Measuring the tail risk: An asymptotic approach

Given a vector of non-negative random variables (X,Y) with marginal survival functions F‾ and G‾, this paper obtains asymptotic approximations for the quantityϕX,Y(p):=E[X|F‾(X)G‾(Y)≤p] as p↓0. Interpreting X and Y as two risks, ϕX,Y(p) defines a novel risk measure where the extreme region as p↓0 is attained when at least one of the two risks, rather than only one of them as in common cases, is large. We conduct our study under wide dependence structures between X and Y, which cover both asymptotic independence and asymptotic dependence cases. Some numerical examples are also shown to analyze the sensitivity of our obtained results with respect to dependence parameters.