Measuring Portfolio Risk Using Machine Learning Techniques: VaR and ES Using Gaussian Process Regression

Abstract

This study explores and addresses the common and practical challenge encountered in regulatory risk modelling - modeling and forecasting time-varying Volatility-Covariance Matrices VCV for regulatory-compliant stress testing processes like Internal Capital Adequacy Assessment Process (ICAAP) and CCAR (Comprehensive Capital Analysis and Review). In practice, this challenge is most acute under forward-looking stress conditions, where model stability and regulatory defensibility take precedence over theoretical elegance. Traditional econometric models often fail to capture the complex, non-linear dependencies and high dimensionality of real-world financial markets, leading to unstable and/or wrong stress test results. Attempts made to address this issue using Multi-Output Gaussian Process Regression (MOGPR) resulted in unstable VCV matrices and convergence issues with the optimizer. Keeping this in mind, this research pivots and proposes the implementation of a novel Hybrid GPR-VCV Framework that leverages the power of Machine Learning (ML) through Univariate Gaussian Process Regression (UGPR) to model individual asset volatilities, while maintaining regulatory tractability by utilizing historical correlation matrices for the covariance structure, for both Value-at-Risk (VaR) and Stressed VaR (SVaR). The framework is dynamically calibrated and tested using a sequential forward-chaining methodology. This approach utilizes an expanding training window-where all historical data from the fixed start date of June 30, 2020, is accumulated-to forecast a fixed-size test set (approximately 252 daily returns, or one trading year) through the total sample period ending June 30, 2025, on seven key global equity indices (Developed and Emerging Markets). This iterative process facilitates the forward-looking stress testing, which subjects the model to three hypothetical, ICAAP-style stress scenarios: West Asia War, Climate Risk, and AI Bubble / Regulatory Burden. The study investigates four primary research problems, including the stability and efficiency of the model, the impact of dynamic paths, and the interpretation of risk drivers using SHAP (SHapley Additive exPlanations). The key finding is that the Hybrid GPR-VCV approach provides statistically stable, out-of-sample volatility forecasts even with a short one-year fit window. This level of stability was not achievable using the standard Multi-Output GPR (MOGPR) framework. The model demonstrates superior resilience and accuracy in projecting market losses under dynamic stress, and the SHAP analysis provides crucial interpretability, revealing the specific non-linear input features driving volatility forecasts. This hybrid methodology represents a significant advancement for risk management, offering a robust, interpretable, and computationally efficient tool for forward-looking stress testing in financial institutions.