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.
3. 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.