*1.1. Motivation*

A bonus-malus system (BMS) is a common method for posteriori ratemaking in nonlife insurance. It is based on partitioning the insurer's portfolio into a finite number of classes: bonus and malus classes. A typical case is automobile third-party liability insurance [1]. In a BMS, policyholders do not have a fixed price for their contracts throughout periods (e.g., the mathematical expectation of claims value per period). Their membership into a concrete BMS class is reviewed each period according to the number of claims in the previous one. Claim-free years are rewarded by discounts or bonuses on a base-premium; at-fault accidents are penalized by surcharges called maluses. Some overviews on how BMS are applied in different countries can be found in [2–4].

Following [5], in most commercial BMSs, by knowing the insured's class in the current period and fitting the statistical distribution for the number of claims per period, it is possible to determine the probabilities of the insured's class in the next period. Therefore, these BMSs are Markovian. For that reason, the academic literature on BMSs uses extensively MCs for their modeling ([1,5–11]). Therefore, a key question in a BMS is fitting the value of the one-step transition probability matrix. Following [12,13], if full knowledge of the probabilities of this matrix is not available, they have to be estimated somehow

**Citation:** Villacorta, P.J.; González-Vila Puchades, L.; de Andrés-Sánchez, J. Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance. *Mathematics* **2021**,*9*, 347. https://doi.org/10.3390/ math9040347

Academic Editor: Michael Voskoglou

Received: 24 December 2020 Accepted: 3 February 2021 Published: 9 February 2021

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with the uncertainty that any estimation procedure involves. Uncertainty may be due to randomness, hazard, vagueness, incomplete information, etc. In our paper, we consider that the claiming process is probabilistic, but the uncertainty about the parameter that governs this random behavior is captured by means of a fuzzy number (FN) and, as a consequence, fuzzy Markov chains (FMCs) will be used.
