A health plan implements an incentive to physicians to control hospitalization under which the physicians will be paid a bonus B equal to c times the amount by which total hospital claims are under [math]400 (0 \leq c \leq 1)[/math] .

The effect the incentive plan will have on underlying hospital claims is modeled by assuming that the new total hospital claims will follow a Pareto distribution with [math]\alpha = 2 [/math] and [math]\theta = 300 [/math].

[math]\operatorname{E}(B) = 100 [/math]

Calculate c.

• 0.44
• 0.48
• 0.52
• 0.56
• 0.60

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

Key: A

[[math]]\begin{aligned} B & =\left\{\begin{array}{cc} c(400-x) & x\lt400 \\ 0 & x \geq 400 \end{array}\right. \\ 100 & =\operatorname{E}(B)=c 400-c \operatorname{E}(X \wedge 400) \\ & =c 400-c 300\left(1-\frac{300}{300+400}\right) \\ & =c\left(400-300 \frac{4}{7}\right) \\ c & =\frac{100}{228.6}=0.44 \end{aligned}[[/math]]

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