Distortion risk measure
# 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 $L^p$ 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:

* Law invariant: If the distribution of $X$ and $Y$ are the same then $\rho _{g}(X)=\rho _{g}(Y)$.
* Monotone with respect to first order stochastic dominance.
   * If $g$ is a concave distortion function, then $\rho _{g}$ is monotone with respect to second order stochastic dominance.
* $g$ 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 $g(x)=x$.
