Distortion risk measure

Mathematical definition

The function \rho_g: L^p \to \mathbb{R} associated with the distortion function g: [0,1] \to [0,1] is a distortion risk measure if for any random variable of gains X \in L^p (where L^p is the Lp space) then : \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 where F_{-X} is the cumulative distribution function for -X and \tilde{g} is the dual distortion function \tilde{g}(u) = 1 - g(1-u).

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

Properties

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

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

Examples

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

References

References

1. (2010). "Handbook of Portfolio Construction".
2. Julia L. Wirch. "Distortion Risk Measures: Coherence and Stochastic Dominance".
3. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability.