Value–at–Risk Models

  • Peter ChristoffersenEmail author


In this chapter, we build first a univariate and then a multivariate filtered historical simulation (FHS) model for financial risk management. Both the univariate and multivariate methods simulate future returns from a model using historical return innovations. While the former relies on portfolio returns filtered by a dynamic variance model, the latter uses individual or base asset return innovations from dynamic variance and correlation models. The univariate model is suitable for passive risk management or risk measurement whereas the multivariate model is useful for active risk management such as optimal portfolio allocation. Both models are constructed in such a way as to capture the stylized facts in daily asset returns and to be simple to estimate. The FHS approach enables the risk manager to easily compute Value-at-Risk and other risk measures including Expected Shortfall for various investment horizons that are conditional on current market conditions. The chapter also lists various alternatives to the suggested FHS approach.


Asset Return GARCH Model Portfolio Return Investment Horizon Historical Simulation 
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  1. Andersen, T.G., Bollerslev, T., Christoffersen, P. and Diebold, F.X. (2006a): Volatility and Correlation Forecasting. In: Elliott, G., Granger, C. and Timmermann, A. (Eds.): Handbook of Economic Forecasting. North-Holland, Amsterdam.Google Scholar
  2. Andersen, T.G., Bollerslev, T., Christoffersen, P. and Diebold, F.X. (2006b): Practical Volatility and Correlation Modeling for Financial Market Risk Management. In: Carey, M. and Stulz, R. (Eds.): The Risks of Financial Institutions. University of Chicago Press.Google Scholar
  3. Barone-Adesi, G., Bourgoin, F. and Giannopoulos, K. (1998): Don't Look Back. Risk 11, 100–104.Google Scholar
  4. Bauwens, L., Laurent, S. and Rombouts, J. (2006): Multivariate GARCH Models: a Survey. Journal of Applied Econometrics 21, 79–109.CrossRefMathSciNetGoogle Scholar
  5. Bodoukh, J., Richardson, M., and Whitelaw, R. (1998): The Best of Both Worlds. Risk 11, 64–67.Google Scholar
  6. Bollerslev, T. (1986): Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics 31, 307–327.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Bollerslev, T. (1987): A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return. Review of Economics and Statistics 69, 542–547.CrossRefGoogle Scholar
  8. Cappiello, L., Engle, R.F. and Sheppard, K. (2004): Asymmetric Dynamics in the Correlations of Global Equity and Bond Returns. Manuscript, Stern School of Business New York University.Google Scholar
  9. Christoffersen, P. (2003): Elements of Financial Risk Management. Academic Press, San Diego.Google Scholar
  10. Christoffersen, P. and Diebold, F. (2000): How Relevant is Volatility Forecasting for Financial Risk Management? Review of Economics and Statistics 82, 1–11.CrossRefGoogle Scholar
  11. Christoffersen, P. and Goncalves, S. (2005): Estimation Risk in Financial Risk Management. Journal of Risk 7, 1–28.Google Scholar
  12. Christoffersen, P., Diebold, F. and Schuermann, T. (1998): Horizon Problems and Extreme Events in Financial Risk Management. Economic Policy Review Federal Reserve Bank of New York, October, 109-118.Google Scholar
  13. Demarta, S., and McNeil, A. J. (2005): The t Copula and Related Copulas. International Statistical Review 73, 111–129.zbMATHGoogle Scholar
  14. Diebold, F.X., Schuermann, T. and Stroughair, J. (1998): Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management. In: Refenes, A.-P. N., Burgess, A.N. and Moody, J.D. (Eds.): Decision Technologies for Computational Finance, 3-12. Kluwer Academic Publishers, Amsterdam.Google Scholar
  15. Duffie, D. and Pan, J. (1997): An Overview of Value at Risk. Journal of Derivatives 4, 7–49.CrossRefGoogle Scholar
  16. Engle, R. (1982): Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of U.K. Inflation. Econometrica 50, 987–1008.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Engle, R. (2002): Dynamic Conditional Correlation - A Simple Class of Multivariate GARCH Models. Journal of Business and Economic Statistics 20, 339–350.CrossRefMathSciNetGoogle Scholar
  18. Engle, R. and Manganelli, S. (2004): CAViaR: Conditional Autoregressive Value at Risk by Quantile Regression. Journal of Business and Economic Statistics 22, 367–381.CrossRefMathSciNetGoogle Scholar
  19. Engle, R. and Ng, V. (1993): Measuring and Testing the Impact of News on Volatility. Journal of Finance 48, 1749–1778.CrossRefGoogle Scholar
  20. Engle, R. F. and Sheppard, K. (2001): Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH. NBER Working Paper 8554.Google Scholar
  21. Gourieroux, C. and Jasiak, J. (2006): Dynamic Quantile Models. Manuscript, University of Toronto.Google Scholar
  22. Hansen, B. (1994): Autoregressive Conditional Density Estimation. International Economic Review 35, 705–730.zbMATHCrossRefGoogle Scholar
  23. Harvey, C.R. and Siddique, A. (1999): Autoregressive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–488.CrossRefGoogle Scholar
  24. Hull, J. and Suo, W. (2002): A methodology for assessing model risk and its application to the implied volatility function model. Journal of Financial and Quantitative Analysis 37, 297–318.CrossRefGoogle Scholar
  25. Hull, J. and White, A. (1998): Incorporating Volatility Updating into the Historical Simulation Method for VaR. Journal of Risk 1, 5–19.Google Scholar
  26. Joe, H. (1997): Multivariate Models and Dependence Concepts. Chapman Hall, London.zbMATHGoogle Scholar
  27. Jondeau, E. and Rockinger, M. (2005): The Copula-GARCH Model of Conditional Dependencies: An International Stock-Market Application. Journal of International Money and Finance forthcoming.Google Scholar
  28. Jorion, P. (2006): Value-at-Risk: The New Benchmark for Managing. Financial Risk. McGraw Hill, New York.Google Scholar
  29. Morgan, J.P. (1996): RiskMetrics – Technical Document 4th Edition. New York.Google Scholar
  30. Lando, D. (2004): Credit Risk Modeling: Theory and Applications Princeton University Press, New Jersey.Google Scholar
  31. Longin, F. and Solnik, B. (2001): Extreme Correlation of International Equity Markets. Journal of Finance 56, 649–676.CrossRefGoogle Scholar
  32. Manganelli, S. (2004): Asset Allocation by Variance Sensitivity Analysis. Journal of Financial Econometrics 2, 370–389.CrossRefGoogle Scholar
  33. McNeil, A. and Frey, R. (2000): Estimation of Tail-Related Risk Measures for Heteroskedastic Financial Time Series: An Extreme Value Approach. Journal of Empirical Finance 7, 271–300.CrossRefGoogle Scholar
  34. Patton, A. (2004): On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation. Journal of Financial Econometrics 2, 130–168.CrossRefGoogle Scholar
  35. Patton, A. (2006): Modeling Asymmetric Exchange Rate Dependence. International Economic Review 47, 527–556.CrossRefMathSciNetGoogle Scholar
  36. Patton, A.J. and Sheppard, K. (2008): Evaluating volatility and Correlation forecasts. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 801–838. Springer Verlag, New York.Google Scholar
  37. Persaud, A. (2003): Liquidity Black Holes: Understanding, Quantifying and Managing Financial Liquidity Risk. Risk Books, London.Google Scholar
  38. Pesaran, H. and Zaffaroni, P. (2004): Model Averaging and Value-at-Risk based Evaluation of Large Multi Asset Volatility Models for Risk Management. Manuscript, University of Cambridge.Google Scholar
  39. Poon, S.-H., Rockinger, M. and Tawn, J. (2004): Extreme Value Dependence in Financial Markets: Diagnostics, Models and Financial Implications. Review of Financial Studies 17, 581–610.CrossRefGoogle Scholar
  40. Pritsker, M. (2001): The Hidden Dangers of Historical Simulation. Finance and Economics Discussion Series 2001–27. Washington: Board of Governors of the Federal Reserve System.Google Scholar
  41. Tse, Y.K. and Tsui, K.C. (2002): A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model with Time-varying Correlations. Journal of Business and Economic Statistics 20, 351–362.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Desautels Faculty of ManagementMcGill UniversityMontrealCanada

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