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We study a finite-depositor version of the Diamond–Dybvig model of financial intermediation in which the bank and all depositors observe withdrawals as they occur. We derive the constrained efficient allocation of resources in closed form and show that this allocation provides liquidity insurance to depositors. The contractual arrangement that decentralizes this allocation resembles a standard bank deposit with a demandable debt-like structure. When withdrawals are unusually high, however, depositors who withdraw relatively late experience significant losses. This contractual arrangement can be fragile, admitting another equilibrium in which depositors run on the bank by withdrawing funds regardless of their liquidity needs. Copyright Springer-Verlag Berlin Heidelberg 2016