Distortion risk measure


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

The function associated with the distortion function is a distortion risk measure if for any random variable of gains 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:
  1. Law invariant: If the distribution of and are the same then.
  2. Monotone with respect to first order stochastic dominance.
  3. # If is a concave distortion function, then is monotone with respect to second order stochastic dominance.
  4. is a concave distortion function if and only if is a coherent risk measure.

    Examples