Thursday, June 2, 2011

Coherent risk measure

Coherent risk measure

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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]
Sub-additivity
\mathrm{If}\; Z_1,Z_2 \in \mathcal{L} ,\; \mathrm{then}\; \rho(Z_1 + Z_2) \leq \rho(Z_1) + \rho(Z_2)
Indeed, the risk of two portfolios together cannot get any worse than adding the two risks separately: this is the diversification principle.
Positive homogeneity
\mathrm{If}\; \alpha \ge 0 \; \mathrm{and} \; Z \in \mathcal{L} ,\; \mathrm{then} \; \rho(\alpha Z) = \alpha \rho(Z)
Loosely speaking, if you double your portfolio then you double your risk.
Translation invariance
\mathrm{If}\; a \in \mathbb{R} \; \mathrm{and} \; Z \in \mathcal{L} ,\;\mathrm{then}\; \rho(Z + a) = \rho(Z) - a
The value a is just adding cash to your portfolio Z, which acts like an insurance: the risk of Z + a is less than the risk of Z, and the difference is exactly the added cash a. In particular, if a = ρ(Z) then ρ(Z + ρ(Z)) = 0.

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