Paper: arXiv 2405.02012

Authors: Sullivan Hué, Christophe Hurlin, Yang Lu

Abstract

We propose an original two-part, duration-severity approach for backtesting Expected Shortfall (ES). While Probability Integral Transform (PIT) based ES backtests have gained popularity, they have yet to allow for separate testing of the frequency and severity of Value-at-Risk (VaR) violations. This is a crucial aspect, as ES measures the average loss in the event of such violations. To overcome this limitation, we introduce a backtesting framework that relies on the sequence of inter-violation durations and the sequence of severities in case of violations. By leveraging the theory of (bivariate) orthogonal polynomials, we derive orthogonal moment conditions satisfied by these two sequences. Our approach includes a straightforward, model-free Wald test, which encompasses various unconditional and conditional coverage backtests for both VaR and ES. This test aids in identifying any mis-specified components of the internal model used by banks to forecast ES. Moreover, it can be extended to analyze other systemic risk measures such as Marginal Expected Shortfall. Simulation experiments indicate that our test exhibits good finite sample properties for realistic sample sizes. Through application to two stock indices, we demonstrate how our methodology provides insights into the reasons for rejections in testing ES validity.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative bivariate orthogonal polynomial framework to disentangle duration and severity in Expected Shortfall backtesting. It combines solid mathematical foundations with robust Monte Carlo simulations and empirical applications to stock indices, offering clear actionable insights for model risk management.

Research Flowchart

  flowchart TD
    A[Research Goal: Backtest Expected Shortfall <br>separating VaR violation frequency and severity] --> B[Input Data: Inter-violation durations <br>& violation severities from stock indices]
    B --> C[Methodology: Derive orthogonal moment conditions <br>using bivariate orthogonal polynomials]
    C --> D[Computation: Construct model-free Wald test <br>for unconditional & conditional coverage]
    D --> E[Validation & Application: Run finite-sample simulations <br>& empirical application to stock indices]
    E --> F[Outcomes: Good finite sample properties & <br>ability to pinpoint specific model mis-specifications]

Cited References (86)

  1. The level and quality of Value-at-Risk disclosure by commercial banks (2010) — link
  2. Coherent Measures of Risk (1999) — link
  3. SRISK : A Conditional Capital Shortfall Measure of Systemic Risk (2016) — link
  4. Testing distributional assumptions: A GMM aproach (2012) — link
  5. Hermite polynomial based expansion of European option prices (2014)
  6. A consistent specification test for dynamic quantile models (2022)
  7. Multivariate Jacobi process with application to smooth transitions (2006)
  8. CAViaR: Conditional autoregressive Value-at-Risk by regression quantiles (2004)
and 78 more
  1. Efficiently Backtesting Conditional Value-at-Risk and Conditional Expected Shortfall (2021)
  2. Improved duration-based backtesting of Value-at-Risk (2006)
  3. Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics (2006)
  4. Monitoring Value-at-Risk and Expected Shortfall forecasts (2023)
  5. On certain distribution problems based on positive definite quadratic functions in normal vectors (1966)
  6. On the distribution of a quadratic form in a multivariate normal sample (1966)
  7. An automatic portmanteau test for serial correlation (2009)
  8. Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach (2002)
  9. Skewness and kurtosis in S&P 500 index returns implied by option prices (1996)
  10. Powerful Backtests for Historical Simulation Expected Shortfall Models (2023)
  11. Measuring systemic risk (2017)
  12. Capital shortfall: A new approach to ranking and regulating systemic risks (2012)
  13. International asset allocation under regime switching, skew, and kurtosis preferences (2008)
  14. Co-skewness and Capital Asset Pricing (1980)
  15. Backtesting Marginal Expected Shortfall and related systemic risk measures (2021)
  16. Elicitability and backtesting: Perspectives for banking regulation (2017)
  17. Testing for autocorrelation and unit roots in the presence of conditional heteroskedasticity of unknown form (2001)
  18. Encompassing tests for Value at Risk and Expected Shortfall multistep forecasts based on inference on the boundary (2023)
  19. A spanning series approach to options (2017)
  20. Quantitative Risk Management: Concepts, Techniques and Tools (2005)
  21. Backtesting Systemic Risk Forecasts Using Multi-Objective Elicitability (2024) — link
  22. Admissible ways of merging p-values under arbitrary dependence (2022)
  23. E-backtesting (2023)
  24. Techniques for Verifying the Accuracy of Risk Measurement Models (1995)
  25. Dynamic semiparametric models for Expected Shortfall (and Value-at-Risk) (2019)
  26. Orthogonal Polynomials (1939)
  27. Forecasting Value-at-Risk and expected shortfall using a semiparametric approach based on the asymmetric Laplace distribution (2019)
  28. A joint quantile and Expected Shortfall regression framework (2019)
  29. Robust forecast evaluation of expected shortfall (2020)
  30. What is the best risk measure in practice? A comparison of standard measures (2015)
  31. Recovery rates, default probabilities, and the credit cycle (2010)
  32. Evaluating Interval Forecasts (1998)
  33. New backtests for unconditional coverage of Expected Shortfall (2018)
  34. Option pricing with orthogonal polynomial expansions (2020)
  35. Robust Wald-type tests of one-sided hypotheses in the linear model (1992)
  36. Evaluating Value-at-Risk models via quantile regression (2011)
  37. Backtesting for risk-based regulatory capital (2004)
  38. Backtesting Value-at-Risk and Expected Shortfall in the Presence of Estimation Error (2023)
  39. Backtesting Expected Shortfall and beyond (2021)
  40. Backtesting Expected Shortfall: Accounting for tail risk (2017)
  41. Backtesting general spectral risk measures with application to Expected Shortfall (2015)
  42. Flexible (panel) regression models for bivariate count–continuous data with an insurance application (2019)
  43. Regression-Based Expected Shortfall Backtesting (2022)
  44. Tests for skewness, kurtosis, and normality for time series data (2005)
  45. Trades and quotes: a bivariate point process (2003)
  46. Duration time-series models with proportional hazard (2008)
  47. Some statistical models for limited dependent variables with application to the demand for durable goods (1971)
  48. Much ado about two: Reconsidering retransformation and the two-part model in health econometrics (1998)
  49. Modeling multivariate extreme events using self-exciting point processes (2014)
  50. Backtesting parametric Value-at-Risk with estimation risk (2010)
  51. Interpreting financial market crashes as earthquakes: A new early warning system for medium term crashes (2015)
  52. On binary and categorical time series models with feedback (2014)
  53. Autoregressive conditional duration: A new model for irregularly spaced transaction data (1998)
  54. Zero-Inflated Autoregressive Conditional Duration Model for Discrete Trade Durations with Excessive Zeros (2018) — link
  55. Estimating Value-at-Risk: a point process approach (2005)
  56. Tests of independence in parametric models with applications and illustrations (1993)
  57. A new set of improved Value-at-Risk backtests (2014)
  58. Evaluating Value-at-Risk models with desk-level data (2011)
  59. Multinomial VaR backtests: A simple implicit approach to backtesting Expected Shortfall (2018)
  60. The Risk Map: A new tool for validating risk models (2013)
  61. Evaluating Value-at-Risk forecasts: A new set of multivariate backtests (2016)
  62. Testing density forecasts, with applications to risk management (2001)
  63. Negative binomial autoregressive process with stochastic intensity (2019)
  64. Fast pricing of basket default swaps (2008)
  65. Backtesting Value-at-Risk: A duration-based approach (2004)
  66. The geometric-VaR backtesting method (2016)
  67. Backtesting Expected Shortfall via multi-quantile regression (2020)
  68. Elicitability of Marginal Expected Shortfall and Related Systemic Risk Measures (2022)
  69. Measures of multivariate skewness and kurtosis with applications (1970)
  70. Backtesting Value-at-Risk: a GMM duration-based test (2011)
  71. Moment-based tests under parameter uncertainty (2019)
  72. Spectral measures of risk: A coherent representation of subjective risk aversion (2002) — link
  73. Spectral backtests of forecast distributions with application to risk management (2020)
  74. Spectral measures of risk: A coherent representation of subjective risk aversion (2002)
  75. Model risk of Expected Shortfall (2019)
  76. Testing normality: a GMM approach (2005)
  77. Backtesting Expected Shortfall (2014)
  78. Basel Framework. (2019)

0 of 86 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.