Surplus Allocation for the Internal Rate of Return Model: Resolving the Unresolved IssueReturn Model:" Resolving the Unresolved Issue

In this paper, it is shown that with a certain definition of risk-based discounted loss reserves and a certain method of surplus allocation, there is an amount of premium for a contract which has the following properties: (1 .) It is the amount of premium required for the contract to neither help nor hurt the insurer's risk-return relation. (2.) It produces an internal rate of return equal to the insurer's target return. If the insurer gets more than this amount of premium, then the insurer can get more return with the same risk by increasing the percentage of the premium for the overall book which is in the segment. Conversely, if the insurer gets less than this amount of premium, the insurer can increase its return by decreasing the percentage of the overall premium which is in the segment. The amount of premium is equal to the risk-based premium in "Pricing to Optimize an Insurer's Risk-Return Relation," (PCAS 1996). The above property 1 of risk-based premium is proven by Theorem 2 of the 1996 PCAS paper and not by the present paper. The present paper proves property 2.