Paper: arXiv 2502.20978

Abstract

A method for quantile-based, semi-parametric historical simulation estimation of multiple step ahead Value-at-Risk (VaR) and Expected Shortfall (ES) models is developed. It uses the quantile loss function, analogous to how the quasi-likelihood is employed by standard historical simulation methods. The returns data are scaled by the estimated quantile series, then resampling is employed to estimate the forecast distribution one and multiple steps ahead, allowing tail risk forecasting. The proposed method is applicable to any data or model where the relationship between VaR and ES does not change over time and can be extended to allow a measurement equation incorporating realized measures, thus including Realized GARCH and Realized CAViaR type models. Its finite sample properties, and its comparison with existing historical simulation methods, are evaluated via a simulation study. A forecasting study assesses the relative accuracy of the 1% and 2.5% VaR and ES one-day-ahead and ten-day-ahead forecasting results for the proposed class of models compared to several competitors.

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 employs advanced econometric theory including quantile regression, elicitability, and semi-parametric filtering methods, placing it in the high mathematical complexity range. It demonstrates strong empirical rigor through a detailed simulation study and a comprehensive forecasting study comparing VaR and ES across multiple horizons and competitors.

Research Flowchart

  flowchart TD
  A["<b>Research Goal</b><br/>Forecast multi-step VaR & ES<br/>via quantile historical simulation"] --> B["<b>Key Methodology</b><br/>Quantile Scaling + Historical Simulation<br/>Estimate Quantile Loss → Scale Returns → Resample"]
  B --> C["<b>Data & Inputs</b><br/>Financial Return Series<br/>Quantile Levels: 1% & 2.5%"]
  C --> D["<b>Computational Process</b><br/>1. Estimate Quantile Series<br/>2. Scale Returns<br/>3. Resample for 1- & 10-Step Ahead Distributions"]
  D --> E["<b>Key Findings</b><br/>Validated via Simulation & Forecasting Studies<br/>Competitive Performance vs. Standard Methods"]

Cited References (216)

  1. Virtual Historical Simulation for estimating the conditional VaR of large portfolios (2020) — link
  2. Minimum capital requirements for market risk (2019) — link
  3. MAR Calculation of RWA for market risk (2023) — link
  4. Regression Shrinkage and Selection via the Lasso (1996)
  5. Risk-parameter estimation in volatility models (2015)
  6. Multivariate Garch Models: A Survey (2006)
  7. Techniques for verifying the accuracy of risk measurement models (1995)
  8. Evaluating interval forecasts (1998)
and 208 more
  1. Evaluating value-at-risk models via quantile regression (2011)
  2. Regression-based expected shortfall backtesting (2022)
  3. Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? (2007) — link
  4. A New Class of Multivariate Skew Densities, With Application to Generalized Autoregressive Conditional Heteroscedasticity Models (2005) — link
  5. GARCH models: structure, statistical inference and financial applications (2019)
  6. A New Class of Multivariate Skew Densities, With Application to Generalized Autoregressive Conditional Heteroscedasticity Models (2005) — link
  7. On Bayesian Modeling of Fat Tails and Skewness (1998)
  8. Optimization of conditional value-at-risk (2000) — link
  9. An alternative approach to portfolio selection (2002)
  10. Forecasting VaR and ES using a joint quantile regression and its implications in portfolio allocation (2021) — link
  11. Computer Science and Applied Mathematics (1990) — link
  12. Computer Science and Applied Mathematics (1984) — link
  13. Dynamic conditional correlation: on properties and estimation (2013)
  14. Expected Shortfall: A Natural Coherent Alternative to Value at Risk (2002) — link
  15. The structure and degree of dependence: A quantile regression approach (2013)
  16. Bayesian tail risk interdependence using quantile regression (2015)
  17. VAR for VaR: Measuring tail dependence using multivariate regression quantiles (2015)
  18. Coherent Measures of Risk (1999) — link
  19. Thinking coherently (1997)
  20. Generalized autoregressive conditional heteroskedasticity (1986)
  21. Quasi-maximum likelihood estimation for conditional quantiles (2005) — link
  22. Evaluation and Combination of Conditional Quantile Forecasts (2005)
  23. Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting (2015) — link
  24. The model confidence set (2011)
  25. Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules (2011) — link
  26. “Backtesting Expected Shortfall. (2014)
  27. Studies of stock market volatility changes (1976)
  28. Conditional heteroskedasticity in asset returns: A new approach (1991)
  29. Elicitability and backtesting: Perspectives for banking regulation (2017) — link
  30. Bayesian modeling and forecasting of Value-at-Risk via threshold realized volatility (2019)
  31. Asymmetric nonlinear smooth transition GARCH models (1999)
  32. On a threshold model (1978)
  33. Non-linear time series: a dynamical system approach (1990)
  34. On a double-threshold autoregressive heteroscedastic time series model (1996)
  35. Volatility forecasting using threshold heteroskedastic models of the intra-day range (2008)
  36. Forecasting volatility with asymmetric smooth transition dynamic range models (2012)
  37. Good volatility, bad volatility: Signed jumps and the persistence of volatility (2015)
  38. Forecasting volatility in financial markets: A review (2003)
  39. Evaluating Value-at-Risk Models via Quantile Regression (2011) — link
  40. Backtesting for risk-based regulatory capital (2004) — link
  41. Bayesian Expected Shortfall Forecasting Incorporating the Intraday Range (2016) — link
  42. Semi-parametric expected shortfall forecasting in financial markets (2017) — link
  43. On the relation between the expected value and the volatility of the nominal excess return on stocks (1993)
  44. Smooth-transition GARCH models (1998)
  45. Goodness of Fit and Related Inference Processes for Quantile Regression (1999)
  46. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach (2000) — link
  47. Quasi-maximum likelihood estimation for conditional quantiles (2005) — link
  48. Semi-parametric dynamic asymmetric laplace models for tail risk forecasting, incorporating realized measures (2020)
  49. Converting Tail-VaR to VaR: An Econometric Study (2012) — link
  50. VAR: Understanding and Applying Value-at Risk (1997) — link
  51. Dynamic semiparametric models for expected shortfall (and Value-at-Risk) (2019) — link
  52. GENERALIZED AUTOREGRESSIVE SCORE MODELS WITH APPLICATIONS (2013) — link
  53. TERES - Tail Event Risk Expectile based Shortfall (2015) — link
  54. Higher order elicitability and Osband’s principle (2016) — [link](https://doi.org/DOI : 10.1214/16-AOS1439)
  55. The combination of forecasts using changing weights (1994)
  56. Bayesian Semi-parametric Realized-CARE Models for Tail Risk Forecasting Incorporating Realized Measures (2016) — link
  57. Dynamic Semiparametric Models for Expected Shortfall (and Value-at-Risk) (2017) — link
  58. Assessing value at risk with CARE, the Conditional Autoregressive Expectile models (2009) — link
  59. Estimating Value at Risk and Expected Shortfall Using Expectiles (2008) — link
  60. A comparison of financial duration models via density forecasts (2004) — link
  61. Regulatory evaluation of value-at-risk models (1997)
  62. Testing density forecasts, with applications to risk management (2001)
  63. Backtesting Value-at-Risk: A Duration-Based Approach (2004) — link
  64. Evaluating Density Forecasts with Applications to Financial Risk Management (1998)
  65. VQuality Control in VaR: Understanding and Applying Value’at’Risk (1997)
  66. Evaluating Value-at-Risk Models with Desk-Level Data (2011) — link
  67. Evaluating interval forecasts (1998)
  68. Conditional Quantile Estimation and Inference for Arch Models (1996)
  69. Backtesting Value-at-Risk: From Dynamic Quantile to Dynamic Binary Tests (2012) — link
  70. Least absolute deviations: theory, applications, and algorithms (1983) — link
  71. Autoregressive conditional duration: a new model for irregularly spaced transaction data (1998)
  72. New frontiers for ARCH models (2002)
  73. Handbook of volatility models and their applications (2012)
  74. Stationarity of GARCH processes and of some nonnegative time series (1992) — link
  75. Strict Stationarity of Generalized Autoregressive Processes (1992)
  76. Intra-daily volume modeling and prediction for algorithmic trading (2011)
  77. Capturing the zero: a new class of zero-augmented distributions and multiplicative error processes (2013)
  78. Financial econometric analysis at ultra-high frequency: Data handling concerns (2006)
  79. Intraday and interday volatility in the Japanese stock market (2000)
  80. A dynamic component model for forecasting high-dimensional realized covariance matrices (2017) — link
  81. Forecasting Comparison of Long Term Component Dynamic Models for Realized Covariance Matrices (2016)
  82. Forecasting intraday volatility in the US equity market. Multiplicative component GARCH (2012) — link
  83. Stochastic vector difference equations with stationary coefficients (1995)
  84. MIDAS regressions: Further results and new directions (2007)
  85. ECONOMETRIC ANALYSIS OF VOLATILITY COMPONENT MODELS (2015)
  86. The spline-GARCH model for low-frequency volatility and its global macroeconomic causes (2008)
  87. A component GARCH model with time varying weights (2009)
  88. Stock market volatility and macroeconomic fundamentals (2013)
  89. A multiple indicators model for volatility using intra-daily data (2006)
  90. Asymptotic theory for the GARCH (1, 1) quasi-maximum likelihood estimator (1994)
  91. Modelling volatility by variance decomposition (2013)
  92. Comparison of volatility measures: a risk management perspective (2010)
  93. Forecasting realized volatility with changing average levels (2015)
  94. Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression (1991)
  95. High Moment Variations and Their Application (2013) — link
  96. CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles (2004) — link
  97. Forecasting financial volatilities with extreme values: the conditional autoregressive range (CARR) model (2005)
  98. Duration, volume and volatility impact of trades (2005)
  99. Vector multiplicative error models: representation and inference (2006)
  100. A mixture multiplicative error model for realized volatility (2006)
  101. Time series modeling of daily log-price ranges for CHF/USD and USD/GBP (2007)
  102. Multiplicative error models (2011)
  103. Modeling and forecasting realized volatility (2003)
  104. A simple approximate long-memory model of realized volatility (2009)
  105. Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility (2007)
  106. Predicting volatility: getting the most out of return data sampled at different frequencies (2006)
  107. A multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries (2008)
  108. Asymmetric effects and long memory in the volatility of Dow Jones stocks (2009)
  109. Forecasting realized volatility using a long-memory stochastic volatility model: estimation, prediction and seasonal adjustment (2006)
  110. Long memory and nonlinearities in realized volatility: a Markov switching approach (2012)
  111. Predicting Bid–Ask Spreads Using Long-Memory Autoregressive Conditional Poisson Models (2013)
  112. VWAP strategies (2002)
  113. Autoregressive conditional density estimation (1994)
  114. Improving VWAP strategies: A dynamic volume approach (2008)
  115. On the bias in flexible functional forms and an essentially unbiased form: the Fourier flexible form (1981)
  116. Fractals and intrinsic time: A challenge to econometricians (1993)
  117. A forecast comparison of volatility models: does anything beat a GARCH (1, 1)? (2005)
  118. Comparing predictive accuracy (1995)
  119. Chapter 36 Large sample estimation and hypothesis testing (1994) — link
  120. A reality check for data snooping (2000)
  121. Stepwise multiple testing as formalized data snooping (2005)
  122. A test for superior predictive ability (2005)
  123. Tests of conditional predictive ability (2006)
  124. Correction to “Automatic block-length selection for the dependent bootstrap” by D. Politis and H. White (2009)
  125. Fitting vast dimensional time-varying covariance models (2021)
  126. Multivariate simultaneous generalized ARCH (1995)
  127. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models (2002)
  128. Inference in Arch and Garch Models with Heavy–Tailed Errors (2003) — link
  129. Dynamic Equicorrelation (2012) — link
  130. Forecasting intraday volatility in the us equity market. multiplicative component GARCH (2012)
  131. Semiparametric vector MEM (2013)
  132. Assessing the risk of liquidity suppliers on the basis of excess demand intensities (2003)
  133. Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution (2019) — link
  134. Asymmetric least squares regression estimation: A nonparametric approach (1996) — link
  135. Estimating Value at Risk and Expected Shortfall Using Expectiles (2008) — link
  136. Forecast combinations for value at risk and expected shortfall (2020)
  137. Modeling Autoregressive Conditional Skewness and Kurtosis with Multi-Quantile CAViaR (2010) — link
  138. The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR (2008) — link
  139. Autoregresive conditional volatility, skewness and kurtosis (2005) — link
  140. Autoregressive Conditional Kurtosis (2005) — link
  141. Quantile Regression (2005)
  142. A semiparametric nonlinear quantile regression model for financial returns (2017) — link
  143. Bayesian Time-Varying Quantile Forecasting for Value-at-Risk in Financial Markets (2011) — link
  144. The Distribution of Realized Exchange Rate Volatility (2001) — link
  145. Bayesian Semi-parametric Realized Conditional Autoregressive Expectile Models for Tail Risk Forecasting (2021)
  146. Dynamic Models for Volatility and Heavy Tails (2013)
  147. Fundamental review of the trading book: A revised market risk framework (2013) — link
  148. Minimum capital requirements for market risk (2019) — link
  149. Semi-parametric Conditional Quantile Models for Financial Returns and Realized Volatility (2014) — link
  150. Dynamic Semiparametric Models for Expected Shortfall (and Value-at-Risk) (2018) — link
  151. On the Estimation of Production Frontiers: Maximum Likelihood Estimation of the Parameters of a Discontinuous Density Function (1976)
  152. Elicitability and backtesting: Perspectives for banking regulation (2017) — link
  153. Backtesting Derivative Portfolios with Filtered Historical Simulation (FHS) — link
  154. Forecasting risk via realized GARCH, incorporating the realized range (2016)
  155. Exponential GARCH modeling with realized measures of volatility (2016)
  156. A conditionally heteroskedastic time series model for speculative prices and rates of return (1987)
  157. Weak convergence and optimal scaling of random walk Metropolis algorithms (1997)
  158. Making and evaluating point forecasts (2011)
  159. Dynamic quantile function models (2017) — link
  160. Constrained statistical inference: Inequality, order and shape restrictions (2005)
  161. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation (1982)
  162. Nonlinear methods in econometrics (1972)
  163. Modeling long memory volatility using realized measures of volatility: A realized HAR GARCH model (2016)
  164. Modelling non-linear economic relationships (1993)
  165. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts (1998)
  166. Bayesian tail-risk forecasting using realized GARCH (2017)
  167. On Bayesian modeling of fat tails and skewness (1998)
  168. Quantile forecasts of financial returns using realized GARCH models (2012)
  169. Realized GARCH: A joint model for returns and realized measures of volatility (2012)
  170. HARK the SHARK: Realized Volatility Modeling with Measurement Errors and Nonlinear Dependencies* (2019) — link
  171. The Volatility of Realized Volatility (2008) — link
  172. Volatility forecast comparison using imperfect volatility proxies (2011) — link
  173. Jump-robust volatility estimation using nearest neighbor truncation (2012)
  174. Econometric analysis of realized volatility and its use in estimating stochastic volatility models (2002)
  175. Jump-robust volatility estimation using nearest neighbor truncation (2012)
  176. A tale of two time scales (2012)
  177. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise (2008)
  178. Financial econometric analysis at ultra-high frequency: Data handling concerns (2006)
  179. Does anything beat 5-minute RV? A comparison of realized measures across multiple asset classes (2015)
  180. The model confidence set (2011)
  181. Exploiting the errors: A simple approach for improved volatility forecasting (2016)
  182. Power and bipower variation with stochastic volatility and jumps (2004)
  183. Forecasting volatility using high frequency data (2011)
  184. Realising the future: forecasting with high-frequency-based volatility (HEAVY) models (2010)
  185. Econometric analysis of multivariate realised QML: efficient positive semi-definite estimators of the covariation of equity prices (2016)
  186. A ROBUST NEIGHBORHOOD TRUNCATION APPROACH TO ESTIMATION OF INTEGRATED QUARTICITY (2014)
  187. The distribution of realized exchange rate volatility (2001)
  188. A tale of two time scales: Determining integrated volatility with noisy high-frequency data (2005)
  189. The relative contribution of jumps to total price variance (2005)
  190. Econometrics of testing for jumps in financial economics using bipower variation (2006)
  191. Realized kernels in practice: Trades and quotes (2009)
  192. Microstructure noise in the continuous case: the pre-averaging approach (2009)
  193. On Information and Sufficiency (1951)
  194. Semi-parametric financial risk forecasting incorporating multiple realized measures — → 2402.09985
  195. CAViaR models for Value-at-Risk and Expected Shortfall with long range dependency features (2023) — link
  196. Tail risk forecasting of realized volatility CAViaR models (2023) — link
  197. Forecasting Expected Shortfall and Value-at-Risk With Cross-Sectional Aggregation (2025) — link
  198. Sensitivity analysis of Values at Risk (2000) — link
  199. Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall (2004) — link
  200. Nonparametric Estimation of Expected Shortfall (2007) — link
  201. Semi-parametric dynamic asymmetric Laplace models for tail risk forecasting, incorporating realized measures (2020) — link
  202. ESTIMATION RISK IN GARCH VaR AND ES ESTIMATES (2008)
  203. A Comparison of Value-at-Risk Models in Finance (2004)
  204. Value at risk models in finance (2001) — link
  205. Estimating multivariate volatility models equation by equation (2016)
  206. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance (2001)
  207. Evaluating Value-at-Risk Models via Quantile Regression (2011)
  208. Asymmetric Least Squares Estimation and Testing (1987)

1 of 216 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.