Expected shortfall estimation and capital coverage under overlapping scenarios

Martin Aichele; Igor Cialenco; Damian Jelito; Marcin Pitera

AI summary · arXiv: 2610.41624v1 · q-fin.RM

Expected shortfall estimation and capital coverage under overlapping scenarios

Martin Aichele, Igor Cialenco, Damian Jelito, Marcin Pitera

Original paper Submitted to arXiv on 30 Oct. 2026 License: NONEXCLUSIVE-DISTRIB 1.0 Also in: fr

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The essentials

Overlapping scenarios systematically lower estimated expected shortfall and weaken capital coverage; conservative weights mitigate, but do not eliminate, the problem.

Question

The paper asks how applying non-parametric ES estimators designed for independent samples to overlapping scenarios affects estimation accuracy and capital coverage. It also examines how these effects interact with serial dependence and heavy-tailed P&L distributions.

Method

The authors compare six order-statistic ES estimators at a 2.5% confidence level and sample size of 250, using simulations across static distributions and GARCH models with Gaussian or Student-t innovations. They evaluate fit, bias, and safety over 1,000,000 Monte Carlo samples, and analyze daily and overlapping 10-day returns from twelve risk-factor series and 99 equity portfolios.

Findings

Result Overlapping sampling is associated with systematic ES underestimation and worse fit, bias, and safety metrics.
See the passage in the paper
Overlapping samples do more than inflate variance through a smaller effective sample size: they are associated with a directional bias, systematically underestimating risk across fit, bias, and safety metrics – the most adverse direction for capital adequacy.
Result Serial dependence further worsens underestimation, while heavy tails amplify the combined estimation and coverage problems.
See the passage in the paper
Serial dependence acts in the same direction, and heavy tails amplify both effects.
Authors’ interpretation Giving more weight to extreme losses improves bias and safety, but does not make one estimator best across all performance criteria.
See the passage in the paper
Overall, as expected, there is no best CRE that would optimize across (symmetric) fit, bias and safety.
Result In market data, median realized coverage thresholds roughly double under overlapping sampling for both risk-factor series and equity portfolios.
See the passage in the paper
Across our twelve risk-factor series and, separately, across our ninety-nine equity portfolios, the confidence level at which the estimated capital actually covers subsequent realized losses, ideally equal to the $2.5\%$ target, roughly doubles in median under the overlapping scheme for every ES estimator, while the realized $\ES_{2.5\%}/\var_{1\%}$ ratio moves comparatively little.
Authors’ interpretation The ES-to-VaR ratio alone can miss inadequate absolute coverage, so the authors recommend considering it alongside coverage-based diagnostics.
See the passage in the paper
A ratio above $0.9$ test on overlapping data therefore says little about the absolute ES estimation error, and should be combined with other diagnostics, such as the CT values.

Limitations

The analysis focuses on a specific regulatory calibration and a selected family of six order-statistic estimators, so its numerical conclusions do not directly cover other sample sizes, confidence levels, or estimator classes.
See the passage in the paper
Furthermore, while we focus on $\ES_{2.5\%}$ and $n=250$, motivated by the FRTB Pillar I market risk framework, the underlying estimation problem is not specific to this regulatory calibration.
In the simulation design, comparing daily observations with overlapping 10-day aggregates combines dependence effects with changes in the target distribution’s shape.
See the passage in the paper
The comparison therefore mixes the effect of serial dependence with that of a target law closer to Gaussian, except the Gaussian case, where the 10-day law remains Gaussian and the effect of dependence is isolated.

Why it matters

The results suggest that historical-simulation ES capital can be materially understated when standard estimators are applied to overlapping scenarios. The authors argue that model validation and monitoring should assess coverage at the overlapping horizon and that estimator construction needs a dedicated theory for overlapping, dependent samples.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Expected shortfall overlapping scenarios serial dependence tail risk market-risk capital

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Comprehensive Summary
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Original abstract

The paper examines whether standard expected shortfall (ES) estimators remain reliable when applied to overlapping multi-day profit-and-loss scenarios, despite being designed for independent observations. The question matters for market-risk capital under the Fundamental Review of the Trading Book (FRTB). Banks commonly estimate 2.5% ES from a one-year window of 250 ten-day scenarios formed by rolling across daily data. Because these scenarios share most of their underlying observations, the nominal sample size overstates the amount of independent information available. The authors investigate how overlap, serial dependence in returns, and heavy tails affect estimation accuracy and loss coverage. They compare six non-parametric, order-statistic ES estimators with a 1% value-at-risk (VaR) benchmark. The ES estimators differ in how they weight the most extreme losses. The authors normalize the weights of three estimators so that all six satisfy the finite-sample coherence conditions considered in the study. Performance is assessed using nine measures grouped into fit, bias, and safety. The safety measures include a “safe confidence threshold” (CT), defined as the confidence level at which the reserve added to future P&L leaves no residual ES. A CT near the target 2.5% indicates adequate coverage. The simulations use samples of 250 observations and one million Monte Carlo repetitions. They cover independent Gaussian, Student-t, generalized Pareto, and normal-inverse Gaussian distributions, as well as GARCH models with Gaussian or Student-t innovations. The authors also examine daily and overlapping ten-day returns for twelve market risk-factor series and 99 diversified equity portfolios. Across the simulations, overlapping samples do more than increase variance: they shift estimates systematically downward, weakening coverage. For Gaussian data, the variance of one estimator based on 250 overlapping observations is roughly equivalent to that based on only 57 independent observations. Under overlap, estimated ES bias ranges from about −5% for Gaussian data to −19% for the heavier-tailed generalized Pareto case with a tail parameter of one-third. Serial dependence worsens the results even without overlap. In one Gaussian GARCH specification, the first estimator’s bias is −10.7%, compared with −6.7% for overlapping independent Gaussian data. Heavy tails and dependence compound the problem. In overlapping GARCH and heavy-tailed settings, underestimation is frequent, and even the most conservative estimators do not always provide adequate coverage. Estimator weights involve a trade-off rather than a universal best choice. Assigning more weight to the worst observed losses generally improves bias, downside error, and safety, but worsens symmetric fit. Across models and estimators, average CT rises by about 104% when sampling shifts from non-overlapping to overlapping observations. More conservative weights mitigate some of the deterioration, but they do not consistently ensure safe capital when adverse features coincide. The market data support the simulation findings. For the ten regular risk-factor series, median CT values increase from roughly 3.2–4.0% with daily data to 7.2–8.0% with overlapping ten-day data, depending on the estimator. These values are approximately twice the target and fall within the paper’s warning zone. The 99 equity portfolios show a similar shift, with median overlapping CT values of about 8–9%. Short-rate series perform worse, with particularly high CTs that the authors associate with major changes in rate regimes. By contrast, the estimated ES-to-VaR ratio changes little under overlap because both measures are affected. The ratio can therefore conceal shared underestimation, leading the authors to recommend using direct coverage diagnostics alongside ratio checks. The paper concludes that estimators based on the independent and identically distributed assumption can materially understate ES capital when applied to overlapping, dependent scenarios. It recommends validating estimators against the actual horizon and dependence structure.

When Overlapping Scenarios Undercut Expected Shortfall

Expected shortfall (ES) is meant to capture the average loss in the worst part of a portfolio’s outcomes. It plays an important role in market-risk capital under the Fundamental Review of the Trading Book (FRTB). But estimates can be misleading if the data used to calculate them do not meet the assumptions behind standard methods.

In “Expected shortfall estimation and capital coverage under overlapping scenarios,” Martin Aichele, Igor Cialenco, Damian Jelito, and Marcin Pitera investigate a common source of trouble: overlapping multi-day scenarios. Their findings suggest that overlap can make ES estimates systematically too low—not merely less precise—and that this can weaken the protection capital is supposed to provide.

Why overlapping data matter

A typical approach uses a one-year window to create 250 ten-day profit-and-loss scenarios. These are formed by rolling across daily data. Because consecutive ten-day scenarios share most of their underlying observations, they are not independent, even if they are treated as though they were.

This distinction matters. A set of 250 overlapping observations may look like a substantial sample, but it contains much less independent information than 250 separate scenarios. The authors examine whether standard non-parametric ES estimators remain reliable under these conditions, and how their performance changes with serial dependence and heavy-tailed losses.

What the study tests

The researchers compare six order-statistic ES estimators, which differ in how they weight the largest observed losses. They also include a 1% value-at-risk (VaR) benchmark. To make the comparison fair, they normalize the weights of three estimators so all six meet the finite-sample coherence conditions considered in the study.

Performance is evaluated in simulations using samples of 250 observations and one million repetitions. The tests cover independent Gaussian, Student-t, generalized Pareto, and normal-inverse Gaussian distributions, as well as GARCH models with Gaussian or Student-t innovations. The authors also analyze daily and overlapping ten-day returns for twelve market risk-factor series and 99 diversified equity portfolios.

Rather than relying on a single measure, the study assesses nine metrics grouped into fit, bias, and safety. One safety measure is the “safe confidence threshold” (CT): the confidence level at which a reserve added to future profit and loss leaves no residual ES. A CT close to the target 2.5% suggests that coverage is adequate; a much higher value signals that the estimated reserve may be insufficient.

Overlap can create systematic underestimation

The central result is that overlapping samples do more than inflate estimation variance. They can shift ES estimates downward, reducing loss coverage. For Gaussian data, the variance of one estimator based on 250 overlapping observations is roughly equivalent to its variance with only 57 independent observations.

Estimated ES bias under overlap ranges from about −5% for Gaussian data to −19% for a heavier-tailed generalized Pareto case. Dependence can make matters worse even without overlap. In one Gaussian GARCH specification, the first estimator has a bias of −10.7%, compared with −6.7% for overlapping independent Gaussian data. When overlap, dependence, and heavy tails coincide, underestimation becomes frequent—and even the most conservative estimators do not always provide adequate coverage.

Choosing an estimator means balancing trade-offs

No single weighting scheme wins in every situation. Giving greater weight to the worst observed losses generally improves bias, downside error, and safety. The trade-off is weaker symmetric fit, which measures how closely estimates match the target across outcomes. More conservative weights can therefore help, but they do not reliably solve the problem under adverse conditions.

Across models and estimators, the average CT rises by about 104% when sampling changes from non-overlapping to overlapping observations. This makes the issue concrete: an estimator that appears acceptable under independent sampling may not provide the same level of protection when applied to rolling multi-day scenarios.

Market data tell a similar story

The empirical results broadly echo the simulations. For ten regular risk-factor series, median CT values rise from roughly 3.2–4.0% with daily data to 7.2–8.0% with overlapping ten-day data, depending on the estimator. These overlapping-scenario values are around twice the 2.5% target and fall within the paper’s warning zone. The 99 equity portfolios show a similar pattern, with median CT values of about 8–9%.

Short-rate series perform worse still, with particularly high CTs. The authors associate these results with major changes in interest-rate regimes. They also find that the ES-to-VaR ratio changes little under overlap because both measures are affected. A stable ratio can therefore hide the fact that both estimates are too low.

What risk managers should take away

The study’s message is not that every standard ES estimator fails in every setting. Rather, reliability depends on whether the estimator’s assumptions match the data and scenario construction actually used. In particular, treating overlapping, dependent scenarios as independent can materially understate ES capital.

Check coverage directly, rather than relying only on ES-to-VaR ratios.
Account for overlap and serial dependence when validating capital estimates.
Test estimators against the relevant tail behavior and market conditions.
Do not assume that more conservative loss weights guarantee safe capital.

For banks applying ES under FRTB, the practical lesson is to validate against the actual horizon and dependence structure. The number of scenarios on a spreadsheet may be 250, but the amount of independent evidence—and the safety of the resulting capital estimate—can be substantially smaller.

Created on 02 Nov. 2026

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Expected shortfall estimation and capital coverage under overlapping scenarios

The essentials

Question

Method

Findings

Limitations

Why it matters

Read further

Definitions

When Overlapping Scenarios Undercut Expected Shortfall

Why overlapping data matter

What the study tests

Overlap can create systematic underestimation

Choosing an estimator means balancing trade-offs

Market data tell a similar story

What risk managers should take away

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