Aggregation issues in operational risk

In this paper we study copula-based models for aggregation of operational risk capital across business lines in a bank. A commonly used method of summation of the value-at-risk (VaR) measures, which relies on a hypothesis of full correlation of losses, becomes inappropriate in the presence of dependence between business lines and may lead to overestimation of the capital charge. The problem can be further aggravated by the persistence of heavy tails in operational loss data; in some cases, the subadditivity property of VaR may fail and the capital charge becomes underestimated. We use α-stable heavy-tailed distributions to model the loss data and then apply the copula approach in which the marginal distributions are consolidated in the symmetric and skewed Student t-copula framework. In our empirical study, we compare VaR and conditional VaR estimates with those obtained under the full correlation assumption. Our results demonstrate a significant reduction in capital when a t-copula is employed. However, the capital reduction is significantly smaller than in cases where a moderately heavy-tailed or thin-tailed distribution is calibrated to loss data. We also show that, when historical weekly data is used, VaR exhibits the superadditivity property for confidence levels below 94% and that, when the loss distribution approach is used, the superadditivity of VaR is observed at a higher confidence level (98%).