Coherent risk measure

A coherent risk measure is a risk measure $$\rho$$ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

Properties

 * Monotonicity
 * $$\rho(X) \leq \rho(Y)$$ whenever $$X \geq Y$$


 * Sub-additivity
 * $$\rho(X_1 + X_2) \leq \rho(X_1) + \rho(X_2)$$


 * Positive homogeneity
 * $$\forall \lambda \ge 0 : \rho(\lambda X) \ = \ \lambda \rho(X)$$

For $$r$$ - interest rate and
 * Translational invariance
 * $$\forall a \in \mathbb{R} : \rho(X + r \cdot a) \ = \ \rho(X) - a$$

Example: Value at Risk
It is well known that Value at risk is not, in general, a coherent risk measure as it does not respect the sub-additivity property. An immediate consequence is that Value at risk might discourage diversification.

Value at risk is, however, coherent, under the assumption of normally distributed losses.