Coherent Risk Measures, Subadditivity, and CVaR Convexity

Summary

The document discusses the properties used to define coherent risk measures and asks how superadditivity relates to convexity and Conditional Value at Risk (CVaR). The response lists monotonicity, translation invariance, positive homogeneity, and subadditivity as the usual coherence conditions. It notes that Value at Risk can fail subadditivity, while CVaR is presented as satisfying it, and connects subadditivity with convexity under the usual assumptions. Replacing subadditivity with superadditivity points toward concavity rather than convexity.

The response also cautions that calling a measure coherent does not settle every question about how risk should be ranked: CVaR rankings may differ from approaches that emphasize the full loss distribution or assign greater weight to extreme losses. Distortion risk measures are mentioned as a framework encompassing both VaR and CVaR. This is a brief conceptual answer, not a derivation; the precise properties depend on definitions and assumptions, and the text does not provide formulas or resolve every interpretation of extreme outcomes.

Key ideas

• Coherent risk measures are commonly defined by monotonicity, translation invariance, positive homogeneity, and subadditivity.
• VaR may fail subadditivity, whereas CVaR is described as subadditive.
• Subadditivity is associated with convexity, while superadditivity points toward concavity under standard assumptions.
• Risk measures can rank distributions differently depending on how they treat the full loss distribution and tail outcomes.
• Distortion risk measures offer a framework that includes both VaR and CVaR.

Tags

# Chorent risk measure with superaddative


# Chorent risk measure with superaddative












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???

## Answer by Magic is in the chain (score 0, accepted)

https://quant.stackexchange.com/a/46602

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.