An individual performs dangerous motorcycle jumps at extreme sports events around the world.

The annual cost of repairs to their motorcycle is modeled by a Pareto distribution with [math]\theta = 5000 [/math] and [math] \alpha = 2 [/math].

An insurance policy reimburses motorcycle repair costs subject to the following provisions:

1. The annual ordinary deductible is 1000.
2. The policyholder pays 20% of repair costs between 1000 and 6000 each year.
3. The policyholder pays 100% of the annual repair costs above 6000 until they have paid 10,000 in out-of-pocket repair costs each year.
4. The policyholder pays 10% of the remaining repair costs each year.

Calculate the expected annual insurance reimbursement.

• 2300
• 2500
• 2700
• 2900
• 3100

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

Key: C

Insurance pays 80% of the portion of annual claim between 6,000 and 1,000, and 90% of the portion of annual claims over 14,000.

The 14,000 breakpoint is where the policyholder has paid 10,000:

1000 = deductible

1000 = 20% of costs between 1000 and 6000

8000 = 100% of costs between 14,000 and 6,000

[[math]] \operatorname{E}(X \wedge x ) = \theta \left( 1 - \frac{\theta}{x + \theta} \right) = \frac{5000x}{x + 5000} [[/math]]

[math]x[/math]	[math]\operatorname{E}(X \wedge x ) [/math]
1000	833.33
6000	2727.27
14000	3684.21
[math]\infty[/math]	5000

[[math]] \begin{aligned} &0.80[ \operatorname{E}[ X \wedge 6000) − \operatorname{E}[ X \wedge 1000)] + 0.90[ \operatorname{E}[ X ) − \operatorname{E}( X \wedge 14000)] \\ &= 0.80[2727, 27 − 833.33] + 0.90[5000 − 3684.21] \\ &= 1515.15 + 1184.21 = 2699.36 \end{aligned} [[/math]]

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