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In some definition of chorent risk measure Superadditive is one of the properties I don't understand Why? With subadditivity and homogeneous CvaR is convex, but if we assume another definition for chorence risk measure with superaddative , what happen to convexity?

And if we get Large outcome, then cvar is superaddative???

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You know the concept of coherent risk measure was introduced/developed by Atrzner et al and Delbaen. It defines the list of properties that such measure satisfy -e.g., monotonicity, translation invariance, positive homogeneity, and sub-additivity. So the VaR does not satisfy the sub-additivity, but CVaR does.

Whilst the Coherent risk measure is a nice name, the desired properties of a good measure are heavily debated - for example, CVaR is not as perfect as the label 'coherent risk measure' might imply as it can produce inconsistent ranking of risk if viewed through another lens that consider the entire distribution. So you can see there are many different views - e.g.,consider the whole distribution, assign large weights to extreme losses.

One measure that would help you reconcile the sub-additivity vs super-additivity is the distortion risk measure (pls google its properties) which both VaR and CVaR fall under.

Re-convexity, as you mentioned sub-additivity roughly means convexity, so super-additivity would mean concavity. The CVaR meaning (and properties) don't not change when one changes the properties of a good risk measure - i.e., CVaR is what it is.

Hope the helps.

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