Bounds on the value-at-risk for the sum of possibly dependent risks

In this paper, explicit lower and upper bounds on the value-at-risk (VaR) for the sum of possibly dependent risks are derived when only partial information is available about the dependence structure and the individual behaviors. When the marginal distributions are known, a reformulation of a result of Embrechts et al. [Finan. Stoch. 7 (2003) 145-167] makes it possible, under some regularity conditions, to compute explicit bounds for the VaR under various dependence scenarios. In the case where only the means and the variances of the risks are available, explicit bounds are obtained from an optimization over all possible values of the correlation matrix associated with the vector of risks. Analytical and numerical investigations are presented in order to investigate the quality of these bounds.