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 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 monotone and normalized set function generated by , i.e. for any the sigma-algebra then .
Properties 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.
Examples 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 .
See also Risk measure Coherent risk measure Deviation risk measure Spectral risk measure
References