Annual prescription drug costs are modeled by a Pareto distribution with [math] \theta = 2000 [/math] and [math]\alpha = 2[/math]

A prescription drug plan pays annual drug costs for an insured member subject to the following provisions:

• The insured pays 100% of costs up to the ordinary annual deductible of 250.
• The insured then pays 25% of the costs between 250 and 2250.
• The insured pays 100% of the costs above 2250 until the insured has paid 3600 in total.
• The insured then pays 5% of the remaining costs.

Calculate the expected annual plan payment.

• 1120
• 1140
• 1160
• 1180
• 1200

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

Key: C

[[math]] \operatorname{E}(X \wedge x) = \frac{\theta}{\alpha -1} \left [ 1-\left( \frac{\theta}{x+\theta}\right)^{\alpha-1}\right] - \frac{2000}{1} \left [ 1-\frac{2000}{x + 2000}\right ] = \frac{2000x}{x + 2000} [[/math]]

[math]x[/math]	[math]\operatorname{E}(X \wedge x)[/math]
[math]\infty[/math]	2000
250	222
2250	1059
5100	1437

[[math]] \begin{aligned} &0.75[ \operatorname{E}[ X \wedge 2250) − \operatorname{E}[ X \wedge 250)] + 0.95 [ \operatorname{E}[ X ) − \operatorname{E}[ X \wedge 5100)] \\ &0.75(1059 − 222) + 0.95(2000 − 1437) = 1162.6\\ \end{aligned} [[/math]]

The 5100 breakpoint was determined by when the insured’s share reaches 3600: 3600 = 250 + 0.25 (2250 – 250) + (5100 – 2250)

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