# Tag Info

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 $$... 1 Translation invariance of a risk measure \rho is defined as$$ \rho(X+k) = \rho(X)-k,  where $X$ is a random variable such that $\rho(X)$ exists and $k$ is a constant. The meaning is that if I add an amount $k$ to my risky positions then the risk is reduced by this amount. For VaR we consider the case that $X$ has a continuous distribution and that it ...