# Coherent risk measure

In the field of financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a risk measure ρ that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance.

Properties
Consider a random outcome X viewed as an element of a linear space $\mathcal{L}$ of measurable functions, defined on an appropriate probability space. A functional $\rho : \mathcal{L}$$\R$ is said to be coherent risk measure for $\mathcal{L}$ if it satisfies the following properties:[1]
Monotonicity
$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} \;\mathrm{and}\; Z_1 \leq Z_2 ,\; \mathrm{then} \; \rho(Z_1) \geq \rho(Z_2)$
That is, if portfolio Z2 always has better values than portfolio Z1 under all scenarios then the risk of Z2 should be less than the risk of Z1.[2]
$\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} ,\; \mathrm{then}\; \rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)$
$\mathrm{If}\; \alpha \ge 0 \; \mathrm{and} \; Z \in \mathcal{L} ,\; \mathrm{then} \; \rho(\alpha Z) = \alpha \rho(Z)$
$\mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z) - a$