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 ${\rho }_g:L^p\to ℝ$ 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

${\rho }_g(X)=-\underset{0}{\overset{1}{\int }}{F}_{-X}^{-1}(p)d\tilde{g}(p)=\underset{-\mathrm{\infty }}{\overset{0}{\int }}\tilde{g}(F_{-X}(x))dx-\underset{0}{\overset{\mathrm{\infty }}{\int }}g(1-F_{-X}(x))dx$

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 $X\le 0$ almost surely then ${\rho }_g$ is given by the Choquet integral, i.e. ${\rho }_g(X)=-\underset{0}{\overset{\mathrm{\infty }}{\int }}g(1-F_{-X}(x))dx.$ Equivalently, ${\rho }_g(X)={𝔼}^{\mathbb{Q}}[-X]$ such that $\mathbb{Q}$ is the monotone and normalized set function generated by $g$, i.e. for any $A\in ℱ$ 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.
   1. If $g$ is a concave distortion function, then ${\rho }_g$ is monotone with respect to second order stochastic dominance.
3. $g$ 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 $g(x)=\{\begin{array}{cc}0& \text{if }0\le x<1-\alpha \\ 1& \text{if }1-\alpha \le x\le 1\end{array}.$
• Conditional value at risk is a distortion risk measure with associated distortion function $g(x)=\{\begin{array}{cc}\frac{x}{1-\alpha }& \text{if }0\le x<1-\alpha \\ 1& \text{if }1-\alpha \le x\le 1\end{array}.$
• The negative expectation is a distortion risk measure with associated distortion function $g(x)=x$.

See also

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