We propose a new approach to assess systemic financial stability of a banking system using standard tools from modern risk management in combination with a network model of interbank loans. We apply our model to a unique data set of all Austrian banks. We find that correlation in banks' asset portfolios dominates contagion as the main source of systemic risk. Contagion is rare but can nonetheless wipe out a major part of the banking system. Low bankruptcy costs and an efficient crisis resolution policy are crucial to limit the systemwide impact of contagious default events. We compute the “value at risk” for a lender of last resort and find that the funds necessary to prevent contagion are surprisingly small.