Why does it take a 100 percent gain to make back a 50 percent loss?
A gain is measured from the lower base the loss left behind, so the way back is longer than the way down.
A 10 percent fall followed by a 10 percent rise does not end where it started. The fall takes $100 to $90. The rise is 10 percent of $90, which is $9, so the account ends at $99.
The reason is the base. A percentage fall is measured from the old, higher value; the percentage rise that follows is measured from the new, lower one. The same dollars are a larger percentage of a smaller number.
The arithmetic
After a fall of a given percent, the gain needed to get back is 1 divided by (1 minus the fall), minus 1. Written out for three falls:
- A 10 percent fall takes $100 to $90. Getting back to $100 takes $10 on $90: an 11.1 percent gain.
- A 25 percent fall takes $100 to $75. Getting back takes $25 on $75: a 33.3 percent gain.
- A 50 percent fall takes $100 to $50. Getting back takes $50 on $50: a 100 percent gain.
The gap is small for small falls and grows quickly for large ones. A 75 percent fall needs a 300 percent gain to recover; a 90 percent fall needs 900 percent.
Worked example
An account worth $40,000 falls 25 percent, to $30,000. To return to $40,000 it needs $10,000 on a base of $30,000, a 33.3 percent gain. If it gained 25 percent instead, it would reach $37,500, still $2,500 short.
Where this shows up
The same arithmetic applies to one holding or to a whole account, and to time as well as size: a deeper fall usually needs either a larger gain or more years of ordinary gains to make up. The lesson on compounding shows how yearly percentages stack over long stretches, and the same stacking works on the way back.
This is a property of percentages, not a forecast. It says nothing about whether any fall will be recovered, or how long that takes; it says what size of gain the recovery requires.