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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?

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  • $\begingroup$ What does your first formula say? This is $\rho$ of something .. this is no property not theorem .. just nothing. $\endgroup$
    – Richi Wa
    Commented Dec 30, 2015 at 14:29
  • $\begingroup$ $\rho$ is a risk measure $\endgroup$
    – Elekko
    Commented Dec 30, 2015 at 15:03
  • $\begingroup$ the first formula is still not a statement ... $\endgroup$
    – Richi Wa
    Commented Dec 30, 2015 at 15:40

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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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  • $\begingroup$ So a convex risk measure satisfies 3 axioms: convexity, subadditivity and homogeniety? $\endgroup$
    – Elekko
    Commented Dec 30, 2015 at 15:05
  • $\begingroup$ 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. $\endgroup$
    – Richi Wa
    Commented Dec 30, 2015 at 15:41
  • $\begingroup$ Every coherent measure is convex. The reverse is not true (but maybe in a lot of cases ... still not in general). $\endgroup$
    – Richi Wa
    Commented Dec 30, 2015 at 15:42
  • $\begingroup$ The entropic risk measure (not to be confused with entropic value-at-risk) is an example of a risk measure that is convex but not coherent. $\endgroup$
    – Jamie Ballingall
    Commented Feb 14, 2023 at 19:34

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