Analyzing the Information Contained in the Skewness and Kurtosis of TEPIX Returns for Forecasting Risk: GARCH Model with Gram-Charlier Expansions for Innovations

Document Type : Original Article

Author

Assistant Professor, Nahavand Higher Education Complex, Bu-Ali Sina University, Hamedan, Iran

Abstract

Purpose: A well-known stylized fact of asset return distributions is the pattern of skewness and kurtosis. Previous studies demonstrate that financial crises and turbulence induce shocks that significantly affect return distributions, resulting in both fat tails and asymmetric tail reactions. Despite the common influence of skewness and kurtosis on tail risk, their significance for risk forecasting has been underexplored in empirical financial research. Developing models for accurate risk forecasting remains a critical focus for policymakers, economists, financial market participants, and researchers. In this study, following Jimenez et al. (2022), a semi-nonparametric approach is adopted to estimate return densities based on the asymptotic properties of Gram-Charlier expansions. This approach allows for evaluating the significance of incorporating Hermite polynomials and their cross-products into Gram-Charlier densities for risk forecasting. Using this semi-nonparametric framework, risk measures can capture all stylized facts of the return series by incorporating new parameters, enabling a comprehensive assessment of skewness, kurtosis, and their interactions as valuable sources of information.
Method: For the first time, this study employs the modified Gram-Charlier (mGC) density function, which includes second and third moments (skewness and kurtosis) and their interactions, to model the risk distribution of daily losses in the TEPIX index. The performance of alternative models based on different Gram-Charlier specifications is evaluated in terms of risk forecasting accuracy using advanced backtesting tests. Specifically, the Value-at-Risk (VaR) criteria and the Median Shortfall (MS) measure—introduced for the first time in this research—are utilized. The sample comprises daily TEPIX index series from May 20, 2008, to August 22, 2023. The loss series is computed as the negative logarithmic differences of prices, focusing on the right tail of the TEPIX distribution. Estimations are conducted using R and MATLAB software through a two-step process:
-The ARMA(1,1)-GARCH(1,1) model is estimated using the quasi-maximum likelihood (QML) approach, assuming Gaussian distribution for the error terms.
-The modified Gram-Charlier expansion and its alternative specifications are estimated using standardized residuals extracted from the first step, with maximum likelihood estimation applied for density fitting. The in-sample estimation window consists of 2,656 observations, and forecasts are updated with one new observation. The remaining 1,000 observations are used for out-of-sample forecasting.
Findings: The in-sample fitting of the ARMA(1,1)-GARCH(1,1) model, under the Gram-Charlier densities for innovations, indicates that skewness, kurtosis, and their interactions are statistically and economically significant. Backtesting results for 99%-VaR and 99%-MS demonstrate that the Gram-Charlier density specification incorporating the skewness parameter significantly improves out-of-sample risk forecasting, particularly for the tails, compared to alternative specifications.
Conclusion: Overall, the findings reveal that incorporating the asymmetry parameter of the return density as a stand-alone feature provides a valuable source of information, enabling accurate risk measures for market participants. These results have practical implications for designing risk management strategies and decision-making under market instability. The novelty of this study lies in applying a semi-nonparametric approach to evaluate risk forecasting for the TEPIX index. Unlike previous studies that relied primarily on parametric and non-parametric distributions, this research offers an innovative application for risk management in the Tehran Stock Exchange. The empirical findings also have potential implications for stabilizing financial markets.

Keywords

References

 Alizadeh, A. H., & Gabrielsen, A. (2013). Dynamics of credit spread moments of European corporate bond indexes. Journal of Banking & Finance, 37(8), 3125-3144.
Amiri, H., Najafi Nejad, M., & Mousavi, S. M. (2021). Estimation of Value at Risk (VaR) Based On Lévy-GARCH Models: Evidence from Tehran Stock Exchange. Journal of Money and Economy, 16(2), 165-186. (In Persian)
Bastanzad, H., & Davoudi, P. (2017). An evaluation of risk transmission over foreign exchange, real estate and stock markets in Iran’s economy (An application of parametric and non-parametric value at risk approach). Journal of Asset Management and Financing, 5(4), 33-50. (In Persian)
Biek Khormizi, M., & Rafei, M. (2020). Modeling Value at Risk of Futures Contract of Bahar Azadi Gold Coin with Considering the Historical Memory in Observations Application of FIAPARCH-CHUNG Models. Journal of Asset Management and Financing, 8(1), 57-82. (In Persian)
Botshekan, M. H., Peymani, M., & Sadredin Karami, M. M. (2019). Estimate and evaluate non-parametric value at risk and expected shortfall based on principal component analysis in Tehran Stock Exchange. Financial Management Perspective, 8(24), 79-102. (In Persian)
Carnero, M. A., León, A., & Ñíguez, T. M. (2023). Skewness in energy returns: estimation, testing and implications for tail risk. The Quarterly Review of Economics and Finance.
Corrado, C. (2007). The hidden martingale restriction in Gram‐Charlier option prices. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 27(6), 517-534.
Cortés, L. M., Mora-Valencia, A., & Perote, J. (2020). Retrieving the implicit risk neutral density of WTI options with a semi-nonparametric approach. The North American Journal of Economics and Finance, 54, 100862.
Dadgar, Y., Dargahi, H., Gholizadeh, S. (2023). The Role of Investor Sentiment and Government Behaviour in Volatility of Tehran Stock Exchange Market: A Behavioural Economics Approach. Quarterly Journal of Applied Theories of Economics, 10(1), 191-214. (In Persian)
Danielsson, J. (2011). Financial risk forecasting: the theory and practice of forecasting market risk with implementation in R and Matlab. John Wiley & Sons.
Del Brio, E. B., Mora-Valencia, A., & Perote, J. (2020). Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall. International Review of Financial Analysis, 70, 101163.
Del Brio, E. B., Mora-Valencia, A., & Perote, J. (2014). Semi-nonparametric VaR forecasts for hedge funds during the recent crisis. Physica A: Statistical Mechanics and its Applications, 401, 330-343.
Del Brio, E. B., Ñíguez, T. M., & Perote, J. (2011). Multivariate semi-nonparametric distributions with dynamic conditional correlations. International Journal of Forecasting, 27(2), 347-364.
Dendramis, Y., Spungin, G. E., & Tzavalis, E. (2014). Forecasting VaR models under different volatility processes and distributions of return innovations. Journal of Forecasting, 33(7), 515-531.
Ergün, A. T., & Jun, J. (2010). Time-varying higher-order conditional moments and forecasting intraday VaR and expected shortfall. The Quarterly Review of Economics and Finance, 50(3), 264-272.
Farhadian, A., Rostami, M., & Nilchi, M. (2020). Compare Canonical stochastic volatility model of focal MSGJR-GARCH to measure the volatility of stock returns and calculating VaR. Financial Management Perspective, 10(32), 131-158. (In Persian)
Gerlach, R., Lu, Z., & Huang, H. (2013). Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting. Journal of Forecasting, 32(6), 534-550.
Hansen, B. E. (1994). Autoregressive conditional density estimation. International Economic Review, 705-730.
Hou, Y., Li, S., & Wen, F. (2019). Time-varying volatility spillover between Chinese fuel oil and stock index futures markets based on a DCC-GARCH model with a semi-nonparametric approach. Energy Economics, 83, 119-143.
Jarrow, R., & Rudd, A. (1982). Approximate option valuation for arbitrary stochastic processes. Journal of financial Economics, 10(3), 347-369.
Jiménez, I., Mora-Valencia, A., & Perote, J. (2020). Risk quantification and validation for Bitcoin. Operations Research Letters, 48(4), 534-541.
Jiménez, I., Mora-Valencia, A., & Perote, J. (2022a). Semi-nonparametric risk assessment with cryptocurrencies. Research in International Business and Finance, 59, 101567.
Jiménez, I., Mora-Valencia, A., & Perote, J. (2022b). Has the interaction between skewness and kurtosis of asset returns information content for risk forecasting?. Finance Research Letters, 49, 103105.
Jondeau, E., & Rockinger, M. (2001). Gram–charlier densities. Journal of Economic Dynamics and Control, 25(10), 1457-1483.
Jurczenko, E., Maillet, B., & Negréa, B. (2004). A note on skewness and kurtosis adjusted option pricing models under the martingale restriction. Quantitative Finance, 4(5), 479-488.
Kendall, M. G. (1977). The advanced theory of statistics. (Vol. I). Griffin & Co, London (Ed. 4).
Kiani, T., Farid, D., & Sadeghi, H. (2015). The Measurement of Risk based on the Criterion of Value at Risk via Model of GARCH (A Study of Stock of Listeَd Companies in Tehran Stock Exchange (TSE) in the Cement Industry). Financial Management Strategy, 3(3), 149-168. (In Persian)
Kou, S., & Peng, X. (2014). Expected shortfall or median shortfall. Journal of Financial Engineering, 1(01), 1450007.
Kou, S., Peng, X., & Heyde, C. C. (2013). External risk measures and Basel accords. Mathematics of Operations Research, 38(3), 393-417.
Labit, B., Furno, I., Fasoli, A., Diallo, A., Müller, S. H., Plyushchev, G., ... & Poli, F. M. (2007). Universal statistical properties of drift-interchange turbulence in TORPEX plasmas. Physical review letters, 98(25), 255002.
Lanne, M., & Pentti, S. (2007). Modeling conditional skewness in stock returns. The European Journal of Finance, 13(8), 691-704.
León-Camacho, B., Mora-Valencia, A., & Perote, J. (2022). Modified variance incorporating high-order moments in risk measure with Gram-Charlier returns. The Engineering Economist, 67(3), 218-233.
León, Á., & Ñíguez, T. M. (2020). Modeling asset returns under time-varying semi-nonparametric distributions. Journal of Banking & Finance, 118, 105870.
León, Á., Mencía, J., & Sentana, E. (2009). Parametric properties of semi-nonparametric distributions, with applications to option valuation. Journal of Business & Economic Statistics, 27(2), 176-192.
León, Á., Rubio, G., & Serna, G. (2005). Autoregresive conditional volatility, skewness and kurtosis. The Quarterly Review of Economics and Finance, 45(4-5), 599-618.
Lin, W. (2023). Essays in quantitative finance (Doctoral dissertation, University of Otago).
Lin, W., & Zhang, J. E. (2022). The valid regions of Gram–Charlier densities with high-order cumulants. Journal of Computational and Applied Mathematics, 407, 113945.
Lin, C. H., Changchien, C. C., Kao, T. C., & Kao, W. S. (2014). High-order moments and extreme value approach for value-at-risk. Journal of Empirical Finance, 29, 421-434.
Mandelbrot, B. B., & Mandelbrot, B. B. (1997). The variation of certain speculative prices (pp. 371-418). Springer New York.
Mauleón, I., & Perote, J. (2000). Testing densities with financial data: an empirical comparison of the EdgeworthSargan density to the Students t. The European Journal of Finance, 6(2), 225-239.
Mauleón, I. (2010). Assessing the value of Hermite densities for predictive distributions. Journal of Forecasting, 29(8), 689-714.
Mehrasa, M., & Mohamadi, T. (2019). Extreme Value Theory and Value at Risk: Application to OPEC Market. Iranian Energy Economics, 8(31), 151-170. (In Persian)
Modarresi, N., Peymani, M., & Darvishi, M. (2021). Estimation of Conditional Value at Risk under Stochastic Volatility Levy Processes for Tehran Stock Market. Financial Management Perspective, 11(34), 69-94. (In Persian)
Mohammadian Amiri, E., & Ebrahimi, S. B. (2018). Multiple-Step-Ahead Forecasting of Value at Risk Based on Holt-Winters Exponential Smoothing Multiplicative Method. Financial Management Strategy, 6(1), 93-114. (In Persian)
Namaki, A., Abbasian, E., & Shafiei, E. (2022). Analyzing of Systemic Risk Contributions of Tehran Stock Exchange Companies by Complexity Approach. Financial Management Strategy, 10(1), 91-112. (In Persian)
Ñíguez, T. M., & Perote, J. (2012). Forecasting heavy‐tailed densities with positive Edgeworth and Gram‐Charlier expansions. Oxford Bulletin of Economics and Statistics, 74(4), 600-627.
Phillips, P. C. (1977). A general theorem in the theory of asymptotic expansions as approximations to the finite sample distributions of econometric estimators. Econometrica: Journal of the Econometric Society, 1517-1534.
Sargan, J. D. (1975). Gram-Charlier approximations applied to t ratios of k-class estimators. Econometrica: Journal of the Econometric Society, 327-346.
Schlögl, E. (2013). Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order. Journal of Economic Dynamics and Control, 37(3), 611-632.
Shekarkhah, J., Bolu, G., & Haghighat, M. (2017). The Impact of Higher Moments and Nonsystematic Volatility on Future Stock Return using Fama-MacBeth Model. Empirical Studies in Financial Accounting, 14(56), 109-133. (In Persian)
Talebi, R., Zanjirdar, M., & pour Fakharan, M. (2021). Analysis and explanation of stock returns based on third and fourth order torques of non-systematic risk and the role of arbitrage constraints and investors' limited attention to it. Advances in Finance and Investment, 2(3), 130-97. (In Persian)
Theodossiou, P. (1998). Financial data and the skewed generalized t distribution. Management Science, 44(12-part-1), 1650-1661.
Tehrani, R., Nabizade, A., & Bolguriyan, M. (2011). Examination of the effect of skewness and excess kurtosis on stock returns description through Capital Asset Pricing Model and Fama & French three factor model. Commercial Strategies, 9(47), 155-162. (In Persian)
Torki, L., Esmaeli, N., & Haghparast, M. (2023). Comparison of GARCH Family Models in Estimating Value at Risk and Conditional Value at Risk on the Tehran Stock Exchange. Quarterly Journal of Quantitative Economics, 19(4), 43-78. (In Persian)
Umar, Z., Usman, M., Choi, S. Y., & Rice, J. (2023). Diversification benefits of NFTs for conventional asset investors: Evidence from CoVaR with higher moments and optimal hedge ratios. Research in International Business and Finance, 65, 101957.
Vacca, G., Zoia, M. G., & Bagnato, L. (2022). Forecasting in GARCH models with polynomially modified innovations. International Journal of Forecasting, 38(1), 117-141.
Vacca, G., & Zoia, M. G. (2019). Kurtosis analysis in GARCH models with Gram–Charlier-like innovations. Economics Letters, 183, 108552.
Wang, T., Liang, F., Huang, Z., & Yan, H. (2022). Do realized higher moments have information content?-VaR forecasting based on the realized GARCH-RSRK model. Economic Modelling, 109, 105781.
Wu, P. T., & Shieh, S. J. (2007). Value-at-Risk analysis for long-term interest rate futures: Fat-tail and long memory in return innovations. Journal of Empirical Finance, 14(2), 248-259.
Yan, J. (2005). Asymmetry, fat-tail, and autoregressive conditional density in financial return data with systems of frequency curves. University of Iowa: Department of Statistics and Actuarial Science.
Zhu, D., & Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics, 157(2), 297-305.
Zoia, M. G., Biffi, P., & Nicolussi, F. (2018). Value at risk and expected shortfall based on Gram-Charlier-like expansions. Journal of Banking & Finance, 93, 92-104.
 
 
 
 
Financial Management Perspective
Volume 14, Issue 45
2024
Pages 149-174

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