Evaluation of Value-at-Risk (VaR) using the Gaussian Mixture Models

References

• Adams M, Thornton B. 2013. Black swans and VaR. J Financ Acc. 14:1–17.
   Google Scholar
• Basel Committee on Banking Supervision 1996. Supervisory framework for the use of “Backtesting” in conjunction with the internal models approach to market risk capital requirements. Basel Committee on Banking Supervision, https://www.bis.org/publ/bcbs22.pdf.
   Google Scholar
• Basel Committee on Banking Supervision 2016. Minimum capital requirements for market risk. Basel Committee on Banking Supervision, https://www.bis.org/bcbs/publ/d352.pdf?fbclid=IwAR3ttD8Rda4FcyhZwDVnYJ9fsNw6wKOhcciMBk5BOcEyIh3p0GRmDEUuAMo#
   Google Scholar
• Beran J, Schell D, Stehlík M. 2014. The harmonic moment tail index estimator: asymptotic distribution and robustness. Ann Inst Stat Math. 66:193–220. doi:10.1007/s10463-013-0412-2.
   Web of Science ®Google Scholar
• Campbell SD. 2005. A review of backtesting and backtesting procedures. FEDS. 2005:1–23. doi:10.17016/FEDS.2005.21.
   Google Scholar
• Capital Requirements Regulation 2013. Article 365: VaR and stressed VaR calculation. Capital Requirements Regulation (575/2013), https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=celex:32013R0575.
   Google Scholar
• Christoffersen PF. 1998. Evaluating interval forecasts. Int Econ Rev. 39:841. doi:10.2307/2527341.
   Web of Science ®Google Scholar
• Contreras RJ. 2014. Market risk measures using finite gaussian mixtures. J Appl Finan Bank. 4:29–45.
   Google Scholar
• Cox, J. 2021. Investors have put more money into stocks in the last 5 months than the previous 12 years combined. https://www.cnbc.com/2021/04/09/investors-have-put-more-money-into-stocks-in-the-last-5-months-than-the-previous-12-years-combined.html.
   Google Scholar
• Cox, J. 2022. Inflation surges 7.5% on an annual basis, even more than expected and highest since 1982. https://www.cnbc.com/2022/02/10/january-2022-cpi-inflation-rises-7point5percent-over-the-past-year-even-more-than-expected.html.
   Google Scholar
• Cuevas-Covarrubias C, Inigo-Martinez J, Jimenez-Padilla R. 2017. Gaussian mixtures and financial returns. Discuss Math Probab Stat. 37:101. doi:10.7151/dmps.1190.
   Google Scholar
• Dey, E, 2022. S&P 500 plunges to nine-month low as Russia invades Ukraine. https://www.bloomberg.com/news/articles/2022-02-24/s-p-500-plunges-to-nine-month-low-as-russia-invades-ukraine.
   Google Scholar
• Eisenberger I. 1964. Genesis of bimodal distributions. Technometrics. 6:357–363. doi:10.1080/00401706.1964.10490199.
   Web of Science ®Google Scholar
• Fama EF, French KR. 1993. Common risk factors in the returns on stocks and bonds. J Financ Econ. 33:3–56. doi:10.1016/0304-405X(93)90023-5.
   Web of Science ®Google Scholar
• Feller, W. 1968. The normal approximation to the binomial distribution. In: An introduction to probability theory and its applications. Vol. 1 New York: Wiley. p. 174–196.
   Google Scholar
• Figueiredo F, Gomes M, Rodrigues L. 2017. Value-at-risk estimation and the port mean-of-order-p methodology. Revstat- Stat J. 15:187–204.
   Web of Science ®Google Scholar
• Haas, M. 2001. New methods in backtesting. https://www.ime.usp.br/rvicente/risco/haas.pdf.
   Google Scholar
• Hartigan AJ, Hartigan MP. 1985. The Dip test of unimodality. Ann Stat. 13:70–84.
   Web of Science ®Google Scholar
• Hull, JC. 2015. London: Pearson education. Value at risk. In: Options, futures, and other derivatives 9th ed., 494–520.
   Google Scholar
• Jordanova P, Fabián Z, Hermann P, Střelec L, Rivera A, Girard S, Torres S, Stehlík M. 2016. Weak properties and robustness of t-Hill estimators. Extremes. 19:591–626. doi:10.1007/s10687-016-0256-2.
   Web of Science ®Google Scholar
• Jorion, P. 2007. Backtesting VaR. In: Value at risk: the new benchmark for managing financial risk, 3rd ed.New York: McGraw-Hill. p. 153–158.
   Google Scholar
• Kupiec PH. 1995. Techniques for verifying the accuracy of risk measurement models. JOD. 3:73–84. doi:10.3905/jod.1995.407942.
   Google Scholar
• Macqueen, J. 1967. Some methods for classification and analysis of multivariate observations. Proc Fifth Berkeley Symp Math Statist Prob. 1:281– 297.
   Google Scholar
• Mandelbrot B. 1963. The variation of certain speculative prices. J BUS. 36:394. doi:10.1086/294632.
   Web of Science ®Google Scholar
• Markowitz H. 1952. Portfolio selection*. J Finance. 7:77–91. doi:10.1111/j.1540-6261.1952.tb01525.x.
   Web of Science ®Google Scholar
• Morgan, JP. 1996. RiskMetrics technical document. https://www.msci.com/documents/10199/5915b101-4206-4ba0-aee2-3449d5c7e95a.
   Google Scholar
• Nieppola, O. 2009. Backtestig Value-at-Risk Models. Master’s Thesis, Helsinki School of Economics. https://aaltodoc.aalto.fi/server/api/core/bitstreams/fafacabc-c1c0-4e52-b2f3-35dd99d03e71/content.
   Google Scholar
• Raschka, S. 2015. Working with unlabeled data-clustering analysis. In: Python machine learning. Birmingham: Packt Publishing, p. 311–340.
   Google Scholar
• Rudin C. 2021. Gaussian mixture models and expectation maximization. https://users.cs.duke.edu/cynthia/teaching.html.
   Google Scholar
• Sarkar D., Bali R., Sharma T. 2018. Backtesting VaR. In: Practical machine learning with Python: a problem-solver’s guide to building real-world intelligent systems. Berkeley (CA): Apress, p. 38–42.
   Google Scholar
• scikit-learn developers. 2017. scikit-learn user guide 0.18.2. https://scikit-learn.org/0.18/_downloads/scikit-learn-docs.pdf.
   Google Scholar
• Seyfi SMS, Sharifi A, Arian H. 2021. Portfolio Value-at-Risk and expected-shortfall using an efficient simulation approach based on Gaussian Mixture Model. Math Comput Simul. 190:1056–1079. doi:10.1016/j.matcom.2021.05.029.
   Web of Science ®Google Scholar
• Silva RCA, Da Silveira Barbedo CH, Araújo GS, Das Neves, MBE. 2006. Internal models validation in Brazil: analysis of VaR backtesting methodologies, https://www.redalyc.org/articulo.oa?id=305824716005.
   Google Scholar
• Taleb, NN. 2008. The Black Swan: the impact of the highly improbable. New York: Random House.
   Google Scholar
• Tan K. 2005. Modeling returns distribution based upon radical normal distributions. J Soc Stud Ind Econ. 46:371–383.
   Google Scholar
• Tan K, Chu M. 2012. Estimation of portfolio return and value at risk using a class of Gaussian mixture distributions. Int J Bus Finance Res. 6:97–107.
   Google Scholar
• Tan K, Tokinaga S. 2007. Approximating probability distribution function based upon mixture distribution optimized by genetic algorithm and its application to tail distribution analysis using importance sampling method. J Polit Econ. 74:183–196.
   Google Scholar
• Wang Z, Ritou M, Da Cunha C, Furet B. 2020. Contextual classification for smart machining based on unsupervised machine learning by Gaussian mixture model. Int J Comput Integr Manuf. 33:1042–1054. doi:10.1080/0951192X.2020.1775302.
   Web of Science ®Google Scholar