1. Insurance mathematics IV. lecture Premium principles Introduction There are a lot of principle which can be used for premium calculation of new product (which has different advantages and disadvantages). Actuaries need to know more principles to be well-trained to choose the most adequate principle regarding the actual environment.

2. Insurance mathematics IV. lecture Premium principles

3. Insurance mathematics IV. lecture Premium principles

4. Insurance mathematics IV. lecture Premium principles Classical principles I. Definition: Π is expected value premium principle with λ parameter, if

5. Insurance mathematics IV. lecture Premium principles Classical principles II.

6. Insurance mathematics IV. lecture Premium principles Classical principles III. Proof: At first we suppose that Π is expected value principle. Then

7. Insurance mathematics IV. lecture Premium principles Classical principles IV. Proof (continued):

8. Insurance mathematics IV. lecture Premium principles Classical principles V. Proof (continued): Let ξ variate which has infinite expected value. Then based on (iii) we get: Because of monotone convergence theorem It means that Π is expected value premium principle. Q.e.d.

9. Insurance mathematics IV. lecture Premium principles Classical principles VI. Definition: Π is maximal loss premium principle with p parameter, if Remark: The huge risk is hazardous, this principle punishes that. Definition: Π is quantile premium principle with ε and p parameters, if

10. Insurance mathematics IV. lecture Premium principles Classical principles VII.

11. Insurance mathematics IV. lecture Premium principles Classical principles VIII. Proof of lemma:

12. Insurance mathematics IV. lecture Premium principles Classical principles IX. Let, because

13. Insurance mathematics IV. lecture Premium principles Classical principles X. Q.e.d. Definition: Π is variance premium principle with β parameter, if

14. Insurance mathematics IV. lecture Premium principles Classical principles XI. Definition: Π is standard deviation premium principle with β parameter, if Remark: The above two principles punish the difference with expected value. But there is a question why we punish if the risk less then expected value. That is why it is more useable the next principle for which it is necessary the following definition.

15. Insurance mathematics IV. lecture Premium principles Classical principles XII. Definition: Let semi-variance of ξ the next formula: Definition: Π is semi-variance premium principle with β parameter, if

16. Insurance mathematics IV. lecture Premium principles Classical principles XIII.

17. Insurance mathematics IV. lecture Premium principles Classical principles XIV. Proof (continued): We use the earlier inequalities for these variate: It follows:

18. Insurance mathematics IV. lecture Premium principles Classical principles XV.

19. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles I. Definition: is risk-loading, if Loading for risk is desirable because one generally requires a premium rule to charge at least the expected payout of the risk ξ, namely E(ξ), in exchange for insuring the risk. Otherwise, the insurer will lose money on average.

20. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles II. Definition: is no unjustified risk-loading, if a risk is equal almost everywhere then If we know for certain (with probability 1) that the insurance payout is c, then we have no reason to charge a risk loading because there is no uncertainty as to the payout. Definition: is no rip-off, if Insured will not pay more than its maximum risk (with positive probability). Definition: has translation invariance property, if If we increase a risk ξ by a fixed amount a, then this property states that the premium for ξ + a should be the premium for ξ increased by that fixed amount a.

21. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles III. Definition: has scale invariance property, if This property essentially states that the premium for doubling a risk is twice the premium of the single risk. One usually uses a no arbitrage argument to justify this rule. Indeed, if the premium for 2ξ were greater than twice the premium of ξ, then one could buy insurance for 2 ξ by buying insurance for ξ with two different insurers, or with the same insurer under two policies. Similarly, if the premium for 2 ξ were less than twice the premium of ξ, then one could buy insurance for 2 ξ, sell insurance on ξ and ξ separately, and thereby make an arbitrage profit. Scale invariance might not be reasonable if the risk ξ is large and the insurer (or insurance market) experiences surplus constraints. In that case, we might expect the premium for 2 ξ to be greater than twice the premium of ξ.

22. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles IV.

23. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles V. Additivity is a stronger form of scale invariance. One can use a similar no- arbitrage argument to justify the additivity property.

24. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles VI. Some actuaries might feel that additivity property is too strong and that the no-arbitrage argument only applies to risks that are independent. They, thereby, avoid the problem of surplus constraints for dependent risks.

25. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles VII. Although the above properties are (more or less) natural the earlier reviewed principles do not satisfy these properties in each case. The next table shows a short summary which principle satisfies which property:

26. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles VIII. Property/ principle Expected ValueVarianceStandard Deviation IndependentYYY Risk loadingYYY Not unjustifiedNYY No rip-offNNN Translation inv.NYY Scale inv.YNY AdditivityYNN Sub-additivityYNN Super-additivityYNN MonotoneYNN

27. Insurance mathematics IV. lecture Premium principles Mathematical properties of premium principles IX. As earlier it can be seen that there is no one useable method for each case, but actuaries has to calculate prudential. What is the most useful process to calculate premium? At first we have to make a list according to the given problem which properties has to be satisfied ( it can be used two categories: „must” and „nice to have” categories). Checking which known principle satisfies the necessary properties. If one exists we find the adequate method. If no one exists we have to define a new principle. In this case there is necessary to check each necessary requirement for the problem and the natural requirements also.

28. Insurance mathematics IV. lecture Gross premium Premium elements: - net premium (due to risk) - costs (commission, maintenance cost, claims handling cost) - safety plus - profit rate GP=NP+C+SP+PR