Beyond Value at Risk: Forecasting Market Risk with Bayesian Networks

Author: Denis Avetisyan

This research explores the potential of dynamic Bayesian networks to improve the accuracy of expected shortfall calculations, a key metric for modern risk management.

A comparative analysis of dynamic Bayesian networks for standard and stressed expected shortfall, highlighting the impact of distributional assumptions and benchmarking against simpler forecasting models.

Accurate measurement of market risk remains a persistent challenge, particularly in capturing extreme events despite increasingly sophisticated regulatory demands. This research, ‘Extending the application of dynamic Bayesian networks in calculating market risk: Standard and stressed expected shortfall’, investigates the potential of Dynamic Bayesian Networks (DBNs) to improve forecasts of Expected Shortfall (ES) and Stressed ES, key metrics under Basel Accords. While DBNs demonstrate a capacity for modeling complex dependencies, the study finds that simpler models often achieve comparable performance, and distributional assumptions exert a significant influence on forecast accuracy. Future work should explore weighting schemes that better integrate the forward-looking predictions of DBNs into estimates of tail risk.

Beyond Conventional Risk Assessment: The Limits of Statistical Normality

Conventional financial risk modeling frequently depends on the Normal Distribution, a statistical bell curve that assumes extreme events are improbable. This reliance, however, presents a significant weakness; the Normal Distribution systematically underestimates the likelihood of truly extreme market movements, often termed “black swan” events. Historical analysis reveals that market crashes and periods of significant volatility consistently deviate from the predictions of this distribution, as observed during events like the 2008 financial crisis or the COVID-19 pandemic-induced market downturn. The issue stems from the Normal Distribution’s thin tails – it predicts infrequent, small deviations from the mean, failing to account for the heavier tails observed in real-world financial data where large losses occur with greater frequency than predicted. Consequently, risk managers using models built on this assumption may underestimate potential losses and allocate insufficient capital to buffer against adverse events, creating systemic vulnerabilities within the financial system.

The efficiency of models like DeltaNormal, frequently employed in financial risk management, hinges on the assumption of normally distributed asset returns. However, real-world market data often exhibits characteristics – such as skewness and kurtosis, or ‘fat tails’ – that significantly deviate from this ideal. When returns aren’t normally distributed, DeltaNormal can underestimate the likelihood of extreme negative events, leading to a miscalculation of Value at Risk (VaR) and Expected Shortfall (ES). This, in turn, results in inadequate capital allocation, potentially leaving institutions unprepared for substantial losses during periods of market stress. Consequently, reliance on such models necessitates careful consideration of their limitations and the potential for underestimation when facing non-normal return distributions, prompting the need for more robust alternatives or adjustments to account for these deviations.

Analysis of the SP500Index reveals return patterns that consistently diverge from the assumptions of normality, a characteristic demanding the implementation of more sophisticated risk assessment methodologies. Despite this non-normality, a recent study demonstrated a surprising result: when forecasting 10-day $97.5\%$ Expected Shortfall (ES) and Stressed Expected Shortfall (SES) – key measures of potential extreme losses – the simpler Normal Distribution frequently outperformed the more complex skewed Student’s t distribution. This counterintuitive finding suggests that, while the underlying returns aren’t normally distributed, the Normal Distribution can still provide reasonably accurate estimates of extreme downside risk in this specific context, challenging the automatic adoption of more elaborate models and highlighting the importance of careful model validation.

Expanding the Risk Toolkit: Advanced Distributions and Modeling Approaches

The Skewed Student’s t-distribution ($t_{\alpha, \beta}$) improves upon the normal distribution’s representation of financial asset returns by incorporating parameters that control both the shape of the distribution and its asymmetry. Traditional financial models often assume normality, which underestimates the probability of extreme events due to its light tails. The Student’s t-distribution, with its degrees of freedom parameter, inherently possesses heavier tails, allowing for a higher probability of observing large price movements. Furthermore, the skewness parameter in the Skewed Student’s t-distribution enables the model to capture the empirical observation that negative returns often have a greater magnitude and frequency than positive returns of the same size, a phenomenon frequently observed in financial markets. This combination of features allows for more accurate risk assessment and portfolio optimization compared to models relying on the normal distribution assumption.

Historical Simulation and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models provide adaptable approaches to risk management by addressing limitations of traditional methods that assume normal distributions and constant volatility. Historical Simulation, a non-parametric technique, directly utilizes past returns to simulate potential future outcomes, effectively capturing non-normality without pre-defined distributional assumptions. GARCH models, conversely, explicitly model time-varying volatility, accounting for volatility clustering-the tendency for periods of high volatility to be followed by periods of high volatility and vice versa. However, both methods can present significant computational burdens. Historical Simulation requires storing and processing large historical datasets, while GARCH models, particularly those of higher order or incorporating exogenous variables, necessitate iterative estimation procedures and can be sensitive to parameter selection, increasing processing time and resource requirements.

Dynamic Bayesian Networks (DBNs) represent a probabilistic graphical modeling approach to risk assessment that integrates causal relationships with empirical data. These networks utilize directed acyclic graphs to model dependencies between variables, allowing for the propagation of uncertainty and the forecasting of risk metrics. While DBNs offer a theoretically robust framework for capturing complex interdependencies, our analysis demonstrated a limited practical benefit in forecasting accuracy. Specifically, comparative testing revealed only a minimal percentage improvement in predictive performance when DBNs were benchmarked against simpler, more computationally efficient models such as time series analysis or GARCH models. This suggests that, despite their sophisticated structure, the added complexity of DBNs may not consistently translate into substantial gains in forecasting capability for the analyzed financial data.

Validating Model Integrity: Backtesting and Performance Evaluation

MinimallyBiasedBacktest and DuEscancianoBacktest are essential techniques for validating risk model forecasts by addressing inherent biases in traditional backtesting procedures. Traditional methods can overestimate model performance due to factors like data snooping and selection bias. MinimallyBiasedBacktest mitigates these issues by employing resampling techniques to create multiple out-of-sample datasets, thereby generating a distribution of performance metrics and providing a more robust assessment. DuEscancianoBacktest specifically addresses the issue of multiple hypothesis testing by controlling the Family-Wise Error Rate (FWER), ensuring that observed performance is statistically significant and not a result of chance. Both methods contribute to increased confidence in the accuracy and reliability of risk forecasts, particularly in regulatory contexts where statistical validity is paramount.

The BCBSTrafficLightTest (TLT) serves as a standardized regulatory assessment used to determine the acceptability of risk models for capital calculation purposes. Developed by the Basel Committee on Banking Supervision (BCBS), the TLT evaluates a model’s performance across various dimensions, including model accuracy, stability, and implementation rigor. The test employs a series of statistical checks designed to identify models that systematically underestimate risk exposures, potentially leading to insufficient capital reserves. Passing the TLT is generally a prerequisite for regulatory approval and ongoing use of a risk model, ensuring financial institutions maintain adequate capital buffers and contribute to overall financial system stability. Failure to meet the TLT criteria triggers a review process and may necessitate model remediation or restriction.

Quantitative evaluation of risk model accuracy relies on several performance metrics; Mean Absolute Error ($MAE$), Root Mean Square Error ($RMSE$), and Mean Absolute Percentage Error ($MAPE$) are commonly used to assess the magnitude of forecast errors. Backtesting results indicated zero breaches for the 10-day 97.5% Stress Event Simulation ($SES$) across all models and distributions tested. In contrast, the 10-day 97.5% Expected Shortfall ($ES$) exhibited a breach range of 0 to 3, suggesting a potentially higher sensitivity or different error profile compared to the $SES$ metric. These results highlight the importance of considering multiple metrics and understanding their individual characteristics when evaluating model performance and identifying areas for refinement.

Adaptive Risk Management: Network Learning for Dynamic Systems

SemiInterleaved HITON (Hierarchical Interventional Test on Networks) is a structure learning algorithm designed for Dynamic Bayesian Networks (DBNs). It employs an iterative process, alternating between Hiton-based interventions and network structure optimization. Core to its functionality are techniques like MaxMinHillClimbing, a local search algorithm that maximizes a scoring function while minimizing the risk of getting stuck in local optima, and PeterClarkStable, a stability selection method used to identify robust network structures. This combination allows SemiInterleaved HITON to efficiently explore the space of possible network structures and identify causal relationships, particularly in DBNs where relationships evolve over time. The algorithm’s hierarchical approach further improves efficiency by focusing on relevant network components, enabling it to scale to more complex datasets and dynamic systems.

SemiInterleavedHITON, utilizing algorithms such as MaxMinHillClimbing and PeterClarkStable, facilitates the discovery of non-linear and time-varying causal dependencies within financial time series data. This capability is crucial for adapting to evolving market dynamics, as relationships between assets and risk factors are rarely static. The algorithms dynamically adjust network structures to reflect these changes, enabling more accurate forecasts by capturing previously unmodeled dependencies. This adaptive learning process contributes to forecast robustness by reducing the impact of structural breaks and ensuring the model remains relevant under shifting market conditions. The resulting dynamic Bayesian Networks provide a more responsive and reliable foundation for risk assessment and portfolio optimization compared to static models.

Integration of network learning techniques with advanced risk measures, specifically Stressed Expected Shortfall (SES) and Expected Shortfall (ES), enables the development of a dynamic risk management framework. However, empirical results demonstrate a performance trade-off when utilizing the skewed Student’s t distribution for calculating these measures. Testing revealed a 9-67% decrease in forecast accuracy for ES and a 9-557% decrease for SES, relative to forecasts based on the Normal distribution. This suggests that while the skewed Student’s t distribution may offer benefits in modeling tail risk under certain conditions, its application within this combined network learning and risk measurement system requires careful consideration due to the observed accuracy degradation.

Towards Proactive Resilience: The Future of Financial Risk Management

Contemporary risk management is evolving beyond reliance on static models, embracing instead dynamic systems powered by the integration of advanced network learning, robust risk measures, and rigorous backtesting. This approach leverages the ability of network learning to identify complex interdependencies and propagate risk assessments across interconnected systems – crucial in today’s globally linked financial landscape. By combining this with established quantitative risk measures – such as Value at Risk and Expected Shortfall – and subjecting these models to stringent historical backtesting, institutions can create systems that not only assess current risk exposure but also adapt to changing market conditions and anticipate emerging threats. The result is a proactive capability to adjust portfolios and hedging strategies in real-time, ultimately bolstering financial stability and resilience against unforeseen events and systemic shocks.

Financial institutions are increasingly focused on shifting from reactive to proactive risk management, recognizing that anticipating potential threats is crucial for maintaining stability. This forward-looking strategy involves employing sophisticated analytical tools to identify emerging risks – from geopolitical shifts and technological disruptions to evolving market behaviors – before they fully materialize. By implementing preemptive measures, such as stress testing and scenario planning, institutions can build resilience and minimize potential losses. This isn’t simply about avoiding negative outcomes; it’s about capitalizing on opportunities that arise from a changing landscape, fostering a more sustainable and secure financial system capable of weathering unforeseen challenges and maintaining public trust.

The convergence of advanced network learning, robust risk measurement, and rigorous backtesting strategies heralds a transformative shift in financial risk management. This isn’t merely an incremental improvement, but a fundamental restructuring of how institutions approach uncertainty. By moving beyond the limitations of static models, these combined techniques facilitate the creation of dynamic systems capable of anticipating and adapting to evolving threats. Such a proactive stance doesn’t simply minimize potential losses; it fosters a more resilient and sustainable financial ecosystem, better equipped to withstand systemic shocks and promote long-term stability. The resulting systems promise to not only safeguard capital but also unlock new opportunities by accurately assessing and managing complex risks, paving the way for innovation and responsible growth.

The research demonstrates a nuanced understanding of model complexity in forecasting market risk. While Dynamic Bayesian Networks offer a sophisticated approach to calculating Expected Shortfall, the study reveals that their performance doesn’t consistently surpass that of simpler models. This echoes a fundamental principle of robust system design – elegance often resides in simplicity. As Marie Curie wisely stated, “Nothing in life is to be feared, it is only to be understood.” The pursuit of understanding, as evidenced by this investigation into forecasting methodologies, suggests that striving for overly complex solutions isn’t always necessary; a clear grasp of core principles and distributional assumptions-as highlighted in the article-often yields the most reliable results.

The Road Ahead

The pursuit of accurate market risk forecasting, as demonstrated by this work with Dynamic Bayesian Networks, continually reveals a fundamental truth: every new dependency is the hidden cost of freedom. While DBNs offer a potentially elegant framework for capturing complex interrelationships, the research suggests that their advantage over simpler, more parsimonious models is not always self-evident. The structural complexity must genuinely justify its computational cost and the increased risk of mis-specification. A model’s ability to appear sophisticated does not guarantee improved predictive power.

The sensitivity of forecast accuracy to distributional assumptions deserves particular attention. This highlights a core issue: the tail, often dismissed as statistical noise, dictates the very nature of risk assessment. Focusing solely on model architecture risks obscuring the critical need for robust, empirically-grounded assumptions about the underlying data-generating process. A beautifully constructed machine learning model built upon flawed foundations will inevitably reflect those flaws.

Future work should therefore prioritize a more holistic view. Rather than striving for ever-increasing model complexity, the field might benefit from a renewed emphasis on fundamental statistical principles and a rigorous examination of the limitations inherent in any attempt to predict the unpredictable. The goal is not to eliminate uncertainty, but to understand its structure and account for its influence, accepting that the most elegant solution is often the simplest one.

Original article: https://arxiv.org/pdf/2512.12334.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/