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Multivariate Frequency-Severity Regression Models in Insurance

Author

Listed:
  • Edward W. Frees

    (School of Business, University of Wisconsin-Madison, 975 University Avenue, Madison, WI 53706, USA)

  • Gee Lee

    (School of Business, University of Wisconsin-Madison, 975 University Avenue, Madison, WI 53706, USA)

  • Lu Yang

    (Department of Statistics, University of Wisconsin-Madison, 1300 University Avenue, Madison, WI 53706, USA)

Abstract

In insurance and related industries including healthcare, it is common to have several outcome measures that the analyst wishes to understand using explanatory variables. For example, in automobile insurance, an accident may result in payments for damage to one’s own vehicle, damage to another party’s vehicle, or personal injury. It is also common to be interested in the frequency of accidents in addition to the severity of the claim amounts. This paper synthesizes and extends the literature on multivariate frequency-severity regression modeling with a focus on insurance industry applications. Regression models for understanding the distribution of each outcome continue to be developed yet there now exists a solid body of literature for the marginal outcomes. This paper contributes to this body of literature by focusing on the use of a copula for modeling the dependence among these outcomes; a major advantage of this tool is that it preserves the body of work established for marginal models. We illustrate this approach using data from the Wisconsin Local Government Property Insurance Fund. This fund offers insurance protection for (i) property; (ii) motor vehicle; and (iii) contractors’ equipment claims. In addition to several claim types and frequency-severity components, outcomes can be further categorized by time and space, requiring complex dependency modeling. We find significant dependencies for these data; specifically, we find that dependencies among lines are stronger than the dependencies between the frequency and average severity within each line.

Suggested Citation

  • Edward W. Frees & Gee Lee & Lu Yang, 2016. "Multivariate Frequency-Severity Regression Models in Insurance," Risks, MDPI, vol. 4(1), pages 1-36, February.
    • Handle: RePEc:gam:jrisks:v:4:y:2016:i:1:p:4-:d:64467
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    References listed on IDEAS

    as
    1. Sun, Jiafeng & Frees, Edward W. & Rosenberg, Marjorie A., 2008. "Heavy-tailed longitudinal data modeling using copulas," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 817-830, April.
    2. Anastasios Panagiotelis & Claudia Czado & Harry Joe, 2012. "Pair Copula Constructions for Multivariate Discrete Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1063-1072, September.
    3. Frees, Edward W. & Meyers, Glenn & Cummings, A. David, 2011. "Summarizing Insurance Scores Using a Gini Index," Journal of the American Statistical Association, American Statistical Association, vol. 106(495), pages 1085-1098.
    4. Kojadinovic, Ivan & Segers, Johan & Yan, Jun, 2011. "Large-sample tests of extreme-value dependence for multivariate copulas," LIDAM Reprints ISBA 2011025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    5. Frees,Edward W., 2004. "Longitudinal and Panel Data," Cambridge Books, Cambridge University Press, number 9780521828284, September.
    6. Shi, Peng, 2012. "Multivariate longitudinal modeling of insurance company expenses," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 204-215.
    7. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    8. Peter Xue‐Kun Song, 2000. "Multivariate Dispersion Models Generated From Gaussian Copula," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(2), pages 305-320, June.
    9. Genest C. & Boies J-C., 2003. "Detecting Dependence With Kendall Plots," The American Statistician, American Statistical Association, vol. 57, pages 275-284, November.
    10. Aristidis Nikoloulopoulos & Dimitris Karlis, 2010. "Regression in a copula model for bivariate count data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(9), pages 1555-1568.
    11. Elif F. Acar & Radu V. Craiu & Fang Yao, 2011. "Dependence Calibration in Conditional Copulas: A Nonparametric Approach," Biometrics, The International Biometric Society, vol. 67(2), pages 445-453, June.
    12. Genest, Christian & Nešlehová, Johanna, 2007. "A Primer on Copulas for Count Data," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 475-515, November.
    13. Hua, Lei, 2015. "Tail negative dependence and its applications for aggregate loss modeling," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 135-145.
    14. de Jong,Piet & Heller,Gillian Z., 2008. "Generalized Linear Models for Insurance Data," Cambridge Books, Cambridge University Press, number 9780521879149, September.
    15. Edward Frees & Yunjie (Winnie) Sun, 2010. "Household Life Insurance Demand," North American Actuarial Journal, Taylor & Francis Journals, vol. 14(3), pages 338-354.
    16. Krämer, Nicole & Brechmann, Eike C. & Silvestrini, Daniel & Czado, Claudia, 2013. "Total loss estimation using copula-based regression models," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 829-839.
    17. Andrew J. Patton, 2006. "Modelling Asymmetric Exchange Rate Dependence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(2), pages 527-556, May.
    18. Edward Frees & Jie Gao & Marjorie Rosenberg, 2011. "Predicting the Frequency and Amount of Health Care Expenditures," North American Actuarial Journal, Taylor & Francis Journals, vol. 15(3), pages 377-392.
    19. Kojadinovic, Jean D. & Segers, Johan & Yan, Yun, 2011. "Large-sample tests of extreme-value dependence for multivariate copulas," LIDAM Discussion Papers ISBA 2011012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. Frees,Edward W., 2004. "Longitudinal and Panel Data," Cambridge Books, Cambridge University Press, number 9780521535380, September.
    21. Edward Frees & Ping Wang, 2005. "Credibility Using Copulas," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 31-48.
    22. Frees, Edward W. & Jin, Xiaoli & Lin, Xiao, 2013. "Actuarial Applications of Multivariate Two-Part Regression Models," Annals of Actuarial Science, Cambridge University Press, vol. 7(2), pages 258-287, September.
    23. Edward Frees & Emiliano Valdez, 1998. "Understanding Relationships Using Copulas," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 1-25.
    24. Shi, Peng & Feng, Xiaoping & Ivantsova, Anastasia, 2015. "Dependent frequency–severity modeling of insurance claims," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 417-428.
    25. Edward W. Frees, 2015. "Analytics of Insurance Markets," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 253-277, December.
    26. Song, Peter X.K. & Fan, Yanqin & Kalbfleisch, John D., 2005. "Maximization by Parts in Likelihood Inference," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1145-1158, December.
    27. Patrick L. Brockett & Linda L. Golden & Montserrat Guillen & Jens Perch Nielsen & Jan Parner & Ana Maria Perez‐Marin, 2008. "Survival Analysis of a Household Portfolio of Insurance Policies: How Much Time Do You Have to Stop Total Customer Defection?," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 75(3), pages 713-737, September.
    28. Frees, Edward W. & Wang, Ping, 2006. "Copula credibility for aggregate loss models," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 360-373, April.
    29. Brechmann, Eike & Czado, Claudia & Paterlini, Sandra, 2014. "Flexible dependence modeling of operational risk losses and its impact on total capital requirements," Journal of Banking & Finance, Elsevier, vol. 40(C), pages 271-285.
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