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Systemic risk measurements are important for the assessment of stability ofcomplex financial systems. Empirical evidence indicates that returns fromvarious financial assets have a heavy-tailed behavior; moreover, such returnsoften exhibit asymptotic tail independence, i.e., extreme values are lesslikely to occur simultaneously. Surprisingly, asymptotic tail independence indimensions larger than two has received limited attention both theoretically,and as well for financial risk modeling. In this paper, we establish the notionof mutual asymptotic tail independence for general $d$-dimensions and compareit with the traditional notion of pairwise asymptotic independence.Furthermore, we consider a financial network model using a bipartite graph ofbanks and assets with portfolios of possibly overlapping heavy-tailed riskyassets exhibiting various asymptotic tail (in)dependence behavior. For suchmodels we provide precise asymptotic expressions for a variety of conditionaltail risk probabilities and associated CoVaR measures for assessing systemicrisk. We also propose an Extremal CoVaR Index (ECI) for capturing the strengthof dependence between risk entities in the network. We focus particularly ontwo well-known dependence structures to capture risk in any general dimension:Gaussian dependence and Marshall-Olkin dependence, both of which exhibitdifferent levels of asymptotic independence.

Systemic risk in financial networks: the effects of asymptotic independence