This paper studies how the monetary policy regime affects the relative importance of nominal exchange rates and inflation rates in shaping the response of real exchange rates to shocks. We document two facts about inflation-targeting countries. First, the current real exchange rate predicts future changes in the nominal exchange rate. Second, the real exchange rate is a poor predictor of future inflation rates. We estimate a medium-size, open-economy DSGE model that accounts quantitatively for these facts as well as other empirical properties of real and nominal exchange rates. The key estimated shocks that drive the dynamics of exchange rates and their covariance with inflation are disturbances to the foreign demand for dollar-denominated bonds.