The
paper proposes a new method of dynamic VaR and CVaR risk measures forecasting.
The method is designed for obtaining the forecast estimates of risk measures
for volatile time series with long range dependence. The method is based on the
heteroskedastic time series model. The FIGARCH model is used for volatility
modeling and forecasting. The model is reduced to the AR model of infinite
order. The reduced system of Yule-Walker equations is solved to find the
autoregression coefficients. The regression equation for the autocorrelation
function based on the definition of a long-range dependence is used to get the autocorrelation
estimates. An optimization procedure is proposed to specify the estimates of
autocorrelation coefficients. The procedure for obtaining of the forecast
values of dynamic risk measures VaR and CVaR is formalized as a multi-step
algorithm. The algorithm includes the following steps: autoregression
forecasting, innovation highlighting, obtaining of the assessments for static
risk measures for residuals of the model, forming of the final forecast using
the proposed formulas, quality analysis of the results. The proposed method is
applied to the time series of the index of the Tokyo stock exchange. The
quality analysis using various tests is conducted and confirmed the high
quality of the obtained estimates.
References
[1]
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk. Mathematical Finance, 9, 203-228. https://doi.org/10.1111/1467-9965.00068
[2]
Tsay, R.S. (2010) Analysis of Financial Time Series. 3rd Edition, John Wiley & Sons, Hoboken. https://doi.org/10.1002/9780470644560
[3]
Yamai, Y. and Yoshiba, T. (2002) Comparative Analysis of Expected Shortfall and Value-at-Risk: Their Estimation Error, Decomposition and Optimization. Monetary and Economic Studies, 20, 57-86.
[4]
Nadarajah, S., Zhang, B. and Chan, S. (2014) Estimation Methods for Expected Shortfall. Quantitative Finance, 14, 271-291. https://doi.org/10.1080/14697688.2013.816767
[5]
Rockafellar, R.T. and Uryasev, S.P. (2000) Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21-42. https://doi.org/10.21314/JOR.2000.038
[6]
Rockafellar, R.T. and Uryasev, S.P. (2002) Conditional Value-at-Risk for General Loss Distributions. Journal of Banking & Finance, 26, 1443-1471.
[7]
Scaillet, O. (2004) Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall. Mathematical, 14, 115-129. https://doi.org/10.1111/j.0960-1627.2004.00184.x
[8]
Chen, S.X. (2008) Nonparametric Estimation of Expected Shortfall. Journal of Financial Econometrics, 6, 87-107. https://doi.org/10.1093/jjfinec/nbm019
[9]
Embrechts, P., Kaufmann, R. and Patie, P. (2005) Strategic Long-Term Financial Risks: Single Risk Factors. Computational Optimization and Applications, 32, 61-90. https://doi.org/10.1007/s10589-005-2054-7
[10]
Kjellson, B. (2013) Forecasting Expected Shortfall. An Extreme Value Approach. Bachelor’s Thesis in Mathematical Sciences, K7, 43.
[11]
Koksal, B. and Orhan, M. (2013) Market Risk of Developed and Emerging Countries during the Global Financial Crisis. Emerging Markets. Finance & Trade, 49, 20-34. https://doi.org/10.2753/REE1540-496X490302
[12]
Lee, T.-H., Bao, Y. and Saltoglu, B. (2006) Evaluating Predictive Performance of Value-at-Risk Models in Emerging Markets: A Reality Check. Journal of Forecasting, 25, 101-128. https://doi.org/10.1002/for.977
[13]
Yoon, S.-M. and Kang, S.-H. (2013) VaR Analysis for the Shanghai Stock Market. The Macrotheme Review, 2, 89-95.
[14]
Zivot, E. and Wang, J. (2003) Modeling Financial Time Series with S-PLUS. Springer-Verlag, New York. https://doi.org/10.1007/978-0-387-21763-5
[15]
Palma, W. (2007) Long-Memory Time Series: Theory and Methods. John Wiley & Sons, Hoboken. https://doi.org/10.1002/9780470131466
[16]
Zrazhevska, N. (2016) Construction and Application of the Classification Scheme of Dynamic Risk Measures Estimating. Eureka: Physics and Engineering, 5, 67-80. https://doi.org/10.21303/2461-4262.2016.00162
[17]
Pankratova, N.D. and Zrazhevska, N.G. (2015) Model of the Autocorrelation Function of the Time Series with Strong Dependence. Problemi Upravleniya Informatik, 5, 102-112.
[18]
Zrazhevska, N.G. (2015) Smoothed Autocorrelation Function Method for Predicting of Variation of Heteroscedastictime. Systemnidoslidzhennya ta informacijnitexnologiyi, 2, 97-108.
[19]
Zrazhevskij, A.G. (2011) System Approach to Restoring Functional Dependencies of Non-Stationary Time Series with Different Structures. Abstract of Dissertation for Scientific Degree of PhD, 20. (In Ukrainian)
[20]
Cochrane, J.H. (1988) How Big Is the Random Walk in GNP? Journal of Political Economy, 96, 893-920. https://doi.org/10.21303/2461-4262.2016.00162
[21]
Zrazhevskaja, N.G. and Zrazhevskij, A.G. (2016) Classification of Methods for Risk Measures VaR and CVaR Calculation and Estimation. Systemnidoslidzhennya ta informacijnitexnologiyi, 3.
[22]
Shang, D., Kuzmenko, V. and Uryasev, S. (2015) Cash Flow Matching with Risks Controlled by bPOE and CVaR. Research Report 2015-3, ISE Dept., University of Florida.
