If a security drops 50%, it needs to rise 100% to break even. Calculating other percentage drops

If a security drops 50%, it needs to rise 100% to break even. What about if it drops by a percentage different than 50%?

For the mathematically inclined, follow what Pythagoras might say:

When something drops 20%, you have to perform an “indirect” calculation.  (This is not an official term, hence the quotation marks.). That is, you start by calculating what 20% is.  Then you subtract that 20% from the starting value.

1.) Understand this first:

Here’s an easier approach: Think about the end result.  If you lose 20%, that’s the same as keeping 80%.

Say you have $X.  If you lose 20%, you’re left with 80%, which is 0.80 * X.  Let’s call that value Y.

Y = 0.80 * X

If you start at Y and want to get back to X, you invert that operation.

X = Y / 0.80

Expressing that as multiplication instead of division:

X = Y * (1 / 0.80)

That is, you multiply by the inverse (of what you’re left with, not the drop).

2.) For total understanding:

Above I expressed a percentage drop (in this case, 20%) as the amount remaining (80%).  Mathematically, that is:

Y = 0.80 * X

To express that as the percentage drop instead: Y = (1 – 0.20) * X

Then, to get from Y back to X:

X = Y / (1 – 0.20)

As multiplication:

X = Y * [1 / (1 – 0.20)]

3.) To conclude:

If the percentage drop is P, to recover, you multiply by [1 / (1 – P)].

Straightforward, right?

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