Quantitative response to the operational risk problems of external data scaling and dependence structure optimization for capital diversification

Abstract

In this study we aim to provide a quantitative response to the two respective operational risk problems of i) external data scaling, and ii) dependence structure optimization for capital diversi cation. The study is hosted at a nancial institute in South Africa which utilizes the Advanced Measurement Approach (AMA) to calculate capital requirements for operational risk. For Problem I on external data scaling, our study tracks the usage of power law transformation in order to gauge the proportional e ects of operational risk losses. We consider an extended technique incorporating a ratio-type scaling technique originating from the basic power law transformation study. Our proposed solution then diverts to quantile regression. We apply said theory to the regression problem of compiling a ratio-type scaling mechanism. We conclude our study on Problem I by providing a scaling mechanism for scaling down internationally-sourced external loss data to South African-based internal loss data allowing direct combination in a pooled loss dataset. Using this loss dataset, we consider an impact study where we note an increase in undiversi ed capital estimates of approximately 9% when comparing to internal loss data only. For Problem II on dependence structure optimization our study considers the utilization of copulas to express the dependence structure of operational risk losses over time. We speci cally investigate the application of factor copulas as derived from (exploratory) factor analysis. Using factor-based copulas allows for signi cant reduction in the dimensions of dependence structures which is a major problem when considering the high dimensionality associated with operational risk categories (ORCs). Our proposed solution enables us to construct dependence structures for a large number of ORCs using only two factors. We build the study around elliptical copulas and investigate dependence structure and diversi cation bene ts when adjusting for the presence and magnitude of tail dependence. We conclude our study on Problem II by providing a two-step method for constructing a dependence structure via factor analysis and then using this lowdimensional dependence structure result to easily construct a high-dimensional copula from which we simulate for capital estimates. Using our scaled and pooled dataset from Problem I, we obtain results con rming the general range of between approximately 30% and 50% for reduction in VaR. However, when extending a two-factor Gaussian copula to a two-factor (Student's) t-copula for more conservative capital estimation, we clearly note how tailedness a ects capital estimates when examining for expected loss and VaR estimates of various t-copulas.