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Stationary probabilities and the monotone likelihood ratio in bonus-malus systems

Published online by Cambridge University Press:  04 September 2025

Kolos Csaba Ágoston Open the ORCID record for Kolos Csaba Ágoston [Opens in a new window]  and
Dávid Papp
Kolos Csaba Ágoston*
Affiliation:
Department of Operations Research and Actuarial Sciences Corvinus University of Budapest H-1093, Fövám tér 13-15., Budapest, Hungary
Dávid Papp
Affiliation:
Department of Mathematics North Carolina State University Raleigh, NC, USA
*
Corresponding author: Kolos Csaba Ágoston; Email: kolos.agoston@uni-corvinus.hu
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Abstract

The bonus-malus system (BMS) is a widely recognized and commonly employed risk management tool. A well-designed BMS can match expected insurance payments with estimated claims even in a diverse group of risks. Although there has been abundant research on improving bonus-malus (BM) systems, one important aspect has been overlooked: the stationary probability of a BMS satisfies the monotone likelihood ratio property. The monotone likelihood ratio for stationary probabilities allows us to better understand how riskier policyholders are more likely to remain in higher premium categories, while less risky policyholders are more likely to move toward lower premiums. This study establishes this property for BMSs that are described by an ergodic Markov chain with one possible claim and a transition rule +1/-d. We derive this result from the linear recurrences that characterize the stationary distribution; this represents a novel analytical approach in this domain. We also illustrate the practical implications of our findings: in the BM design problem, the premium scale is automatically monotonic.

Keywords

Bonus-malus designMonotone likelihood ratio propertyD81D86G22C61

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , First View , pp. 1 - 12
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The International Actuarial Association

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References

Abbring, J.H., Heckman, J.J., Chiappori, P.-A. and Pinquet, J. (2003) Adverse selection and moral hazard in insurance: Can dynamic data help to distinguish?. Journal of the European Economic Association, 1(2–3), 512–521.10.1162/154247603322391152CrossRefGoogle Scholar
Ágoston, K. and Gyetvai, M. (2020) Joint optimization of transition rules and the premium scale in a Bonus-Malus system. ASTIN Bulletin, 50(3), 743–776.10.1017/asb.2020.27CrossRefGoogle Scholar
Ágoston, K. and Gyetvai, M. (2022) On monotone likelihood ratio of stationary probabilities in bonus-malus systems. PU.M.A., 30, 43–53.Google Scholar
Boucher, J.-P. (2022) Multiple Bonus–Malus scale models for insureds of different sizes. Risks, 50(3), 2–16.Google Scholar
Caswell, H. (2013) Sensitivity analysis of discrete Markov chains via matrix calculus. Linear Algebra and its Applications, 438(4), 1727–1745.10.1016/j.laa.2011.07.046CrossRefGoogle Scholar
Coene, G. and Doray, L.G. (1996) A financially Balanced Bonus/Malus System. ASTIN Bulletin, 26(1), 107–116.10.2143/AST.26.1.563236CrossRefGoogle Scholar
Dellaert, N.P., Frenk, J.B.G., Kouwenhoven, A. and Van Der Laan, B.S. (1990) Optimal claim behaviour for third-party liability insurances or to claim or not to claim: That is the question. Insurance: Mathematics and Economics, 9(1), 59–76.Google Scholar
Frangos, N.E. and Vrontos, S.D. (2001) Design of optimal Bonus-Malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin, 31(1), 1–22.10.2143/AST.31.1.991CrossRefGoogle Scholar
Funderlic, R.E. and Meyer, C.D. (1986) Sensitivity of the stationary distribution vector for an ergodic Markov chain. Linear Algebra and its Applications, 76, 1–17.10.1016/0024-3795(86)90210-7CrossRefGoogle Scholar
Golub, G.H. and Meyer, C.D. (1986) Using the QR factorization and group inversion to compute, differentiate, and estimate the sensitivity of stationary probabilities for markov chains. SIAM Journal on Algebraic and Discrete Methods, 7, 273–281.10.1137/0607031CrossRefGoogle Scholar
Heras, A.T., Gil, J.A., Garca-Pineda, P. and Vilar, J.L. (2004) An application of linear programming to bonus Malus system design. ASTIN Bulletin: The Journal of the IAA, 34(2), 435–456.10.2143/AST.34.2.505152CrossRefGoogle Scholar
Holtan, J. (2001) Optimal loss financing under Bonus-Malus contracts. ASTIN Bulletin, 31(1), 161–173.10.2143/AST.31.1.1000CrossRefGoogle Scholar
Jewitt, I. (1991) Applications of likelihood ratio orderings in economics. Lecture Notes-Monograph Series, 19, 174–189.10.1214/lnms/1215459856CrossRefGoogle Scholar
Kaas, R., Goovaerts, M., Dhaene, J. and Denuit, M. (2001) Modern Actuarial Risk Theory Using R. Heidelberg: Springer.Google Scholar
Karlin, S. and Rubin, H. (1956) Distributions possessing a monotone likelihood ratio. Journal of the American Statistical Association, 51(276), 637–643.10.1080/01621459.1956.10501355CrossRefGoogle Scholar
Kemeny, J.G. and Snell, J.L. (1976) Finite Markov Chains. Springer-Verlag New York.Google Scholar
Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance. Boston: Kluwer Academic Publisher.CrossRefGoogle Scholar
Martinek, L. and Arató, M. (2019) An approach to merit rating by means of autoregressive sequences. Insurance: Mathematics and Economics, 85, 205–217.Google Scholar
Mensch, J. (2021) Rational inattention and the monotone likelihood ratio property. Journal of Economic Theory, 196, 1–30.10.1016/j.jet.2021.105284CrossRefGoogle Scholar
Milgrom, P.R. (1981) Good news and bad news: Representation theorems and applications. The Bell Journal of Economics, 12(2), 380–391.10.2307/3003562CrossRefGoogle Scholar
Oh, R., Lee, K.S., Park, S.C. and Ahn, J.Y. (2020) Double-counting problem of the Bonus–Malus system Insurance: Mathematics and Economics, 93, 141–155.Google Scholar
Pinquet, J. (1997) Allowance for cost of claims in Bonus-Malus systems. ASTIN Bulletin, 27(1), 33–57.10.2143/AST.27.1.542066CrossRefGoogle Scholar
Pitrebois, S., Denuit, M. and Walhin, J.F. (2003) Setting a bonus-malus scale in the presence of other rating factors: Taylor’s work revisited. ASTIN Bulletin, 33(2), 419–436.10.2143/AST.33.2.503701CrossRefGoogle Scholar
Rogerson, W.P. (1985) The first-order approach to principal-agent problems. Econometrica, 53(6), 1357–1367.10.2307/1913212CrossRefGoogle Scholar
Rukhin, A., Priebe, C.E. and Healy, D.M. (2009) On the monotone likelihood ratio property for the convolution of independent binomial random variables. Discrete Applied Mathematics, 157(11), 2562–2564.10.1016/j.dam.2009.03.002CrossRefGoogle Scholar
Sammartini, G. (1990) A ‘bonus/malus’ system with ‘conditioned’ bonus. Insurance: Mathematics and Economics, 9(2–3), 163–169.Google Scholar
Schweitzer, P.J. (1968) Perturbation theory and finite Markov Chains. Journal of Applied Probability, 5(2), 401–413.10.2307/3212261CrossRefGoogle Scholar
Tan, C.I. (2015) Optimal design of a bonus-malus system: Linear relativities revisited. Annals of Actuarial Science, 10(1), 52–64.10.1017/S1748499515000111CrossRefGoogle Scholar
Tan, C.I., Li, J., Li, J.S. and Balasooriya, U. (2015) Optimal relativities and transition rules of a Bonus–Malus system. Insurance: Mathematics and Economics, 61, 255–263.Google Scholar
Taylor, G. (1997) Setting a Bonus-Malus scale in the presence of other rating factors. ASTIN Bulletin, 27(2), 319–327.10.2143/AST.27.2.542055CrossRefGoogle Scholar
Tremblay, L. (1992) Using the Poisson inverse Gaussian in Bonus-Malus systems. ASTIN Bulletin, 22(1), 97–106.10.2143/AST.22.1.2005129CrossRefGoogle Scholar
Tzougas, G., Vrontos, S. and Frangos, N. (2014) Optimal Bonus-Malus systems using finite mixture models. ASTIN Bulletin, 44(2), 417–444.10.1017/asb.2013.31CrossRefGoogle Scholar
Vandebroek, M. (1993) Bonus-malus system or partial coverage to oppose moral hazard problems? Insurance: Mathematics and Economics, 13(1), 1–5.Google Scholar
von Lanzenauer, C.H. (1974) Optimal claim decisions by policyholders in automobile insurance with merit-rating structures. Operations Research, 22(5), 979–990.10.1287/opre.22.5.979CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (1992) Using mixed Poisson processes in connection with Bonus-Malus systems. ASTIN Bulletin, 29(1), 81–99.10.2143/AST.29.1.504607CrossRefGoogle Scholar
Winter, R.A. (1992) Moral hazard and insurance contracts. In Contributions to Insurance Economics (ed. Dionne, G.). Dordrecht: Springer.Google Scholar