A Fall Needs a Bigger Rise

§ 01The calculation that surprises people

Start with 100. Lose 50 percent: you have 50. Now gain 50 percent on that 50: you have 75. You are not back where you started. You are down a quarter. The percentages are identical, the result is not, because the base changed.

This is not a trick. It follows directly from how growth compounds as a product of factors, not a sum of percentages. A loss of 50 percent is a multiplier of 0.5. A gain of 50 percent is a multiplier of 1.5. Apply them in sequence: 0.5 × 1.5 = 0.75. The order makes no difference — 1.5 × 0.5 is the same — but the result is always short of 1.0 whenever the percentage fall and the percentage rise are equal.

The recovery gain required is always larger than the original loss, and the gap widens sharply as losses deepen. A 10 percent fall needs roughly an 11 percent gain to recover. A 25 percent fall needs 33 percent. A 50 percent fall needs 100 percent. A 75 percent fall needs 300 percent. The relationship is not linear; it accelerates because the base keeps shrinking.

The general rule: if something falls by a fraction f, it must subsequently rise by f ÷ (1 − f) to return to its starting level. Lose 50 percent: 0.5 ÷ 0.5 = 1.0, meaning a 100 percent gain is required. Lose 75 percent: 0.75 ÷ 0.25 = 3.0, meaning 300 percent. The denominator shrinks as the loss grows, which is what makes deep losses so structurally difficult to reverse — not psychologically, arithmetically.

Historical manias make the point plainly. Assets that fell 80 or 90 percent during the collapses of the South Sea bubble, the railway boom's worst promotions, or the tulip trade required extraordinary subsequent gains just to reach par — gains that, in many cases, the original holders never lived to see. The arithmetic was always the constraint. The sentiment came later.

The relationship is not linear; it accelerates because the base keeps shrinking.