For a certain health insurance policy, losses are uniformly distributed on the interval [0, 450]. The policy has a deductible of [math]d[/math] and the expected value of the unreimbursed portion of a loss is 56. Calculate [math]d[/math].

• 60
• 87
• 112
• 169
• 224

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

Solution: A

The expected unreimbursed loss is

[[math]] \begin{align*} \int_0^d x \frac{1}{450} dx + \int_d^{450} d \frac{1}{450} dx = \frac{d^2}{900} + d \frac{450-d}{450} = \frac{1}{900}(900d -d^2) = 56 \\ d^2 - 900d + 50400 = 0 \\ d = \frac{900 ± \sqrt{900^2 - 201600}}{2} = \frac{900-780}{2} = 60. \end{align*} [[/math]]

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