We derive bounds on the distribution function, therefore also on the Value-at-Risk, of \varphi(\mathbf X) where \varphi is an aggregation function and \mathbf X = (X_1, ... \dots ,X_d) is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as  distributions of partial minima and maxima of \mathbf X are known . In order to include this information in the computation of Value-at-Risk bounds, we establish a reduction principle that relates this problem to an optimization problem over a standard Fr\'echet class. Second, we assume that the copula of \mathbf X is known only on a subset of its domain, and finally we consider the case where the copula of \mathbf X lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fr\'echet-Hoeffding bounds on copulas so as to include the additional information . Then, we relate the improved Fr\'echet-Hoeffding bounds to Value-at-Risk using the improved standard bounds of Embrechts et al. In numerical examples we illustrate that the additional information may lead to a considerable improvement of the bounds compared to the marginals-only case.
We derive bounds on the distribution function, therefore also on the Value-at-Risk, of \varphi(\mathbf X) where \varphi is an aggregation function and \mathbf X = (X_1, \dots ,X_d) is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of \mathbf X , is available . In order to include this information in the computation of Value-at-Risk bounds, we establish a reduction principle that relates this problem to an optimization problem over a standard Fr\'echet class. Second, we assume that the copula of \mathbf X is known on a subset of its domain, and finally we consider the case where the copula of \mathbf X lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fr\'echet-Hoeffding bounds on copulas so as to include this additional information on the dependence structure . Then, we relate the improved Fr\'echet-Hoeffding bounds to Value-at-Risk using the improved standard bounds of Embrechts et al. In numerical examples we illustrate that the additional information typically leads to a considerable improvement of the bounds compared to the marginals-only case.
D
...
A
the
R
are known
, is available
D
only
R
the additional information
this additional information on the dependence structure
R
may lead
typically leads