Monitoring a Volatility-Targeted Loss Limit: A Case Study in Estimator-Induced Miscalibration

Suppose a desk caps each position's 21-day loss at twice the estimated volatility. If the estimate were exact, the cap would be exceeded (breached) in 2.275% of windows, the Gaussian tail beyond two standard deviations. On a panel of U.S. equities that keeps stocks through delisting, with 2019–2025 held out and scored once, a fixed-sample coverage backtest tells five volatility estimators apart only after three dependencies are corrected: stocks sharing a date, consecutive windows sharing twenty days, and the five tested at once. Held out, limits sized with the three high–low range estimators, which see only intraday prices and so omit the overnight return, are breached 1.9 times as often as designed (and above the other two in every development year); limits sized with close-to-close or Yang–Zhang, which include the overnight return, are not rejected. Split into a level error and a noise term with the noise scale fixed in advance, about 98% of that excess is level error, an upper bound by construction. Run instead as a sequential monitor, the same breach stream drives a test martingale, a betting process whose wealth is an e-value, so its reciprocal bounds the false-alarm probability at any stopping time. On one pre-specified tiling of the calendar into disjoint 21-day windows (the 21 offsets of a 21-day window), the range estimators' wealth passes 20, the 5% level, after four to five years and the level covering all five estimators on all 21 tilings after about six, about a year before a Bonferroni-corrected monthly re-test and no earlier under other bet choices (exploratory, development period); a martingale on the range family's breach rate minus close-to-close's never crosses. The paper proposes no e-value method; it offers a documented miscalibration as a case for monitors that stay valid under repeated looks.