The paper argues that the reason world fixed exchange rate regimes usually have finite bands instead of completely fixed exchange rates between realignments is that exchange rate bands, counter to the textbook result, give central banks some monetary independence, even with free international capital mobility. The nature and amount of monetary independence is specified, informally and in a formal model, and quantified with Swedish krona data. Altogether the amount of monetary independence appears sizeable. For instance, an increase in the Swedish krona band from zero to about ±2 percent may reduce krona interest rate's standard deviation by about 1/2.