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Backtesting Value-at-Risk: A Duration-Based Approach

Listed author(s):
  • Peter Christoffersen
  • Denis Pelletier

Financial risk model evaluation or backtesting is a key part of the internal model's approach to market risk management as laid out by the Basle Commitee on Banking Supervision (1996). However, existing backtesting methods such as those developed in Christoffersen (1998), have relatively small power in realistic small sample settings. Methods suggested in Berkowitz (2001) fare better, but rely on information such as the shape of the left tail of the portfolio return distribution, which is often not available. By far the most common risk measure is Value-at-Risk (VaR), which is defined as a conditional quantile of the return distribution, and it says nothing about the shape of the tail to the left of the quantile. Our contribution is the exploration of a new tool for backtesting based on the duration of days between the violations of the VaR. The chief insight is that if the VaR model is correctly specified for coverage rate, p, then the conditional expected duration between violations should be a constant 1/p days. We suggest various ways of testing this null hypothesis and we conduct a Monte Carlo analysis which compares the new tests to those currently available. Our results show that in realistic situations, the duration based tests have better power properties than the previously suggested tests. The size of the tests is easily controlled using the Monte Carlo technique of Dufour (2000). L'évaluation des modèles de risque financier, ou test inversé, est une partie importante de l'approche avec modèle interne pour la gestion de risque tel qu'établie par le Comité de Basle pour la supervision bancaire (1996). Toutefois, les procédures existantes de tests inversés telles que celles développées dans Christoffersen (1998), ont une puissance relativement faible pour des tailles d'échantillon réalistes. Les méthodes suggérées dans Berkowitz (2001) performe mieux mais sont basées sur de l'information, telle que la forme de la queue gauche de la distribution des rendements du portefeuille, qui n'est pas toujours disponible. La mesure de risque de loin la plus courante est la Valeur-à-Risque (VaR), qui est définie comme un quantile de la distribution conditionnelle du rendement, et elle ne dit rien à-propos de la forme de la distribution à gauche du quantile. Notre contribution est l'exploration d'un nouvel outil pour les tests inversés basé sur la durée en jours entre les violations de la VaR. L'intuition est que si le modèle de VaR est correctement spécifié pour un taux de couverture p, alors la durée espérée conditionnelle entre les violations devrait être une constante 1/p jours. Nous proposons diverses façons de tester cette hypothèse nulle et nous effectuons une analyse Monte Carlo où l'on compare ces nouveaux tests à ceux présentement disponibles. Nos résultats montrent que pour des situations réalistes, les tests basés sur les durées ont de meilleures propriétés en termes de puissance que ceux précédemment proposés. La taille des tests est facilement contrôlée en utilisant la technique Monte Carlo de Dufour (2000).

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2003s-05.

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Length: 29 pages
Date of creation: 01 Feb 2003
Handle: RePEc:cir:cirwor:2003s-05
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  1. Jeremy Berkowitz & James O'Brien, 2002. "How Accurate Are Value-at-Risk Models at Commercial Banks?," Journal of Finance, American Finance Association, vol. 57(3), pages 1093-1111, June.
  2. James D. Hamilton & Oscar Jorda, 2002. "A Model of the Federal Funds Rate Target," Journal of Political Economy, University of Chicago Press, vol. 110(5), pages 1135-1167, October.
  3. Kiefer, Nicholas M, 1988. "Economic Duration Data and Hazard Functions," Journal of Economic Literature, American Economic Association, vol. 26(2), pages 646-679, June.
  4. Francis X. Diebold & Todd A. Gunther & Anthony S. Tay, "undated". "Evaluating Density Forecasts," CARESS Working Papres 97-18, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
  5. Paul H. Kupiec, 1995. "Techniques for verifying the accuracy of risk measurement models," Finance and Economics Discussion Series 95-24, Board of Governors of the Federal Reserve System (U.S.).
  6. M.J.B. Hall, 1996. "The amendment to the capital accord to incorporate market risk," Banca Nazionale del Lavoro Quarterly Review, Banca Nazionale del Lavoro, vol. 49(197), pages 271-277.
  7. Dufour, Jean-Marie, 2006. "Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics," Journal of Econometrics, Elsevier, vol. 133(2), pages 443-477, August.
  8. Dale J. Poirier, 1995. "Intermediate Statistics and Econometrics: A Comparative Approach," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262161494, January.
  9. Christoffersen, Peter F, 1998. "Evaluating Interval Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 841-862, November.
  10. 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.
  11. Christoffersen, Peter & Hahn, Jinyong & Inoue, Atsushi, 2001. "Testing and comparing Value-at-Risk measures," Journal of Empirical Finance, Elsevier, vol. 8(3), pages 325-342, July.
  12. Darryll Hendricks, 1996. "Evaluation of value-at-risk models using historical data," Economic Policy Review, Federal Reserve Bank of New York, issue Apr, pages 39-69.
  13. 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 705-730, August.
  14. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
  15. Matthew Pritsker, 1997. "Evaluating Value at Risk Methodologies: Accuracy versus Computational Time," Journal of Financial Services Research, Springer;Western Finance Association, vol. 12(2), pages 201-242, October.
  16. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
  17. Gourieroux,Christian, 2000. "Econometrics of Qualitative Dependent Variables," Cambridge Books, Cambridge University Press, number 9780521331494, December.
  18. Gourieroux,Christian, 2000. "Econometrics of Qualitative Dependent Variables," Cambridge Books, Cambridge University Press, number 9780521589857, December.
  19. Matthew Pritsker, 2001. "The hidden dangers of historical simulation," Finance and Economics Discussion Series 2001-27, Board of Governors of the Federal Reserve System (U.S.).
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