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 $\rho_g: L^p \to \mathbb$ associated with the distortion function g: ,1\to ,1/math> 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
: $\rho_g(X) = -\int_0^1 F_^(p) d\tilde(p) = \int_^0 \tilde(F_(x))dx - \int_0^ g(1 - F_(x)) dx$
where $F_$ is the cumulative distribution function for $-X$ and $\tilde$ is the dual distortion function $\tilde(u) = 1 - g(1-u)$.

If $X \leq 0$ almost surely then $\rho_g$ is given by the Choquet integral, i.e. $\rho_g(X) = -\int_0^ g(1 - F_(x)) dx.$ Equivalently, \rho_g(X) = \mathbb^ X/math> such that $\mathbb$ is the probability measure generated by $g$, i.e. for any $A \in \mathcal$ the sigma-algebra then $\mathbb(A) = g(\mathbb(A))$.

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 $g(x) = \begin0 & \text0 \leq x < 1-\alpha\\ 1 & \text1-\alpha \leq x \leq 1\end.$
* Conditional value at risk is a distortion risk measure with associated distortion function $g(x) = \begin\frac & \text0 \leq x < 1-\alpha\\ 1 & \text1-\alpha \leq x \leq 1\end.$
* 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

References

*{{cite journal, last=Wu, first=Xianyi, author2=Xian Zhou, title=A new characterization of distortion premiums via countable additivity for comonotonic risks, journal=Insurance: Mathematics and Economics, date=April 7, 2006, volume=38, issue=2, pages=324–334, doi=10.1016/j.insmatheco.2005.09.002

Financial risk modeling