Abstract
We investigate the probability equivalent level of Value at Risk and nth-order Expected Shortfall (called PELVEn), which can be considered as a variant of the notion of the probability equivalent level of Value at Risk and Expected Shortfall (called PELVE) due to Li and Wang (2022). We study the finiteness, uniqueness and several properties of PELVEn, we calculate PELVEn of some notable distributions, PELVE2 of a random variable having generalized Pareto excess distribution, and we describe the asymptotic behaviour of PELVE2 of regularly varying distributions as the level tends to 0. Some properties of nth-order Expected Shortfall are also investigated. Among others, it turns out that the Gini Shortfall at some level p∈[0,1) corresponding to a (loading) parameter λ⩾0 is the linear combination of the Expected Shortfall at level p and the 2nd-order Expected Shortfall at level p with coefficients 1−2λ and 2λ, respectively.
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