Losses in Year 1 follow a Pareto distribution with [math]\alpha = 2[/math] and [math]\theta = 5[/math]. Losses in Year 2 are uniformly 20% higher than in Year 1. An insurance covers each loss subject to an ordinary deductible of 10.

Calculate the Loss Elimination Ratio in Year 2.

• 0.567
• 0.625
• 0.667
• 0.750
• 0.800

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

Key: B

A Pareto [math](\alpha = 2,\theta = 5)[/math] distribution with 20% inflation becomes Pareto with [math]\alpha = 2,\theta = 5(1.2) = 6 [/math] . In Year 2

[[math]] \begin{aligned} &\operatorname{E}(X) = \frac{6}{2-1} = 6 \\ &\operatorname{E}(X \wedge 10) = \frac{6}{2-1} \left [ 1-(\frac{6}{10+6})^{2-1}\right] = 3.75 \\ &\operatorname{E}[( X − 10)_+ ]_+ = \operatorname{E}[ X ) − \operatorname{E}[ X \wedge 10) = 6 − 3.75 − 2.25 \\ & LER = 1- \frac{\operatorname{E}[(X-10)_+]}{\operatorname{E}(X)} = 1 - \frac{2.25}{6} = 0.625 \end{aligned} [[/math]]

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