Finance:Distortion risk measure

In financial mathematics and economics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.

Mathematical definition

The function [math]\displaystyle{ \rho_g: L^p \to \mathbb{R} }[/math] associated with the distortion function [math]\displaystyle{ g: [0,1] \to [0,1] }[/math] is a distortion risk measure if for any random variable of gains [math]\displaystyle{ X \in L^p }[/math] (where [math]\displaystyle{ L^p }[/math] is the L^p space) then

[math]\displaystyle{ \rho_g(X) = -\int_0^1 F_{-X}^{-1}(p) d\tilde{g}(p) = \int_{-\infty}^0 \tilde{g}(F_{-X}(x))dx - \int_0^{\infty} g(1 - F_{-X}(x)) dx }[/math]

where [math]\displaystyle{ F_{-X} }[/math] is the cumulative distribution function for [math]\displaystyle{ -X }[/math] and [math]\displaystyle{ \tilde{g} }[/math] is the dual distortion function [math]\displaystyle{ \tilde{g}(u) = 1 - g(1-u) }[/math].

If [math]\displaystyle{ X \leq 0 }[/math] almost surely then [math]\displaystyle{ \rho_g }[/math] is given by the Choquet integral, i.e. [math]\displaystyle{ \rho_g(X) = -\int_0^{\infty} g(1 - F_{-X}(x)) dx. }[/math] Equivalently, [math]\displaystyle{ \rho_g(X) = \mathbb{E}^{\mathbb{Q}}[-X] }[/math] such that [math]\displaystyle{ \mathbb{Q} }[/math] is the probability measure generated by [math]\displaystyle{ g }[/math], i.e. for any [math]\displaystyle{ A \in \mathcal{F} }[/math] the sigma-algebra then [math]\displaystyle{ \mathbb{Q}(A) = g(\mathbb{P}(A)) }[/math].

Properties

In addition to the properties of general risk measures, distortion risk measures also have:

1. Law invariant: If the distribution of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are the same then [math]\displaystyle{ \rho_g(X) = \rho_g(Y) }[/math].
2. Monotone with respect to first order stochastic dominance.
   1. If [math]\displaystyle{ g }[/math] is a concave distortion function, then [math]\displaystyle{ \rho_g }[/math] is monotone with respect to second order stochastic dominance.
3. [math]\displaystyle{ g }[/math] is a concave distortion function if and only if [math]\displaystyle{ \rho_g }[/math] is a coherent risk measure.

Examples

• Value at risk is a distortion risk measure with associated distortion function [math]\displaystyle{ g(x) = \begin{cases}0 & \text{if }0 \leq x \lt 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}. }[/math]
• Conditional value at risk is a distortion risk measure with associated distortion function [math]\displaystyle{ g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x \lt 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}. }[/math]
• The negative expectation is a distortion risk measure with associated distortion function [math]\displaystyle{ g(x) = x }[/math].

See also

• Risk measure
• Coherent risk measure
• Deviation risk measure
• Spectral risk measure

References

• Wu, Xianyi; Xian Zhou (April 7, 2006). "A new characterization of distortion premiums via countable additivity for comonotonic risks". Insurance: Mathematics and Economics 38 (2): 324–334. doi:10.1016/j.insmatheco.2005.09.002.