Exam C Practice Problem 19 – Buhlmann Credibility Estimates

Problem 19-A

The number of claims in a year for an insurance policy in a large pool of insurance policies has a distribution with mean \theta and variance \lambda.

The following provides more information about the large pool of insurance policies.

  • For half of the insurance policies in the large pool \theta=1, while for the other half \theta=0.5.
  • For three-quarters of the insurance policies in the large pool \lambda=0.5, while for the other one-quarter \lambda=0.375.

An insurance policy is randomly selected from the large pool. Insurance company records indicate that there are 6 claims in last 5 years.

Determine the Buhlmann credibility estimate of the number of claims for the selected insurance policy in the next year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.82

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.93

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.02

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Problem 19-B

The number of claims in a year for an insurance policy in a large pool of insurance policies has a distribution with mean \theta and variance \lambda.

The following provides more information about the large pool of insurance policies.

  • For three-quarters of the insurance policies in the large pool \theta=1, while for the other one-quarter \theta=0.5.
  • For one-quarter of the insurance policies in the large pool \lambda=0.5, while for the other three-quarters \lambda=0.375.

An insurance policy is randomly selected from the large pool.

Determine the Buhlmann credibility factor assigned to 5 years of claim data from the selected insurance policy.

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{15}{41}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{49}{133}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{19}{41}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{133}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{130}{133}

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\copyright \ 2013 \ \ \text{Dan Ma}

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