### Practice Problem Set 4 – Pareto Distribution

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The previous post is a discussion of the Pareto distribution as well as a side-by-side comparison of the two types of Pareto distribution. This post has several practice problems to reinforce the concepts in the previous post.

 Practice Problem 4A The random variable $X$ is an insurer’s annual hurricane-related loss. Suppose that the density function of $X$ is: $\displaystyle f(x)=\left\{ \begin{array}{ll} \displaystyle \frac{2.2 \ (250)^{2.2}}{x^{3.2}} &\ X > 250 \\ \text{ } & \text{ } \\ \displaystyle 0 &\ \text{otherwise} \end{array} \right.$ Calculate the inter-quartile range of annual hurricane-related loss. Note that the inter-quartile range of a random variable is the difference between the first quartile (25th percentile) and the third quartile (75th percentile).
 Practice Problem 4B Claim size for an auto insurance coverage follows a Pareto Type II Lomax distribution with mean 7.5 and variance 243.75. Determine the probability that a randomly selected claim will be greater than 10.
 Practice Problem 4C Losses follow a Pareto Type II distribution with shape parameter $\alpha>1$ and scale parameter $\theta$. The value of the mean excess loss function at $x=8$ is 32. The value of the mean excess loss function at $x=16$ is 48. Determine the value of the mean excess loss function at $x=32$.
 Practice Problem 4D For a large portfolio of insurance policies, the underlying distribution for losses in the current year has a Pareto Type II distribution with shape parameter $\alpha=2.9$ and scale parameter $\theta=12.5$. All losses in the next year are expected to increases by 5%. For the losses in the next year, determine the value-at-risk at the security level 95%.
 Practice Problem 4E (Continuation of 4D) For a large portfolio of insurance policies, the underlying distribution for losses in the current year has a Pareto Type II distribution with shape parameter $\alpha=2.9$ and scale parameter $\theta=12.5$. All losses in the next year are expected to increases by 5%. For the losses in the next year, determine the tail-value-at-risk at the security level 95%.
 Practice Problem 4F For a large portfolio of insurance policies, losses follow a Pareto Type II distribution with shape parameter $\alpha=3.5$ and scale parameter $\theta=5000$. An insurance policy covers losses subject to an ordinary deductible of 500. Given that a loss has occurred, determine the average amount paid by the insurer.
 Practice Problem 4G The claim severity for an auto liability insurance coverage is modeled by a Pareto Type I distribution with shape parameter $\alpha=2.5$ and scale parameter $\theta=1000$. The insurance coverage pays up to a limit of 1200 per claim. Determine the expected insurance payment under this coverage for one claim.
 Practice Problem 4H For an auto insurance company, liability losses follow a Pareto Type I distribution. Let $X$ be the random variable for these losses. Suppose that $\text{VaR}_{0.90}(X)=3162.28$ and $\text{VaR}_{0.95}(X)=4472.14$. Determine $\text{VaR}_{0.99}(X)$.
 Practice Problem 4I For a property and casualty insurance company, losses follow a mixture of two Pareto Type II distributions with equal weights, with the first Pareto distribution having parameters $\alpha=1$ and $\theta=500$ and the second Pareto distribution having parameters $\alpha=2$ and $\theta=500$. Determine the value-at-risk at the security level of 95%.
 Practice Problem 4J The claim severity for a line of property liability insurance is modeled as a mixture of two Pareto Type II distributions with the first distribution having $\alpha=1$ and $\theta=2500$ and the second distribution having $\alpha=2$ and $\theta=1250$. These two distributions have equal weights. Determine the limited expected value of claim severity at claim size 1000.

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4A 184.54
4B 0.20681035
4C 80
4D 23.7499
4E 43.1577
4F 1575.97
4G 1159.51615
4H 10,000
4I 4,958.04
4J 698.3681

$\copyright$ 2017 – Dan Ma