### Practice Problem Set 4 – Pareto Distribution

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 is an insurer’s annual hurricane-related loss. Suppose that the density function of is:
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 and scale parameter . The value of the mean excess loss function at is 32. The value of the mean excess loss function at is 48. Determine the value of the mean excess loss function at . |

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 and scale parameter . 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%. |

(Continuation of 4D)Practice Problem 4E |

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 and scale parameter . 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 and scale parameter . 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 and scale parameter . 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 be the random variable for these losses. Suppose that and . Determine . |

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 and and the second Pareto distribution having parameters and . 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 and and the second distribution having and . These two distributions have equal weights. Determine the limited expected value of claim severity at claim size 1000. |

Problem | Answer |
---|---|

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 |

2017 – Dan Ma

October 29, 2017 at 11:25 pm

[…] Type I and Pareto Type II. It also discusses the two Pareto types from a calculations stand point. This blog post has practice problems on the two Pareto […]

October 29, 2017 at 11:40 pm

[…] Update (10/29/2017). This blog post has updated information on Pareto distribution. It also has links to more detailed contents on Pareto distribution in two companion blogs. These links are also given here: more detailed post on Pareto, Pareto Type I and Type II and practice problems on Pareto. […]