A company insures a fleet of vehicles. Aggregate losses have a compound Poisson distribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with [math] \theta = 200.[/math]

To reduce the cost of the insurance, two modifications are to be made:

1. A certain type of vehicle will not be insured. It is estimated that this will reduce loss frequency by 20%.
2. A deductible of 100 per loss will be imposed.

Calculate the expected aggregate amount paid by the insurer after the modifications.

• 1600
• 1940
• 2520
• 3200
• 3880

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

By the memoryless property, the distribution of amounts paid in excess of 100 is still exponential with mean 200.

With the deductible, the probability that the amount paid is 0 is [math]F (100) = 1 − e−100/200 = 0.393 .[/math]

Thus the average amount paid per loss is (0.393)(0) + (0.607)(200) = 121.4 The expected number of losses is (20)(0.8) = 16. The expected amount paid is (16)(121.4) = 1942.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.