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Risk quantification in turmoil markets

Risk Management volume 19, pages 202–224 (2017)Cite this article

Abstract

The aim of this paper is to examine the performance of the Value-at-Risk measure under different distributional models in the highly demanding context of the recent financial crisis. This task is one of the main challenges of the financial industry. In addition to the normal and Student’s t distributions, we analyze three distributions especially appropriate for capturing tail risk: the generalized Pareto distribution (GPD), the α-stable distribution, and the g-and-h distribution. We also address the problem of efficiently estimating the parameters of these distributions. Our backtesting analysis shows that GPD and α-stable distributions perform well for this risk measurement purpose.

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References

  • Angelidis, T., A. Benos, and S. Degiannakis. 2004. The use of GARCH models in VaR estimation. Statistical Methodology 1 (1–2): 105–128.

    Article  Google Scholar 

  • Araújo Santos, P., and Fraga Alves, M.I. 2013. Forecasting Value-at-Risk with a duration-based POT method. Mathematics and Computers in Simulation 94 (C), 295–309.

  • Araújo Santos, P., I. Fraga-Alves, and S. Hammoudeh. 2013. High quantiles estimation with Quasi-PORT and DPOT: an application to value-at-risk for financial variables. North American Journal of Economics and Finance 26: 487–496.

    Article  Google Scholar 

  • Ausín, M.C., P. Galeano, and P. Ghosh. 2014. A semiparametric Bayesian approach to the analysis of financial time series with applications to value at risk estimation. European Journal of Operational Research 232 (2): 350–358.

    Article  Google Scholar 

  • Bank for International Settlements .2016. Minimum capital requirements for market risk. Basel Committee on Banking Supervision. January.

  • Beirlant, J. Goegebeur, Y., Segers, J., and Teugels, J. 2004. Statistics of extremes: theory and applications. New York: Wiley.

  • Bekiros, S.D., and D.A. Georgoutsos. 2005. Estimation of Value-at-Risk by exteme value and conventional methods: a comparative evaluation of their predictive performance. Journal of International Financial Markets, Institutions and Money 15: 209–228.

    Article  Google Scholar 

  • Berkowitz, J., and J. O’Brien. 2002. How accurate are Value-at-Risk models at commercial banks? Journal of Finance 57: 1093–1111.

    Article  Google Scholar 

  • Byström, H.N.E. 2004. Managing extreme risks in tranquil and volatile markets using conditional extreme value theory. International Review of Financial Analysis 13 (2): 133–152.

    Article  Google Scholar 

  • Carrasco, M., and Florens J. P. 2002. Efficient GMM Estimation Using the Empirical Characteristic Function. Working Paper, Institut d’Economie Industrielle, Toulouse.

  • Čížek, P., W.K. Härdle, and R. Weron. 2011. Statistical Tools for Finance and Insurance. Berlin: Springer.

    Google Scholar 

  • Chavez-Demoulin, V. 1999. Two problems in environmental statistics: Capture-recapture analysis and smooth extremal models. Ph.D. thesis. Department of Mathematics, Swiss Federal Institute of Technology, Lausanne.

  • Chavez-Demoulin, V., and J.A. McGill. 2012. High-frequency financial data modeling using Hawkes processes. Journal of Banking & Finance 36 (12): 3415–3426.

    Article  Google Scholar 

  • Christoffersen, P. 1998. Evaluating Interval Forecasts. International Economic Review 39: 841–862.

    Article  Google Scholar 

  • Christoffersen, P. 2003. Elements of financial risk management. San Diego, CA: Academic Press.

    Google Scholar 

  • Cont, R. 2001. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1: 223–236.

    Article  Google Scholar 

  • Danielsson, J. 2011. Financial risk forecasting: the theory and practice of forecasting market risk with implementation in R and Matlab. Blackwell: Wiley.

  • Davison, A.C., and R.L. Smith. 1990. Models for exceedances over high thresholds. Journal of the Royal Statistical Society 52: 393–442.

    Google Scholar 

  • De Jesús, R., E. Ortiz, and A. Cabello. 2013. Long run peso/dollar exchange rates and extreme value behavior: value at risk modeling. The North American Journal of Economics and Finance 24: 139–152.

    Article  Google Scholar 

  • Degen, M., P. Embrechts, and D. Lambrigger. 2007. The quantitative modeling of operational risk: between g-and-h and EVT. ASTIN Bulletin 37 (2): 265–291.

    Article  Google Scholar 

  • Del Brio, E.B., A. Mora-Valencia, and J. Perote. 2014. VaR performance during the subprime and sovereign debt crises: an application to emerging markets. Emerging Markets Review 20: 23–41.

    Article  Google Scholar 

  • Diebold, F.X., Schuermann, and T., Stroughair, J.D. 1998. Pitfalls and opportunities in the use of extreme value theory in risk management. Working Paper, 98–10, Wharton School, University of Pennsylvania.

  • Dutta, K., and Babbel D.F. 2002. On measuring skewness and kurtosis in short rate distributions: The Case of the US Dollar London Inter Bank óffer Rates. Wharton Financial Institutions Center Working Paper.

  • Dutta, K., and Perry, J. 2006. A tale of tails: An empirical analysis of loss distribution models for estimating operational risk capital. Federal Reserve Bank of Boston, Working Paper No. 06-13.

  • Embrechts, P., C. Klüppelberg, and T. Mikosch. 1997. Modelling extremal events for insurance and finance. Berlin: Springer.

    Book  Google Scholar 

  • Ergen, I. 2015. Two-step methods in VaR prediction and the importance of fat tails. Quantitative Finance 15 (6): 1013–1030.

    Article  Google Scholar 

  • Fama, E.F., and R. Roll. 1971. Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association 66: 331–338.

    Article  Google Scholar 

  • Ghosh, S., and S. Resnick. 2010. A discussion on mean excess plots. Stochastic Processes and their Applications 120: 1492–1517.

    Article  Google Scholar 

  • Goldberg, L.G.M., and J. Weinstein. 2008. Beyond value at risk: forecasting portfolio loss at multiple horizons. Journal of Investment Management 6 (2): 73–98.

    Google Scholar 

  • Granger, C.W.J., and Z. Ding. 1995. Some properties of absolute returns. An alternative measure of risk. Annales d’économie et de statistique 40: 67–91.

    Article  Google Scholar 

  • Grigoletto, M., and F. Lisi. 2009. Looking for skewness in financial time series. The Econometrics Journal 12: 310–323.

    Article  Google Scholar 

  • Grigoletto, M., and F. Lisi. 2011. Practical implications of higher moments in risk management. Statistical Methods and Applications 20: 487–506.

    Article  Google Scholar 

  • Güner, B., Rachev, S., Edelman, D., and Fabozzi, F.J. 2010. Bayesian inference for hedge funds with stable distribution of returns. En Rethinking risk measurement and reporting: Volume II (Ed. K. Böcker), 95–136.

  • Haas, M. 2009. Value-at-Risk via mixture distributions reconsidered. Applied Mathematics and Computation 215 (6): 2103–2119.

    Article  Google Scholar 

  • Haynes, M., and K. Mengersen. 2005. Bayesian estimation of g-and-k distributions using MCMC. Computational Statistics 20 (1): 7–30.

    Article  Google Scholar 

  • Herrera, R., and B. Schipp. 2014. Statistics of extreme events in risk management: the impact of the subprime and global financial crisis on the German stock market. The North American Journal of Economics and Finance 29: 218–238.

    Article  Google Scholar 

  • Hoaglin, D.C. 1985. Summarizing shape numerically: The g-and-h distributions. Chapter 11 en exploring data tables trends, and shapes. Eds. Hoaglin, Mosteller, and Tukey. New York, NY: John Wiley.

  • Jalal, A., and M. Rockinger. 2008. Predicting tail-related risk measures: the consequences of using GARCH filters for non GARCH data. Journal of Empirical Finance 15 (5): 868–877.

    Article  Google Scholar 

  • Jondeau, E., Poon, S.-H., and Rockinger M. 2007. Financial modeling under non-gaussian distributions, London: Springer

  • Khindanova, I., S. Rachev, and E. Schwartz. 2001. Stable modeling of value at risk. Mathematical and Computer Modelling 34 (9–11): 1223–1259.

    Article  Google Scholar 

  • Kiesel, R., Perraudin, W., and Taylor, A. 2003. An extreme analysis of VaRs for emerging market benchmark bonds. En: Credit Risk: Measurement, Evaluation and Management, eds: G.Bol, et al., Physica.

  • Koutrouvelis, I.A. 1980. Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association 75: 918–928.

    Article  Google Scholar 

  • Kuester, K., S. Mittnik, and M.S. Paolella. 2006. Value-at-risk prediction: a comparison of alternative strategies. Journal of Financial Econometrics 4: 53–89.

    Article  Google Scholar 

  • Kupiec, P. 1995. Techniques for verifying the accuracy of risk management models. Journal of Derivatives 3: 73–84.

    Article  Google Scholar 

  • Lauridsen, S. 2000. Estimation of value at risk by extreme value methods. Extremes 3 (2): 107–144.

    Article  Google Scholar 

  • Leadbetter, M.R. 1991. On a basis for ‘peaks over thresholds’ modeling. Statistics & Probability Letters 4: 357–362.

    Article  Google Scholar 

  • Li, W.K., S. Ling, and M. McAleer. 2002. Recent theoretical results for time series models with GARCH errors. Journal of Economic Surveys 16 (3): 245–269.

    Article  Google Scholar 

  • Lunde, A., and P.R. Hansen. 2005. A forecast comparison of volatility models: does anything beat a garch(1,1)? Journal of Applied Econometrics 20 (7): 873–889.

    Article  Google Scholar 

  • Marinelli, C., S. D’Addona, and S. Rachev. 2007. A comparison of some univariate models for Value-at-Risk and expected shortfall. International Journal of Theoretical and Applied Finance 10 (6): 1043–1075.

    Article  Google Scholar 

  • Martínez, J. 1981. Some Applications of robust scale estimators, Doctoral Thesis, Temple University. Philadelphia: Department of Statistics.

    Google Scholar 

  • Martínez, J., and B. Iglewicz. 1984. Some properties of the Tukey g and h family of distributions. Communications in Statistics: Theory and Methods 13 (3): 353–369.

    Article  Google Scholar 

  • McCulloch, J.H. 1986. Simple consistent estimators of stable distribution parameters. Communications in Statistics—Simulations 15: 1109–1136.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • McNeil, A.J., R. Frey, and P. Embrechts. 2005. Quantitative risk management: concepts, techniques and tools. Pricenton: Princeton University Press.

    Google Scholar 

  • Mittnik, S., S.T. Rachev, T. Doganoglu, and D. Chenyao. 1999. Maximum likelihood estimation of stable Paretian models. Mathematical and Computer Modelling 29: 275–293.

    Article  Google Scholar 

  • Nolan, J.P. 2001. Maximum likelihood estimation and diagnostics for stable distributions. In Lévy Processes, ed. E. Barndorff-Nielsen, T. Mikosch, and S. Resnick. Boston: Brikhäuser.

    Google Scholar 

  • Ourir, A., and W. Snoussi. 2012. Markets liquidity risk under extremal dependence: analysis with VaRs methods. Economic Modelling 29 (5): 1830–1836.

    Article  Google Scholar 

  • Perignon, C., Z.Y. Deng, and Z.J. Wang. 2008. Do banks overstate their Value-at-Risk? Journal of Banking & Finance 32: 783–794.

    Article  Google Scholar 

  • Polanski, A., and E. Stoja. 2010. Incorporating higher moments into value-at-risk forecasting. Journal of Forecasting 29: 523–535.

    Article  Google Scholar 

  • Rachev, S., Racheva-Iotova, B., and Stoyanov, S. 2010. Capturing fat tails. Risk Magazine, May 2010, pp. 72–77.

  • Rayner, G.D., and H.L. MacGillivray. 2002. Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions. Statistics and Computing 12: 57–75.

    Article  Google Scholar 

  • Rydén, T., T. Teräsvirta, and S. Asbrink. 1998. Stylized facts of daily return series and the hidden markov model. Journal of Applied Econometrics 13: 217–244.

    Article  Google Scholar 

  • Stavroyiannis, S., I. Makris, V. Nikolaidis, and L. Zarangas. 2012. Econometric modeling and value-at-risk using the Pearson type-IV distribution. International Review of Financial Analysis 22: 10–17.

    Article  Google Scholar 

  • Smith, R.L. 1989. Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Statistical Science 4: 367–393.

    Article  Google Scholar 

  • Teräsvirta, T., and Z. Zhao. 2011. Stylized facts of return series, robust estimates and three popular models of volatility. Applied Financial Economics 21: 67–94.

    Article  Google Scholar 

  • Wei, Y., W. Cheng, and Y. Lin. 2013. Measuring daily Value-at-Risk of SSEC index: a new approach based on multifractal analysis and extreme value theory. Physica A: Statistical Mechanics and its Applications 392 (9): 2163–2174.

    Article  Google Scholar 

  • Zhang, M.-H., and Q.-S. Cheng. 2005. An Approach to VaR for capital markets with Gaussian mixture. Applied Mathematics and Computation 168: 1079–1085.

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Spanish Ministerio de Ciencia e Innovación (ECO2014-59664-P), Spanish Ministerio de Economía y Competitividad (MTM2016-77501-P), Junta de Comunidades de Castilla-La Mancha (PEII-2014-019-P) and FAPA-Uniandes (P16.100322.001). Any errors are solely the responsibility of the authors.

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Authors and Affiliations

  1. Department of Economics and Finance, Universidad de Castilla-La Mancha, Plaza de la Universidad 1., 02071, Albacete, Spain

    Antonio Díaz

  2. Department of Economics and Finance and Instituto de Desarrollo Regional, Universidad de Castilla-La Mancha, Plaza de la Universidad 1., 02071, Albacete, Spain

    Gonzalo García-Donato

  3. School of Management, Universidad de los Andes, Calle 21 No. 1-20, Bogotá, Colombia

    Andrés Mora-Valencia

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  1. Antonio Díaz
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  2. Gonzalo García-Donato
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  3. Andrés Mora-Valencia
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Correspondence to Antonio Díaz.

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Díaz, A., García-Donato, G. & Mora-Valencia, A. Risk quantification in turmoil markets. Risk Manag 19, 202–224 (2017). https://doi.org/10.1057/s41283-017-0018-8

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Keywords

  • Backtesting
  • VaR
  • g-and-h
  • Alpha-stable
  • EVT-POT