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.

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

[edit] Properties

Monotonicity: $\rho(X) \leq \rho(Y)$ whenever $Y \leq X$

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

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

Translational invariance

For $r$ - interest rate and

$\forall a \in \mathbb{R} : \rho(X + r \cdot a) \ = \ \rho(X) - a$

[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.

[edit] References

Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, David Heath (1997). Thinking Coherently, RISK, 10, 68-71.
Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, David Heath (1999) Coherent Measures of Risk, Mathematical Finance 9 no. 3, 203-228