Why Convex Risk Measures Are Not Necessarily Coherent

Summary

The document distinguishes convex risk measures from coherent risk measures and corrects a proposed argument that every convex measure must be coherent. Convexity bounds the risk of a weighted combination of positions by the corresponding weighted risks. Coherence includes additional properties, notably subadditivity and positive homogeneity, which together imply convexity when the risk measure satisfies them.

The key logical direction is therefore from coherence to convexity, not the reverse. Convexity alone does not establish subadditivity or positive homogeneity, so those properties cannot be inferred from the convexity inequality. The exchange answer gives the implication using the two coherence axioms but does not provide a counterexample or discuss other risk-measure axioms, such as monotonicity and cash invariance. The displayed question contains a malformed combination expression, so its notation should be read as the usual weighted sum of two positions.

Key ideas

• Coherence requires more than convexity, including subadditivity and positive homogeneity.
• Subadditivity and positive homogeneity together imply the convexity inequality for weighted positions.
• A convex risk measure need not be coherent because convexity alone does not guarantee either additional property.

Tags

# Convex risk measure and a coherent risk measure?


# Convex risk measure and a coherent risk measure?












A coherent risk measure is:

$\rho(\lambda X_1+(1-\lambda X_2))$

How can it be shown that everey convex risk measure is indeed a coherent risk measure?

I assume that it is enough to show that a convex risk measure is coherent by using, subadditivity, positive homogeniety. So we get: $\rho(\lambda X_1+(1-\lambda X_2))=\rho(\lambda X_1)+\rho((1-\lambda)X_2)=\lambda \rho(X_1)+(1-\lambda)\rho(X_2))$ right?

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/22524

We define a convex risk measure as $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2), $$ for $\lambda \in(0,1) $.

A coherent risk measure is subadditive and homogeneous thus for coherent $\rho$ we get: $$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \rho( \lambda X_1) + \rho( (1-\lambda) X_2) $$ by subadditivity and $$ \rho( \lambda X_1) + \rho( (1-\lambda) = \lambda \rho(X_1) + (1-\lambda)\rho(X_2) $$ by homogeneity. Thus a coherent risk measure is convex. The reverse is not true in general.

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