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 $\mathbf X = (X_1,...,X_d)$ 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.
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