Models for risk aggregation and sensitivity analysis: An application to bank economic capital

A challenge in enterprise risk measurement for diversified financial institutions is developing a coherent approach to aggregating different risk types. This has been motivated by rapid financial innovation, developments in supervisory standards (Basel 2) and recent financial turmoil. The main risks faced - market, credit and operational - have distinct distributional properties, and historically have been modeled in differing frameworks. We contribute to the modeling effort by providing tools and insights to practitioners and regulators. First, we extend the scope of the analysis to liquidity and interest rate risk, having Basel Pillar II of Basel implications. Second, we utilize data from major banking institutions' loss experience from supervisory call reports, which allows us to explore the impact of business mix and inter-risk correlations on total risk. Third, we estimate and compare alternative established frameworks for risk aggregation (including copula models) on the same data-sets across banks, comparing absolute total risk measures (Value-at-Risk - VaR and proportional diversification benefits-PDB), goodness-of-fit (GOF) of the model as data as well as the variability of the VaR estimate with respect to sampling error in parameter. This benchmarking and sensitivity analysis suggests that practitioners consider implementing a simple non-parametric methodology (empirical copula simulation- ECS) in order to quantify integrated risk, in that it is found to be more conservatism and stable than the other models. We observe that ECS produces 20% to 30% higher VaR relative to the standard Gaussian copula simulation (GCS), while the variance-covariance approximation (VCA) is much lower. ECS yields the highest PDBs than other methodologies (127% to 243%), while Archimadean Gumbel copula simulation (AGCS) is the lowest (10-21%). Across the five largest banks we fail to find the effect of business mix to exert a directionally consistent impact on total integrated diversification benefits. In the GOF tests, we find mixed results, that in many cases most of the copula methods exhibit poor fit to the data relative to the ECS, with the Archimadean copulas fitting worse than the Gaussian or Student-T copulas. In a bootstrapping experiment, we find the variability of the VaR to be significantly lowest (highest) for the ECS (VCA), and that the contribution of the sampling error in the parameters of the marginal distributions to be an order or magnitude greater than that of the correlation matrices.