You are given:

1. Losses follow an exponential distribution with the same mean in all years.
2. The loss elimination ratio this year is 70%.
3. The ordinary deductible for the coming year is 4/3 of the current deductible.

Calculate the loss elimination ratio for the coming year.

• 70%
• 75%
• 80%
• 85%
• 90%

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

Key: C

[math]\mathrm{LER}=\frac{\operatorname{E}(X \wedge d)}{\operatorname{E}(X)}=\frac{\theta\left(1-e^{-d / \theta}\right)}{\theta}=1-e^{-d / \theta}[/math]

Last year: [math]\quad 0.70=1-e^{-d / \theta} \Rightarrow-d=\theta \log (0.30)[/math]

Next year: [math]\quad-d_{\text {new }}=\theta \log \left(1-\mathrm{LER}_{\text {new }}\right)[/math]

Hence [math]\theta \log \left(1-\mathrm{LER}_{\text {new }}\right)=-d_{\text {new }}=\frac{4}{3} \theta \log (0.30)[/math]

[math]\log \left(1-\mathrm{LER}_{\text {new }}\right)=-1.6053[/math]

[math]\left(1-\mathrm{LER}_{\text {new }}\right)=e^{-1.6053}=0.20[/math]

[math]\mathrm{LER}_{\text {new }}=0.80[/math]

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