1. Insurance mathematics V. lecture Bonus systems, premium refund Introduction The basic principle of bonus systems is if somebody has less claims in the past then there is probable that in the future he/she has less claims also. Insurers have varied solutions to manage this theme – we will consider the most common types of solutions.

2. Insurance mathematics V. lecture Premium refund in case of less claims I.

3. Insurance mathematics V. lecture Premium refund in case of less claims II.

4. Insurance mathematics V. lecture Premium refund in case of less claims III.

5. Insurance mathematics V. lecture Premium refund in case of less claims IV. Remarks: - with λ adequate giving there is possible to adjust adequate cost and security level; - from formula it can be seen that the premium is increasing, because without premium refund the premium would be

6. Insurance mathematics V. lecture Premium refund in case of less claims V.

7. Insurance mathematics V. lecture Premium refund in case of less claims VI.

8. Insurance mathematics V. lecture Premium refund in case of less claims VII. Proof: Because of net premium was calculated with variance principle: It means that

9. Insurance mathematics V. lecture Claim-free allowance I. Insurer gives allowance in advance for next policy year based on earlier claim-free claim experience. There are k different sections, 1,2,…k. Usually the new insured goes to 1. section and after every claim-free policy year he/she goes 1. step forward (till k-th section). Every claim causes one step backward (till 1. section).

10. Insurance mathematics V. lecture Claim-free allowance II. Advantages: - insured is interested to be claim-free; - low claims will not be reported; - premiums are more adequate (related to actual risk). Disadvantages: - in case of liability insurance the insured can try to avoid his/her liability; - more complicated premium calculation, reserving and administration.

11. Insurance mathematics V. lecture Bonus-malus systems I. For consideration there is necessary to take into account the probabilities.

12. Insurance mathematics V. lecture Bonus-malus systems II. Usually we assume that Z(n) is Markov chain, i.e. In this case there is enough to consider the next probabilities

13. Insurance mathematics V. lecture Bonus-malus systems III.

14. Insurance mathematics V. lecture Bonus-malus systems IV.

15. Insurance mathematics V. lecture Bonus-malus systems V.

16. Insurance mathematics V. lecture Bonus-malus systems VI.

17. Insurance mathematics V. lecture Bonus-malus systems VII.

18. Insurance mathematics V. lecture Bonus-malus systems VIII. Example (Hungarian bonus-malus system in MTPL): The new insured goes to A0 section and after every claim-free policy year he/she goes 1. step forward (till B10 section). Every claim causes two steps backward (till M4 section). If one insured causes 4 or more claims in one year then he/she goes to M4. If ‚being claim-free’ and ‚causing claim’ have both positive probability then the Markov chain will be irreducible and aperiodic. (B10 has 1 period, it follows that the Markov chain is aperiodic. From any section can getting to any other section with positive probability, it follows the Markov chain is irreducible.) If the number of claim is Poisson distribution then the transition probability matrix will be as follows:

19. Insurance mathematics V. lecture Bonus-malus systems IX. Example (continued): The most common used bonus-malus factors are as follows: M4M3M2M1A0B1B2B3B4B5B6B7B8B9B10 21,61,351,1510,950,90,850,80,750,70,650,60,550,5

20. Insurance mathematics V. lecture Bonus-malus systems X. Example (continued): In this case we can calculate stationer premium and distribution if we add claim frequency. For example if claim frequency is 10% then the stationer premium will be 0,5223. It means that the average premium will converge the 52,23% of A0 premium with these assumptions. The stationer distribution will be as follows: 0,01630,02220,09050,08190,779

21. Insurance mathematics V. lecture Comparison of bonus-malus systems I.

22. Insurance mathematics V. lecture Comparison of bonus-malus systems II.

23. Insurance mathematics V. lecture Comparison of bonus-malus systems III.

24. Insurance mathematics V. lecture Comparison of bonus-malus systems IV.

25. Insurance mathematics V. lecture Comparison of bonus-malus systems V. Related to Hungarian bonus-malus system η(λ)=0,064 (if we assume 10% claim frequency).

26. Insurance mathematics V. lecture Bonus-malus systems Remarks We did not consider some other factors in this chapter. Just some of them: - effect of entering, exiting; - optimum of self-part; - calculation distribution of claim number and claim amount in different bonus sections; - consideration of Markov attribution and homogenity.