Main Article Content
Abstract
The increasing number of motor vehicles in Sumatera has heightened accident risks, emphasizing the need for motor vehicle insurance to distribute risk between policyholders and insurers. Determining fair and risk-based premium requires consideration of each policyholder’s claim history. This study aimed to determine motor vehicle insurance premiums using the optimal bonus-malus system based on claim data for the minibus category with comprehensive coverage in Sumatera during 2022. The proposed model extended the Bayesian bonus-malus framework by incorporating the trust region reflective (TRR) method for estimating claim severity and the Newton-Raphson method for estimating claim frequency, thereby enhancing parameter estimation accuracy and numerical stability. This approach offers a more equitable and precise premium adjustment mechanism aligned with individual risk levels, contributing to improved risk-based pricing, reduced underwriting losses, and greater transparency for policyholders. The results showed that the claim frequency followed the Poisson-Lindley distribution, while claim severity followed the lognormal-gamma distribution. Based on these models, the premium was computed by multiplying the basic premium by the relative value of the subsequent year and dividing it by the base relative value. Premium decrease in the absence of claims and increase when claims occur.
Keywords
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References
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References
Badan Pusat Statistik, “Jumlah Kendaraan Bermotor Menurut Provinsi dan Jenis Kendaraan (unit), 2023.” [Online]. Available: https://www.bps.go.id/id/statistics-table/3/
VjJ3NGRGa3dkRk5MTlU1bVNFOTVVbmQyVURSTVFUMDkjMw==/jumlah-kendaraan-bermotor-menurut-provinsi-dan-jenis-kendaraan--unit---2023.html
R. Fitriani and Gunardi, “Implementasi metode bayes pada perhitungan premi kendaraan bermotor,” J. Fundam. Math. Appl. (JFMA), vol. 3, no. 2, pp. 112–123, Nov. 2020, doi: 10.14710/jfma.v3i2.8257.
H. Halim, M. Tumpu, F.E.P. Lapian, P.R. Rangan, A. Yunianta, and H. Yusuf, “Patterns and sociodemograohic determinants of traffic accidents in Makassar City: A study toward targeted road safety interventions,” Eng. Technol. Appl. Sci. Res., vol. 15, no. 6, pp. 29077–29083, Dec. 2025, doi: 10.48084/etasr.13119.
B.N Rizkia and A.K. Mutaqin, “Bonus-malus premium for third party liability insurance with Poisson-Lindley distribution claim frequency and exponential-inverse gamma distribution claim severity,” J. Stat., vol. 18, no.1, pp. 887–896, Jul. 2025, doi: 10.36456/jstat.vol18.no1.a10327
I.P. Batubara and R. Syahriza, “Analisis klaim asuransi kendaraan bermotor pada PT Asuransi Jasindo Kantor Cabang Medan,” J. Soc. Reas., vol. 1, no. 9, pp 1026–1035, Sep. 2022, doi: 10.55324/josr.v1i9.62
E. Stepanes and C.S. Basani, “Legal protection for health insurance policyholders against claim denials due to non-transparent information in insurance policies,” J. Indones. Sos. Sains, vol. 7, no. 1, pp 323–335, 2026, doi: 10.59141/jiss.v7i1.2184
H.A. Shah and A. Ahmed, “Mitigating catastrophic events: Assessing the effectiveness of reinsurance risk management mechanisms,” Int. J. Acad. Res. Bus. Soc. Sci., vol. 14, no. 6, pp. 1558–1563, 2024, doi: 10.6007/IJARBSS/v14-i6/21967.
L. Girald, “Analysis of factors influencing automobile insurance premiums in France,” J. Stat. Actuar. Res., vol. 8, no. 1, pp. 1 – 10, 2024, doi: 10.47604/jsar.2761.
R. Oh, J.H.T. Kim, and J.Y. Ahn, “Designing a bonus-malus system reflecting the claim size under the dependent frequency–severity model,” Probability Eng. Inf. Sci., vol. 3 6, no. 4, pp. 963–987, Oct. 2022, doi: 10.1017/S0269964821000188.
A. Moumeesri, W. Klongdee, and T. Pongsart, “Bayesian bonus-malus premium with Poisson-Lindley distributed claim frequency and lognormal-gamma distributed claim severity in automobile insurance,” WSEAS Tran. Math., vol. 19, pp. 443–451, 2020, Art. no #46, doi: 10.37394/23206.2020.19.46.
G.R. Sevina and J. Purwadi, “Bonus malus system for motorized vehicle insurance using geometric distributions and Weibull distributions,” J. Fundam. Math. Appl., vol. 6, no. 1, pp. 71–79, Jun. 2023, doi: 10.14710/jfma.v6i1.16505.
R.E. Walpole, R.H. Myers, S.L. Myers, and K. Ye, Probability & Statistics for Engineers & Scientist. Boston, MA, USA: Perason Education, 2016.
A. Aijaz, S.Q. Ul Ain, A. Afaq, and R. Tripathi, “Poisson area-biased Ailamujia distribution and its applications in environmental and medical sciences,” J. Stat. Transit. New Ser., vol. 23, no. 3, pp. 167–184, Sep. 2022, doi: 10.2478/stattrans-2022-036.
M. Shakil, A. Khadim, A. Saghir, M. Ahsanullah, B.M.G. Kibria, and M.I. Bhatti, “Ratio of two independent lindley random variables,” J. Stat. Theory Appl., vol. 21, no. 4, pp. 217–241, Oct. 2022, doi: 10.1007/s44199-022-00050-4
S.A. Klugman, H.H. Panjer, and G.E. Willmot, Loss Models: From Data to Decisions, 5th ed. Hoboken, NJ, USA: Wiley, 2019.
A. Sutrisno, D. Aziz, A. Hasanah, T. Ruby, and N. Mega, “Comparison of Gauss-Seidel method, Newton-Raphson method, and Broyden method in solving nonlinear equation system,” Int. J. Adv. Soc. Sci. Edu., vol. 3, no. 2, pp. 47–58, Feb. 2025, doi: 10.59890/ijasse.v3i1.292
K.A. Dalal, et al. “Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems,” AIMS Math., vol 9, no. 11, pp. 32014–32044, 2024, doi: 10.3934/math.20241538.
L. Endreyani, Kinetic Data Analysis. Toronto, Canada: Springer, 1981.
T.M. Reshid, “Monte Carlo simulation and derivation of chi-square statistics,” Am. J. Theor. Appl. Stat., vol. 12, no 3, pp. 51–65, 2023, doi: 10.11648/j.ajtas.20231203.13.
A.A Mir, A.A. Bhat, S.P. Ahmad, B.S. Alnssyan, A. Alsubie, and Y.S. Raghav, “Statistical inference and goodness-of-fit assessment using the AAP-X probability framework with symmetric and asymmetric properties: Applications to medical and reliability data,” J. Symmetry, vol. 17, no. 6, 2025, Art. no 863, doi: 10.3390/sym17060863.
M. Denuit, X. Maréchal, S. Pitrebois, and J.F. Walhin, Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems, Belgium: John Wiley & Sons, Ltd, 2007.