You are given the following information about a homeowners insurance policy:

1. The deductible is 400 for claims up to 500.
2. For claims of 500 or more, the deductible disappears linearly, until completely disappearing for claims of 2100 and beyond. #There is no policy limit.

The policyholder submits a claim of X and receives a payment of 0.6X.

Calculate X.

• 618
• 808
• 1000
• 1300
• 1420

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

Key: B

We need to find the value of the claim X for which the deductible is 40% of the claim. Clearly the deductible is at least 80% of the claim for X < 500 and is 0% of the claim for X > 2100 . So we consider only the interval 500 < X < 2100. Since the deductible is linear on this interval, 400 for a claim of 500, and 0 for a claim of 2100, the deductible is given by the linear expression

[[math]] 400 + (X-500) \left( \frac{0 − 400}{ 2100-500} \right) = 400 − 0.25( X − 500) = 525 − 0.25 X [[/math]]

Therefore, we have [math]525 − 0.25 X = 0.4 X [/math] , which gives [math]X = 807.69.[/math]

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