Model uncertainty and VaR aggregation

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. Since the early nineties, Value-at-Risk (VaR) has established itself as a (if not the) key metric for the calculation of regulatory capital within the financial industry. Furthermore, VaR is increasingly used as a risk management constraint within portfolio optimization. Whereas books like Jorion (2006) prize VaR as the industry standard, numerous papers have pointed out many of the (most obvious) shortcomings of VaR as a risk measure; see for instance McNeil et al. (2005) and the references therein, but also the recent Basel Committee on Banking Supervision (2012), already referred to as Basel 3.5. A very informative overview on the use of VaR technology within the banking industry is Pérignon and Smith (2010). As so often, a middle-of-the-road point of view is advisable: there is no doubt that the construction and understanding of the P&L distribution of a bank's trading book is of the utmost importance. The latter includes the availability of data warehouses, independent pricing tools and a complete risk factor mapping. And of course Corporate Governance decisions may have a major impact on the P&L, like for instance in the case of strategic decisions. In that sense, VaR, as a number, is just the peak of the risk management iceberg. Nonetheless, once the number leaves the IT system of the CRO, all too often it starts a life of its own and one often forgets the numerous warnings about its proper interpretation. Moreover, once several VaRs are involved, the temptation is there …