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Does the 4% rule of thumb assume that you die with zero funds or are left with the principal intact?

Taking out 4% and the effects of inflation are not the only things going on. After the 4% and inflation is taken out, the remaining funds earn some assumed rate. So you take three steps back when you withdraw funds but one step forward with a return on the remaining balance. This process repeats every year. Depending on the rate of return, you probably will lose some years and catch up in other years. But I'd like to know whether this rule of thumb assumes that the balance at death is closer to zero or to the original principal. If you want to leave some money for your children and this rule of thumb assumes a zero balance, you might want to consider some other rule of thumb in that case.
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