Down 40%, what gain gets you back to even?

Question

A position drops 40\%. Solve in your head. What percentage gain from there just brings it back to where it started? Aim: under 10 seconds.

Accepted Answer

After a 40\% drop the price is 0.60 of the original. To get back to 1.00 you need to multiply by \frac{1}{0.60} = \frac{5}{3} \approx 1.6667, a gain of about 66.7\%. The general rule: a loss of x needs a recovery gain of \dfrac{x}{1 - x}, because you must climb back on the smaller post-loss base. 	ext{down } 20\% 	o +25\%, \quad 	ext{down } 33	frac{1}{3}\% 	o +50\%, \quad 	ext{down } 50\% 	o +100\%. Takeaway. Recovery gains always exceed the loss and blow up as losses deepen: -50\% needs +100\%, -40\% needs about +67\%, and the required gain is \frac{x}{1-x}. This asymmetry is why avoi…

Down 40%, what gain gets you back to even?

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