As defined in Entropic Value-at-Risk: A New Coherent Risk Measure (reference) by A.Ahmadi-Javid, the entropic value-at-risk (EVaR) of $x\in L_{M^{+}}$ with confidence level $1-\alpha$ is: $$EVaR_{1-\alpha}(X) := \inf_{z>0}\left\{z^{-1}\ln\left(\frac{M_{X}}{\alpha}\right)\right\}$$ where $M_X$ is the moment generating function.
I was wondering can EVaR fit in the framework of distortion risk measures? And in that case how to calculate the distortion function for EVaR?
You may find the answer based on the Kusuoka representation of the EV@R given in the following papers:
Freddy Delbaen, Remark on the Paper "Entropic Value-at-Risk: A New Coherent Risk Measure" by Amir Ahmadi-Javid. https://arxiv.org/abs/1504.00640
Amir Ahmadi-JavidEmail authorAlois Pichler. An analytical study of norms and Banach spaces induced by the entropic value-at-risk. https://link.springer.com/article/10.1007/s11579-017-0197-9
See the equivalent form of this representation in Remark 4.16 of the second paper.