The quantification of diversification benefit plays a critical role in quantitative risk models, especially within the context of regulatory and economic capital. However, the complexity of today's risk landscape, together with the associated uncertainty surrounding the modeling of dependencies makes quantitative analysis of diversification a challenging task. By using well-known mathematical results, we provide a lower bound on the risk concentration RC (which we define as 1 minus the diversification benefit) in the presence of dependent underlying risk factors. Our methodology quantifies the modeling uncertainty by providing a lower bound for the RC, and hence a theoretical lower bound for the firm-wide advanced measurement approach (AMA) capital. In other words, we quantify the RC under VaRa for the least conservative case, even when we cannot define what the least conservative scenario would be. Finally, we apply this framework to operational risk and provide bank supervisors a potential tool to benchmark and rank order the RC that the banks report for their operational risk AMA capital. We show that the aggregation methods are overly reliant on modeling choices and assumptions, and that some of the banks have very aggressive reduction in firm-wide operational risk capital.
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