Abstract
In this paper we consider a discrete-time risk model, which allows the premium to be adjusted according to claims experience. This model is inspired by the well-known bonus-malus system in the non-life insurance industry. Two strategies of adjusting periodic premiums are considered: aggregate claims or claim frequency. Recursive formulae are derived to compute the finite-time ruin probabilities, and Lundberg-type upper bounds are also derived to evaluate the ultimate-time ruin probabilities. In addition, we extend the risk model by considering an external Markovian environment in which the claims distributions are governed by an external Markov process so that the periodic premium adjustments vary when the external environment state changes. We then study the joint distribution of premium level and environment state at ruin given ruin occurs. Two numerical examples are provided at the end of this paper to illustrate the impact of the initial external environment state, the initial premium level and the initial surplus on the ruin probability.
Highlights
The commonly adopted bonus-malus system in the general insurance industry is based on a principal that insurance premiums can be adjusted based on the historical claims record of individual policyholders
In this paper we considered a discrete-time risk model, which allows the premium to be adjusted according to claims experience
The premium correction was based on the well-known bonus-malus system and the claims experience was assumed to depend on an external Markovian environment
Summary
The commonly adopted bonus-malus system in the general insurance industry is based on a principal that insurance premiums can be adjusted based on the historical claims record of individual policyholders. Policyholders who make more claims than the given thresholds in the current policy year may need to pay higher premiums ( called ‘malus’) if they decide to renew their policies. The bonus-malus system plays an important role in the insurance industry, in particular in motor vehicle insurance sector, because this system defines risk specific premium levels that help to sustain the total premium pool in covering all motor insurance claims of the given insurance portfolio. The discount in premium acts as an incentive to retain low-risk policyholders and to attract new customers; on the other hand, the malus component prevents high-risk policyholders from taking advantage of the low-risk policyholders by receiving disproportionate insurance benefits. In terms of commercial purpose, the bonus-malus system is widely adopted in pricing by motor vehicle insurance companies to enhance their competitiveness in the insurance market
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