Aggregate losses for a portfolio of policies are modeled as follows:

1. The number of losses before any coverage modifications follows a Poisson distribution with mean [math]\lambda[/math] .
2. The severity of each loss before any coverage modifications is uniformly distributed between 0 and b.

The insurer would like to model the effect of imposing an ordinary deductible, d (0 < d < b) , on each loss and reimbursing only a percentage, c (0 < c < 1) , of each loss in excess of the deductible.

It is assumed that the coverage modifications will not affect the loss distribution.

The insurer models its claims with modified frequency and severity distributions. The modified claim amount is uniformly distributed on the interval [0, c(b − d )] .

Determine the mean of the modified frequency distribution.

• [math]\lambda [/math]
• [math]\lambda c[/math]
• [math] \lambda \frac{d}{b}[/math]
• [math]\lambda \frac{b-d}{b} [/math]
• [math]\lambda c \frac{b-d}{b}[/math]

A group dental policy has a negative binomial claim count distribution with mean 300 and variance 800.

Ground-up severity is given by the following table:

You expect severity to increase 50% with no change in frequency. You decide to impose a per claim deductible of 100.

Calculate the expected total claim payment after these changes.

• Less than 18,000
• At least 18,000, but less than 20,000
• At least 20,000, but less than 22,000
• At least 22,000, but less than 24,000
• At least 24,000

You own a light bulb factory. Your workforce is a bit clumsy – they keep dropping boxes of light bulbs. The boxes have varying numbers of light bulbs in them, and when dropped, the entire box is destroyed.

You are given:

• Expected number of boxes dropped per month: 50
• Variance of the number of boxes dropped per month: 100
• Expected value per box: 200
• Variance of the value per box: 400

You pay your employees a bonus if the value of light bulbs destroyed in a month is less than 8000.

Assuming independence and using the normal approximation, calculate the probability that you will pay your employees a bonus next month.

• 0.16
• 0.19
• 0.23
• 0.27
• 0.31

A company insures a fleet of vehicles. Aggregate losses have a compound Poisson distribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with [math] \theta = 200.[/math]

To reduce the cost of the insurance, two modifications are to be made:

1. A certain type of vehicle will not be insured. It is estimated that this will reduce loss frequency by 20%.
2. A deductible of 100 per loss will be imposed.

Calculate the expected aggregate amount paid by the insurer after the modifications.

• 1600
• 1940
• 2520
• 3200
• 3880

The number of annual losses has a Poisson distribution with a mean of 5. The size of each loss has a Pareto distribution with [math]\theta = 10 [/math] and [math] \alpha = 2.5 [/math]. An insurance policy for the losses has an ordinary deductible of 5 per loss.

Calculate the expected value of the aggregate annual payments for this policy.

• 8
• 13
• 18
• 23
• 28

Aggregate losses are modeled as follows:

1. The number of losses has a Poisson distribution with [math]\lambda = 3 [/math]
2. The amount of each loss has a Burr distribution with [math]\alpha = 3, \theta = 2, \gamma = 1 [/math] .
3. The number of losses and the amounts of the losses are mutually independent.

Calculate the variance of aggregate losses.

• 12
• 14
• 16
• 18
• 20

For an aggregate loss distribution S:

1. The number of claims has a negative binomial distribution with [math]r = 16[/math] and [math]\beta = 6[/math] .
2. The claim amounts are uniformly distributed on the interval (0, 8).
3. The number of claims and claim amounts are mutually independent.

Calculate the premium such that the probability that aggregate losses will exceed the premium is 5% using the normal approximation for aggregate losses.

• 500
• 520
• 540
• 560
• 580

You are given:

Claim sizes are independent.

Calculate the variance of the aggregate loss.

• 4,050
• 8,100
• 10,500
• 12,510
• 15,612

A towing company provides all towing services to members of the City Automobile Club. You are given:

1. The automobile owner must pay 10% of the cost and the remainder is paid by the City Automobile Club.
2. The number of towings has a Poisson distribution with mean of 1000 per year.
3. The number of towings and the costs of individual towings are all mutually independent.

Calculate the probability that the City Automobile Club pays more than 90,000 in any given year using the normal approximation for the distribution of aggregate towing costs.

• 3%
• 10%
• 50%
• 90%
• 97%

Two types of insurance claims are made to an insurance company. For each type, the number of claims follows a Poisson distribution and the amount of each claim is uniformly distributed as follows:

The numbers of claims of the two types are independent and the claim amounts and claim numbers are independent.

Calculate the normal approximation to the probability that the total of claim amounts in one year exceeds 18.

• 0.37
• 0.39
• 0.41
• 0.43
• 0.45