The instability of loss-distribution-approach operational-risk capital measured on independent synthetic histories of known truth

A bank that estimates its 99.9 percent operational-risk capital from five or ten years of internal losses cannot say how far it is from the true quantile, which is never observed. We measure that distance on a generator with a known annual loss distribution: a compound Poisson sum of generalised Pareto losses calibrated to the 2008 Loss Data Collection Exercise, at four tail indices (0.6 to 1.1), two history lengths and two collection thresholds, with 1000 independent histories in each of 16 cells. Ten estimation rungs (nine counted below) and the standardised approach are run by one engine on the same histories and divided by the true quantile. The interquartile range of the splice capital is 1.35 (95 percent interval 1.13 to 1.61) to 3.30 (2.39 to 5.26) times the truth; the capital stability test is red in all 144 rung-by-cell rows; and 26.8 to 34.4 percent of the histories at tail index 0.96 conclude an infinite mean that does not exist, while 8.8 to 26.8 percent at 1.1 conclude a finite one. The hypothesis written before the robust fits were run, that they beat the splice in spread and in bias, is inconclusive over the 16 cells and refuted at tail indices 0.96 and 1.1. A delta-method theory predicts the dispersion of the single loss approximation within 4.7 percent and puts the history needed for the stability limit at 97 to 272 years. The synthetic severity belongs to the family the estimators assume, which favours them. Repository and reproducible pipeline (MIT): https://github.com/fsantibanezleal/CAOS_Contraste .