Some asymptotic properties of bonus systems

Abstract

In motorcar insurance—at least in Europe—is widely used a merit rating system called bonus (or malus) system, characterized among others by the fact that only the number of claims occurred does modify the premium. A great variety of different bonus systems is in use. Each country—if not each company—seems to prefer a different bonus system and there is a lack of agreement whether one of them is better than the other. Efforts to construct a theoretically optimal bonus system have not been able to change the situation. General methods to evaluate and compare different existing or proposed bonus systems are therefore wanted.In the present paper the theory of Markow chains is used to develope formulas for some asymptotic properties of bonus systems. Quantities: efficiency of a bonus system, discrimination power of bonus rules and minimum variance bonus scale are introduced. The last of them gives an asymptotical solution for the problem to find locally “best” bonus scales for given bonus rules. Finally the Danish bonus system is used as an illustration.We call a merit rating system bonus system if the following assumptions are valid:i. All policies of a given risk group can be devided into a finite number of classes so that the premium of a policy for a given period depends solely on the class for that period.ii. The actual class is uniquely defined by the class for the previous period, and the number of claims occurred (regulated) during the period.