Down 20%, then up 20%, where are you?

Question

A stock falls 20\% one day and rises 20\% the next. Solve in your head. What is the net change from the start? Aim: under 10 seconds.

Accepted Answer

Multiply the two factors: 0.80 	imes 1.20 = 0.96. The net change is a loss of 4\%. Multiplication commutes, so falling first then rising gives the same result as rising first then falling: order does not matter. Using the identity for a matched move, (1 - r)(1 + r) = 1 - r^2 with r = 0.20: 1 - 0.04 = 0.96, down 4\%. Takeaway. A 20\% move each way, in either order, leaves you down 4\% ( = r^2). The loss comes from the base changing between the two steps, not from which step happens first; equal-percentage round trips always end below where they started.

Down 20%, then up 20%, where are you?

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