Annual losses in Year 1 follow an exponential distribution with mean θ . An inflation factor of 20% applies to all Year 2 losses. The ordinary deductible for Year 1 is 0.25θ . The deductible is doubled in Year 2.

Calculate the percentage increase in the loss elimination ratio from Year 1 to Year 2.

• 19%
• 28%
• 37%
• 54%
• 78%

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

Key: D

For Year 1, we have:

[[math]] \operatorname{E}( X ) = \theta , \operatorname{E}[ X \wedge 0.25\theta ) = \theta (1 − e ^{−0.25\theta / \theta} ) = \theta (1 − e^{−0.25} ) [[/math]]

Therefore,

[[math]] LER_{1} = \frac{\operatorname{E}(X \wedge 0.25 \theta)}{\operatorname{E}(X)} = 1-e^{-0.25} = 0.2212 [[/math]]

For Year 2,

[[math]] \operatorname{E}(Y) = 1.20\theta, \, \operatorname{E}(Y \wedge 0.5 \theta ) = 1.2 \operatorname{E}( X \wedge \frac{0.5\theta}{1.20}) = 1.2\theta (1-e^{-5/12}) [[/math]]

and

[[math]] LER_{2} = \frac{\operatorname{E}(Y \wedge 0.5 \theta)}{\operatorname{E}(Y)} = 1-e^{-5/12} = 0.34076.[[/math]]

[[math]] \frac{LER_{2}}{LER_{1}} = \frac{0.34076}{0.22120} = 1.54 \Rightarrow 54\% \quad \textrm{increase} [[/math]]

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