Paper: arXiv 2610.17681

Authors: Ruodu Wang, Jingcheng Yu

Abstract

Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR structure. Under the stated assumptions, both control false rejections at any stopping time. For the time-series method, validity requires a fixed parameter–quantile pair representing the conditional VaR path and a positive report margin. The time-series statistic takes an infimum over possible parameter and VaR values. We choose the betting process $(λ_t)$ by maximizing an empirical growth rate averaged across candidate parameters. We compute the statistic by continuous VaR profiling on a dense parameter grid. For grid approximations, we prove convergence to the continuous infimum, together with convergence of the probability of false rejection, as the grid is refined over a fixed monitoring period. Lower and upper bounds for the VaR infimum on the parameter grid control the accuracy of the numerical profiling. Simulations, including ES reports that change from over-reporting to under-reporting, and an application to daily NASDAQ losses provide numerical comparisons with VaR–ES e-processes and fixed-sample ES backtests, assessing detection frequencies and the timing of rejections.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and rigorous approach to a challenging problem in risk management. The novel e-backtesting methodology for Expected Shortfall without explicit VaR forecasts, especially for time-series models, is a significant contribution. The paper demonstrates strong empirical validation through simulations and a real-world application.

Research Flowchart

  flowchart TD
    A[Research Goal: E-Backtest ES without VaR Forecasts] --> B{Methodology: Two E-Backtesting Methods};
    B -- For i.i.d. Losses --> C1[Method 1: i.i.d. Losses Only];
    B -- For Time Series --> C2[Method 2: AR-GARCH Conditional VaR];
    C2 -- Requires: Fixed Param-Quantile Pair & Positive Report Margin --> D[Computational Process: Infimum over Params & VaR];
    D -- Betting Process ($\lambda_t$) --> E[Maximize Empirical Growth Rate (Avg. across Params)];
    E -- VaR Profiling --> F[Compute Statistic: Continuous VaR Profiling on Dense Grid];
    F -- Inputs: Observed Losses, ES Forecasts --> G{Outcomes: Valid Backtests, Numerical Comparisons};
    G -- Control False Rejections at Any Stopping Time --> H[Convergence to Continuous Infimum & Lower/Upper Bounds for VaR Infimum];
    H --> I[Simulations & NASDAQ Application: Assess Detection Frequencies & Rejection Timing];