# 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?

• What does your first formula say? This is $\rho$ of something .. this is no property not theorem .. just nothing. – Ric Dec 30 '15 at 14:29
• $\rho$ is a risk measure – Elekko Dec 30 '15 at 15:03
• the first formula is still not a statement ... – Ric Dec 30 '15 at 15:40

## 1 Answer

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.

• So a convex risk measure satisfies 3 axioms: convexity, subadditivity and homogeniety? – Elekko Dec 30 '15 at 15:05
• no, but if you have subadditivity and homogeneity then you automatically have a convex risk measure. This is what "implied" means. Coherent implies convex. Thus convex is more general. – Ric Dec 30 '15 at 15:41
• Every coherent measure is convex. The reverse is not true (but maybe in a lot of cases ... still not in general). – Ric Dec 30 '15 at 15:42