Multivariate shortfall risk allocation and systemic risk
The ongoing concern about systemic risk since the outburst of the global financial crisis has highlighted the need for risk measures at the level of sets of interconnected financial components, such as portfolios, institutions or members of clearing houses. The two main issues in systemic risk measurement are the computation of an overall reserve level and its allocation to the different components according to their systemic relevance. In this work, we develop a pragmatic approach to systemic risk measurement and allocation based on multivariate shortfall risk measures, where acceptable allocations are first computed and then aggregated in order to minimize costs. We provide existence and uniqueness results for an optimal allocation under mild conditions, as well as a characterization of the optimal allocation which is useful for numerical computations. We also analyze the interplay between the loss function in the multivariate shortfall risk measure and the dependence structure of the random vector, and highlight the relevance as an indicator of systemic risk. Moreover, we provide and test various numerical schemes to assess the risk allocation in high dimensions.