The fixed rate tender is one of the main procedures used by central banks in the implementation of their monetary policies. While academic research has largely dismissed the procedure owing to its tendency to encourage overbidding, central banks such as the ECB and the Bank of England have continued using it. We investigate this apparent conflict by considering an auction-theoretic setting with private information about declining marginal valuations. Since overbidding entails exposure risk, an equilibrium may exist even if bids are costless and the intended volume is pre-announced. In fact, the allotment quota may be strictly below one with certainty. Also with adaptive expectations, overbidding need not escalate. However, the resulting allocation is typically inefficient. Empirical proxies of exposure risk are significant in both euro and sterling operations. Our findings have implications, in particular, for the potential reintroduction of pro rata allotment in the main refinancing operations of the Eurosystem.