We study efficiency properties of competitive economies in which banks provide liquidity insurance and interact on secondary asset markets. While all banks are subject to extrinsic risk, a bank's portfolio choice determines whether it is prone to a bank run in one of the extrinsic states. Asset prices determine the value of bank assets and thus how to structure run-proof portfolios. Except for very large sunspot probabilities, equilibria with trivial sunspots exist, where asset prices are state-dependent, bank runs do not occur and the efficient allocation obtains. Interbank asset markets are also a new source of multiplicity of equilibrium. For low sunspot probabilities, there are equilibria in which all banks are run-prone. For high sunspot probabilities, there is no equilibrium with run-prone banks but consumption can be indeterminate. If the sunspot probability is neither high nor low, equilibria may exist in which some banks are run-prone and others are run-proof.