Model uncertainty and VaR aggregation

- We derive an algorithm for the calculation of bounds on the best and worst VaR.
- The numerical results are compared with available analytical bounds.
- Worst dependence scenarios are derived using the notion of complete mixability.
- Examples show one can handle portfolios with several hundreds of risk factors.
- The results obtained highlight VaR-based model uncertainty.

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.