European Actuarial Journal vol:2 issue:2 pages:319-328
Distorted expectations can be expressed as weighted averages of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in
the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate
expressions for distorted expectations in terms of quantiles.