Distortion risk measure

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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[edit]

The function associated with the distortion function is a distortion risk measure if for any random variable of gains (where is the Lp space) then

where is the cumulative distribution function for and is the dual distortion function .[1]

If almost surely then is given by the Choquet integral, i.e. [1][2] Equivalently, [2] such that is the probability measure generated by , i.e. for any the sigma-algebra then .[3]


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

  1. Law invariant: If the distribution of and are the same then .
  2. Monotone with respect to first order stochastic dominance.
    1. If is a concave distortion function, then is monotone with respect to second order stochastic dominance.
  3. is a concave distortion function if and only if is a coherent risk measure.[1][2]


  • Value at risk is a distortion risk measure with associated distortion function [2][3]
  • Conditional value at risk is a distortion risk measure with associated distortion function [2][3]
  • The negative expectation is a distortion risk measure with associated distortion function .[1]

See also[edit]


  1. ^ a b c d 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. 
  2. ^ a b c d e Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012. 
  3. ^ a b c 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.