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CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles

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  • Robert F. Engle
  • Simone Manganelli

Abstract

Value at Risk (VaR) has become the standard measure of market risk employed by financial institutions for both internal and regulatory purposes. VaR is defined as the value that a portfolio will lose with a given probability, over a certain time horizon (usually one or ten days). Despite its conceptual simplicity, its measurement is a very challenging statistical problem and none of the methodologies developed so far give satisfactory solutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation which does not require any of the extreme assumptions invoked by existing methodologies (such as normality or i.i.d. returns). The Conditional Autoregressive Value-at-Risk or CAViaR model moves the focus of attention from the distribution of returns directly to the behavior of the quantile. We specify the evolution of the quantile over time using a special type of autoregressive process and use the regression quantile framework introduced by Koenker and Bassett to determine the unknown parameters. Since the objective function is not differentiable, we use a differential evolutionary genetic algorithm for the numerical optimization. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the Dynamic Quantile test. Applications to simulated and real data provide empirical support to this methodology and illustrate the ability of these algorithms to adapt to new risk environments.

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Bibliographic Info

Article provided by American Statistical Association in its journal Journal of Business and Economic Statistics.

Volume (Year): 22 (2004)
Issue (Month): (October)
Pages: 367-381

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Handle: RePEc:bes:jnlbes:v:22:y:2004:p:367-381

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References

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  1. Andrews, Donald W.K., 1988. "Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables," Econometric Theory, Cambridge University Press, vol. 4(03), pages 458-467, December.
  2. Granger, C. W. J. & White, Halbert & Kamstra, Mark, 1989. "Interval forecasting : An analysis based upon ARCH-quantile estimators," Journal of Econometrics, Elsevier, vol. 40(1), pages 87-96, January.
  3. 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 841-62, November.
  4. Jon Danielsson & Casper G. de Vries, 1998. "Beyond the Sample: Extreme Quantile and Probability Estimation," FMG Discussion Papers dp298, Financial Markets Group.
  5. Weiss, Andrew A., 1991. "Estimating Nonlinear Dynamic Models Using Least Absolute Error Estimation," Econometric Theory, Cambridge University Press, vol. 7(01), pages 46-68, March.
  6. White, Halbert & Domowitz, Ian, 1984. "Nonlinear Regression with Dependent Observations," Econometrica, Econometric Society, vol. 52(1), pages 143-61, January.
  7. repec:cup:etheor:v:7:y:1991:i:1:p:46-68 is not listed on IDEAS
  8. Foresi, S. & Paracchi, F., 1992. "The Conditional Distribution of Excess Returns: An Empirical Analysis," Working Papers 92-49, C.V. Starr Center for Applied Economics, New York University.
  9. Robert F. Engle & Victor K. Ng, 1991. "Measuring and Testing the Impact of News on Volatility," NBER Working Papers 3681, National Bureau of Economic Research, Inc.
  10. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
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  12. Powell, James L., 1986. "Censored regression quantiles," Journal of Econometrics, Elsevier, vol. 32(1), pages 143-155, June.
  13. Powell, James L., 1984. "Least absolute deviations estimation for the censored regression model," Journal of Econometrics, Elsevier, vol. 25(3), pages 303-325, July.
  14. repec:cup:etheor:v:6:y:1990:i:3:p:295-317 is not listed on IDEAS
  15. Newey, Whitney K. & Powell, James L., 1990. "Efficient Estimation of Linear and Type I Censored Regression Models Under Conditional Quantile Restrictions," Econometric Theory, Cambridge University Press, vol. 6(03), pages 295-317, September.
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