Models for Risk Aggregation and Sensitivity Analysis: An Application to Bank Economic Capital

Abstract

A central challenge to the practice of enterprise risk measurement and management faced by diversified financial institutions (e.g., an internationally active bank or insurance company) 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 along four dimensions, providing tools and insights to practitioners and regulators. First, we extend the scope of the analysis for these three risks that have been the focus of the literature, analyzing proxies for liquidity and interest rate risk, which has implications for Pillar II of Basel IRB framework. Second, we utilize actual data representative of major banking institutions’ loss experience, extracted from call reports, submitted by banks to supervisory agencies. This allows us to explore the impact of business mix and inter-risk correlations on total risk. Third, we estimate alternative copula models, an established framework for capturing realistic distributional features of different risk types (e.g., non-normality) and cohesively combining such, on the same data-set. We then compare our models to several conventional approaches to computing risk amongst practitioners. Finally, we apply recently developed goodness-of-fit (GOF) tests to the various copula models, which has been a largely neglected in theory and practice. Overall, our results constitute a sensitivity analysis that is evidence for practitioners to consider implementing a simple non-parametric methodology in order to quantify integrated risk. We observe a wide divergence in measured VaR, diversification benefits as well as the sampling variation across different risk aggregation methodologies and types of institutions (the five largest banks by book value of assets as of 4Q08), We observe in our sample that, contrary to asymptotic theory, empirical copula simulation (ECS) tends to produce the highest absolute magnitudes of VaR relative to standard copula formulations (e.g. Gaussian copula simulation - GCS) on the order of about 20% to 30%; while the variance-covariance approximation (VCA) tends to under-state risk. The proportional diversification benefits, as measured by the relative VaR reduction vis a vis the assumption of perfect correlation, exhibit wide variation across banks and aggregation techniques. The ECS generally yields the highest values relative the other methodologies (127% to 243%), while the Archimadean Gumbel copula (AGCS) is the lowest (10-21%). We conclude that while the ECS may over-state absolute risk relative, proportional diversification benefits may be understated by standard methodologies on the order of about 15% to 30%. Through differences observed across the five largest banks, we fail to find the effect of business mix to exert a directionally consistent an 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, in which we are able to measure the inherent variability in the VaR integrated risk and proportional diversification benefit measures, we find the variability of the VaR to be significantly lowest (highest) for the ECS (VCA) as compared to other copula formulations. We also find 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. Given the conservatism and stability of the ECS methodology, the poor performance of the standard methodologies in GOF analysis, and the lack of consensus upon the best copula to use, we believe that we have an argument for consideration of the ECS method.