Percentage Changes and Reversing Them

A stock drops 50%. How much does it need to gain to get back to where it started? Almost everyone's gut answer is 50%, and almost everyone is wrong: the correct answer is 100%. Percentage changes are not symmetric, because a percentage is always taken of whatever value you currently have, and that base shifts every time the value moves. Getting this wrong isn't a rounding error — in the example above it's the difference between still being underwater and actually breaking even.

The rule: percentages compose multiplicatively, not additively

A change of $+r\%$ multiplies your value by $(1 + r/100)$; a change of $-r\%$ multiplies it by $(1 - r/100)$. To find the percentage change $g$ that exactly reverses a move of $r\%$, you need:

$$(1 - r/100)(1 + g/100) = 1 .$$

In plain English: whatever fraction you multiplied by to go down, you need to multiply by its reciprocal to come back up — and because the reciprocal of a number less than 1 is always more than 1 by a larger margin, the up-move required is always bigger than the down-move that caused it.

Solving for $g$:

$$g = \frac{r}{100 - r} \times 100 .$$

Worked example 1: the 50% drop

Plug $r = 50$ into the formula: $g = \dfrac{50}{100-50}\times 100 = \dfrac{50}{50}\times 100 = 100$. A 50% loss requires a 100% gain to recover — you'd need the stock to double just to get back to even, on top of recovering the original drop. Sanity check it directly: start at $100, drop 50% to $50, then gain 100% of $50 (i.e. $50) to get back to $100. Checks out.

Worked example 2: a 20% drawdown and a compounding sequence

A fund loses 20% in a bad month. What gain is needed to recover? $g = \dfrac{20}{100-20}\times 100 = \dfrac{20}{80}\times 100 = 25\%$. Now extend it: suppose the fund then loses another 10% the following month before recovering. After the two losses, $100 becomes $100 \times 0.8 \times 0.9 = 72$, i.e. $72. To get back to $100 from $72 requires a gain of $\frac{100 - 72}{72} \times 100 \approx 38.9\%$ — noticeably more than simply adding the 20% and 10% losses (30%) would suggest, because each loss compounds on an already-shrunken base.

What this means in practice

This asymmetry is why drawdowns are so dangerous for compounding capital: the deeper the hole, the disproportionately larger the climb back out, which is exactly why risk management focuses on avoiding large losses rather than trusting that "it'll average out." It also shows up constantly in interview brainteasers about prices, discounts, and returns — any time a question chains two percentage moves together, resist the instinct to just add or subtract percentages, and instead multiply the factors.