Abstract
We develop central limit theory for tail risk forecasts in general location–scale models. We do so for a wide range of risk measures, viz. distortion risk measures (DRMs) and expectiles. Two popular members of the class of DRMs are the Value-at-Risk and the Expected Shortfall. The forecasts we consider are motivated by a Pareto-type tail assumption for the innovations and allow for extrapolation beyond the range of available observations. Simulations reveal adequate coverage of the forecast intervals derived from the limit theory. An empirical application demonstrates that our estimators outperform nonparametric alternatives when forecasting extreme risk in sufficiently large samples.
In this paper, we investigate representation results for set-valued law invariant coherent and convex risk measures, which can be considered as a set-valued extension of the multivariate scalar law invariant coherent and convex risk measures studied in the literature. We further introduce a new class of set-valued risk measures, named set-valued distortion risk measures, which can be considered as a set-valued version of multivariate scalar distortion risk measures introduced in the literature. The relationship between set-valued distortion risk measures and set-valued weighted value at risk is also given.
Initial premium, aggregate claims and distortion risk measures in XL reinsurance with reinstatements
