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 \FH bounds on copulas so as to include this additional information on the dependence structure. Then, we relate the improved Fr\'echet-Hoeffding bounds to\FH  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.
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 , which can then be solved by means of the well-known standard bounds or the rearrangement algorithm . 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 \FH bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved\FH 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 considerable improvement of the bounds compared to the marginals-only case.
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, which can then be solved by means of the well-known standard bounds or the rearrangement algorithm
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Fr\'echet-Hoeffding
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relate the improved Fr\'echet-Hoeffding bounds to
translate the improved
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bounds to bounds on the
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so-called
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of Embrechts et al