zbMATH — the first resource for mathematics

A conditional-SGT-VaR approach with alternative GARCH models. (English) Zbl 1132.91455
Summary: This paper proposes a conditional technique for the estimation of VaR and expected shortfall measures based on the skewed generalized \(t\) (SGT) distribution. The estimation of the conditional mean and conditional variance of returns is based on ten popular variations of the GARCH model. The results indicate that the TS-GARCH and EGARCH models have the best overall performance. The remaining GARCH specifications, except in a few cases, produce acceptable results. An unconditional SGT-VaR performs well on an in-sample evaluation and fails the tests on an out-of-sample evaluation. The latter indicates the need to incorporate time-varying mean and volatility estimates in the computation of VaR and expected shortfall measures.

91G70 Statistical methods; risk measures
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI
[1] Andersen, T.G. (1994). ”Stochastic Autoregressive Volatility: A Framework for Volatility Modeling.” Mathematical Finance, 4, 75–102. · Zbl 0884.90013
[2] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1997). ”Thinking Coherently.” Risk, 10, 68–71.
[3] Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). ”Coherent Measures of Risk.” Mathematical Finance 9, 203–228. · Zbl 0980.91042
[4] Bai, J. and S. Ng. (2005). ”Tests for Skewness, Kurtosis, and Normality for Time-Series Data.” Journal of Business and Economic Statistics, 23, 49–60.
[5] Bali, T.G. (2000). ”Testing the Empirical Performance of Stochastic Volatility Models of the Short Term Interest Rate.” Journal of Financial and Quantitative Analysis, 35, 191–215.
[6] Bali, T.G. (2003). ”An Extreme Value Approach to Estimating Volatility and Value at Risk.” Journal of Business, 76, 83–108.
[7] Bali, T.G. and S. Neftci. (2003). ”Disturbing Extremal Behavior of Spot Rate Dynamics.” Journal of Empirical Finance, 10, 455–477.
[8] Bera, A.K. and M.L. Higgins (1995). ”On ARCH Models: Properties, Estimation, and Testing.” In L. Exley, D.A.R. George, C.J. Roberts and S. Sawyer (eds.), Survey in Econometrics. Oxford: Basil Blackwell.
[9] Berkowitz, J. (2001). ”Testing Density Forecasts with Applications to Risk Management.” Journal of Business and Economic Statistics, 19, 465–474. · Zbl 04569369
[10] Bollerslev, T. (1986). ”Generalized Autoregressive Conditional Heteroscedasticity.” Journal of Econometrics, 31, 307–327. · Zbl 0616.62119
[11] Bollerslev, T. (1987). ”A Conditionally Heteroscedastic Time Series Model of Security Prices and Rates of Return Data.” Review of Economics and Statistics, 59, 542–547.
[12] Bollerslev, T., R.Y. Chou, and K.F. Kroner. (1992). ”ARCH Modeling in Finance: A Review of the Theory and Empirical Evidence.” Journal of Econometrics, 52, 5–59. · Zbl 0825.90057
[13] Bollerslev, T., R.F. Engle, and D.B. Nelson. (1994). ”ARCH Models.” In R.F. Engle, and D.L. McFadden (eds.), Handbook of Econometrics, Vol 4, pp. 2959–3038, Amsterdam: Elsevier.
[14] Box, G., and G.C. Tiao. (1962). ”A Further Look at Robustness Via Bayes Theorem,” Biometrika, 49, 419–432. · Zbl 0114.34903
[15] Christoffersen, P.F. (1998). ”Evaluating Interval Forecasts.” International Economic Review, 39, 841–862.
[16] Christoffersen, P.F., and F.X., Diebold. (2000). ”How Relevant is Volatility Forecasting for Financial Risk Management.” Review of Economics and Statistics, 82, 12–22.
[17] Cox, J.C., J. Ingersoll, and S. Ross. (1985). ”A Theory of the Term Structure of Interest Rates.” Econometrica, 53, 385–407. · Zbl 1274.91447
[18] Cummins, J.D., G. Dionne, J.B. McDonald, and B.M. Pritchett. (1990). ”Applications of the GB2 Family of Distributions in Modeling Insurance Loss Processes.” Insurance: Mathematics and Economics, 9, 257–272.
[19] De Ceuster, M. and D. Trappers (1992). ”Diagnostic Checking of Estimation with a Student-\(t\) Error Density.” Working Paper UFSIA, Centrum voor Bedrijfeconomie en Bedrijfeconometrie.
[20] Delbaen, F. 1998. ”Coherent Risk Measures on General Probability Spaces.” Working Paper, ETH Zurich.
[21] Ding, Z., C.W. Granger, and R.F. Engle. (1993). ”A Long Memory Property of Stock Market Returns and a New Model.” Journal of Empirical Finance, 1, 83–106.
[22] Duan, J.-C. (1997). ”Augmented GARCH(p,q) Process and its Diffusion Limit.” Journal of Econometrics, 79, 97–127. · Zbl 0898.62141
[23] Embrechts, P. (2000). ”Extreme Value Theory: Potential and Limitations as an Integrated Risk Management Tool.” Working Paper, ETH Zurich.
[24] Engle, R.F. (1982). ”Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica, 50, 987–1007. · Zbl 0491.62099
[25] Engle, R.F. (1990). ”Discussion: Stock Market Volatility and the Crash of ’87.” Review of Financial Studies, 3, 103–106.
[26] Engle, R.F. and T. Bollerslev. (1986). ”Modeling the Persistence of Conditional Variances.”Econometric Reviews, 5, 1–50. · Zbl 0619.62105
[27] Engle, R.F. and S. Manganelli. (2004). ”CAViaR: conditional autoregressive Value at Risk by Regression Quantiles.” Journal of Business and Economic Statistics, 22, 367–381.
[28] Engle, R.F. and V.K. Ng. (1993). ”Measuring and Testing the Impact of News on Volatility.” Journal of Finance, 48, 1749–1778.
[29] Giot, P. and S. Laurent. (2003). ”Value at Risk for Long and Short Positions.” Journal of Applied Econometrics, 18, 641–663.
[30] Glosten, L.R., R. Jagannathan, and D.E. Runkle (1993). ”On the Relation Between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” Journal of Finance, 48, 1779–1801.
[31] Hansen, B.E. (1994). ”Autoregressive Conditional Density Estimation,” International Economic Review, 35, 705–730. · Zbl 0807.62090
[32] Hansen, C.B., J.B. McDonald, and P. Theodossiou. (2001). ”Some Flexible Parametric Models and Leptokurtic Data.” Working Paper, Brigham Young University.
[33] Hardouvelis, G. and P. Theodossiou. (2002). ”The Asymmetric Relation Between Margin Requirements and Stock Market Volatility Across Bull and Bear Markets,” Review of Financial Studies, 15, 1525–1159.
[34] Heikkinen, V.P. and A. Kanto. (2002). ”Value-at-Risk Estimation Using Noninteger Degrees of Freedom of Student’s Distribution.” Journal of Risk, 4, 77–84.
[35] Hentschel, L.E. (1995). ”All in the Family: Nesting Symmetric and Asymmetric GARCH Models.” Journal of Financial Economics, 39, 71–104.
[36] Heston, S. and S. Nandi. (1999). ”Pricing Bonds and Interest Rate Derivatives Under a Two-Factor Model of Interest Rates with GARCH Volatility: Analytical Solutions and Their Applications.” Working Paper 99-20 (November), Federal Reserve Bank of Atlanta.
[37] Higgins, M.L. and A.K. Bera. (1992). ”A Class of Nonlinear Arch Models.” International Economic Review, 33, 137–158. · Zbl 0744.62152
[38] Jondeau, E. and M. Rockinger. (2003). ”Conditional Volatility, Skewness, and Kurtosis: Existence and Persistence.” Journal of Economic Dynamics and Control, forthcoming. · Zbl 1178.91226
[39] Krokhmal, P., J. Palmquist, and S. Uryasev. (2002). ”Portfolio Optimization with Conditional Value-at-Risk Objective and Constraints.” Journal of Risk, 4, 43–68.
[40] Krokhmal, P., S. Uryasev, and G. Zrazhevsky. (2002). ”Risk Management for Hedge Fund Portfolios: A Comparative Analysis of Linear Balancing Strategies.” Journal of Alternative Investments, 5, 10–29.
[41] Kupiec, P.H. (1995). ”Techniques for Verifying the Accuracy of Risk Measurement Models.” Journal of Derivatives, 3, 73–84.
[42] Longin, F.M. (2000). ”From Value at Risk to Stress Testing: The Extreme Value Approach.” Journal of Banking and Finance, 24, 1097–1130.
[43] McDonald, J.B. and W.K. Newey (1988). ”Partially Adaptive Estimation of Regression Models Via the Generalized \(t\) Distribution” Econometric Theory, 4, 428–457.
[44] McNeil, A.J. and R. Frey (2000). ”Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: An Extreme Value Approach.” Journal of Empirical Finance, 7, 271–300.
[45] Mittnik, S., and M. Paolella. (2000). ”Conditional Density and Value-at-Risk Prediction of Asian Currency Exchange Rates.” Journal of Forecasting, 19, 313–333.
[46] Nelson, D. (1991). ”Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica, 59, 347–370. · Zbl 0722.62069
[47] Pagan, A. (1996). ”The Econometrics of Financial Markets.” Journal of Empirical Finance, 3, 15–102.
[48] Piero, A. (1999). ”Skewness in Financial Returns.” Journal of Banking and Finance, 23, 847–862.
[49] Rockafeller, R.T. and S. Uryasev. (2000). ”Optimization of Conditional Value at Risk.” Journal of Risk, 2, 21–41.
[50] Schwert, G.W. (1989). ”Why Does Stock Market Volatility Change Over Time?” Journal of Finance, 44, 1115–1153.
[51] Sentana, E. (1995). ”Quadratic ARCH Models.” Review of Economic Studies, 62, 639–661. · Zbl 0847.90035
[52] Seymour, A.J. and D.A. Polakow. (2003). ”A Coupling of Extreme-Value Theory and Volatility Updating with Value-at-Risk Estimation in Emerging Markets: A South African Test.” Multinational Finance Journal, 7, 3–23.
[53] Subbotin, M.T.H. (1923). ”On the Law of Frequency of Error.” Matematicheskii Sbornik, 31, 296–301. · JFM 49.0370.01
[54] Taylor, S. (1986). Modeling Financial Time Series. New York: Wiley. · Zbl 1130.91345
[55] Theodossiou, P. (2001). ”Skewness and Kurtosis in Financial Data and the Pricing of Options.” Working Paper, Rutgers University.
[56] Theodossiou, P. (1998). ”Financial Data and the Skewed Generalized \(t\) Distribution.” Management Science, 44, 1650–1661. · Zbl 1001.91051
[57] Topaloglou, N., Vladimirou, H., and S.A. Zenios. (2002). ”CVaR models with selective hedging for international asset allocation.” Journal of Banking and Finance, 26, 1535–1561.
[58] Wu, G. and Z. Xiao. (2002). ”An Analysis of Risk Measures.” Journal of Risk, 4, 53–75.
[59] Zakoian, J.-M. (1994). ”Threshold Heteroscedastic Models.” Journal of Economic Dynamics and Control 18, 931–995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.