We introduce SRISK to measure the systemic risk contribution of a financial firm. SRISK measures the capital shortfall of a firm conditional on a severe market decline, and is a function of its size, leverage and risk. We use the measure to study top US financial institutions in the recent financial crisis. SRISK delivers useful rankings of systemic institutions at various stages of the crisis and identifies Fannie Mae, Freddie Mac, Morgan Stanley, Bear Stearns and Lehman Brothers as top contributors as early as 2005-Q1. Moreover, aggregate SRISK provides early warning signals of distress in indicators of real activity.

In this paper we propose an empirical methodology to measure systemic risk. Building up on Acharya et al. (2010), we think of the systemic risk of a financial institution as its contribution to the total capital shortfall of the financial system that can be expected in a future crisis. We propose a systemic risk measure (SRISK) that captures the expected capital shortage of a firm given its degree of leverage and Marginal Expected Shortfall (MES). MES is the expected loss an equity investor in a financial firm would experience if the overall market declined substantially. To construct MES predictions, we introduce a dynamic model for the market and firm returns. This bivariate process is characterized by time varying volatility and correlation, which in turn are estimated by the familiar TARCH and DCC. The innovation distribution of the system is left unspecified and we rely on flexible methods for inference, allowing for potential tail dependence in the shocks. The model is extrapolated to estimate the equity losses of a firm in a future crisis and consequently the capital shortage that would be experienced depending on the initial leverage. The empirical application on a set of top US financial firms finds that the methodology provides useful rankings of systemically risky firms at various stages of the financial crisis. One year and a half before the Lehman bankruptcy, eight companies out of the SRISK top ten turned out to be troubled institutions. Results also highlight the deterioration of the capitalization of the financial system starting from January 2007 and that as of July 2010 the financial system does not appear fully recovered.

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The purpose of this paper is to discuss empirical risk minimization when the losses are not necessarily bounded and may have a distribution with heavy tails. In such situations, usual empirical averages may fail to provide reliable estimates and empirical risk minimization may provide large excess risk. However, some robust mean estimators proposed in the literature may be used to replace empirical means. In this paper, we investigate empirical risk minimization based on a robust estimate proposed by Catoni. We develop performance bounds based on chaining arguments tailored to Catoni's mean estimator.Comment: Published at http://dx.doi.org/10.1214/15-AOS1350 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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