For a medical insurance company, you are given:

1. Losses for a new product are assumed to follow a lognormal distribution with parameters μ = 6 and σ = 1.5.
2. The new product has a per-loss deductible that results in a loss elimination ratio of 0.33.

In a review of the business after five years of experience, it is determined that:

1. Losses for this product actually followed an exponential distribution.
2. The initial mean for the exponential distribution is the same as the initial mean under the lognormal assumption.
3. Since it was introduced, the expected value of a loss for this product increased at an annual compound rate of 4%.
4. The per-loss deductible required to target the same loss elimination ratio is d.

Calculate d.

• 605
• 659
• 722
• 775
• 852

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

Key: A

Under the initial assumptions, losses, [math]X[/math], have [math]\operatorname{E}(X) = \exp(6 + \frac{1.5^2}{2}) = 1242.65 [/math]

After 5 years: exponential distribution with mean 1242.65(1.045 ) = 1511.87 and LER = 0.33, so

[[math]] 0.33 = \frac{\operatorname{E}[Y \wedge d )}{\operatorname{E}(Y)} = \frac{\theta(1-e^{-d/\theta}}{\theta} = 1- e^{-d/1511.87} \Rightarrow d = −1511.87 \ln(1 − 0.33) = 605.47 [[/math]]

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