Up 20%, Down 20%

Question

A stock rises 20% one day and falls 20% the next. What is the net move, in percent? Sign matters, and so does the general pattern. State both.

Accepted Answer

Sequential returns compound multiplicatively: 1.20 times 0.80 = 0.96 ;to; mathbf-4%. The trap is adding +20 - 20 = 0. The governing identity is (1+x)(1-x) = 1 - x^2, so a symmetric up-then-down round trip always ends underwater, by the square of the move: ±10% → −1%; ±20% → −4%; ±50% → −25%. Order is irrelevant, because multiplication commutes. Companion reflexes worth hard-wiring: - Same-direction chains: +20% twice is 1.2^2 = 1.44, i.e. +44%, not +40%, the expansion (1+x)^2 = 1 + 2x + x^2 shows the naive sum drops the x^2. - Recovery asymmetry: after -20% you need frac10.8 - 1 = +25% to break even; after -50%, +100%. Drawdowns are geometrically heavier than the rallies that repair them. - Small-move cross term: (1+x)(1+y) approx 1 + x + y + xy. So +5% then +4% is +9.2% (1.05 times 1.04 = 1.092), the xy correction is exactly what an interviewer listens for. This question is a favorite at Flow Traders and IMC precisely because the wrong answer (0%) arrives instantly and confidently. The permanent fix is one habit: convert every percentage change into its multiplier before chaining anything.

Up 20%, Down 20%

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