Why a 30% Drop Needs a 43% Rise to Break Even

Down 30, up 30, and still short

Start with 100. Take 30% off and you have 70. Now put 30% back on: 70 × 1.3 = 91. You applied the same number twice, once in each direction, and ended up 9% poorer than you started.

This is not a rounding curiosity. It is the reason a portfolio that halved needs to double, the reason a 25% price rise does not cut your money's value by 25%, and the reason removing sales tax is a division and not a subtraction. One arithmetic fact underneath all three.

The base moves under your feet

A percentage is never a quantity on its own. It is a quantity divided by whatever you happened to be standing on when you measured it. The first 30% was 30% of 100, so it was 30. The second 30% was 30% of 70, so it was 21. Nine points of the original went missing between those two sentences, and that missing nine is the whole story.

So percentage changes do not add. They multiply: 0.7 × 1.3 = 0.91. Chain any sequence of moves and only the product matters, which also means the order of the moves does not — a 30% loss followed by a 30% gain lands exactly where a 30% gain followed by a 30% loss does.

To get back to even after losing a fraction d of your money, you need a gain of d ÷ (1 − d). Lose 30% and 0.3 ÷ 0.7 = 42.9%. The formula is small, but the shape it describes is not gentle.

Up to about 20% the intuition is nearly harmless — a 20% loss needs 25%, and most people would shrug at five points. Past 50% the curve leaves the room. A 70% loss needs 233%, and a 90% loss needs 900%. The asymmetry is not a matter of degree; it changes what recovery even means.

Where it costs money: averaging down

Say you hold 100 shares bought at 100 each. Cost 10,000, average price 100. The stock is now 70, so you are down 3,000 — a 30% loss, and you need 42.9% to get back to your entry.

The usual response is to buy more at 70 to pull the average down. That works, and the arithmetic of how much it works is worth seeing before you do it. To move your average from 100 to a target, buying at 70, the shares you need are qty × (avg − target) ÷ (target − 70).

Shaving the first five points off your average costs 1,400. Getting to 75 — still above the current price — costs 35,000, three and a half times the original position. And no amount of buying at 70 will ever bring the average to 70 or below; the denominator target − 70 goes to zero and the required quantity runs off to infinity.

That is the same asymmetry wearing different clothes. Averaging down does not repair a loss, it enlarges the position that has to recover. Whether that is a good trade depends entirely on what you think the asset is worth, which is a judgement no calculator makes for you.

Inflation eats less than the headline says

US consumer prices rose 24.4% between 2020 and 2025. Most people read that as a quarter of their savings gone. It is not: the loss of purchasing power is 1 − 1 ÷ 1.244 = 19.6%. Prices rose by 24.4% of the old level; your money lost 19.6% of its old value. Two different bases, two different numbers, both correct.

The gap widens as you look further back, because ratios compound while percentages of the original do not.

Over thirty years prices rather more than doubled — up 111.3% — while cash under the mattress lost 52.7% of what it could buy. Notice you cannot lose more than 100% of your purchasing power no matter how high prices go, while the price rise has no ceiling at all. The two columns are not two views of one number; they answer different questions.

The last column is the one worth carrying around. A 111.3% total rise sounds violent and a 2.52% annual rise sounds like nothing, and they are the same three decades. Annual rates are geometric means, so they do not add either: thirty years at 2.52% is 1.0252 to the thirtieth power, not 30 × 2.52%.

Removing a tax is a division

A receipt says 110 including 10% sales tax. The net price is not 110 × 0.9 = 99. It is 110 ÷ 1.1 = 100, and the tax is 10. That one-unit error is the same base problem again: the 10% was charged on 100, not on 110, so you cannot take it off 110.

The rule generalises. To add a rate you multiply by 1 + r. To strip it you divide by 1 + r. Multiplying by 1 − r is a different operation that answers a different question, and the two only agree when r is zero.

At a 10% rate the wrong method understates the net price by 1%. At a 20% VAT rate it understates it by 4%: strip 120 correctly and the base is 100, strip it by multiplying by 0.8 and you get 96. For anyone issuing invoices at volume, that is the difference between books that balance and books that do not.

One rule to carry

Percentages are ratios, and ratios multiply. Whenever two or more percentage moves land on the same quantity, convert each into a multiplier — down 30% is × 0.7, up 30% is × 1.3, plus 10% tax is × 1.1 — then multiply the chain and look at the product.

Doing it that way makes the surprises disappear before they cost anything. It tells you that 0.7 × 1.3 = 0.91, that stripping tax means dividing by 1.1, and that the answer never depends on the order you apply the moves in. Everything above is that one habit, applied to four different bills.