Compounding with Withdrawals

Compounding applies each period’s return to the balance as it then stands. A withdrawal makes that balance smaller, so every later period’s return is worked on less. The cost of a withdrawal is therefore more than the amount taken: it is that amount and everything it would have gone on to add. The last line of the working puts a figure on that gap.

With a fixed amount there is one dividing line: the gain of the first period. Take out less than that and the balance still ends each period higher than it began, only more slowly. Take out exactly that and it stays level. Take out more and it falls, a little faster each period, because the gain shrinks with the balance while the withdrawal does not. The page names the period in which nothing is left.

A share of the gain behaves differently. It can slow the balance or hold it level, but it cannot make it fall, because in a period without a gain nothing is taken. A fixed amount is taken whatever the period brought.

Everything here follows from one number that you type in: the return for a period, repeated unchanged. No trading result arrives like that. Real results vary from one period to the next and some are losses, and a fixed withdrawal taken after a loss removes a larger share of what is left. The page is arithmetic on an assumption, not a forecast of what an account will do or of what can be withdrawn from one.