×

On fitting dependent nonhomogeneous loss models to unearned premium risk. (English) Zbl 1489.91220

In this paper, the authors use a non-homogeneous Poisson process and an individual model capturing between-coverage dependence to study the risk attributed to unearned premium in non-life insurance. Specifically, the authors develop a generalized linear model for claim occurrences and use copulas to model loss amounts. Claim seasonality and multiple coverage frequency are incorporated. Proposition 1 gives the conditional expectation and the conditional variance of a risk variable linked to unearned premium. An algorithm for computing the reserve for unearned premium risk is presented in Section 3.4.1. Data analysis and model fitting are presented in Section 4. Specifically, a Poisson piecewise constant model with an adjusted exposure measure is used to model the intensity of claim frequency. The generalized linear models for coverages are fitted by the maximum likelihood estimation method. Using rank-based methods and a tail function, copulas are fitted to pairs of coverages. The empirical results based on Ontario auto databases are presented, and practical implications of the empirical results are discussed.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

VGAM; actuar; CopulaModel; DCL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdallah, A.; Boucher, J.-P.; Cossette., H., Sarmanov family of multivariate distributions for bivariate dynamic claim counts model, Insurance: Mathematics and Economics, 68, 120-33 (2016) · Zbl 1373.62507 · doi:10.1016/j.insmatheco.2016.01.003
[2] Agbeko, T.; Hiabu, M.; Martínez-Miranda, M.; Nielsen, J. P.; Verrall., R. J., Validating the double chain ladder stochastic claims reserving model, Variance: Advancing the Science of Risk, 8, 2, 138-60 (2014)
[3] Antonio, K., v Micro-level stochastic loss reserving for general insurance, Scandinavian Actuarial Journal, 7, 649-69 (2014) · Zbl 1401.91091 · doi:10.1080/03461238.2012.755938
[4] Barnett, G.; Zehnwirth, B.; Dubbossarsky., E., When can accident years be regarded as development years?, Proceedings of the Casualty Actuarial Society, 92, 177, 239-56 (2005)
[5] Bornhuetter, R. L.; Ferguson., R. E., The actuary and IBNR, Proceedings of the Casualty Actuarial Society, 59, 112, 181-95 (1972)
[6] Boucher, J.-P.; Denuit, M.; Guillén., M., Risk classification for claim counts: A comparative analysis of various zero-inflated mixed Poisson and Hurdle models, North American Actuarial Journal, 11, 4, 110-31 (2007) · Zbl 1480.91187 · doi:10.1080/10920277.2007.10597487
[7] Boucher, J.-P.; Denuit, M.; Guillén, M., Models of insurance claim counts with time dependence based on generalization of Poisson and negative binomial distributions, Variance, 2, 1, 135-62 (2008)
[8] Canadian Institute of Actuaries. 2014. Premium liabilities. Educational Notes 214114: 4-5.
[9] Cantin, C., and Trahan, P.. 1999. A study note on the actuarial evaluation of premium liabilities. Arlington VA: Casualty Actuarial Society, CAS Study Notes.
[10] Collins, E.; Hu, S., Practical considerations in valuing premium liabilities (2003)
[11] Cossette, H.; Côté, M.-P.; Marceau, E.; Moutanabbir., K., Multivariate distribution defined with Farlie-Gumbel-Morgenstern copula and mixed Erlang marginals: Aggregation and capital allocation, Insurance: Mathematics and Economics, 52, 3, 560-72 (2013) · Zbl 1284.60027 · doi:10.1016/j.insmatheco.2013.03.006
[12] Cossette, H.; Mailhot, M.; Marceau, É.; Mesfioui., M., Bivariate lower and upper orthant Value-at-Risk, European Actuarial Journal, 3, 2, 321-57 (2013) · Zbl 1304.91097 · doi:10.1007/s13385-013-0079-3
[13] Côté, M.-P.; Genest, C.; Abdallah., A., Rank-based methods for modeling dependence between loss triangles, European Actuarial Journal, 6, 2, 377-08 (2016) · Zbl 1394.91205 · doi:10.1007/s13385-016-0134-y
[14] Goulet, V., S. Auclair, C. Dutang, N. Langevin, X. Milhaud, T. Ouellet, A. Parent, M. Pigeon, L.-P. Pouliot, and J. A. Ryan.2008. actuar: An r package for actuarial science. Journal of Statistical Software25 (7):1-37. doi:
[15] Feldblum, S., NAIC property/casualty insurance company risk-based capital requirements, Proceedings of the Casualty Actuarial Society, 83, 159, 300-89 (1996)
[16] Financial Services Commission of Ontario, What do the coverages mean (2016)
[17] Frees, E. W., Dependent insurance risks, In Encyclopedia of Quantitative Risk Analysis and Assessment (2008)
[18] Frees, E. W.; Valdez., E. A., Hierarchical insurance claims modeling, Journal of the American Statistical Association, 103, 484, 1457-469 (2008) · Zbl 1286.62087 · doi:10.1198/016214508000000823
[19] Friedland, J.2010. Estimating unpaid claims using basic techniques. Arlington ,VA: Casualty Actuarial Society.
[20] Joe, H., Dependence modeling with copulas (2014), Chapman and Hall/CRC · Zbl 1346.62001
[21] Kaye, P. (2005)
[22] Kendall, M. G.1948. Rank correlation methods. Charles Green & Co. · Zbl 0032.17602
[23] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss models: From data to decisions (2012), John Wiley & Sons · Zbl 1272.62002
[24] Li, J., Prediction error of the future claims component of premium liabilities under the loss ratio approach, Variance, 4, 2, 155-69 (2010)
[25] Mack, T., Distribution-free calculation of the standard error of chain ladder reserve estimates, ASTIN Bulletin, 23, 2, 213-25 (1993) · doi:10.2143/AST.23.2.2005092
[26] Martínez-Miranda, M. D.; Nielsen, J. P.; Verrall., R., Double chain ladder, ASTIN Bulletin, 42, 1, 59-76 (2012) · Zbl 1277.91092
[27] Martínez-Miranda, M. D.; Nielsen, J. P.; Verrall., R., Double chain ladder and Bornhuetter-Ferguson, North American Actuarial Journal, 17, 2, 101-13 (2013) · Zbl 1515.91137 · doi:10.1080/10920277.2013.793158
[28] Meyers, G., Correlations between accident years in loss reserve models, The Actuarial Review, 40, 1, 27-28 (2013)
[29] Office of the Superintendent of Financial Institutions (2018)
[30] Office of the Superintendent of Financial Institutions (2018)
[31] Pechon, F.; Denuit, M.; Trufin., J., Multivariate modeling of multiple guarantees in motor insurance of a household, European Actuarial Journal, 1-28 (2019) · Zbl 1433.91143 · doi:10.1007/s13385-019-00201-5
[32] Pigeon, M.; Antonio, K.; Denuit., M., Individual loss reserving with the multivariate skew normal framework, ASTIN Bulletin, 43, 3, 399-428 (2013) · Zbl 1284.91263 · doi:10.1017/asb.2013.20
[33] Priest, C., Premium liability correlations and systemic risk, Australian Actuarial Journal, 18, 1, 1-66 (2012)
[34] Shi, P.; Feng, X.; Boucher, J.-P., Multilevel modeling of insurance claims using copulas, The Annals of Applied Statistics, 10, 2, 834-63 (2016) · Zbl 1400.62238 · doi:10.1214/16-AOAS914
[35] Shi, P.; Zhang, W.; Boucher., J.-P., Dynamic moral hazard: A longitudinal examination of automobile insurance in canada, Journal of Risk and Insurance, 85, 4, 939-58 (2018) · doi:10.1111/jori.12172
[36] Sklar, A., Random variables, joint distribution functions, and copulas, Kybernetika, 9, 6, 449-60 (1973) · Zbl 0292.60036
[37] Venter, G. G., Tails of copulas, Proceedings of the Casualty Actuarial Society, 89, 171, 68-113 (2002)
[38] Verrall, R.; Wüthrich., M., Understanding reporting delay in general insurance, Risks, 4, 3 (2016) · doi:10.3390/risks4030025
[39] Wüthrich, M. V.; Merz, M., Stochastic claims reserving methods in insurance (2008), John Wiley & Sons · Zbl 1273.91011
[40] Yang, L.; Shi., P., Multiperil rate making for property insurance using longitudinal data, Journal of the Royal Statistical Society: Series A (Statistics in Society), 182, 2, 647-68 (2019) · doi:10.1111/rssa.12419
[41] Yee, T., and C. Moler. (2010). The vgam package for categorical data analysis. Journal of Statistical Software, 32(10):1-34. doi:
[42] Zhao, X.; Zhou, X., Applying copula models to individual claim loss reserving methods, Insurance: Mathematics and Economics, 46, 2, 290-99 (2010) · Zbl 1231.91260 · doi:10.1016/j.insmatheco.2009.11.001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.