Asymptotics of Multivariate Conditional Risk Measures for Gaussian Risks

This paper is concerned with approximations of marginal moment excess, marginal moment shortfall and conditional tail moment in the framework of multivariate Gaussian system risks. Super-exponential and polynomial convergence speed of those quantities of risk contagions are obtained by its dimension reduction property via the quadratic programming. A sufficient and necessary condition is given to alternate the convergence speed via the risk threshold, correlation matrix and moment order. An interesting application involved in the minimal additional risk capital assigned by risk regulators is given to avoid infinite risk contagions with an illustrating example. Our theoretical results have potential applications in quantitative risk management, including risk allocations and multivariate conditional risk measures of tail covariance, tail skewness with dependence and extremal risk contagions under consideration.