Coherent risk measure

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A coherent risk measure is a risk measure ρ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. Consider a random outcome $Z i$ viewed as an element of a linear space $\Z$ of measurable functions, defined on an appropriate sample space. According to [1], a function $\rho : \Z$ → $\R$ is said to be coherent risk measure for Z if it satisfies the following properties.

1 Properties
2 Example: Value at Risk
3 References
4 See also

[edit] Properties

Monotonicity: $If Z_1,Z_2 \in \Z \ and \ Z_1 \geq Z_2 ,\ then \ \rho(Z_1) \geq \rho(Z_2)$

Sub-additivity: $If Z_1,Z_2 \in \Z ,\ then \ \rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)$

Positive Homogeneity: $If \alpha \ge 0 \ and \ Z \in \Z ,\ then \ \rho(\alpha Z) \ = \ \alpha \rho(Z)$

Translation Invariance: $If \ a \ \in \mathbb{R} \ and \ Z \in \Z ,\ then \ \rho(Z + a) \ = \ \rho(Z) \ + \ a$

[edit] 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.