Distortion risk measure

In financial mathematics, 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 ${\displaystyle \rho _{g}:L^{p}\to \mathbb {R} }$ associated with the distortion function ${\displaystyle g:[0,1]\to [0,1]}$ is a distortion risk measure if for any random variable of gains ${\displaystyle X\in L^{p}}$ (where ${\displaystyle L^{p}}$ is the L^p space) then

${\displaystyle \rho _{g}(X)=-\int _{0}^{1}F_{-X}^{-1}(p)d{\tilde {g}}(p)=\int _{-\infty }^{0}{\tilde {g}}(F_{-X}(x))dx-\int _{0}^{\infty }g(1-F_{-X}(x))dx}$

where ${\displaystyle F_{-X}}$ is the cumulative distribution function for ${\displaystyle -X}$ and ${\displaystyle {\tilde {g}}}$ is the dual distortion function ${\displaystyle {\tilde {g}}(u)=1-g(1-u)}$.[1]

If ${\displaystyle X\leq 0}$ almost surely then ${\displaystyle \rho _{g}}$ is given by the Choquet integral, i.e. ${\displaystyle \rho _{g}(X)=-\int _{0}^{\infty }g(1-F_{-X}(x))dx.}$[1][2] Equivalently, ${\displaystyle \rho _{g}(X)=\mathbb {E} ^{\mathbb {Q} }[-X]}$[2] such that ${\displaystyle \mathbb {Q} }$ is the probability measure generated by ${\displaystyle g}$, i.e. for any ${\displaystyle A\in {\mathcal {F}}}$ the sigma-algebra then ${\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (A))}$.[3]

Properties

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

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

Examples

• Value at risk is a distortion risk measure with associated distortion function ${\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}$[2][3]
• Conditional value at risk is a distortion risk measure with associated distortion function ${\displaystyle g(x)={\begin{cases}{\frac {x}{1-\alpha }}&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.}$[2][3]
• The negative expectation is a distortion risk measure with associated distortion function ${\displaystyle g(x)=x}$.[1]

See also

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

References

1. Sereda, E. N.; Bronshtein, E. M.; Rachev, S. T.; Fabozzi, F. J.; Sun, W.; Stoyanov, S. V. (2010). "Distortion Risk Measures in Portfolio Optimization". Handbook of Portfolio Construction. p. 649. CiteSeerX 10.1.1.316.1053. doi:10.1007/978-0-387-77439-8_25. ISBN 978-0-387-77438-1.
2. Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (PDF). Archived from the original (PDF) on July 5, 2016. Retrieved March 10, 2012.
3. Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability. 11 (3): 385. doi:10.1007/s11009-008-9089-z. hdl:10016/14071.

• Wu, Xianyi; Xian Zhou (April 7, 2006). "A new characterization of distortion premiums via countable additivity for comonotonic risks". Insurance: Mathematics and Economics. 38 (2): 324–334. doi:10.1016/j.insmatheco.2005.09.002.