question : calculate the retention limit that minimizes the probability of the total cost exceeding 8.250.000. assume that the limit is between 30.000 and 50.000.

Example 2.5.3 A life insurance company covers 16,000 lives for 1-year term life insurance in amounts shown below. Benefit Amount Number Covered 10 000 8 000 20 000 3 500 30 000 2 500 50 000 1 500 100 000 500 The probability of a claim q for each of the 16,000 lives, assumed to be mutually independent, is 0.02. The company wants to set a retention limit. For each life, the retention limit is the amount below which this (the ceding) company will retain the insurance and above which it will purchase reinsurance coverage from another (the reinsuring) company. For example, assume the retention limit is 20,000. The company will retain up to 20,000 on each life and purchase reinsurance for the difference between the benefit amount and 20,000 for each of the 4,500 individuals with benefit amounts in excess of 20,000. As a decision criterion, the company wants to minimize the probability that retained claims plus the amount that it pays for reinsurance will exceed 8,250,000. Reinsurance is available at a cost of 0.025 per unit of coverage (i.e., at 125% of the expected claim amount per unit, 0.02). We will consider the block of business as closed. New policies sold during the year are not to enter this decision process. C