Adjusted Higher-Order Expected Shortfall

How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, called as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual high-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) high-order Expected Shortfall is also given