Combining a Large Pool of Forecasts of Value-at-Risk and Expected Shortfall

Paper: arXiv 2508.16919

Authors: James W. Taylor, Chao Wang

Abstract

We consider the combination of value-at-risk (VaR) and expected shortfall (ES) forecasts when a large pool of candidate forecasts is available. Given the limited literature in this area, we implement a variety of new combining methods. In terms of simplistic methods, in addition to the mean, we consider the median and mode. As a complement to the previously proposed performance-based weighted combinations, we use regularisation to reduce overfitting in the presence of many weights. Treating VaR and ES forecasts jointly as interval forecasts allows the application of adapted interval forecast combination methods, including trimmed means and a mixtures approach based on inferred probability distributions. In an empirical study involving 90 forecasting methods, trimmed mean combinations, the mixtures method, and performance-based weighting delivered particularly strong results. However, greater forecasting accuracy resulted for a pool of just six methods, chosen to ensure diversity, with performance-based weighting producing the best overall performance.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a comprehensive and empirically rigorous framework for combining Value-at-Risk (VaR) and Expected Shortfall (ES) forecasts across a massive candidate pool of 90 methods. It leverages sophisticated joint scoring functions (Fissler-Ziegel / Asymmetric Laplace) alongside regularized and distribution-mixture combination methods. The extensive empirical evaluation across six global stock indices provides robust, highly actionable insights for financial risk managers.

Research Flowchart

  flowchart TD
    A["Goal: Combine Large Pool of VaR & ES Forecasts"] --> B["Data: 90 Base Forecasting Models (Cross-Asset)"]
    
    subgraph C["Combination Methodologies"]
        C1["Simple: Mean, Median, Mode"]
        C2["Regularization: Ridge / Lasso Weighting"]
        C3["Interval & Distributional: Trimmed Means & Mixtures"]
    end

    B --> C
    
    C --> D["Empirical Evaluation: 90 Models vs. 6 Diverse Models"]
    
    D --> E1["Outcome (Large Pool): Trimmed Means, Mixtures & Performance-Weights Excel"]
    D --> E2["Key Finding: Smaller, Diverse Pool (6 Models) + Performance-Based Weighting Yields Best Accuracy"]