Assessment of dependent risk using extreme value theory in a time-varying framework

Year 2023, Volume: 52 Issue: 1, 248 - 267, 15.02.2023

Bükre Yıldırım Külekci , Uğur Karabey , Sevtap Selcuk-kestel

https://doi.org/10.15672/hujms.992699

Cited By: 1

Abstract

Several extreme events in history have shown that the low probability and high impact extreme values may result in catastrophic losses. In this paper, we propose the use of extreme value theory with a time-varying framework to model the bivariate dependent insurance occurrences and provide more reliable risk measures, such as value at risk and expected shortfall. In this paper three models are considered; time series for the underlying volatility of the data, extreme value theory for the tail estimation, and copula to model the dependence structure are combined. The performance of the proposed generalized Pareto-GARCH-Copula model is tested using the violation numbers and backtesting methods. We then aim to assess the combined model in terms of its effectiveness in reducing the ruin probability. Results show that, compared to well-known traditional methods, which may underestimate the extreme risks, the dynamic generalized Pareto-GARCH-Copula model captures better the real-life data's behavior and results in lower ruin probabilities for heavy-tailed and non-conventional dependent insurance data.

Keywords

Extreme value theory, risk measures, copula, backtesting, ruin probability

References

• [1] A.A. Balkema and L.D. Haan, Residual life time at great age, Ann. Probab. 792–804, 1974.
• [2] O. Beelders and D. Colarossi, Modelling mortality risk with extreme value theory: The case of swiss res mortality-indexed bond, Global Association of Risk Professionals, 4(July/August), 26–30, 2004.
• [3] C.N. Behrens, H.F. Lopes and D. Gamerman, Bayesian analysis of extreme events with threshold estimation, Stat Modelling 4 (3), 227–244, 2004.
• [4] J. Beirlant and J.L. Teugels, Modeling large claims in non-life insurance, Insur Math Econ 11 (1), 17–29, 1992.
• [5] J. Bravo and P. Real, Modeling longevity risk using extreme value theory: An empirical investigation using portuguese and spanish population data, Portuguese Finance Network, 2012.
• [6] V. Chavez-Demoulin, P. Embrechts and S. Sardy, Extreme-quantile tracking for financial time series, J Econom 181 (1), 44–52, 2014.
• [7] P. Christoffersen and D. Pelletier, Backtesting value-at-risk: A duration-based approach, J. Financial Econ. 2 (1), 84–108, 2004.
• [8] P.F. Christoffersen, Evaluating interval forecasts, Int. Econ. Rev. 841–862, 1998.
• [9] S.G. Coles and E.A. Powell, Bayesian methods in extreme value modelling: a review and new developments, Int Stat Rev 119–136, 1996.
• [10] W.H. DuMouchel, Estimating the stable index α in order to measure tail thickness: a critique, Ann. Stat. 1019–1031, 1983.
• [11] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: for Insurance and Finance, Volume 33, Springer Science & Business Media, 2013.
• [12] P. Embrechts, A.Höing and A. Juri, Using copulae to bound the value-at-risk for functions of dependent risks, Finance Stoch. 7 (2), 145–167, 2003.
• [13] P. Embrechts, S.I. Resnick and G. Samorodnitsky, Extreme value theory as a risk management tool, N. Am. Actuar. J. 3 (2), 30–41, 1999.
• [14] P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insur Math Econ 1 (1), 55–72, 1982.
• [15] M. Gilli, An application of extreme value theory for measuring financial risk, Comput. Econ. 27 (2-3), 207–228, 2006.
• [16] R. Kaas, M. Goovaerts, J. Dhaene and M. Denuit, Modern Actuarial Risk Theory: Using R, Volume 128, Springer Science & Business Media, 2008.
• [17] V. Kalashnikov and R. Norberg, Power tailed ruin probabilities in the presence of risky investments, Stoch Process Their Appl 98 (2), 211–228, 2002.
• [18] A.Y. Khintchine, Mathematisches uber die erwortung vor einemoffenthchen schalter, Mat. Sb. 39, 73–84, 1932.
• [19] D.G. Konstantinides, Extremal subexponentiality in ruin probabilities, Comm. Statist. Theory Methods 40 (16), 2907–2918, 2011.
• [20] M. Kratz, Introduction to Extreme Value Theory: Applications to Risk Analysis and Management, 2017 MATRIX Annals 591–636. Springer, 2019.
• [21] P. Kupiec, Techniques for verifying the accuracy of risk measurement models, J. Deriv. 3 (2), 1995.
• [22] A.J. McNeil and T. Saladin, The peaks over thresholds method for estimating high quantiles of loss distributions, In Proceedings of 28th International ASTIN Colloquium 23–43, 1997.
• [23] J. Pickands, Statistical inference using extreme order statistics, Ann. Stat. 3 (1), 119–131, 1975.
• [24] F. Pollaczek, Über eine aufgabe der wahrscheinlichkeitstheorie. I., Math. Z. 32 (1), 64–100, 1930.
• [25] C. Scarrott and A. MacDonald, A review of extreme value threshold estimation and uncertainty quantification, Revstat Stat. J. 10 (1), 33–60, 2012.
• [26] A. Sklar, Distribution functions of n dimensions and margins, Publications of the Institute of Statistics of the University of Paris 8:229–231, 1959.
• [27] M.K.P. So and L.H. Philip, Empirical analysis of garch models in value at risk estimation, J. Int. Financial Mark. Inst. Money, 16 (2), 180–197, 2006.
• [28] M.S. Tsai and L.C. Chen, The calculation of capital requirement using extreme value theory, Econ. Model. 28 (1-2), 390–395, 2011.
• [29] B. Yıldırım Külekci, Risk Measurement Using Time Varying Extreme Value Copulas, Ph.D. thesis, Hacettepe University, Department of Actuarial Sciences, 2021.
• [30] X. Zhao, C.J. Scarrott, L. Oxley and M. Reale, Garch dependence in extreme value models with bayesian inference, Math Comput Simul 81 (7), 1430–1440, 2011.
• [31] E. Zivot and J. Wang, Vector Autoregressive Models for Multivariate Time Series, Modeling Financial Time Series with S-Plus® 385–429, 2006.