In this paper, we derive upper and lower bounds on the Range Value-at-Risk of the portfolio loss when we only know its mean and variance, and its feature of unimodality. In a first step, we use some classic results on stochastic ordering to reduce this optimization problem to a parametric one,...

For any random vector X = (X1; ...;Xn) on a given probability space (Ω;F;Pr), one can always instruct comonotonic modi cations of X, which are de ned as random vectors with the same marginals as X but with the comonotonic copula describing their dependency structure. In this short note, we...