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Keywords:

  • Value at Risk;
  • distribution functions;
  • quantiles;
  • law invariant risk measures;
  • quasi-convex functions;
  • dual representation

We propose a generalization of the classical notion of the V@Rλ that takes into account not only the probability of the losses, but the balance between such probability and the amount of the loss. This is obtained by defining a new class of law invariant risk measures based on an appropriate family of acceptance sets. The V@Rλ and other known law invariant risk measures turn out to be special cases of our proposal. We further prove the dual representation of Risk Measures on inline image.