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

Losses in Year 1 follow a Pareto distribution with [math]\alpha = 2[/math] and [math]\theta = 5[/math]. Losses in Year 2 are uniformly 20% higher than in Year 1. An insurance covers each loss subject to an ordinary deductible of 10.

Calculate the Loss Elimination Ratio in Year 2.

• 0.567
• 0.625
• 0.667
• 0.750
• 0.800

The distribution of a loss, [math]X[/math], is a two-point mixture:

1. With probability 0.8, [math]X[/math] has a Pareto distribution with [math]\alpha = 2[/math] and [math]\theta = 100[/math].
2. With probability 0.2, [math]X[/math] has a Pareto distribution with with [math]\alpha = 4[/math] and [math]\theta = 3000 [/math].

Calculate [math]\operatorname{P}( X \leq 200).[/math]

• 0.76
• 0.79
• 0.82
• 0.85
• 0.88

For a medical insurance company, you are given:

1. Losses for a new product are assumed to follow a lognormal distribution with parameters μ = 6 and σ = 1.5.
2. The new product has a per-loss deductible that results in a loss elimination ratio of 0.33.

In a review of the business after five years of experience, it is determined that:

1. Losses for this product actually followed an exponential distribution.
2. The initial mean for the exponential distribution is the same as the initial mean under the lognormal assumption.
3. Since it was introduced, the expected value of a loss for this product increased at an annual compound rate of 4%.
4. The per-loss deductible required to target the same loss elimination ratio is d.

Calculate d.

• 605
• 659
• 722
• 775
• 852

You are given:

Assume a uniform distribution of claim sizes within each interval.

Calculate [math]\operatorname{E}(X^2) - \operatorname{E}( (X \wedge 150)^2 ][/math]

• Less than 200
• At least 200, but less than 300
• At least 300, but less than 400
• At least 400, but less than 500
• At least 500

You are given:

1. Losses follow an exponential distribution with the same mean in all years.
2. The loss elimination ratio this year is 70%.
3. The ordinary deductible for the coming year is 4/3 of the current deductible.

Calculate the loss elimination ratio for the coming year.

• 70%
• 75%
• 80%
• 85%
• 90%

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

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

Annual losses in Year 1 follow an exponential distribution with mean θ . An inflation factor of 20% applies to all Year 2 losses. The ordinary deductible for Year 1 is 0.25θ . The deductible is doubled in Year 2.

Calculate the percentage increase in the loss elimination ratio from Year 1 to Year 2.

• 19%
• 28%
• 37%
• 54%
• 78%