Segmentation and Estimation of Claim Severity in Motor Third-party Liability Insurance through Contrast Analysis

• Autorzy: Marian Reiff , Erik Šoltés , Silvia Komara , Tatiana Šoltésová , Silvia Zelinová
• Języki publikacji: EN

Abstrakty

EN

Research background: Using the marginal means and contrast analysis of the target variable, e.g., claim severity (CS), the actuary can perform an in-depth analysis of the portfolio and fully use the general linear models potential. These analyses are mainly used in natural sciences, medicine, and psychology, but so far, it has not been given adequate attention in the actuarial field.
Purpose of the article: The article's primary purpose is to point out the possibilities of contrast analysis for the segmentation of policyholders and estimation of CS in motor third-party liability insurance. The article focuses on using contrast analysis to redefine individual relevant factors to ensure the segmentation of policyholders in terms of actuarial fairness and statistical correctness. The aim of the article is also to reveal the possibilities of using contrast analysis for adequate segmentation in case of interaction of factors and the subsequent estimation of CS.
Methods: The article uses the general linear model and associated least squares means. Contrast analysis is being implemented through testing and estimating linear combinations of model parameters. Equations of estimable functions reveal how to interpret the results correctly.
Findings & value added: The article shows that contrast analysis is a valuable tool for segmenting policyholders in motor insurance. The segmentation's validity is statistically verifiable and is well applicable to the main effects. Suppose the significance of cross effects is proved during segmentation. In that case, the actuary must take into account the risk that even if the partial segmentation factors are set adequately, statistically proven, this may not apply to the interaction of these factors. The article also provides a procedure for segmentation in case of interaction of factors and the procedure for estimation of the segment's CS. Empirical research has shown that CS is significantly influenced by weight, engine power, age and brand of the car, policyholder's age, and district. The pattern of age's influence on CS differs in different categories of car brands. The significantly highest CS was revealed in the youngest age category and the category of luxury car brands. (original abstract)

Słowa kluczowe

PL

Modele liniowe; Ubezpieczenia komunikacyjne OC; Odszkodowania

EN

Linear models; Motor insurance; Compensation

• Czasopismo: Equilibrium
• Rocznik: 2022
• Tom: 17
• Numer: nr 3
• Strony: 803-842

Twórcy

autor

Marian Reiff

• University of Economics in Bratislava, Slovak Republic

autor

Erik Šoltés

• University of Economics in Bratislava, Slovakia

autor

Silvia Komara

• University of Economics, Slovakia

autor

Tatiana Šoltésová

• University of Economics, Slovakia

autor

Silvia Zelinová

• University of Economics in Bratislava

Bibliografia

• Agresti, A. (2015). Foundations of linear and generalized linear models. New York: John Wiley & Sons.
• Alemany, R., Bolancé, C., Rodrigo, R., & Vernic, R. (2020). Bivariate mixed Poisson and Normal Generalised Linear models with Sarmanov dependence-an application to model claim frequency and optimal transformed average severity. Mathematics, 9(1), 73. doi: 10.3390/math9010073.
• Ayuso, M., Guillen, M., & Nielsen, J. P. (2019). Improving automobile insurance ratemaking using telematics: incorporating mileage and driver behaviour data. Transportation, 46(3), 735-752. doi: 10.1007/s11116-018-9890-7.
• Bae, J., Kim, Y. Y., & Lee, J. S. (2017). Factors associated with subjective life expectancy: comparison with actuarial life expectancy. Journal of Preventive Medicine and Public Health, 50(4), 240. doi: 10.3961/jpmph.17.036.
• Bergelt, M., Fung Yuan, V., O'Brien, R., Middleton, L. E., & Martins dos Santos, W. (2020). Moderate aerobic exercise, but not anticipation of exercise, improves cognitive control. PloS One, 15(11), e0242270. doi: 10.1371/journal.pone.0242270.
• Burka, D., Kovács, L., & Szepesváry, L. (2021). Modelling MTPL insurance claim events: can machine learning methods overperform the traditional GLM approach? Hungarian Statistical Review, 4(2), 34-69. doi: 10.35618/hsr2021.02.en034.
• Byrne, K. M., Adler, P. B., & Lauenroth, W. K. (2017). Contrasting effects of precipitation manipulations in two Great Plains plant communities. Journal of Vegetation Science, 28(2), 238-249. doi: 10.1111/jvs.12486.
• Cai, W. (2014). Making comparisons fair: how LS-means unify the analysis of linear models. SAS Institute Inc. Paper SA, S060-2014.
• Colin, T., Bruce, J., Meikle, W. G., & Barron, A. B. (2018). The development of honey bee colonies assessed using a new semi-automated brood counting method: CombCount. PLoS One, 13(10), e0205816. doi: 10.1371/journal.pone.0205816.
• Darlington, R. B., & Hayes, A. F. (2016). Regression analysis and linear models: concepts, applications, and implementation. Guilford Publications.
• David, M. (2015). Auto insurance premium calculation using generalized linear models. Procedia Economics and Finance, 20, 147-156. doi: 10.1016/S2212-5671(15)00059-3.
• de Azevedo, F. C., Oliveira, T. A., & Oliveira, A. (2016). Modeling non-life insurance price for risk without historical information. REVSTAT-Statistical Journal, 14(2), 171-192. doi: 10.57805/revstat.v14i2.185.
• de Jong, P., & Heller, G. Z. (2008). Generalized linear models for insurance data. Cambridge Books.
• de Sá, J. P. M. (2007). Applied statistics using SPSS, Statistica, MatLab and R. Springer Science & Business Media.
• Dean, A., Voss, D., & Draguljić, D. (2017). Design and analysis of experiments Springer, Cham.
• Duan, Z., Chang, Y., Wang, Q., Chen, T., & Zhao, Q. (2018). A logistic regression based auto insurance rate-making model designed for the insurance rate reform. International Journal of Financial Studies, 6(1), 18. doi: 10.3390/ijfs6010018.
• Elswick Jr, R. K., Gennings, C., Chinchilli, V. M., & Dawson, K. S. (1991). A simple approach for finding estimable functions in linear models. American Statistician, 45(1), 51-53. doi: 10.1080/00031305.1991.10475766.
• Ennour-Idrissi, K., Têtu, B., Maunsell, E., Poirier, B., Montoni, A., Rochette, P. J., & Diorio, C. (2016). Association of telomere length with breast cancer prognostic factors. PLoS One, 11(8), e0161903. doi: 10.1371/journal.pone.0161903.
• Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.
• Frees, E. W., Derrig, R. A., & Meyers, G. (Eds.) (2014). Predictive modelling applications in actuarial science (Vol. 1). Cambridge University Press.
• Frees, E. W., Lee, G., & Yang, L. (2016). Multivariate frequency-severity regression models in insurance. Risks, 4(1), 4. doi: 10.3390/risks4010004.
• Fung, T. C., Badescu, A. L., & Lin, X. S. (2021). A new class of severity regression models with an application to IBNR prediction. North American Actuarial Journal, 25(2), 206-231. doi: 10.1080/10920277.2020.1729813.
• George, D., & Mallery, P. (2019). IBM SPSS statistics 26 step by step: a simple guide and reference. Routledge.
• Goldburd, M., Khare, A., Tevet, D., & Guller, D. (2016). Generalized linear models for insurance rating. Casualty Actuarial Society, CAS Monographs Series, 5.
• Goodnight, J. H, & Harvey, W. R (1997). SAS technical report R-103. Least Squares Means in the Fixed Effects General Model. Cary, NC: SAS Institute Inc.
• Haans, A. (2018). Contrast analysis: a tutorial. Practical Assessment, Research, and Evaluation, 23(1), 9. doi: 10.7275/7dey-zd62.
• Henckaerts, R., Antonio, K., Clijsters, M., & Verbelen, R. (2018). A data driven binning strategy for the construction of insurance tariff classes. Scandinavian Actuarial Journal, 8, 681-705. doi: 10.1080/03461238.2018.1429300.
• Henckaerts, R., Côté, M. P., Antonio, K., & Verbelen, R. (2021). Boosting insights in insurance tariff plans with tree-based machine learning methods. North American Actuarial Journal, 25(2), 255-285. doi: 10.1080/10920277.2020.1745656.
• Henckaerts, R., & Antonio, K. (2022). The added value of dynamically updating motor insurance prices with telematics collected driving behavior data. Insurance: Mathematics and Economics, 105, 79-95. doi: 10.1016/j.insmatheco.2022.03.011.
• Herberich, E., Sikorski, J., & Hothorn, T. (2010). A robust procedure for comparing multiple means under heteroscedasticity in unbalanced designs. PloS one, 5(3), e9788. doi: 10.1371/journal.pone.0009788.
• Huzar-Novakowiski, J., & Dorrance, A. E. (2018). Genetic diversity and population structure of Pythium irregulare from soybean and corn production fields in Ohio. Plant Disease, 102(10), 1989-2000. doi: 10.1094/PDIS-11-17-1725-RE.
• Kafková, S., & Křivánková, L. (2014). Generalized linear models in vehicle insurance. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 62(2), 383-388. doi: 10.11118/actaun201462020383.
• Kafková, S. (2015). Bonus-malus systems in vehicle insurance. Procedia Economics and Finance, 23, 216-222. doi: 10.1016/S2212-5671(15)00354-8.
• Kim, K., & Timm, N. (2006). Univariate and multivariate general linear models: theory and applications with SAS. Chapman and Hall/CRC.
• Kim, J. H. (2019). Multicollinearity and misleading statistical results. Korean Journal of Anesthesiology, 72(6), 558. doi: 10.4097/kja.19087.
• Kuznetsova. A., Brockhoff. P. B., & Christensen. R. H. B. (2017). lmerTest package: tests in linear mixed effects models. Journal of Statistical Software. 82(13), 1-26. doi: 10.18637/jss.v082.i13.
• LaMotte, L. R. (2020). A formula for Type III sums of squares. Communications in Statistics-Theory and Methods, 49(13), 3126-3136. doi: 10.1080/03610926.2019.1586933.
• Lee, S., & Lee, D. K. (2018). What is the proper way to apply the multiple comparison test? Korean Journal of Anesthesiology, 71(5), 353. doi: 10.4097/kja.d.18.00242.
• Lenth, R., V. (2016). Least-squares means: the R package lsmeans. Journal of Statistical Software, 69(1), 1-33. doi: 10.18637/jss.v069.i01.
• Lenth, R., Buerkner, P., Herve, M., Love, J., Miguez, F., Riebl, H., & Singmann, H. (2022). Estimated marginal means, aka least-squares means. R package 'emmeans', version 1.7.2. Retrieved from https://cran.r-project.org/web/packages/emmeans/emmeans.pdf (15.03.2022).
• Littell, R. C., Stroup, W. W., & Freund, R. J. (2010). SAS for linear models. Cary, NC: SAS Institute Inc.
• McFarquhar, M. (2016). Testable hypotheses for unbalanced neuroimaging data. Frontiers in Neuroscience, 10, 270. doi: 10.3389/fnins.2016.00270.
• O'Brien, R. M. (2014). Estimable functions in age-period-cohort models: a unified approach. Quality & Quantity, 48(1), 457-474. doi: 10.1007/s11135-012-9780-6.
• Olivera-La Rosa, A., Chuquichambi, E. G., & Ingram, G. P. (2020). Keep your (social) distance: pathogen concerns and social perception in the time of COVID-19. Personality and Individual Differences, 166, 110200. doi: 10.1016/j.paid.2020.110200.
• Ordaz, J. A., del Carmen Melgar, M., & Khan, M. K. (2011). An analysis of Spanish accidents in automobile insurance: the use of the Probit model and the theoretical potential of other econometric tools. Equilibrium. Equilibrium. Quarterly Journal of Economics and Economic Policy, 6(3), 117-134. doi: 10.12775/EQUIL2011.024.
• Poline, J. B., Kherif, F., Pallier, C., & Penny, W. (2007). Contrasts and classical inference. In W. D. Penny, K. J. Friston, J. T. Ashburner, S. J. Kiebel & T. E. Nichols (Eds.) (2011). Statistical parametric mapping: the analysis of functional brain images (126-139). Elsevier.
• Rafter, J. A., Abell, M. L., & Braselton, J. P. (2002). Multiple comparison methods for means. Siam Review, 44(2), 259-278. doi: 10.1137/S0036144501357233.
• Rivers, J. W., Newberry, G. N., Schwarz, C. J., & Ardia, D. R. (2017). Success despite the stress: violet-green swallows increase glucocorticoids and maintain reproductive output despite experimental increases in flight costs. Functional Ecology, 31(1), 235-244. doi: 10.1111/1365-2435.12719.
• Rahardja, D. (2020). Multiple comparison procedures for the differences of proportion parameters in over-reported multiple-sample binomial data. Stats, 3(1), 56-67. doi: 10.3390/stats3010006.
• Quigley, M. Y., Rivers, M. L., & Kravchenko, A. N. (2018). Patterns and sources of spatial heterogeneity in soil matrix from contrasting long term management practices. Frontiers in Environmental Science, 6, 28. doi: 10.3390/stats3010006
• SAS Institute Inc. (2017). The four types of estimable functions. In SAS/STAT® 14.3 User's Guide. Cary, NC: SAS Institute Inc.
• SAS Institute Inc. (2018). SAS/STAT® 15.1 User's Guide. The GLM Procedure. Cary, NC: SAS Institute Inc.
• Schad, D. J., Vasishth, S., Hohenstein, S., & Kliegl, R. (2020). How to capitalize on a priori contrasts in linear (mixed) models: a tutorial. Journal of Memory and Language, 110, 104038. doi: 10.1016/j.jml.2019.104038.
• Searle, S. R., & Gruber, M. H. J. (2017). Linear models. John Wiley & Sons.
• Searle, S. R., Speed, F. M., & Milliken, G. A. (1980). Population marginal means in the linear model: an alternative to least squares means. American Statistician, 34(4), 216-221. doi: 10.1080/00031305.1980.10483031.
• Shi, P., Feng, X., & Ivantsova, A. (2015). Dependent frequency-severity modelling of insurance claims. Insurance Mathematics and Economics, 64, 417-428. doi: 10.1016/j.insmatheco.2015.07.006.
• Singh, N., Wang, C., & Cooper, R. (2015). Role of vision and mechanoreception in bed bug, Cimex lectularius L. behavior. PLoS one, 10(3), e0118855. doi: 10.1371/journal.pone.0118855.
• Spilbergs, A., Fomins, A., Krastins, M. (2021). Impact of Covid-19 on the dynamics of MTPL insurance premiums and claims paid in Latvia. WSEAS Transactions on Computer Research, 9, 33-42. doi: 10.37394/232018.2021.9.5
• Spilbergs, A., Fomins, A., & Krastins, M. (2022). Road traffic accidents risk drivers' analysis - multivariate modelling based on Latvian motor third party liability insurance data. In D. Tipuric, A. Krajnovic & N. Recker (Eds.). Economic and social development: book of proceedings (pp. 246-264). Varazdin, Croatia: Varazdin Development and Entrepreneurship Agency.
• Statgraphics Technologies Inc. (2017). General linear models. Statgraphics centurion 18.
• Staudt, Y., & Wagner, J. (2021). Assessing the performance of random forests for modeling claim severity in collision car insurance. Risks, 9(3), 53. doi: 10.3390/risks9030053.
• Su, X., & Bai, M. (2020). Stochastic gradient boosting frequency-severity model of insurance claims. PloS one, 15(8), e0238000. doi: 10.1371/journal.pone.0238000.
• Suzuki, M., Taniguchi, T., Furihata, R., Yoshita, K., Arai, Y., Yoshiike, N., & Uchiyama, M. (2019). Seasonal changes in sleep duration and sleep problems: a prospective study in Japanese community residents. PLoS One, 14(4), e0215345. doi: 10.1371/journal.pone.0215345.
• Šoltés, E., Zelinová, S., & Bilíková, M. (2019). General linear model: an effective tool for analysis of claim severity in motor third party liability insurance. Statistics in Transition New Series, 20(4), 13-31, doi: 10.21307/stattrans-2019-032.
• Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics. Boston, MA: Pearson.
• Tattar, P. N., Ramaiah, S., & Manjunath, B. G. (2016). A course in statistics with R. John Wiley & Sons.
• Thompson, P. A. (2006). The "handy-dandy, quick-n-dirty" automated contrast generator-A SAS/IML R с macro to support the GLM, MIXED, and GENMOD procedures. SUGI 31 Statistics and data Analysis. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.176.736&rep=rep1&type=pdf (11.12.2021).
• Ugarte, M. D., Militino, A. F., & Arnholt, A. T. (2008). Probability and statistics with R. CRC press.
• Wang, B., Wu, P., Kwan, B., Tu, M. X., & Feng, Ch. (2018). Simpson's paradox: examples. Shanghai Archives of Psychiatry, 30(2), 139. doi: 10.11919/j.issn.1002-0829.218026.
• Westfall, P. H., & Tobias, R. D. (2007). Multiple testing of general contrasts: Truncated closure and the extended Shaffer-Royen method. Journal of the American Statistical Association, 102(478), 487-494. doi: 10.1198/016214506000001338.
• Wicklin R. (2018). Generalized inverses for matrices. Retrieved from https://blogs.sas.com/content/iml/2018/11/21/generalized-inverses-for-matrices.html (23.02. 2022).
• Wilcox, R. R. (2003). Applying contemporary statistical techniques. Elsevier.
• Wooldridge, J. M. (2013). Introductory econometrics: a modern approach. Mason: South-Western.
• Zahi, J. (2021). Non-life insurance ratemaking techniques. International Journal of Accounting, Finance, Auditing, Management and Economics, 2(1), 344-361. doi: 10.5281/zenodo.4474479.
• Zhao, J., Wang, C., Totton, S. C., Cullen, J. N., & O'Connor, A. M. (2019). Reporting and analysis of repeated measurements in preclinical animals experiments. PloS one, 14(8), e0220879. doi: 10.1371/journal.pone.0220879.

Identyfikatory

DOI

10.24136/eq.2022.028