A dynamic risk form of entropic value at risk

In this work, we show that the entropic value at risk (EVaR), which is not a dynamic risk measure in general, can be a finitely-valued dynamic risk measure for at least one scenario. Specifically, for any distribution where the moment generating function (mgf) exists, then there exists a confidence parameter such that EVaR can be a finitely-valued dynamic risk measure. This separates EVaR from other risk measures, such as the value at risk (VaR), which may not be a finitely-valued dynamic risk for any allowable constant confidence parameter value. This result is not unique to EVaR, as we also show that there exists a confidence parameter such that the average value at risk (AVaR)–which does not require the mgf to exist–is also a finite dynamic risk measure. The main focus of this work is EVaR as it has a stronger stochastic ordering guarantee than AVaR. Future work will examine whether EVaR can be a dynamic risk measure for any other cases.