General insurance deductible ratemaking. (English) Zbl 1414.91211

Summary: Insurance claims have deductibles, which must be considered when pricing for insurance premium. The deductible may cause censoring and truncation to the insurance claims. However, modeling the unobserved response variable using maximum likelihood in this setting may be a challenge in practice. For this reason, a practitioner may perform a regression using the observed response, in order to calculate the deductible rates using the regression coefficients. A natural question is how well this approach performs, and how it compares to the theoretically correct approach to rating the deductibles. Also, a practitioner would be interested in a systematic review of the approaches to modeling the deductible rates. In this article, an overview of deductible ratemaking is provided, and the pros and cons of two deductible ratemaking approaches are compared: the regression approach and the maximum likelihood approach. The regression approach turns out to have an advantage in predicting aggregate claims, whereas the maximum likelihood approach has an advantage when calculating theoretically correct relativities for deductible levels beyond those observed by empirical data. For demonstration, loss models are fit to the Wisconsin Local Government Property Insurance Fund data, and examples are provided for the ratemaking of per-loss deductibles offered by the fund. The article discovers that the regression approach is actually a single-parameter approximation to the true relativity curve. A comparison of selected models from the generalized beta family discovers that the usage of long-tail severity distributions may improve the deductible rating, while advanced frequency models such as 01-inflated models may have limited advantages due to estimation issues under censoring and truncation. In addition, in this article, models for specific peril types are combined to improve the ratemaking.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI


[1] Aban, I. B.; Meerschaert, M. M.; Panorska, A. K., Parameter estimation for the truncated pareto distribution, Journal of the American Statistical Association, 101, 473, 270-277, (2006) · Zbl 1118.62312
[2] Anscombe, F. J., Large-sample theory of sequential estimation, Mathematical Proceedings of the Cambridge Philosophical Society, 48, 600-607, (1952) · Zbl 0047.13401
[3] Bahnemann, D., Distributions for Actuaries, (2015)
[4] Barr, D. R.; Sherrill, T., Mean and variance of truncated normal distributions, American Statistician, 53, 4, 357-361, (1999)
[5] Bernegger, S., The Swiss Re exposure curves and the MBBEFD distribution class, ASTIN Bulletin, 27, 1, 99-111, (1997)
[6] Brown, R. L.; Lennox, W. S., Ratemaking and Loss Reserving for Property and Casualty Insurance, (2015)
[7] Chapman, D. G., Estimating the parameters of a truncated gamma distribution, Annals of Mathematical Statistics, 27, 2, 498-506, (1956) · Zbl 0072.36208
[8] Chavez-Demoulin, V.; Embrechts, P.; Hofert, M., An extreme value approach for modeling operational risk losses depending on covariates, The Journal of Risk and Insurance, 83, 3, 735-776, (2016)
[9] Cohen, A.; Einav, L., Estimating risk preferences from deductible choice, American Economic Review, 97, 3, 745-788, (2007)
[10] Cummings, D., Practical glm analysis of homeowners, Presentation, State Farm Insurance Companies, (2005)
[11] Einav, L.; Finkelstein, A.; Pascu, I.; Cullen, M. R., How general are risk preferences? choices under uncertainty in different domains, American Economic Review, 102, 6, 2606-2638, (2012)
[12] Fasen, V.; Kluppelberg, C., Large insurance losses distributions, Wiley StatsRef: Statistics Reference Online, (2014)
[13] Finkelstein, D. M.; Wolfe, R. A., A semiparametric model for regression analysis of interval-censored failure time data, Biometrics, 41, 4, 933-945, (1985) · Zbl 0655.62101
[14] Frees, E. W., Frequency and severity models, Predictive Modeling Applications in Actuarial Science, (2014)
[15] Frees, E. W.; Lee, G.; Yang, L., Multivariate frequency-severity regression models in insurance, Risks, 4, 1, (2015)
[16] Frees, E. W.; Lee, G. Y., Rating endorsements using generalized linear models, Variance, 10, 1, 51-74, (2016)
[17] Gray, R. J.; Pitts, S. M., Risk Modelling in General Insurance: From Principles to Practice, (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1250.91057
[18] Guiahi, F., Fitting loss distributions with emphasis on rating variables, CAS Specialty Seminar, (2001) · Zbl 1070.91509
[19] Gurnecki, K.; Kukla, G.; Weron, R., Modelling catastrophe claims with left-truncated severity distributions, Computational Statistics, 21, 3, 537-555, (2006) · Zbl 1164.62411
[20] Halek, M.; Eisenhauer, J., Demography of risk aversion, Journal of Risk and Insurance, 68, 1, 1-24, (2001)
[21] Holt, C. A.; Laury, S. K., Risk aversion and incentive effects, American Economic Review, 92, 5, 00-00, (2002)
[22] Kalbfleisch, J. D.; Prentice, R. L., The Statistical Analysis of Failure Time Data, Second Edition, (2002), New Jersey: John Wiley & Sons, Inc, New Jersey · Zbl 1012.62104
[23] Kaplan, E. L.; Meier, P., Nonparametric estimation from incomplete observations, Journal of the American Statistical Association, 53, 457-481, (1958) · Zbl 0089.14801
[24] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: From Data to Decisions, Fourth Edition, (2012), New Jersey: John Wiley & Sons, New Jersey · Zbl 1272.62002
[25] Koszegi, B.; Rabin, M., A model of reference-dependent preferences, Quarterly Journal of Economics, 121, 4, 1133-1165, (2006) · Zbl 1179.91059
[26] Lai, T. L.; Ying, Z., Estimating a distribution function with truncated and censored data, Annals of Statistics, 19, 1, 417-442, (1991) · Zbl 0741.62037
[27] Ludwig, S. J., An exposure rating approach to pricing property excess-of-loss reinsurance, Casualty Actuarial Society, 78, (1991)
[28] Mas-Colell, A.; Whinston, M. D.; Green, J., Microeconomic Theory, First Edition, (1995), Oxford: Oxford University Press, Oxford
[29] Plackett, R. L., The truncated poisson distribution, International Biometric Society, 9, 4, 485-488, (1953)
[30] Rothschild, M.; Stiglitz, J., Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quarterly Journal of Economics, 90, 4, 629-649, (1976)
[31] Shi, P., Insurance ratemaking using a copula-based multivariate tweedie model, Scandinavian Actuarial Journal, 2016, 3, 198-215, (2016) · Zbl 1401.91194
[32] Siegmund, D., The sequential probability ratio test, Sequential Analysis: Tests and Confidence Intervals, (1985)
[33] Sydnor, J., (Over)insuring modest risks, American Economic Journal: Applied Economics, 2, 4, 177-199, (2010)
[34] Tse, Y.-K., Nonlife Actuarial Models: Theory, Methods and Evaluation, First Edition, (2009), Cambridge: Cambridge University Press, Cambridge · Zbl 1207.91007
[35] Verbelen, R.; Claeskens, G., Multivariate mixtures of erlangs for density estimation under censoring and truncation, Social Science Research Network, (2014)
[36] White, S., Property ratemaking — an advanced approach: Exposure rating, CAS Seminar on Reinsurance, (2005)
[37] White, S.; Mrazek, K., Advanced exposure rating: Beyond the basics, CAS Seminar on Reinsurance, (2004)
[38] Woodroofe, M., Estimating a distribution function with truncated data, Annals of Statistics, 13, 1, 163-177, (1985) · Zbl 0574.62040
[39] Wu, S.; Cai, X., Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac statistical entropies in a d-dimensional stationary axisymmetry space-time, Cornell University Library, (1999)
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