Systemic risk in financial networks: the effects of asymptotic independence

Systemic risk measurements are important for the assessment of stability of complex financial systems. Empirical evidence indicates that returns from various financial assets have a heavy-tailed behavior; moreover, such returns often exhibit asymptotic tail independence, i.e., extreme values are less likely to occur simultaneously. Surprisingly, asymptotic tail independence in dimensions larger than two has received limited attention both theoretically, and as well for financial risk modeling. In this paper, we establish the notion of mutual asymptotic tail independence for general $d$-dimensions and compare it with the traditional notion of pairwise asymptotic independence. Furthermore, we consider a financial network model using a bipartite graph of banks and assets with portfolios of possibly overlapping heavy-tailed risky assets exhibiting various asymptotic tail (in)dependence behavior. For such models we provide precise asymptotic expressions for a variety of conditional tail risk probabilities and associated CoVaR measures for assessing systemic risk. We also propose an Extremal CoVaR Index (ECI) for capturing the strength of dependence between risk entities in the network. We focus particularly on two well-known dependence structures to capture risk in any general dimension: Gaussian dependence and Marshall-Olkin dependence, both of which exhibit different levels of asymptotic independence.