The Use of GARCH Models in VaR Estimation

Angelidis, Timotheos and Benos, Alexandros and Degiannakis, Stavros (2004): The Use of GARCH Models in VaR Estimation. Published in: Statistical Methodology , Vol. 2, No. 1 (2004): pp. 105-128.

Abstract

We evaluate the performance of an extensive family of ARCH models in modelling daily Valueat-Risk (VaR) of perfectly diversified portfolios in five stock indices, using a number of distributional assumptions and sample sizes. We find, first, that leptokurtic distributions are able to produce better one-step-ahead VaR forecasts; second, the choice of sample size is important for the accuracy of the forecast, whereas the specification of the conditional mean is indifferent. Finally, the ARCH structure producing the most accurate forecasts is different for every portfolio and specific to each equity index.

• Item Type: MPRA Paper
• Language: English
• Keywords: Value at Risk, GARCH estimation, Backtesting, Volatility forecasting, Quantile Loss Function
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; G - Financial Economics > G1 - General Financial Markets > G15 - International Financial Markets

References

H. Akaike, Information theory and the extension of the maximum likelihood principle, Proceeding of the Second International Symposium on Information Theory, Budapest (1973) 267-281.

A.K. Bera, M.L. Higgins, ARCH models: Properties, estimation and testing, Journal of Economic Surveys, 7 (1993) 305-366.

M. Billio, L. Pelizzon, Value-at-Risk: A multivariate switching regime approach, Journal of Empirical Finance, 7 (2000) 531-554.

F. Black, Studies of stock market volatility changes, Proceedings of the American Statistical Association, Business and Economic Statistics Section (1976) 177-181.

T. Bollerslev, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31 (1986) 307-327.

T. Bollerslev, A conditional heteroscedastic time series model for speculative prices and rates of return, Review of Economics and Statistics, 69 (1987) 542-547.

T. Bollerslev, J.M. Wooldridge, Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Reviews, 11 (1992) 143-172.

T. Bollerslev, R. Chou, K.F. Kroner, ARCH modelling in Finance: A review of the theory and empirical evidence, Journal of Econometrics, 52 (1992) 5-59.

T, Bollerslev, R,F, Engle, D, Nelson, ARCH models, in: R.F., Engle and D., McFadden (Eds.), Handbook of Econometrics, vol. 4, Elsevier, Amsterdam, 1994, pp. 2959-3038.

C, Brooks, S,P, Burke, Information criteria for Garch model selection, The European Journal of Finance, 9 (2003) 557-580.

C, Brooks, G, Persand, Model choise and Value-at-Risk performance, Financial Analysts Journal, 58,5 (2002) 87-98.

C. Brooks, G. Persand, The effect of asymmetries on stock index return Value-at-Risk estimates, The Journal of Risk Finance, Winter (2003) 29-42.

C. Brooks, G. Persand, Volatility forecasting for risk management, Journal of Forecasting, 22 (2003) 1-22.

D.J. Cabedo, I. Moya, Estimating oil price ’Value at Risk’ using the historical simulation approach, Energy Economics, 25 (2003) 239-253.

P. Christoffersen, Evaluating interval forecasts, International Economic Review, 39, 1998, 841-862.

M. Crouhy, D. Galai, R. Mark, Risk Management, McGraw Hill, New York, 2001.

S. Danielsson, Multivariate stochastic volatility models: estimation and a comparison with VGARCH models, Department of Economics, University of Iceland, (1996) Mimeo.

J. Dan´ielsson, The emperor has no clothes: Limits to risk modelling, Journal of Banking & Finance, 26 (2002) 1273-1296.

S. Degiannakis, E. Xekalaki, Autoregressive conditional heteroskedasticity models: A review, Quality Technology and Quantitative Management, (2004), 1(2), 271-324.

F.X. Diebold, R. Mariano, Comparing predictive accuracy, Journal of Business and Economic Statistics, 13,3 (1995) 253-263.

K. Dowd, Beyond Value at Risk: The new science of risk management, John Wiley & Sons, New York, 1998.

R.F. Engle, Autoregressive Conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica, 50 (1982) 987-1008.

R.F. Engle, C.H. Hong, A. Kane, J. Noh, Arbitrage valuation of variance forecasts with simulated options, Advances in Futures and Options Research, 6 (1993) 393-415.

R.F. Engle, A.J. Patton, What good is a volatility model? Quantitative Finance, 1,2 (2001), 237-245.

R. Frey, P. Michaud, The effect of GARCH-type volatilities on prices and payoff-distributions of derivative assets - a simulation study, ETH Zurich, (1997) Working Paper.

P. Giot, S. Laurent, Value-at-Risk for long and short trading positions, Journal of Applied Econometrics, (2004) forthcoming.

P. Giot, S. Laurent, Market risk in commodity markets: a VaR approach, Energy Economics, 25 (2003) 435-457.

C. Guermat, R.D.F. Harris, Forecasting value at risk allowing for time variation in the variance and kurtosis portfolio returns, International Journal of Forecasting, 18 (2002) 409-419.

J.D. Hamilton, Time series analysis, New Jersey: Princeton University Press, 1994.

P.R. Hansen, A. Lunde, A forecast comparison of volatility models: Does anything beat a GARCH(1,1)? Brown University, (2003) Working Paper.

D. Hendricks, Evaluation of value-at-risk models using historical data, Federal Reserve Bank of New York, Economic Police Review, 2 (1996) 39-70.

R. Hoppe, VAR and the unreal world, Risk, 11 (1998) 45-50.

D.A. Hsieh, Modelling heteroscedasticity in daily foreign-exchange rates, Journal of Business and Economic Statistics, 7 (1989) 307-317.

P. Jackson, D.J. Maude, W. Perraudin, Testing Value-at-Risk approaches to capital adequacy, Bank of England Quarterly Bulletin, 38 (1998) 256-266.

P. Jorion, On jump processes in the foreign exchange and stock markets, Review of Financial Studies, 1 (1988) 427-445.

P. Jorion, Value at Risk. The new benchmark for controlling derivatives risk, McGraw Hill, New York, 1997.

F. Klaassen, Improving GARCH volatility forecasts with regime-switching GARCH, in J.D.

Hamilton and B. Raj (Eds.), Advances in Markov-Switching Models, Physica Verlag, New York, 2002, pp. 223-254.

P.H. Kupiec, Techniques for verifying the accuracy of risk measurement models, The Journal of Derivatives, 3 (1995) 73-84.

G. Lambadiaris, L. Papadopoulou, G. Skiadopoulos, Y. Zoulis, VAR: history or simulation?, Risk, September 2003.

P. Lambert, S. Laurent, Modelling skewness dynamics in series of financial data, Institutde Statistique, Louvain-la-Neuve, (2000) Discussion Paper.

P. Lambert, S. Laurent, Modelling financial time series using GARCH-type models and a skewed student density, Universite de Liege, (2001) Mimeo.

T.K.Y. Lee, Y.K. Tse, Term structure of interest rates in the Singapore Asian dollar market, Journal of Applied Econometrics, 6 (1991) 143-152.

A. Lo, A.C. MacKinlay, Stock market prices do not follow random walks: Evidence from a simple specification test. Review of Financial Studies, 1 (1988) 41-66.

J. Longerstay, An improved methodology for measuring VaR, RiskMetrics, Monitor Second Quarter, J.P. Morgan, 1996.

J.A. Lopez, Methods for evaluating Value-at-Risk estimates. Federal Reserve Bank of New York, Economic Policy Review, 1998.

B. Mandelbrot, The Variation of certain speculative prices, Journal of Business, 36 (1963), 394-419.

D.W. Marquardt, An algorithm for least squares estimation of non-linear parameters, Journal of the Society for Industrial and Applied Mathematics, 11 (1963) 431-441.

D. Nelson, Conditional heteroskedasticity in asset returns: A new approach, Econometrica, 59 (1991) 347-370.

W. Newey, K. West, A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica, 55 (1987) 703-708.

A.R. Pagan, G.W. Schwert, Alternative models for conditional stock volatility, Journal of Econometrics, 45 (1990) 267-290.

RiskMetrics, Technical document, J.P. Morgan, New York, USA, 1995.

M. Sarma, S. Thomas, A. Shah, Selection of VaR models, Journal of Forecasting, 22,4,(2003) 337-358.

M. Scholes J. Williams, Estimating betas from non-synchronous data, Journal of Financial Economics, 5 (1977) 309-328.

G. Schwarz, Estimating the dimension of a model, Annals of Statistics, 6 (1978) 461-464.

S. Venkataraman, Value at risk for a mixture of normal distributions: The use of quasiBayesian estimation techniques, Economic Perspectives, Federal Reserve Bank of Chicago, March/April 1997, 2-13.

P. Vlaar, Value at Risk models for Dutch bond portfolios, Journal of Banking and Finance, 24 (2000) 131-154.

H. White, A heteroskedasticity-consistent covariance matrix and a direct test for heteroskedasticity, Econometrica, 48 (1980) 817-838.

P. Zangari, An improved methodology for measuring VAR, RiskMetrics Monitor, Reuters/JP Morgan, 1996.