 20%, the company has not cut costs by 30%, or by 15%, or by any figure derivable from those two numbers alone — you need the actual budgets. Averaging them unweighted is the most common version of the error, and it silently assumes the two bases are equal. The same trap sits inside every survey summary that reports percentages by subgroup and then quotes an overall figure: unless the subgroups are the same size, the overall percentage is not the average of the subgroup percentages, and the discrepancy is sometimes large enough to reverse the conclusion.

The related wording trap is "of" versus "more than". A hundred and fifty percent of 80 is 1.5 × 80 = 120. A hundred and fifty percent more than 80 is 80 + 1.5 × 80 = 200. The two differ by the original amount, and the gap grows with the multiplier, which makes the phrasing worth reading twice in any contract or quotation. The same applies at the small end: "twice as fast" and "a hundred percent faster" mean the same thing, while "two hundred percent faster" means three times as fast, not twice. When in doubt, restate the claim as a multiplier — that form has no ambiguity at all.

Basis points: the fix finance actually adopted

Markets move rates in small increments and cannot afford a sentence that reads two ways, so they invented a unit that has only one reading. One basis point is one hundredth of a percentage point: 100 basis points is one percentage point, 25 basis points is a quarter of a point. A central bank raising rates by 25 basis points takes 4.00% to 4.25%, unambiguously — and that is a relative rise of 6.25%, which is precisely the number that would have been argued about had anyone written "a quarter of a percent". The unit is additive on the scale by construction, so basis points can be added and subtracted without thought, which is exactly what a trading desk needs.

The size of what a basis point buys is worth internalising. Twenty-five basis points on a million of principal is 2,500 a year: 4.00% pays 40,000 and 4.25% pays 42,500. That is a small number on the rate and a real number on the cash flow, and the mismatch is the whole reason the unit exists. You can borrow the same discipline outside finance for free. Report conversion rates, error rates, market shares and vaccination coverage in points when you mean the difference on the scale, in percent when you mean the relative change, and say which one you mean in the same sentence. It costs three extra words and removes an entire class of argument.

Frequently asked questions

What is the difference between a percentage point and a percent?

A percentage point is the arithmetic difference between two percentages; a percent change is the relative difference. Going from 4% to 5% is a rise of one percentage point, because 5 − 4 = 1, and simultaneously a rise of twenty-five percent, because 1 ÷ 4 = 0.25. Both descriptions are correct and they answer different questions: the first asks how far the rate moved on the scale, the second asks how much larger it is than before. The gap between the two readings widens as the base gets smaller — one point is a hundred percent from a base of 1%, fifty percent from 2%, twenty-five percent from 4% and only two percent from 50%. Whenever you report a change in a rate, name the measure explicitly, because the reader cannot infer it and the two numbers can differ by an order of magnitude.

If a price goes up 50% and then down 50%, is it back where it started?

No. It ends at 75% of the original, a loss of a quarter. A rise of x multiplies by (1 + x) and a fall of x multiplies by (1 − x), so the pair multiplies by (1 + x)(1 − x) = 1 − x². For x = 0.5 that is 1 − 0.25 = 0.75. The shortfall is always the square of the move, so it is small for small moves and brutal for large ones: 10% up and down leaves 0.99, 20% leaves 0.96, 50% leaves 0.75 and 80% leaves 0.36. Order makes no difference, because multiplication commutes. The corollary is the recovery rule: undoing a fall of d needs a rise of d ÷ (1 − d), so a 20% fall needs a 25% rise, a 25% fall needs 33.33%, and a 50% fall needs 100% — a doubling — to get back to level.

What is a basis point?

One hundredth of a percentage point, so 100 basis points equals one percentage point and 25 basis points equals a quarter of a point. The unit exists precisely to kill the ambiguity this article is about: a basis point is always a difference on the scale and never a relative change, so it can be added and subtracted without any risk of being misread. A rate rising 25 basis points from 4.00% becomes 4.25%, which is also a relative rise of 6.25% — and that second number is exactly the one that would have caused an argument if someone had written "a quarter of a percent". The financial stakes are what drove the convention: 25 basis points on a million of principal is 2,500 a year, since 4.00% pays 40,000 and 4.25% pays 42,500. You can adopt the same discipline in any field that reports rates.

Can I add or average percentages?

Only when they share the same base. Two percentages computed against different denominators are not commensurable, so if one department cuts costs 10% and another cuts 20%, the company has not cut 30% or 15% — you cannot answer without the actual budgets. Averaging unweighted silently assumes the bases are equal, which is the single most common form of the error, and it is why an overall survey percentage is generally not the average of the subgroup percentages unless the subgroups happen to be the same size. Sequential percentage changes are worse: they must be multiplied, not added. Three consecutive 10% falls leave 0.9³ = 0.729, a total drop of 27.1%, not 30%. To average a series of percentage changes, multiply the factors and take the nth root — the geometric mean — rather than averaging the percentages themselves.

Does "150% of" mean the same as "150% more than"?

No, and the two differ by exactly the original amount. A hundred and fifty percent of 80 is 1.5 × 80 = 120. A hundred and fifty percent more than 80 is 80 + 1.5 × 80 = 200. The first is a multiplier applied to the base; the second is an increase added on top of the base. The gap widens as the percentage grows, so the phrasing is worth reading twice in any contract, quotation or performance claim. The same distinction lurks in comparisons of speed and size: "twice as fast" and "a hundred percent faster" agree, but "two hundred percent faster" means three times as fast, not twice. The safest habit when writing is to state a multiplier instead — "1.5 times" or "2.5 times" — because a multiplier has exactly one reading and needs no convention to interpret.

How much does a one-point rate rise actually cost on a loan?

Far more than one percent, and the answer depends on which number you look at. Take a thirty-year loan of $300,000. At 4% the monthly payment is $1,432.25; at 5% it is $1,610.46. That is one percentage point on the rate, an extra $178.22 a month, and a 12.4% rise in the payment. Look at total interest over the full term and the picture changes again: $215,608.52 at 4% against $279,767.35 at 5%, an extra $64,158.83, a rise of 29.8%. One move, three legitimate percentages, each supporting a different sentence — and this is exactly why a headline that says rates rose "only one point" and one that says borrowing costs "jumped thirty percent" can describe the same event without either being false. Before signing anything, compute the monthly figure and the lifetime figure, because they answer different questions about the same loan.