Abstract
Markov chains (MCs) are widely used to model a great deal of financial and actuarial problems. Likewise, they are also used in many other fields ranging from economics, management, agricultural sciences, engineering or informatics to medicine. This paper focuses on the use of MCs for the design of non-life bonus-malus systems (BMSs). It proposes quantifying the uncertainty of transition probabilities in BMSs by using fuzzy numbers (FNs). To do so, Fuzzy MCs (FMCs) as defined by Buckley and Eslami in 2002 are used, thus giving rise to the concept of Fuzzy BMSs (FBMSs). More concretely, we describe in detail the common BMS where the number of claims follows a Poisson distribution under the hypothesis that its characteristic parameter is not a real but a triangular FN (TFN). Moreover, we reflect on how to fit that parameter by using several fuzzy data analysis tools and discuss the goodness of triangular approximates to fuzzy transition probabilities, the fuzzy stationary state, and the fuzzy mean asymptotic premium. The use of FMCs in a BMS allows obtaining not only point estimates of all these variables, but also a structured set of their possible values whose reliability is given by means of a possibility measure. Although our analysis is circumscribed to non-life insurance, all of its findings can easily be extended to any of the abovementioned fields with slight modifications.
Highlights
This paper focuses on the use of Markov chains (MCs) for the design of non-life bonus-malus systems (BMSs)
BMSs are often modelled by means of MCs with crisp probabilities
It is considered that transition probabilities of Markovian BMSs are not crisp but uncertain
Summary
A bonus-malus system (BMS) is a common method for posteriori ratemaking in nonlife insurance. 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). 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 period. SinonSse.ction 6, the work ends with a summary of its main contributions and potential extensions
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