Up 10% then down 10%, where do you end up?

Question

A stock rises 10%, then falls 10%. It feels like a wash, but it isn't. Solve in your head. After +10% then −10%, what is the net percentage change? Aim: under 8 seconds.

Accepted Answer

Multiply the two retention factors rather than adding the percentages: 1.10 	imes 0.90 = 0.99. So you end at 99% of where you started, a net change of \mathbf{-1\%}. Mentally, 1.1 	imes 0.9 = (1)^2 - (0.1)^2 = 1 - 0.01 = 0.99 (difference of squares again). The -1\% is the cross term: (1+r)(1-r) = 1 - r^2, and r^2 = 0.01 is your leakage. Takeaway. Equal up-then-down percentages always lose r^2 of the value. Percentage moves compound multiplicatively, never additively.

Up 10% then down 10%, where do you end up?

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