Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T03:50:21.861Z Has data issue: false hasContentIssue false
  • English
  • Français

AGGREGATE CLAIM ESTIMATION USING BIVARIATE HIDDEN MARKOV MODEL

Published online by Cambridge University Press:  29 November 2018

Zarina Nukeshtayeva Oflaz,
Ceylan Yozgatligil and
A. Sevtap Selcuk-Kestel
Zarina Nukeshtayeva Oflaz
Affiliation:
Department of Statistics, Middle East Technical University, Ankara, TurkeyDepartment of Insurance and Social Security, KTO Karatay University, Konya, Turkey E-Mail: nukesh.zar@mail.ru
Ceylan Yozgatligil
Affiliation:
Department of Statistics, Middle East Technical University, Ankara, Turkey E-Mail: ceylan@metu.edu.tr
A. Sevtap Selcuk-Kestel*
Affiliation:
Institute of Applied Mathematics, Actuarial Sciences, Middle East Technical University, Ankara, Turkey E-Mail: skestel@metu.edu.tr
*
E-Mail: skestel@metu.edu.tr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we propose an approach for modeling claim dependence, with the assumption that the claim numbers and the aggregate claim amounts are mutually and serially dependent through an underlying hidden state and can be characterized by a hidden finite state Markov chain using bivariate Hidden Markov Model (BHMM). We construct three different BHMMs, namely Poisson–Normal HMM, Poisson–Gamma HMM, and Negative Binomial–Gamma HMM, stemming from the most commonly used distributions in insurance studies. Expectation Maximization algorithm is implemented and for the maximization of the state-dependent part of log-likelihood of BHMMs, the estimates are derived analytically. To illustrate the proposed model, motor third-party liability claims in Istanbul, Turkey, are employed in the frame of Poisson–Normal HMM under a different number of states. In addition, we derive the forecast distribution, calculate state predictions, and determine the most likely sequence of states. The results indicate that the dependence under indirect factors can be captured in terms of different states, namely low, medium, and high states.

Keywords

Claim estimationbivariate Hidden Markov modelEM algorithmViterbi algorithmMTPL

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 1 , January 2019, pp. 189 - 215
Copyright
Copyright © Astin Bulletin 2018 

References

Ambagaspitiya, R.S. (1999) On the distributions of two classes of correlated aggregate claims. Insurance: Mathematics and Economics, 24(3), 301–308.Google Scholar
Äuerle, N. and Müller, A. (1998) Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bulletin: The Journal of the IAA, 28(1), 59–76.Google Scholar
Avanzi, B., Wong, B. and Yang, X. (2016) A micro-level claim count model with overdispersion and reporting delays. Insurance: Mathematics and Economics, 71, 1–14.Google Scholar
Badescu, A.L., Lin, X.S. and Tang, D. (2016) A marked Cox model for the number of IBNR claims: Theory. Insurance: Mathematics and Economics, 69, 29–37.Google Scholar
Boudreault, M., Cossette, H. and Marceau, Ã. (2014) Risk models with dependence between claim occurrences and severities for Atlantic hurricanes. Insurance: Mathematics and Economics, 54, 123–132.Google Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics. Shaumburg, IL: The Society of Actuaries.Google Scholar
Cheung, L.W.K. (2004) Use of runs statistics for pattern recognition in genomic DNA sequences. Journal of Computational Biology, 11(1), 107–124.CrossRefGoogle ScholarPubMed
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1), 1–38.CrossRefGoogle Scholar
Denuit, M., Lefèvre, C. and Utev, S. (2002) Measuring the impact of dependence between claims occurrences. Insurance: Mathematics and Economics, 30(1), 1–19.Google Scholar
Dhaene, J. and Goovaerts, M.J. (1997) On the dependency of risks in the individual life model. Insurance: Mathematics and Economics, 19(3), 243–253.Google Scholar
Dias, J.G. and Ramos, S.B. (2014) Dynamic clustering of energy markets: An extended hidden Markov approach. Expert Systems with Applications, 41(17), 7722–7729.CrossRefGoogle Scholar
Forney, G.D. (1973) The Viterbi algorithm. Proceedings of the IEEE, 61(3), 268–278.CrossRefGoogle Scholar
Garanti Securities (2016) Turkish Non-life Insurance Sector - Waiting for the Dust to Settle. Istanbul: Garanti Securities.Google Scholar
Garrido, J., Genest, C. and Schulz, J. (2016) Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Mathematics and Economics, 70, 205–215.Google Scholar
Hassan, M.R. and Nath, B. (2005) Stock market forecasting using hidden Markov model: A new approach. Proceedings of the 5th International Conference on Intelligent Systems Design and Applications, 2005, ISDA’05, pp. 192–196. IEEE, Warshaw, Poland.Google Scholar
Hudecová, Š. and Pešta, M. (2013) Modeling dependencies in claims reserving with GEE. Insurance: Mathematics and Economics, 53(3), 786–794.Google Scholar
Inflation rates (2016), http://www.bigpara.com/haberler/enflasyon-verileri/, last accessed 13 July 2016.Google Scholar
JLT Sigorta ve Reasurans Brokerligi A.S. (2017) Turkish Insurance Market Outlook 2016-17. Istanbul: JLT Sigorta ve Reasurans Brokerligi A.S.Google Scholar
Krogh, A., Brown, M., Mian, I.S., Sjölander, K. and Haussler, D. (1994) Hidden Markov models in computational biology: Applications to protein modeling. Journal of molecular biology, 235(5), 1501–1531.CrossRefGoogle ScholarPubMed
Little, R.J. and Rubin, D.B. (2014) Statistical Analysis with Missing Data. Hoboken: John Wiley & Sons.Google ScholarPubMed
Oflaz, Z. (2016) Bivariate Hidden Markov model to capture the claim dependency. Unpublished M.Sc. Thesis, METU, Department of Statistics.Google Scholar
Paroli, R., Redaelli, G. and Spezia, L. (2000) Poisson hidden Markov models for time series of overdispersed insurance counts. Proceedings of the XXXI International ASTIN Colloquium (Porto Cervo, 18–21 September 2000). Roma: Instituto Italiano degli Attuari.Google Scholar
Pešta, M. and Okhrin, O. (2014) Conditional least squares and copulae in claims reserving for a single line of business. Insurance: Mathematics and Economics, 56, 28–37.Google Scholar
Pinquet, J. (2000) Experience rating through heterogeneous models. In Handbook of Insurance, pp. 459–500. Netherlands: Springer.CrossRefGoogle Scholar
Rabiner, L.R. (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257–286.CrossRefGoogle Scholar
Rabiner, L.R. and Juang, B.H. (1993) Fundamentals of Speech Recognition, Vol. 14. Englewood Cliffs: PTR Prentice Hall.Google Scholar
Ren, J. (2012) A multivariate aggregate loss model. Insurance: Mathematics and Economics, 51(2), 402–408.Google Scholar
Rocca, G. (2016) Insurance Objectives, studio D, http://work.chron.com/insurance-objectives-23793.html, last accessed 12 July 2016.Google Scholar
Tse, Y.K. (2009) Nonlife Actuarial Models. Theory, Methods and Evaluation. New York: Cambridge University Press.CrossRefGoogle Scholar
Viterbi, A. (1967) Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Transactions on Information Theory, 13(2), 260–269.CrossRefGoogle Scholar
Wang, G. and Yuen, K.C. (2005) On a correlated aggregate claims model with thinning-dependence structure. Insurance: Mathematics and Economics, 36(3), 456–468.Google Scholar
Yuen, K.C., Guo, J. and Wu, X. (2002) On a correlated aggregate claims model with Poisson and Erlang risk processes. Insurance: Mathematics and Economics, 31(2), 205–214.Google Scholar
Zucchini, W. and Guttorp, P. (1991) A hidden Markov model for space -time precipitation. Water Resources Research, 27(8), 1917–1923.CrossRefGoogle Scholar
Zucchini, W. and MacDonald, I. L. (2009) Hidden Markov Models for Time Series: An Introduction Using R. Boca Raton: Chapman and Hall, CRC.CrossRefGoogle Scholar