The question you opened
Down 50 percent, back to even
Your fund drops 50 percent in March.
By what percent does it have to rise to get back to where it started?
100 percent
Halving takes 100 down to 50. Climbing from 50 back to 100 is a gain of 50 on a base of 50, and that is 100 percent.
Put a round number under it and the whole thing opens up. Say the fund held 100 dollars in February. A 50 percent drop removes half of it, so 50 dollars are left. To get back to 100 the fund has to add 50 dollars. That 50 is now being measured against a base of 50, so it is a gain of 100 percent. The fall and the recovery move the same amount of money and carry completely different labels.
The reason is that a percent is always a fraction of something, and here the something changes halfway through. On the way down the base is 100. On the way up the base is 50. A percent with no stated base is an unfinished sentence, and this is the cheapest way to see it. The same asymmetry runs through every loss. Down 20 percent needs 25 percent to recover. Down 80 percent needs 400 percent. Down 90 percent needs 900 percent.
The rule behind all of them is short: if the fund keeps the fraction k of its value, the recovery it needs is 1 divided by k, minus 1. At k equal to one half that gives 1, which reads as 100 percent. This shows up well outside markets. A shop that cuts a price 30 percent has to raise it about 43 percent to get back to the old sticker.
A team that loses 40 percent of its people needs to grow the survivors by about 67 percent to reach the old headcount. A battery that has lost 50 percent of its capacity does not get back to full on a 50 percent improvement. The habit worth keeping is small. Whenever a percent appears, say out loud what it is a percent of. If the answer to that question changes between two sentences, the two percents cannot be added, subtracted, or compared.
Technique: Every percent needs its base