Up 10%, then down 10%, where are you?

Question

A stock rises 10\% one day and falls 10\% the next. Solve in your head. What is the net change from the start? Aim: under 10 seconds.

Accepted Answer

Multiply the two factors: 1.10 	imes 0.90 = 0.99. The net change is a loss of 1\%. It is not zero, even though the percentages match, because the 10\% drop applies to the larger post-gain price. The clean way to see it: an up-then-down move by the same r always gives (1 + r)(1 - r) = 1 - r^2. Here r = 0.10, so 1 - 0.01 = 0.99, down 1\%. Takeaway. Equal up-and-down percentage moves never cancel; they lose r^2 of value. A matched 10\% round trip costs 1\%, a matched 20\% round trip costs 4\%, and the gap grows with the square of the move.

Up 10%, then down 10%, where are you?

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