Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities

In quantitative risk management, it is important and challenging to find sharp bounds for the distribution of the sum of dependent risks with given marginal distributions, but an unspecified dependence structure. These bounds are directly related to the problem of obtaining the worst Value-at-Risk of the total risk. Using the idea of the complete mixability, we provide a new lower bound for any given marginal distributions and give a necessary and sufficient condition for the sharpness of this new bound. For the sum of dependent risks with an identical distribution, which has either a monotone density or a tail-monotone density, the explicit values of the worst Value-at-Risk and the bounds on the distribution of the total risk are obtained. Some examples are given to illustrate the new results.