Losing the VaR forecast costs about 25 points of detection power in the paper's cleanest test. With iid Normal losses, ES understated by 10% and 500 monitoring days, detection at threshold 10 falls from 56.7% with the true VaR and model parameters to 31.8% without them. The betting process is the same in both cases; profiling accounts for the gap. Wang and Yu accept that cost to monitor ES forecasts day by day without a VaR forecast. Their least settled implementation choice is the parameter-grid spacing: the paper gives no bound on its discretization error.

Betting on ES alone

The test starts with the ES e-statistic of Wang, Wang and Ziegel. It takes the realized loss above a VaR level z and divides it by (1-p) times the gap between reported ES and z. With the true VaR at z and an adequate ES report, the ratio has conditional mean at most one. Each day, a fraction λ_t, capped at 1-p, is bet on that ratio, and wealth compounds. During monitoring, the null puts the chance of wealth ever reaching c at no more than 1/c. Expected wealth at any stopping time is at most one.

An estimated VaR cannot simply replace z. The authors show that "simply inserting an estimated VaR into the ES e-statistic does not in general give an e-process for the ES null alone". An error in z can raise the expected betting factor even when the ES report is exact.

Instead, they calculate a wealth path for every candidate AR(1)-GARCH(1,1) parameter vector θ = (φ, α, β, v) and every feasible residual quantile z, then report the minimum. If a fixed pair (θ0, z0) is true, that minimum remains below the pair's wealth path, which is a genuine supermartingale. The authors make the scope of validity clear in their abstract: one fixed parameter-quantile pair must generate the conditional VaR path. They require neither iid residuals nor a specified innovation distribution. The residual distribution may drift, provided its 97.5% quantile remains z0 throughout monitoring.

Validity also needs a report margin ε = 0.01: standardized ES must exceed residual VaR by at least 0.01. The authors acknowledge that even an exact forecast can miss this condition. The profiled statistic is an e-process, although it need not itself be a supermartingale; Example 1 shows the latter property failing.

The simulations run 1000 paths per design at p = 0.975. Time-series cases have 200 calibration days and 500 monitoring days. For the empirical test, the authors monitor daily NASDAQ Composite losses from January 3, 2005 to December 31, 2021. ES forecasts use rolling 500-day AR-GARCH fits with Normal, Student t, skewed t and empirical distributions.

Where the numerical choices bite

Most settings are specified:

The 0.01 parameter grid is the worry.

Minimizing over fewer points can only raise the result above the continuous infimum. The authors' convergence result has grid values decreasing toward the true statistic, which means coarse spacing errs toward false alarms. They say their numerical bounds "do not bound the error from discretizing the AR-GARCH parameters." Tightening the spacing from 0.02 to 0.01 removes detections on up to 1.0% of the 1,000 paths in a cell. Median log capital drops 0.047; the drop is 0.028 for the 0.01 to 0.005 step. Those comparisons use selected paths only.

We did not find a sensitivity check on ε. When no grid point admits the current report, the fallback bet is 1-p. That occurred on 0.43% of simulated days.

NASDAQ alarms: how early?

For Normal ES forecasts, the ES-only process reaches threshold 10 on monitoring day 363. GREE, GREL and GREM, the three VaR-ES e-processes, reach it on days 714, 480 and 541. Student t forecasts give the ES-only process another day 363 crossing; the corresponding benchmark days are 942, 541 and 541. With skewed t forecasts, ES-only crosses thresholds 2/5/10 on days 354/363/541. GREE takes 718/3174/3478, while GREL takes 363/541/1345. At threshold 2, GREL's day 363 crossing trails ES-only by nine days.

Those dates compare complete procedures with different betting rules and initial histories, as the authors stress. The benchmarks use a rolling 500-day window to place bets. Normal and t ES-only crossings also coincide on days 342/358/363, suggesting that one episode drives both paths. The authors add that "interpreting these detections as evidence of ES under-reporting requires the conditional VaR structure and report margin condition in Section 3.1."

Even the 10%-inflated skewed-t forecast reaches threshold 2 on day 541 with ES-only. None of the benchmarks crosses any threshold for that forecast.

Simulations make the advantage less uniform. At threshold 5 with Normal innovations, ES-only detects 44% in GARCH against GREE's 48%. In AR-GARCH with t3 innovations, it detects 12% against GREE's 7%, despite GREE receiving the true conditional VaR. When reports change from over-statement to under-statement after 300 days, threshold-10 detection with Normal innovations is 23% for ES-only and 12 to 14% for the benchmarks.

The price of an unknown VaR

In the iid Normal case, with ES understated by 10% over 500 days, detection at thresholds 2/5/10 is 71.8/46.3/31.8% when both θ and z are unknown. Knowing θ raises it to 77.3/53.1/37.3%; knowing both raises it to 86.4/70.3/56.7%. At threshold 10, the missing VaR accounts for 19.4 of the 24.9-point gap. The costs are closer under iid t3 losses: detection is 4.9%, then 6.3% with θ known, then 7.6% with both known.

Heavy tails weaken detection further. With iid t3 losses, the same test detects 26/9/5%. In AR-GARCH with t3 innovations, ES-only detects 28/12/6%. The ES regression (ESR) backtest of Bayer and Dimitriadis rejects 57/16/4% there at levels 1/2, 1/5 and 1/10. For iid t3, ESR rejects 65/29/9%, ahead of ES-only at all three levels. The paper calls these ESR results "rejection frequencies at a fixed sample size, not probabilities of detection during the monitoring period." The sequential test allows an earlier stop. False alarms remain at most 2% at threshold 10 across time-series simulations with exact or conservative reports. After the iid simulations, the authors write that "heavier tails may require longer monitoring."

Across t3 designs with 10% ES under-reporting, threshold-10 detection in 500 monitoring days ranges from 4 to 6%: AR 4, GARCH 4, AR-GARCH 6 and iid 5. A fat-tailed book understating ES by 10% will therefore usually go 500 trading days, about two years, without a threshold-10 alarm. Even the oracle with true θ0 and z0 reaches only 7.5 to 10.6% at threshold 10. Most of the power problem comes from the tail.

QQQ needs a trading test

The paper reports no trading returns. Its evidence consists of detection frequencies and crossing dates for an index; QQQ may track that index imperfectly. The reported days cannot simply be carried over.

Our proposed rule applies the ES-only process to daily QQQ losses using ES forecasts built as in the paper, then cuts exposure when log capital crosses log 5. We chose both that threshold and the restart after a de-risk. The paper's guarantee applies to a single crossing in a fixed monitoring period. Alarms are reactive by construction: the e-statistic grows only when realized losses exceed candidate VaR levels. Whether the NASDAQ lead of 117 to 351 trading days has value in a portfolio remains open. Our backtest has to answer that question.