| A bonus-malus system plays a very important role in actuarial mathematics through determining its relativity premium, which is extensively used in automobile insurance. In insurance, BMS is a system that adjusts the premium paid by a policyholder according to his individual claim history. Bonus usually is a discount in the premium which is given on the renewal of the policy if no claim is made in the previous year. Malus usually imposes some punishment on those policyholders whose claims are relatively serious.There are many ways including Bayesian estimator and ordinary linear estimator to calculate the relativity premium. No doubt that Bayesian estimator is the most accurate estimator, however, it is undesirable for commercial purposes for its rather irregular pattern. This paper aims to introduce an optimal linear estimator for relativity premium, which has a simple pattern and is obtained under the quadratic loss function such that the result is as close as the Bayesian estimator. The Loimaranta efficiency of such an optimal linear estimator has been studied compared with two methods mentioned above. And some simulations and demonstrations allowing for severity of the claims of each policyholder are also presented in this paper. |
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