Some Results on Measures of Interaction among Risks

It has become a common understanding that financial risk can spread rapidly from one institution to another, and the stressful status of one institution may finally result in a systemic crisis. One popular method to assess and quantify the risk of contagion is employing the co-risk measures and risk contribution measures. It is interesting and important to understand how the underlining dependence structure and magnitude of random risks jointly affect systemic risk measures. In this paper, we mainly focus on the conditional value-at-risk, conditional expected shortfall, the delta conditional value-at-risk, and the delta conditional expected shortfall. Existing studies mainly focus on the situation with two random risks, and this paper makes some contributions by considering the scenario with possibly more than two random risks. By employing the tools of stochastic order, positive dependence concepts and arrangement monotonicity, several results concerning the usual stochastic order, increasing convex order, dispersive order and excess wealth order are presented. Concisely speaking, it is found that for a large enough stress level, a larger random risk tends to lead to a more severe systemic risk. We also performed some Monte Carlo experiments as illustrations for the theoretical findings.