On stability of operational risk estimates by LDA: From causes to approaches

The stability of estimates is critical when applying advanced measurement approaches (AMA) such as loss distribution approach (LDA) for operational risk capital modeling. Recent studies have identified issues associated with capital estimates by applying the maximum likelihood estimation (MLE) method for truncated distributions: significant upward mean-bias, considerable uncertainty about the estimates, and non-robustness to both small and large losses. Although alternative estimation approaches have been proposed, there has not been any comprehensive study of how alternative approaches perform compared to the MLE method. This paper is the first comprehensive study on the performance of various potentially promising alternative approaches (including minimum distance approach, quantile distance approach, scaling-based bias correction, upward scaling of lower quantiles, and right-truncated distributions) as compared to MLE with regards to accuracy, precision and robustness. More importantly, based on the properties of each estimator, we propose a right-truncation with probability weighted least squares method, by combining the right-truncated distribution and minimizing a probability weighted distance (i.e., the quadratic upper-tail Anderson-Darling distance), and we find it significantly reduces the bias and volatility of capital estimates and improves the robustness of capital estimates to small losses near the threshold or moving the threshold, demonstrated by both simulation results and real data application.