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