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Bootstrap prediction intervals for VaR and ES in the context of GARCH models

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  • María Rosa Nieto

    ()

  • Esther Ruiz

    ()

Abstract

In this paper, we propose a new bootstrap procedure to obtain prediction intervals of future Value at Risk (VaR) and Expected Shortfall (ES) in the context of univariate GARCH models. These intervals incorporate the parameter uncertainty associated with the estimation of the conditional variance of returns. Furthermore, they do not depend on any particular assumption on the error distribution. Alternative bootstrap intervals previously proposed in the literature incorporate the first but not the second source of uncertainty when computing the VaR and ES. We also consider an iterated smoothed bootstrap with better properties than traditional ones when computing prediction intervals for quantiles. However, this latter procedure depends on parameters that have to be arbitrarily chosen and is very complicated computationally. We analyze the finite sample performance of the proposed procedure and show that the coverage of our proposed procedure is closer to the nominal than that of the alternatives. All the results are illustrated by obtaining one-step-ahead prediction intervals of the VaR and ES of several real time series of financial returns.

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Bibliographic Info

Paper provided by Universidad Carlos III, Departamento de Estadística y Econometría in its series Statistics and Econometrics Working Papers with number ws102814.

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Date of creation: May 2010
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Handle: RePEc:cte:wsrepe:ws102814

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Keywords: Expected Shortfall; Feasible Historical Simulation; Hill estimator; Parameter uncertainty; Quantile intervals; Value at Risk;

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  1. Bams, Dennis & Lehnert, Thorsten & Wolff, Christian C.P., 2005. "An evaluation framework for alternative VaR-models," Journal of International Money and Finance, Elsevier, vol. 24(6), pages 944-958, October.
  2. Basak, Suleyman & Shapiro, Alexander, 2001. "Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices," Review of Financial Studies, Society for Financial Studies, vol. 14(2), pages 371-405.
  3. Hartz, Christoph & Mittnik, Stefan & Paolella, Marc, 2006. "Accurate value-at-risk forecasting based on the normal-GARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 51(4), pages 2295-2312, December.
  4. Manfred Gilli & Evis këllezi, 2006. "An Application of Extreme Value Theory for Measuring Financial Risk," Computational Economics, Society for Computational Economics, vol. 27(2), pages 207-228, May.
  5. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-30, August.
  6. Hang Chan, Ngai & Deng, Shi-Jie & Peng, Liang & Xia, Zhendong, 2007. "Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 137(2), pages 556-576, April.
  7. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
  8. Carlo Acerbi & Dirk Tasche, 2001. "Expected Shortfall: a natural coherent alternative to Value at Risk," Papers cond-mat/0105191, arXiv.org.
  9. Peter Christoffersen & Sílvia Gonçalves, 2004. "Estimation Risk in Financial Risk Management," CIRANO Working Papers 2004s-15, CIRANO.
  10. Pascual, Lorenzo & Romo, Juan & Ruiz, Esther, 2006. "Bootstrap prediction for returns and volatilities in GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 50(9), pages 2293-2312, May.
  11. Yamai, Yasuhiro & Yoshiba, Toshinao, 2005. "Value-at-risk versus expected shortfall: A practical perspective," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 997-1015, April.
  12. Peter Hall & Qiwei Yao, 2003. "Data tilting for time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 425-442.
  13. Jeremy Berkowitz & Lutz Kilian, 1996. "Recent developments in bootstrapping time series," Finance and Economics Discussion Series 96-45, Board of Governors of the Federal Reserve System (U.S.).
  14. Song Xi Chen, 2005. "Nonparametric Inference of Value-at-Risk for Dependent Financial Returns," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 3(2), pages 227-255.
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