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 $X$ viewed as an element of a linear space $\mathcal{L}$ of measurable functions, defined on an appropriate probability space. According to [1], a function $\rho : \mathcal{L}$ → $\R$ is said to be coherent risk measure for $L$ if it satisfies the following properties.

1 Properties
2 Convex Risk Measures
3 Example: Value at Risk
4 Example: Average Value at Risk
5 Example: Entropic Risk Measure
6 References
7 External links
8 See also

Properties

Monotonicity: $If \ Z_1,Z_2 \in \mathcal{L} \ and \ Z_1 \geq Z_2 ,\ then \ \rho(Z_1) \geq \rho(Z_2)$

Sub-additivity: $If \ Z_1,Z_2 \in \mathcal{L} ,\ then \ \rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)$

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

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

Convex Risk Measures

The notion of coherence was subsequently relaxed by [2] to the notion of convexity, where Sub-additivity and Positive Homogeneity have been replaced by

Convexity: $If \ Z_1,Z_2 \in \mathcal{L} ,\ and \ \lambda \in [0,1] \ then \ \rho(\lambda Z_1 + (1-\lambda) Z_2) \leq \lambda \rho(Z_1) + (1-\lambda) \rho(Z_2)$

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.

Example: Average Value at Risk

The Average Value at Risk (sometimes called Expected Shortfall or Conditional Value-at-Risk) is a coherent risk measure.

Example: Entropic Risk Measure

The Entropic Risk Measure is a convex risk measure which is not coherent. It is related to the exponential utility.

References

[1] Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, David Heath (1999). Coherent Measures of Risk, Mathematical Finance 9 no. 3, 203-228

[2] Föllmer, H., Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics 6(4), 429-447.

External links

A list of important papers on Coherent and Convex Risk Measures