Anytime-valid e-process monitors for VaR back-tests at a cost in power, and a PD monitor that loses its size under correlated defaults and its power to a chi-square rule

Value-at-risk (VaR) back-tests and default-probability (PD) calibration tests are valid when read once on a fixed window, but validation reads them every day or year: read daily over 1,000 days, the Kupiec test on the last 250 days rejects a correct 99 % VaR in 65.7 % of the series. The proposed monitors are mixtures of likelihood-ratio e-processes over alternatives fixed before the data, for the VaR exceedance rate, exceedance clustering and the defaults of each rating grade; each rejects when its process reaches 20, anytime-valid size 5 %. The paper proves validity and a power bound at a fixed horizon. Ten other VaR rules and the engine's PD tests run on simulated streams, six market-risk windows and five loan cohorts. The rate monitor rejects 2.4 % (1.8 to 3.2 %) of correct series read daily and detects a hit rate four times too high at a median of 128 days. The delay and power-cost hypotheses set before the experiments fail at three times the rate (251 days, 247 to 255; 12.4 ± 0.19 points) and hold at four times; an upper-side variant chosen after these results meets both (228 days, 6.7 ± 0.17 points) and needs a prospective test. The PD monitor rejects 91.0 % of correct histories at asset correlation 0.03, where the engine's Vasicek test with the true correlation rejects 1.4 %, and at the monitor's size (0.4 %) a chi-square rule read every year detects a 5 % PD error in 23.2 % of histories against 2.9 % for the monitor (53.2 % at a nominal 5 %). Repository and reproducible pipeline (MIT): https://github.com/fsantibanezleal/CAOS_Contraste .