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.

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 probability measure 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 

Financial risk modeling