### Practice Problem Set 7 – Expected Insurance Payment

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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 basic problems on calculating average insurance payment (per loss or per payment).

Additional problem set: Practice Problem Set 8 and Practice Problem Set 9.

 Practice Problem 7A Losses follow a uniform distribution on the interval $(0,50000)$. An insurance policy has an ordinary deductible of 10,000. Determine the expected payment per loss. Suppose that in addition to the deductible of 10,000, the coverage has a policy limit of 30,000 (i.e. the maximum covered loss is 40,000). Determine the expected payment per loss.
 Practice Problem 7B Losses for the current year follow a uniform distribution on the interval $(0,50000)$. Further suppose that inflation of 25% impacts all losses uniformly from the current year to the next year. Determine the expect insurance payment per loss for the next year assuming that the policy has a deductible of 10,000. Determine the expect insurance payment per loss for the next year assuming that the policy has a deductible of 10,000 and a policy limit of 30,000 (maximum covered loss = 40,000).
 Practice Problem 7C Losses follow an exponential distribution with mean 5,000. An insurance policy covers losses subject to a franchise deductible of 2,000. Determine the expected insurance payment per loss.
 Practice Problem 7D Liability claim sizes follow a Pareto distribution with shape parameter $\alpha=1.2$ and scale parameter $\theta=10000$. Suppose that the insurance coverage pays claims subject to an ordinary deductible of 20,000 per loss. Given that a loss exceeds the deductible, determine the expected insurance payment.
 Practice Problem 7E Losses follow a lognormal distribution with $\mu=7.5$ and $\sigma=1$. For losses below 1,000, no payment is made. For losses exceeding 1,000, the amount in excess of the deductible is paid by the insurer. Determine the expected insurance payment per loss.
 Practice Problem 7F Losses in the current exposure period follow a lognormal distribution with $\mu=7.5$ and $\sigma=1$. Losses in the next exposure period are expected to experience 12% inflation over the current year. Determine the expected insurance payment per loss if the insurance contract has an ordinary deductible of 1,000.
 Practice Problem 7G Losses follow an exponential distribution with mean 2,500. An insurance contract will pay the amount of each claim in excess of a deductible of 750. Determine the standard deviation of the insurance payment for one claim such that a claim includes the possibility that the amount paid is zero.
 Practice Problem 7H Liability losses for auto insurance policies follow a Pareto distribution with $\alpha=3$ and $\theta=5000$. These insurance policies have an ordinary deductible of 1,250. Determine the expected payment made by these insurance policies per loss.
 Practice Problem 7I Liability losses for auto insurance policies follow a Pareto distribution with $\alpha=3$ and $\theta=5000$. These insurance policies make no payment for any loss below 1,250. For any loss greater than 1,250, the insurance policies pay the loss amount in excess of 1,250 up to a limit of 5,000. Determine the expected payment made by these insurance policies per loss.
 Practice Problem 7J Losses follow a lognormal distribution with $\mu=7.5$ and $\sigma=1$. For losses below 1,000, no payment is made. For losses exceeding 1,000, the amount in excess of the deductible is paid by the insurer. Determine the average insurance payment for all the losses that exceed 1,000.

All normal probabilities are obtained by using the normal distribution table found here. $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

7A
• 16,000
• 15,000
7B
• 22,050
• 18,000
7C
• 4,692.24
7D
• 150,000
7E
• $0.9441 e^8-722.4 = 2091.92$
7F
• $1.071168 e^8-761.1 = 2432.01$
7G
• 2414.571397
7H
• 1,600
7I
• 1,106.17284
7J
• $\displaystyle \frac{0.9441 e^8-722.4}{0.7224}= 2895.80$

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