Exact Finite-Sample Power Ceilings for Expected Shortfall Backtests

The Basel FRTB framework validates internal-model Expected Shortfall (ES) with backtests that are widely reported to suffer from both distorted size and negligible power at the 97.5% level, with one regulatory year of data (T = 250, about six violations). We settle both questions computationally and exactly, without asymptotics: the finite-sample null distribution of every candidate statistic is obtained by a T-fold circular FFT convolution of the per-day statistic on a 106-node probability-integral-transform (PIT) grid, so every test is calibrated to a size of exactly 5%, and every reported number is a probability, not a simulation estimate. Four results follow. First, at matched exact 5% size, the Acerbi–Sz´ekely test has power 0.0687 against a 5% ES understatement, 0.0906 at 10% and 0.1422 at 20%; a mixture-of-likelihood-ratio e-value test attains 0.1488, 0.3542 and 0.7847 — 2.2×, 3.9× and 5.5× higher — while its distributionfree Ville calibration keeps size below 5% with no null assumptions at all. Second, we prove and compute the exact finite-sample power ceiling (Neyman–Pearson bound, lifted to the losses by the data-processing inequality): no test whatsoever can exceed power 0.2223 at a 5% ES understatement, 0.5363 at 10% and 0.9873 at 20%; the literature’s implicit 80% power standard is therefore unattainable for a 5% ES error at T = 250, and is first reached at a 14.2% error (oracle) or 21.1% (our test). Third, we measure exactly, by order-statistic quadrature, the noise that Monte-Carlo estimation of ES injects into the backtest: at M = 1000 simulation paths the variance inflation is r = 0.0222 (fixed denominator) or r = 0.0918 (realised-count denominator) under the leading-order (centred) decomposition, and r = 0.0302 / r = 0.0954 under the exact law of the simulation error; the implied size of the Acerbi–Sz´ekely test rises from a nominal 5.00% to 5.37% and 6.34% (centred) or to 5.55% and 6.73% (exact law). The implied variance share on the fixed-denominator measure is 2.2% under the centred decomposition and 2.9% under the exact law — an order of magnitude below the 26–35% variance share recently claimed in the literature, which we correct. Fourth, we close the loop on that literature by exhibiting, exactly, why a single-bet e-value cannot recover the lost power at this horizon (its expected log growth is 0.049 nats against the log 20 = 3.00 nats a Ville threshold demands). Every claim is verified three independent ways: grid convergence bounded at 2 × 10-4, a Monte-Carlo simulation that uses no part of the convolution engine (including a 2×106-path check of the power ceiling), and independent quadrature cross-checks of every estimator moment. All computations are handwritten C++17 with no external dependencies; all six software defects found during verification are reported with their fixes.