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.
The function ρ g : L p → R associated with the distortion function g : [ 0 , 1 ] → [ 0 , 1 ] is a distortion risk measure if for any random variable of gains X ∈ L p (where L p is the Lp space) then
ρ g ( X ) = − ∫ 0 1 F − X − 1 ( p ) d g ~ ( p ) = ∫ − ∞ 0 g ~ ( F − X ( x ) ) d x − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x where F − X is the cumulative distribution function for − X and g ~ is the dual distortion function g ~ ( u ) = 1 − g ( 1 − u ) .
If X ≤ 0 almost surely then ρ g is given by the Choquet integral, i.e. ρ g ( X ) = − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x . Equivalently, ρ g ( X ) = E Q [ − X ] such that Q is the probability measure generated by g , i.e. for any A ∈ F the sigma-algebra then Q ( A ) = g ( P ( A ) ) .
In addition to the properties of general risk measures, distortion risk measures also have:
- Law invariant: If the distribution of X and Y are the same then ρ g ( X ) = ρ g ( Y ) .
- Monotone with respect to first order stochastic dominance.
- If g is a concave distortion function, then ρ g is monotone with respect to second order stochastic dominance.
- g is a concave distortion function if and only if ρ g is a coherent risk measure.
Value at risk is a distortion risk measure with associated distortion function g ( x ) = { 0 if 0 ≤ x < 1 − α 1 if 1 − α ≤ x ≤ 1 . Conditional value at risk is a distortion risk measure with associated distortion function g ( x ) = { x 1 − α if 0 ≤ x < 1 − α 1 if 1 − α ≤ x ≤ 1 . The negative expectation is a distortion risk measure with associated distortion function g ( x ) = x .