Interconnections among financial institutions create potential channels for contagion and amplification of shocks to the financial system. We estimate the extent to which interconnections increase expected losses, with minimal information about network topology, under a wide range of shock distributions. Expected losses from network effects are small without substantial heterogeneity in bank sizes and a high degree of reliance on interbank funding. They are also small unless shocks are magnified by some mechanism beyond simple spillover effects; these include bankruptcy costs, fire sales, and mark-to-market revaluations of assets. We illustrate the results with data on the European banking system.