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Concept
Distortion risk measure
Summary
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 .
Official source
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