<![CDATA[Motor vehicle usage in Indonesia ranks among the highest globally, reaching approximately 141,992,573 units. The growing variety and number of automobiles contribute significantly to traffic congestion and heightened risks to public safety. Given the inherent dangers associated with motorized transportation, including auto theft and accidents, efforts to shift these risks to insurance companies have become crucial. The fundamental idea of insurance is to establish a pool in which policyholders can manage their risk, with premiums determined by the amount of risk that each participant adds to the group. Actuaries in the field of motor vehicle insurance must generate a reasonable premium rate utilizing a variety of methodologies, including the Bonus-Malus approach. The latter, a widely utilized approach, classifies policyholders based on their claims history, incentivizing safe driving. Examining the internal dynamics of the Bonus-Malus system necessitates studying mathematics, particularly algebra, and the use of linear algebra in transition matrices is critical in anticipating changes in bonus-malus rates over time. This research is a quantitative descriptive analysis that explores the implementation of the Bonus-Malus system using a transition matrix framework. It aims to investigate the collaboration of algebra and actuarial science in a real-world application of the Bonus-Malus scheme for motor vehicle insurance, focusing on the use of the transition matrix in premium computation, utilizing secondary data from PT. Jasa Raharja Kota Semarang for the years 2021–2022. The transition matrix analysis shows that Model 2 allows for smoother class transition, lowers the possibility of high-risk class recurrence, and provides more consistent premium adjustments. This demonstrates the model's ability to create a balanced incentive structure while interpreting claim trends. Furthermore, Model 2 has a greater expected value of Loimaranta efficiency than Model 1, supporting findings that added status improves Bonus-Malus system efficiency.]]>
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