An insurance policy is written to cover a loss, [math]X[/math], where [math]X[/math] has a uniform distribution on [0, 1000]. The policy has a deductible, [math]d[/math], and the expected payment under the policy is 25% of what it would be with no deductible.

Calculate [math]d[/math].

• 250
• 375
• 500
• 625
• 750

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

Solution: C

Let Y represent the payment made to the policyholder for a loss subject to a deductible D. That is

[[math]] Y = \begin{cases} 0, \quad 0 \leq X \leq D \\ x-D, \quad D \lt x \leq 1 \end{cases} [[/math]]

Then since [math]\operatorname{E}[X] = 500 [/math], we want to choose D so that

[[math]] \frac{500}{4} = \int_{D}^{1000} \frac{1}{1000} (x-D) dx = \frac{1}{1000} \frac{(x-D)^2}{2} \Big |_D^{1000} = \frac{(1000-D)^2}{2000} [[/math]]

which implies that [math]D=500[/math] (or [math]D =1500 [/math] which is extraneous).

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