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 utilize a reduction principle that relates this problem to an optimization problem over a standard Frechet class, which can then be solved by means of the rearrangement algorithm or using analytical results. 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 Frechet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Frechet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.
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