Joint Tail Risk in Dependent Default Systems with Systematic Risk and Common Shocks

Music City Center 

We consider two systems of firms that may default, both exposed to a common systematic risk factor in addition to their respective system-specific common shock factor. While much of the existing literature focuses on a single system, we investigate joint risk due to defaults across two interdependent systems, allowing for varying dominance among the risk factors. Specifically, we explore three scenarios: (1) both systems are driven primarily by the shared systematic risk factor, (2) each system is dominated by its own common shock factor, and (3) a scenario where all three factors – the systematic risk and the two common shocks – are asymptotically dependent and jointly contribute to defaults. For each scenario, we analyze the conditional probability of large default losses in one system given large default losses in the other, which serves as a measure of contagion between the systems. The analysis is conducted under the limiting regime where individual default probabilities tend to zero. For all three scenarios, we derive asymptotic equivalences showing that the conditional probability remains strictly positive even as individual default probabilities vanish. Sensitivity analysis highlights that when systematic risk dominates, its heavy-tailedness drives the conditional probability, whereas when common shocks dominance, or when all factors joint dominate, the dependence among risk factors becomes the determining force.

Portfolio default losses

Asymptotic analysis

Multivariate regular variation