Distortion risk measure explained

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)} = - \int ^{1} _{0} F ^{-1} _{- X} (p) d\tilde{g}(p) = \int ^{0} _{-inf ty} \tilde{g}(F _{-X} (x))dx - \int ^{inf ty} _{0} 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 ^{inf ty} _{0} g( 1 - F _{-X} (x)) dx. ^{[1] [2]} Equivalently, \rho _{g(X)} = E ^{Q} [ - X] ^[2] such that Q is the probability measure generated by g , i.e. for any A \in l{F} the sigma-algebra then Q (A) = g( 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

X and Y are the same then \rho _{g(X)} = \rho _{g(Y)} .

1. 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.g is a concave distortion function if and only if \rho _{g} is a coherent risk measure.^{[1] [2]}

Examples

• Value at risk is a distortion risk measure with associated distortion function

g(x) = \begin{cases} 0 & if 0 \leq x < 1- \alpha\ 1 & if 1- \alpha \leq x \leq 1 \end{cases}. ^{[2] [3]}

• Conditional value at risk is a distortion risk measure with associated distortion function

g(x) = \begin{cases} (x)/(1- \alpha) & if 0 \leq x < 1- \alpha\ 1 & if 1- \alpha \leq x \leq 1 \end{cases}. ^{[2] [3]}

• The negative expectation is a distortion risk measure with associated distortion function

g(x) = x .^[1]

See also

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

References

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

Notes and References

1. Book: Sereda . E. N. . Bronshtein . E. M. . Rachev . S. T. . Fabozzi . F. J. . Sun . W. . Stoyanov . S. V. . Distortion Risk Measures in Portfolio Optimization . 10.1007/978-0-387-77439-8_25 . Handbook of Portfolio Construction . 649 . 2010 . 978-0-387-77438-1 . 10.1.1.316.1053 .
2. Web site: Distortion Risk Measures: Coherence and Stochastic Dominance. Julia L. Wirch. Mary R. Hardy. March 10, 2012. https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf. July 5, 2016. dead.
3. Balbás . A. . Garrido . J. . Mayoral . S. . 10.1007/s11009-008-9089-z . Properties of Distortion Risk Measures . Methodology and Computing in Applied Probability . 11 . 3 . 385 . 2008 . 10016/14071 . 53327887 . free .