Financial institution under inspection using e-values

Is Your Bank Hiding Something? Unveiling the Secrets of Risk Forecasts

Reese Marlowe in Business & Economy December 2025 • 4 min read.

"New backtesting methods use e-values and e-processes to monitor the accuracy of financial risk models and safeguard against underreported risks."

In today's complex financial world, accurately forecasting risk is critical for financial institutions. These forecasts determine the capital reserves needed to manage potential losses and ensure stability. Regulatory bodies are tasked with monitoring these risk forecasts through rigorous testing, known as backtesting, to ensure forecasts are reliable and institutions are not underestimating their true risk exposure.

Traditional backtesting methods, however, face unique challenges. Financial data is rarely straightforward; losses and risk predictions are often neither independent nor identically distributed. Regulators are particularly concerned with underestimation of risk, which could lead to insolvency, while overly conservative forecasts are less of a concern. Moreover, regulators often lack precise knowledge of the sophisticated models used by financial institutions, making it difficult to assess the accuracy of their predictions.

A common approach to backtesting Value-at-Risk (VaR) involves a three-zone method based on binomial tests. This model-free approach directly tests risk forecasts without relying on specific models. However, the recent shift towards Expected Shortfall (ES) as the primary regulatory measure for market risk has created new hurdles. ES, unlike VaR, is not easily backtested, and current methods often require specific assumptions about loss distributions or are limited to fixed data sizes.

E-Values and E-Processes: A New Era in Risk Management

Financial institution under inspection using e-values

To overcome the limitations of traditional backtesting methods, researchers are exploring innovative techniques using e-values and e-processes. These methods, offering advantages over classical statistical tests, provide a powerful framework for regulators to assess the accuracy and reliability of risk forecasts. E-tests, based on e-values, offer several benefits, including greater robustness, the ability to handle complex data dependencies, and the capacity for anytime-valid inference. E-processes can be updated continuously as new data becomes available, providing an ongoing assessment of risk model performance.

The new method defines backtest e-statistics and then constructs e-processes to backtest ES and other risk measures. Backtest e-statistics, formulated into e-processes, uniquely characterize Value-at-Risk (VaR) and Expected Shortfall (ES) using recent results on identification functions. A few criteria for optimally constructing the e-processes are studied for any given backtest e-statistic. A regulator can detect underestimation and can deliberately rewards risk prudence.

E-values: An e-variable for a hypothesis is a random variable whose expected value is less than or equal to 1 under the null hypothesis. An e-test rejects the hypothesis if a realized e-variable, called an e-value, is larger than a given threshold.
E-processes: An e-process is a non-negative stochastic process where the expected value at any stopping time is less than or equal to 1 under the null hypothesis.
Backtest e-statistics: A backtest e-statistic for a risk measure is a function that, when applied to a realized loss and its forecast, produces an e-variable.

In particular settings the approach uses similar forms as mixtures between the constant function equal to 1 and a simple backtest e-statistic. The simulation and data analysis in sections 7 and 8 demonstrate detailed procedures of backtesting VaR and ES using e-values for practical operations of financial regulations.

The Future of Risk Assessment

The e-backtesting method represents a significant advancement in risk management, offering a model-free and non-asymptotic approach to backtesting ES. By leveraging e-values and e-processes, regulators can gain a more robust and timely assessment of risk forecasts, safeguarding financial institutions and the broader economy. As financial models become increasingly complex, these innovative techniques will be essential for maintaining stability and trust in the financial system.

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This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2209.00991,

Title: E-Backtesting

Subject: q-fin.rm math.st stat.me stat.th

Authors: Qiuqi Wang, Ruodu Wang, Johanna Ziegel

Published: 26-08-2022

Everything You Need To Know

What are e-values and e-processes, and how do they improve the accuracy of risk model assessments?

E-values are random variables with an expected value less than or equal to 1 under the null hypothesis; an e-test rejects the hypothesis if a realized e-variable, the e-value, exceeds a threshold. E-processes are non-negative stochastic processes with an expected value at any stopping time less than or equal to 1 under the null hypothesis. These methods enhance risk model assessments by offering greater robustness, the ability to handle complex data dependencies, and capacity for continuous updating as new data becomes available. By formulating backtest e-statistics into e-processes, regulators can more effectively characterize Value-at-Risk (VaR) and Expected Shortfall (ES).

Why are traditional backtesting methods sometimes inadequate for assessing financial risk, especially with the shift towards Expected Shortfall (ES)?

Traditional backtesting methods often struggle because financial data is rarely independent or identically distributed. Underestimation of risk poses a significant concern, potentially leading to insolvency, while regulators may lack comprehensive insight into the sophisticated models used by financial institutions. The shift towards Expected Shortfall (ES) as a primary regulatory measure introduces new challenges because, unlike Value-at-Risk (VaR), ES is not easily backtested. Current methods often require specific assumptions about loss distributions or are limited to fixed data sizes, making them less versatile and reliable for modern risk assessment.

How do backtest e-statistics and e-processes help regulators detect underestimation of risk and encourage prudent risk management?

Backtest e-statistics, when applied to a realized loss and its forecast, produce an e-variable that helps characterize Value-at-Risk (VaR) and Expected Shortfall (ES). Formulating these statistics into e-processes allows regulators to continuously monitor risk model performance. By using these e-processes, regulators can detect underestimation of risk and, crucially, can reward risk prudence. The criteria for optimally constructing these e-processes from backtest e-statistics enable a nuanced approach, allowing for both the identification of potential dangers and the encouragement of responsible forecasting.

What are the key advantages of using e-values and e-processes over classical statistical tests in the context of financial risk management?

E-values and e-processes offer several key advantages over classical statistical tests in financial risk management. These include greater robustness, the ability to handle complex data dependencies, and the capacity for anytime-valid inference. Unlike traditional methods that may require specific assumptions about data distributions or independence, e-values and e-processes provide a model-free and non-asymptotic approach. This is particularly important in the context of Expected Shortfall (ES) backtesting, where classical methods may fall short due to the challenges in directly backtesting ES without making strong assumptions.

How does the e-backtesting method contribute to the future of risk assessment, especially considering the increasing complexity of financial models?

The e-backtesting method, which leverages e-values and e-processes, represents a significant advancement by providing a model-free and non-asymptotic approach to backtesting Expected Shortfall (ES). This method offers regulators a more robust and timely assessment of risk forecasts, thereby safeguarding financial institutions and the broader economy. As financial models grow more complex, the ability to continuously and reliably assess risk becomes essential. By enabling ongoing assessment and adaptation, e-backtesting will be crucial for maintaining stability and trust in the financial system, offering a pathway to address the challenges posed by increasingly sophisticated financial instruments and market dynamics.