A primary liability insurer has a book of business with the following characteristics:

• All policies have a policy limit of 500,000
• The expected loss ratio is 60% on premiums of 4,000,000

A reinsurer provides an excess of loss treaty for the layer 300,000 in excess of 100,000. The following table of increased limits factors is available:

Limit	ILF
100,000	1.00
200,000	1.25
300,000	1.45
400,000	1.60
500,000	1.70

Calculate the reinsurer’s expected losses for this coverage (answer to the nearest 000s).

• 840,000
• 847,000
• 850,000
• 862,000
• 871,000

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

Key: B

The expected losses for the primary insurer are 0.6(4,000,000) = 2,400,000. The expected proportion of losses in the treaty layer is (1.6/1.7 – 1/1.7 = 0.352941). The expected cost is 0.352941(2,400,000) = 847,058.

The relative cost of the layer can be derived using formulas from Loss Models as follows:

[[math]] \begin{aligned} &\frac{\operatorname{E}[ X \wedge 400, 000) − \operatorname{E}[ X \wedge 100, 000)}{\operatorname{E}[ X \wedge 500, 000)} \\ &= \frac{\operatorname{E}[ X \wedge 400, 000) / \operatorname{E}[ X \wedge 100, 000) − \operatorname{E}[ X \wedge 100, 000) / \operatorname{E}[ X \wedge 100, 000)}{\operatorname{E}[ X \wedge 500, 000) / \operatorname{E}[ X \wedge 100, 000)} \\ &= \frac{ILF (400, 000) − ILF (100, 000)}{ILF (500, 000)} = \frac{1.60 − 1.00}{1.70} = 0.352941. \end{aligned} [[/math]]

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