Paper: arXiv 2407.06619

Authors: Federico Gatta, Fabrizio Lillo, Piero Mazzarisi

Abstract

In financial risk management, Value at Risk (VaR) is widely used to estimate potential portfolio losses. VaR’s limitation is its inability to account for the magnitude of losses beyond a certain threshold. Expected Shortfall (ES) addresses this by providing the conditional expectation of such exceedances, offering a more comprehensive measure of tail risk. Despite its benefits, ES is not elicitable on its own, complicating its direct estimation. However, joint elicitability with VaR allows for their combined estimation. Building on this, we propose a new methodology named Conditional Autoregressive Expected Shortfall (CAESar), inspired by the CAViaR model. CAESar handles dynamic patterns flexibly and includes heteroskedastic effects for both VaR and ES, with no distributional assumption on price returns. CAESar involves a three-step process: estimating VaR via CAViaR regression, formulating ES in an autoregressive manner, and jointly estimating VaR and ES while ensuring a monotonicity constraint to avoid crossing quantiles. By employing various backtesting procedures, we show the effectiveness of CAESar through extensive simulations and empirical testing on daily financial data. Our results demonstrate that CAESar outperforms existing regression methods in terms of forecasting performance, making it a robust tool for financial risk management.

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 introduces CAESar, a novel three-step joint estimation framework for VaR and ES that effectively solves quantile crossing via soft monotonicity constraints. It features significant statistical rigor, incorporating comprehensive backtesting protocols and robust comparative empirical evaluations against state-of-the-art benchmarks.

Research Flowchart

  flowchart TD
    A[Goal: Enhance Tail Risk Forecasting<br/>via Joint VaR & ES Estimation] --> B[Input: Daily Financial Returns & Portfolio Data]
    B --> C[Step 1: Estimate VaR via CAViaR Regression]
    C --> D[Step 2: Formulate Autoregressive ES<br/>with Monotonicity Constraint]
    D --> E[Step 3: Joint Optimization<br/>via Joint Elicitability Function]
    E --> F[Validation: Simulations & Backtesting Procedures]
    F --> G[Outcome: CAESar Outperforms Existing<br/>Methods Without Distribution Assumptions]

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1 of 54 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.