Bonus-malus systems are the most applicable method of posterior ratemak- ing in the area of automobile insurance, which mainly designed to motivate drivers to drive better. These systems are applied extensively and adjusted by di erent researchers through the years. Lemaire (1995), Dionne and Vanase (1989), Frangos and Vrontos (2001), Denuit et al, (2007) and Mah- moudvand and Hassani (2009) and references therein are the most related studies on these types of BMS's. Here, we focus on BMS based on fre- quency components and would like to make the BMS penalty depending on the timing of the claim in the year. Generally speaking, BMS based on frequency component can be designed by determining the Number of bonus-malus (BM) classes, Transition rules among class and Relativity for each class. A good reference for nding the way of determining these components is Dennuit, et al (2007). In this paper, we introduce a new adjusted form of BMS, which add a new component into the current systems based on the time of claims. According to this system, both insurer and policyholders can have more satisfactions. This system will increase the incentive of policyholders to (1) extend their contract and (2) drive better to receive no claim discount. On the other hand, the business model for insurers is to collect more in premium and investment income than is paid out in losses. Therefore, paying out losses soon for insurers is not okay. The designed system of this paper covers this issue and therefore it can be accepted by insurers, too.
