The problem of establishing reliable estimates or bounds for the (T)VaR of a joint risk portfolio is a relevant subject in connection with the computation of total economic capital in the Basel regulatory framework for the finance sector as well as with the Solvency regulations for the insurance sector. In the computation of total economic capital, a financial institution faces a considerable amount of model uncertainty related to the estimation of the interdependence amongst the marginal risks. In this paper, we propose to apply a clustering procedure in order to partition a risk portfolio into independent subgroups of positively dependent risks. Based on available data, the portfolio partition so obtained can be statistically validated and allows for a reduction of capital and the corresponding model uncertainty. We illustrate the proposed methodology in a simulation study and two case studies considering an Operational and a Market Risk portfolio. A rule of thumb stems from the various examples proposed: in a mathematical model where the risk portfolio is split into independent subsets with comonotonic dependence within, the smallest VaR-based capital estimate (at the high regulatory probability levels typically used) is produced by assuming that the infinite-mean risks are comonotonic and the finite-mean risks are independent. The largest VaR estimate is instead generated by obtaining the maximum number of independent infinite-mean sums.

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.