Distortion risk measure
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where is the cumulative distribution function for and is the dual distortion function .
In addition to the properties of general risk measures, distortion risk measures also have:
- Law invariant: If the distribution of and are the same then .
- Monotone with respect to first order stochastic dominance.
- If is a concave distortion function, then is monotone with respect to second order stochastic dominance.
- is a concave distortion function if and only if is a coherent risk measure.
- Value at risk is a distortion risk measure with associated distortion function 
- Conditional value at risk is a distortion risk measure with associated distortion function 
- The negative expectation is a distortion risk measure with associated distortion function .
- 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. doi:10.1007/978-0-387-77439-8_25. ISBN 978-0-387-77438-1.
- Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.
- 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.
- 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. Retrieved March 14, 2012.