📈 economics

Does the Choice of Lookback Window Create Model Risk in Historical Value-at-Risk?

This paper demonstrates that the choice of lookback window in Historical Value-at-Risk creates material model risk by producing significant, non-monotonic variations in VaR estimates despite passing regulatory backtesting, thereby necessitating the disclosure of window length and interpolation conventions as critical model choices.

Original authors: Mehul Mehta, Prabesh Bista, Moon Shimray, Hrishita Harsh Kumar

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mehul Mehta, Prabesh Bista, Moon Shimray, Hrishita Harsh Kumar

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the high-stakes world of banking, risk managers need a way to answer a single, terrifying question: how much money could we lose tomorrow? To answer this, they use a tool called Value-at-Risk, or VaR. Think of it as a weather forecast for a financial portfolio. Just as a meteorologist might say there is a one-in-a-hundred chance of a storm, a bank calculates the maximum loss it expects to face on 99 out of 100 trading days. This number is not just a guess; it determines how much capital a bank must keep in reserve to survive a crisis, and it is the figure that boards of directors and government regulators scrutinize most closely.

The most popular method for creating this forecast is called historical simulation. It is a straightforward approach: the bank looks at how its portfolio performed in the past and assumes the future will look similar. To do this, it needs to decide how far back to look. This is the crucial choice. A bank could look at the last six months of data, or the last five years. The rules set by global regulators say the bank must look back at least one year, but beyond that minimum, the bank is free to choose its own window. The question that has long lingered in the minds of risk experts is whether this freedom creates a hidden danger. If two banks look at the same portfolio but choose different time windows, will they get the same answer? Or will their choices create a blind spot that makes the risk look smaller or larger than it truly is?

A team of researchers set out to test this uncertainty using a realistic, million-dollar portfolio made up of four distinct assets: large American companies, government bonds, gold, and technology stocks. They gathered ten years of daily price data, a period that included both calm markets and moments of intense stress. They then ran their risk model four different times, each time using a different lookback window: 125 days, 250 days, 500 days, and 1,000 days. All four windows met the regulatory requirement of looking back at least a year, and all were applied to the exact same portfolio on the exact same days.

The results revealed a significant problem. On the very last day of their study, the four different windows produced four different risk numbers. The 250-day window estimated a potential loss of $20,504. The 500-day window estimated $21,378. The 1,000-day window estimated $18,220. The difference between the highest and lowest estimates was $3,158. This gap is not a rounding error; it is a substantial sum of money that could alter a bank's capital requirements or its internal trading limits. More importantly, the researchers found that the size of the risk estimate did not simply grow or shrink as the window got longer. Instead, the numbers jumped around in a way that defied a simple pattern, proving that the choice of window is not a neutral technical detail but a decision that materially changes the outcome.

The study also put these different windows through a series of rigorous tests to see which ones were actually reliable. These tests checked how often the model's predictions were wrong. At a 99% confidence level, the 125-day window failed the tests, producing too many surprises and showing that its errors were clustered in time rather than spread out randomly. The other three windows, however, passed the tests. This created a paradox: three different models, all of which passed the regulatory checks, all agreed that the risk was real, yet they still disagreed on the exact dollar amount by thousands. The tests could tell the researchers that a model was working, but they could not tell them which of the working models was the most accurate.

The researchers also examined how the model handled the severity of losses. They found that longer windows tended to predict that when a loss did happen, it would be more severe. This suggests that looking further back captures more extreme events, but it also means the model might be slower to react to sudden changes in the market. Shorter windows react quickly to new information but can be unstable because they rely on very few data points to estimate rare events. The study highlighted that the way the computer calculates the final number from the data also matters. A specific rule used to pick the exact value from a list of past returns can determine whether a model passes or fails a test, yet this rule is rarely disclosed by banks.

Ultimately, the paper concludes that the choice of the lookback window creates a genuine model risk. It is a risk that exists even when the bank follows all the rules. Because there is no single "correct" window that works best in every market condition, the choice becomes a matter of judgment rather than a solved mathematical problem. The researchers recommend that banks should not just pick a number and move on. Instead, they should disclose exactly which window they are using and which calculation rules they apply. They suggest that banks should report the range of risk estimates across different windows, treating the spread between them as a daily measure of their own uncertainty. By acknowledging that different valid choices lead to different dollar amounts, banks can better understand the limits of their own forecasts and avoid the false confidence that comes from thinking there is only one right answer.

Technical Summary: Does the Choice of Lookback Window Create Model Risk in Historical Value-at-Risk?

Problem Statement
Historical Simulation (HS) is the dominant method for calculating Value-at-Risk (VaR) in banking due to its transparency and lack of distributional assumptions. However, HS possesses exactly one free parameter: the length of the rolling lookback window ( $W$ ). While regulatory frameworks (e.g., Basel) mandate a minimum of one year of data, they leave the specific choice of $W$ to the firm's discretion, provided it undergoes supervisory review. In practice, firms utilize windows ranging from six months to four years.

The central problem addressed is whether this regulatory latitude creates material model risk. Following the definition by Danielsson et al. (2016), this is not the risk that a model is "wrong," but the risk that multiple equally admissible, compliant specifications produce materially different risk numbers with no data-driven mechanism to select the superior one. The paper investigates whether the choice of $W$ alone contributes to significant discrepancies in VaR estimates and backtesting outcomes.

Methodology
The study employs a practical, out-of-sample design using a $1,000,000 equal-weighted portfolio of four exchange-traded funds (SPY, BND, GLD, QQQ) representing US large-cap equity, aggregate bonds, gold, and technology equity. The dataset spans ten years of daily data (September 2016 – September 2026).

Estimators: The authors compute one-day HS-VaR and Expected Shortfall (ES) at 95% and 99% confidence levels using four distinct lookback windows: $W \in \{125, 250, 500, 1000\}$ days.
Quantile Convention: The study explicitly adopts linear interpolation for calculating empirical quantiles, noting that the choice of interpolation convention is critical, particularly for short windows and deep tails.
Backtesting Framework: The models are evaluated on a common 1,512-day out-of-sample period (following a 1,000-day burn-in). The authors apply a standard supervisor's battery of tests:
- Kupiec (1995) Unconditional Coverage Test.
- Christoffersen (1998) Conditional Coverage and Independence Tests.
- Basel Traffic Light Test (specifically at 99% confidence).
- A severity test (McNeil & Frey, 2000) is discussed, though the authors note implementation errors in their initial severity analysis that preclude reporting definitive severity results in the main text.

Key Results

Material Dollar Disagreement: Even among windows that pass all regulatory backtests, the estimated VaR on the final day of the sample differs significantly. At 99% confidence, the 250-, 500-, and 1,000-day windows report VaRs of $20,504, $21,378, and $18,220, respectively. This represents a spread of $3,158 (17% of the lowest estimate) that is not monotonic with window length.
Differential Test Performance: The choice of $W$ $W$ determines whether a model passes regulatory backtests.
- The 125-day window fails both the Kupiec and Conditional Coverage tests at 99% confidence (p-values of 0.0007 and 0.0028, respectively) and sits in the Basel yellow zone on the final 250 days of the sample. Its exceptions are clustered rather than independent.
- The 250-, 500-, and 1,000-day windows pass all tests at 99% confidence and fall in the green zone.
- At 95% confidence, the 125-day window also fails the Conditional Coverage test, while the others pass. (Note: At 95%, all windows fall into the red zone due to the test's calibration for 99% VaR, making the red readings an artifact of the threshold rather than evidence of model failure).
Bias-Variance Trade-off: The results illustrate the theoretical trade-off: shorter windows (e.g., 125 days) are more responsive to recent volatility but suffer from high estimation noise in the deep tail (few observations), leading to backtest failures. Longer windows (e.g., 1,000 days) reduce estimation noise but introduce bias by averaging over diverse volatility states, potentially smoothing out current risk.
Tail Severity Modeling: The modelled ratio of ES to VaR increases with window length at 99% confidence (rising from 1.20 at 125 days to 1.40 at 1,000 days), indicating that longer windows model a fatter tail beyond the VaR threshold.
Quantile Convention Sensitivity: The paper highlights that the 125-day window's failure is driven by the linear interpolation convention used. At $W=125$ and $\alpha=1\%$ , the quantile interpolates between only the first and second worst observations; thus, the indexing rule dictates the result. The authors did not test alternative conventions (e.g., plotting-position) on this data but flag this as a critical disclosure requirement.
Limitations: The study acknowledges that a single portfolio over a decade contains too few crisis days (only 21 surviving the burn-in) to perform robust conditional coverage testing within stressed regimes. Consequently, the analysis cannot definitively determine which window performs best specifically during crises.

Significance and Recommendations
The paper concludes that the lookback window is a source of significant model risk. The choice of $W$ is not merely a technical tuning parameter but a model specification that determines both the magnitude of the risk capital and the pass/fail status of regulatory compliance.

The authors propose four recommendations, prioritized by implementation cost:

Disclose the Quantile Convention: Firms must explicitly state the interpolation method used, as it can determine backtest outcomes, particularly for short windows.
Test Severity and Frequency: Firms should complement exception counts (frequency) with severity tests (e.g., McNeil-Frey) to ensure the model captures the magnitude of breaches, not just their occurrence.
Treat $W$ as a Model Parameter: Firms should report the "envelope" of VaR estimates across multiple windows (e.g., 250, 500, 1,000) as a daily model-risk indicator. Changes to $W$ should be treated as model changes.
Prefer Conditional Models: The paper argues that the "hard problem" of window selection arises because unweighted HS attempts to condition on the current market state using an unweighted average. The authors suggest that volatility-filtered models (e.g., Filtered Historical Simulation) are superior because they explicitly condition risk forecasts on current volatility, rendering the specific choice of a static lookback window less critical.

The authors maintain a modest scope, noting that their findings are based on one portfolio over one decade and that broader cross-asset or simulation studies are required to generalize these findings across different market regimes.

Technical Summary: Does the Choice of Lookback Window Create Model Risk in Historical Value-at-Risk?

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