Coherent risk measures are defined by number of axioms (see e.g. Coherent Risk Measure) but a question that does not seem well studied is how to use them. Let's take a coherent risk measure $\rho$ and a position X in the book. The risk is $\rho(X)$ and suppose that you are extend with another asset Y. So, in a sense we could define the value of the asset to be $\rho(X + \{Y\}) - \rho(X)$.
The problem is that if we do that we obtain some results which are inconsistent. For example coherent risk measures such as conditional value-at-risk and entropic value-at-risk depend on a threshold $\alpha$. The $CVaR_{\alpha}$ and $EVaR_{\alpha}$ are decreasing with $\alpha$ just as one might expect.
However, when computing the difference $XVaR_{\alpha}(X + \{Y\}) - XVaR_{\alpha}(X)$ (with $X=C$ or $X=E$) are not decreasing with $\alpha$. This is sensible since a difference of decreasing functions is not necessarily decreasing. To be precise the $CVaR$ is really highly variable from one threshold to the next. The $EVaR$ seems to be decreasing with $\alpha$ and then increasing which is less random but still not what one would expect.
Does this mean that coherent risk measures should not be used that way? If so, what would be the right way to do valuation? Is there a generalization of coherent risk measures that allow to resolve that issue?
