A 20% drop needs what gain to get back to even?

Question

A stock falls 20\%. Solve in your head. What percentage gain is needed to return to the original price? Aim: under 10 seconds.

Accepted Answer

After a 20\% drop the price is 0.80 of its start. The gain g that restores it satisfies 0.80 	imes (1 + g) = 1 \implies 1 + g = \frac{1}{0.80} = 1.25, so g = 0.25, a 25\% gain, not 20\%. Takeaway. Recovering a loss takes a bigger percentage than the loss itself, because the gain works on a smaller base: down 20\% needs up 25\%, down 50\% needs up 100\%. The recovery gain is 	frac{1}{1 - 	ext{loss}} - 1.

A 20% drop needs what gain to get back to even?

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