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 $$\rho_g: L^p \to \mathbb{R}$$ 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 probability measure 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 $$g(x) = \begin{cases}0 & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$$
 * Conditional value at risk is a distortion risk measure with associated distortion function $$g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$$
 * The negative expectation is a distortion risk measure with associated distortion function $$g(x) = x$$.