The expectile loses its precision edge once capital requirements are matched.
At matched nominal levels, the 1% conditional expectile produces prediction intervals measuring 66.3% to 89.8% of the length of the 1% value-at-risk interval. Relative to the expected-shortfall interval, the range is 35.2% to 45.0%. Those figures belong to the paper.
Fernandes, Henriques and Mendes then repeat the comparison under comparable capital requirements. VaR remains at 1%. ES shifts to 2.5%, following the BIS guidance used in the paper, while the expectile moves to the tau that reproduces the 1% VaR. Its intervals are then wider on average. BTC records 7.64 against 4.50 and 5.79, ETH 8.38 against 6.86 and 6.88, and EUR 0.54 against 0.35 and 0.41. BNB reverses the ranking, with 7.21 for the expectile against 7.86 for VaR and 6.33 for ES. The paper reports both comparisons. The matched-capital version captures the choice a risk manager would actually confront.
The object and its estimator
An expectile at level tau minimizes an asymmetric squared loss. Following Newey and Powell, squared deviations below the chosen point receive weight (1 - tau), while those above it receive weight tau. The quadratic loss makes each shortfall's size part of the estimate, alongside its frequency. Value at risk stops after counting breaches. Expected shortfall averages severity inside the tail.
The paper states the cost plainly: ES yields very imprecise estimates in finite samples for large values of alpha, especially under heavy-tailed loss distributions. Its summary of the trade-off is equally direct: "the ES is too conservative because it is conditional only on tail events, the VaR is too lenient because it does not account for their severity."
Following Ziegel, expectiles are also the only law-invariant coherent risk measure that is elicitable. Backtesting can therefore proceed by minimizing an expected score.
Estimation takes two steps. Returns follow y_t = sigma_t(theta_0) eta_t, a pure scale model with iid innovations. The GARCH-type volatility is fitted by Gaussian quasi-maximum likelihood, producing standardized residuals. Asymmetric least squares then estimates their unconditional expectile at level tau. Multiplying that estimate by the one-step-ahead volatility forecast gives the conditional expectile, since positive homogeneity turns the innovation expectile into a scale multiple.
Francq and Zakoïan provide this recipe for VaR. Fernandes, Henriques and Mendes carry it to a different functional and derive the error bar.
Why first-step error survives
The innovations are unobserved, so estimation uses residuals carrying QML sampling error. Whether that error remains in the second step's limit distribution depends on the expectile regime.
It disappears for extreme expectiles. Girard, Stupfler and Usseglio-Carleve examine tau_n approaching one as n grows, where the estimator converges at sqrt(n(1 - tau_n)). The GARCH parameter error contracts at the faster sqrt(n) rate and becomes asymptotically invisible.
A fixed tau removes that convenience. The paper's own sentence is exact: "the estimation of expectiles at a fixed level tau converges at the same sqrt(n)-rate as the estimation of the conditional mean and variance parameters."
The authors expose the interaction through a Bahadur representation. This expansion separates the second-step estimator into a clean sample average and an error inherited from the first step. They combine it with a joint martingale CLT for the score and the QML error, obtaining a closed-form asymptotic variance and plug-in estimators for its components.
Three terms enter the variance: the oracle variance available with true innovations, a scale-parameter covariance term multiplied by the squared innovation expectile, and a cross term. The last term carries the substance. Sampling error from the volatility fit correlates with sampling error in the residual expectile. Omitting that covariance can make the interval too wide or too narrow, according to its sign.
One result cuts against the obvious expectation. The conditional expectile, which is the forecasted quantity, achieves better finite-sample Wald coverage than its unconditional innovation input. Coverage for the 95% intervals spans 0.927 to 0.952 for the conditional object, versus 0.907 to 0.951 for the unconditional one. In the paper's words, "the influence function of the conditional expectile attenuates part of the first-step estimation error that remains visible in the unconditional estimator."
Where theory is supposed to fail
The simulation uses ten thousand replications across GARCH(1,1) and GJR-GARCH(1,1). Omega is calibrated to 20% annualized volatility on 252 days, and alpha is fixed at 0.05. Persistence is either 0.90 or 0.98. Samples range from 500 to 5,000 after a 500-observation burn-in. The innovations are Gaussian, t8 and t4, each standardized to unit variance. The expectile level is tau = 0.05, and every design correctly specifies the volatility model.
The regular cases behave as the asymptotics predict. At n = 5,000, with Gaussian innovations and low persistence, the unconditional expectile bias for the symmetric GARCH design is -0.0002 and the RMSE is 0.0166. In the corresponding GJR column, the figures are -0.0003 and 0.0166. Across the designs in Table 1, the Wald statistic for the unconditional expectile has a standard deviation ranging from 0.994 to 1.125.
The t4 design carries more information because it violates the assumptions. The asymptotics require fourth moments for both the innovations and the derivative envelopes. t4 has none. Fernandes, Henriques and Mendes acknowledge the failure and retain the case as a stress test.
For the symmetric GARCH design at n = 500 with high persistence, coverage drops to 0.907, bias reaches -0.0820 and RMSE reaches 2.0999. The matching GJR column reports 0.908, -0.0512 and 1.4201. At n = 2,500 in the same GARCH design, bias has fallen to -0.0101, coverage is 0.922 and RMSE is 0.0450. Even though the t4 designs violate the r = 4 conditions, the estimator still delivers. That matters for the application, where sample kurtosis reaches 20.03 for BNB and 14.82 for BTC.
Four markets, four out-of-sample years
The empirical section uses daily Yahoo Finance prices for BTC, ETH, BNB and EUR/USD. BTC and EUR begin in January 2016. BNB and ETH begin in November 2017. Every series ends in December 2023.
Forecasts come from a rolling GJR-GARCH(1,1) fitted on windows of 1,000 observations. The out-of-sample period extends from January 2, 2020 to December 30, 2023. BTC contributes 1,921 windows, while the remaining series contribute between 1,082 to 1,243.
At the 1% level, BNB has 10 VaR exceedances against 12.43 expected. BTC has 20 against 19.21, ETH 8 against 12.43, and EUR 14 against 10.82. Every unconditional and conditional coverage test passes at 5%.
The duration test examines the spacing between breaches and rejects for BNB (p = 0.03) and ETH (p = 0.02). Acceptable counts therefore coexist with breach clustering. McNeil-Frey residual tests for the expectile forecasts return 0.48, 0.79, 0.06 and 0.98. ETH remains borderline.
The presentation contains one discrepancy. The text gives exceedance rates of 1.05% for BNB and 0.72% for ETH, while Table 4 reports 0.80% and 0.64% for the same series. The difference is small enough to leave the results intact, though readers reconciling the table and prose will see it.
The capital match reverses the result
Setting tau = alpha = 0.01 places all three measures on the same statistical level and produces the expectile's tight intervals. Comparable capital requires a different exercise. Solving for the tau that reproduces the 1% VaR gives an average of 0.32% to 0.47% across the four series, far into the tail relative to the nominal 1%.
At that matched level, BTC has the widest expectile interval: 7.64 against 4.50 and 5.79. ETH gives 8.38 against 6.86 and 6.88, while EUR gives 0.54 against 0.35 and 0.41. BNB remains the exception, at 7.21 against 7.86 for VaR and 6.33 for ES.
The simulations produce the same ordering. At larger sample sizes, matched-tau intervals are 20% to 25% longer than those for VaR and ES. They are roughly comparable at n = 500. The authors state the reason directly: tau(alpha) is much smaller than alpha.
Table 5 weakens the practical framing from another direction. The average capital buffer equals the requirement minus the realized loss. For BNB, the average is 13.97 under VaR and the matched expectile, compared with 14.70 for ES. BTC records 10.69 and 10.69, compared with 11.24. The difference between ES(2.5%) and VaR(1%) is 73 bps for BNB, 55 bps for BTC, 2 bps for ETH and 9 bps for EUR.
These four series therefore require almost the same money under all three measures. Estimation precision separates them. Across the four series in Table 5, the matched expectile is estimated least precisely for three. BNB again supplies the exception, with 7.21 against 7.86 for VaR.
Limits of the claim
This remains estimation theory with an empirical demonstration attached. No returns, no Sharpe, no costs, no P&L. The theory assumes zero conditional mean. Fernandes, Henriques and Mendes identify the nonzero-mean extension as future work and cite Francq and Zakoïan.
The framework stays univariate, leaving the high-dimensional portfolio case open. Subadditivity motivates the exercise, yet the paper never measures a diversification benefit. In the empirical work, GJR-GARCH(1,1) is imposed without model selection or a misspecification check. The tau(alpha) mapping is estimated from VaR and ES on the same data, and the reported intervals omit the resulting estimation error.
We could not run this ourselves. The universe contains three cryptocurrencies and one FX rate, and we hold no price history for either asset class. Substituting equities would test a different estimator on a different scale process.
The contribution is genuine and narrow. Anyone forecasting conditional expectiles from GARCH residuals has standard errors that ignore the first stage. This paper quantifies the omission and supplies a consistent plug-in correction. Its prediction intervals achieve 0.927 to 0.952 coverage in simulation.
A matched-capital comparison in which the expectile intervals remain tighter would change my view of the practical case. Reaching that result requires a way to pin down tau without first estimating the quantile the expectile is meant to replace. Until such a method appears, the expectile's claimed precision advantage comes from comparing 1% with 1%.
We have written before about premia that survive only in the units selected by their authors (the correlation rotation note). The same problem appears here. Fernandes, Henriques and Mendes deserve credit for reporting the reversal themselves.