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, that relies on a hypothesis of full correlation of losses, becomes inappropriate in the presence of dependence between business lines and may lead to over-estimation 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 value-at-risk 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 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 for confidence levels below 94% VaR exhibits the super-additivity property.