Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

- Study the tail subadditivity for distortion risk measures.
- Obtain sufficient and necessary conditions for the tail subadditivity.
- Propose multivariate tail distortion (MTD) risk measures and give their properties.
- Applications of MTD risk measures in capital allocations for a portfolio of risks.
- Explore the impacts of dependence and extreme tail events on allocations.

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.