Practice Problem Set 9 – Expected Insurance Payment – Additional Problems

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This practice problem set is to reinforce the 3-part discussion on insurance payment models (Part 1, Part 2 and Part 3). The practice problems in this post are additional practice problems in addition to Practice Problem Set 7 and Practice Problem Set 8.

Practice Problem 9A

Losses follow a distribution that is a mixture of two equally weighted Pareto distributions, one with parameters \alpha=2 and \theta=2000 and the other with with parameters \alpha=2 and \theta=4000. An insurance coverage for these losses has an ordinary deductible of 1000.

Calculate the expected payment per loss.

Practice Problem 9B
Losses, prior to any deductible being applied, follow an exponential distribution with mean 17.5. An insurance coverage has a deductible of 8. Inflation of 15% impacts all claims uniformly from the current year to next year.

Determine the percentage change in the expected claim cost per loss from the current year to next year.

Practice Problem 9C
Losses follow a distribution that has the following density function.

    \displaystyle  f(x)=\left\{ \begin{array}{ll}                     \displaystyle  0.15 &\ \ \ \ 0<x < 2 \\           \text{ } & \text{ } \\           \displaystyle  0.10 &\ \ \ \ 2 \le x < 5 \\           \text{ } & \text{ } \\           \displaystyle  0.08 &\ \ \ \ 5 \le x < 10           \end{array} \right.

An insurance policy is purchased to cover these losses. The policy has a deductible of 3.

Calculate the expected insurance payment per payment.

Practice Problem 9D
Losses follow a distribution with the following density function.

    \displaystyle f(x)=\frac{1250}{(x+1250)^2} \ \ \ \ \ \ \ \ \ x>0

An insurance coverage pays losses up to a maximum of 100,000. Determine the average payment per loss.

Practice Problem 9E

You are given the following information.

  • Losses, prior to any application of a deductible, follow a Pareto distribution with \alpha=3 and \theta=5000.
  • An insurance coverage is purchased to cover these losses.
  • If the size of a loss is between 5,000 and 15,000, the coverage pays for the loss in excess of 5,000. Otherwise, the coverage pays nothing.

Determine the average insurance payment per loss.

Practice Problem 9F

You are given the following information.

  • An insurance coverage is purchased to cover a certain type of liability losses.
  • The coverage has a deductible of 1000.
  • Other than the deductible, there are no other coverage modifications.
  • If the deductible is an ordinary deductible, the expected insurance payment per loss is 1,215.
  • If the deductible is a franchise deductible, the expected insurance payment per loss is 1,820.

Determine the proportion of the losses that exceed 1,000.

Practice Problem 9G

Losses follow a uniform distribution on the interval (0, 1000). The insurance coverage has a deductible of 250.

Determine the variance of the insurance payment per loss.

Practice Problem 9H

Losses follow an exponential distribution with mean 500. An insurance coverage that is designed to cover these losses has a deductible of 1,000.

Determine the coefficient of variation of the insurance payment per loss.

Practice Problem 9I
Losses are modeled by an exponential distribution with mean 3,000. An insurance policy covers these losses according to the following provisions.

  • The insured pays 100% of the loss up to 1,000.
  • For the loss amount between 1,000 and 10,000, the insurance pays 80%.
  • The loss amount above 10,000 is paid by the insured until the insured has paid 10,000 in total.
  • For the remaining part of the loss, the insurance pays 90%.

Determine the expected insurance payment per loss.

Practice Problem 9J

You are given the following information.

  • The underlying loss distribution for a block of insurance policies is a Pareto distribution with \alpha=2 and \theta=5000.
  • In the next calendar year, all claims in this block of policies are expected to be impacted uniformly by an inflation rate of 25%.
  • In the next calendar year, the insurance company plans to purchase an excess-of-loss reinsurance policy that caps the insurer’s loss at 10,000 per claim.

Determine the insurance company’s expected claim cost per claim after the effective date of the reinsurance policy.

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Problem Answer
9A
  • \displaystyle \frac{6800}{3}=2266.67
9B
  • \displaystyle 1.15 e^{1.2/20.125}=1.22
  • About 22% increase
9C
  • \displaystyle \frac{10}{3}=3.33
9D
  • 5493.061443
9E
  • 312.5
9F
  • 0.605
9G
  • 61523.4375
9H
  • \sqrt{2 e^2-1}=3.77887956
9I
  • 1642.795495
9J
  • \displaystyle \frac{50000}{13}=3846.15

Daniel Ma actuarial

Dan Ma actuarial

\copyright 2017 – Dan Ma

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