Tag Archives: Poisson Claim Frequency

Exam C Practice Problem 9 – Examples of Claim Frequency Models

Problem 9-A

A portfolio consists of independent risks divided into two classes. Eighty percent of the risks are in Class 1 and twenty percent are in Class 2.

The following provides more information about these risks:

  • For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean \theta such that \theta follows a Gamma distribution with mean 1.6 and variance 1.28.
  • For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean \delta such that \delta follows a Gamma distribution with mean 2.5 and variance 3.125.

An actuary is hired to examine the claim experience of the risks in this portfolio. What proportion of the risks can be expected to incur exactly 1 claim in one year?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28

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Problem 9-B

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and fourty percent are in Class 2.

The following provides more information about the two classes of risks:

  • For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean \theta such that \theta follows a Gamma distribution with mean 2.4 and variance \displaystyle \frac{48}{25}.
  • For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean \delta such that \delta follows a Gamma distribution with mean 3.75 and variance \displaystyle \frac{75}{16}.

An actuary is hired to examine the claim experience of the risks in this portfolio. Of the risks that incur exactly 2 claims in a year, what proportion of the risks can be expected to come from Class 2?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.38

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

Exam c Practice Problem 7 – Working with Buhlmann Credibility

Problem 7-A

You are given the following information:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The risk parameter \theta follows a Gamma distribution whose coefficient of variation is 0.5.
    • After observing the given risk for 5 calendar years, twelve claims are observed.
    • Based on the observed data, the posterior distribution of \theta is a continuous distribution whose mean is 2.0.

What is the Buhlmann credibility used in estimating the expected claim frequency for the given risk in the next period?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.500

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.600

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.625

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.650

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.667

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Problem 7-B

You are given the following information:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The risk parameter \theta follows a Gamma distribution with mean 1.5.
    • The value of Buhlmann’s k is 8.
    • After observing the given risk for 4 calendar years, the posterior distribution of \theta is a continuous distribution whose mean is 1.75

What is the number of claims observed in the observation period?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11

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

Exam C Practice Problem 6 – Working with Posterior Distributions

Problem 6-A

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the Gamma distribution with mean 2 and variance 1.

After observing this risk for five calendar years, a total of 12 claims are observed.

Which of the following is the moment generating function of the posterior distribution of \theta?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{2}{2-t}\biggr)^{16} \ \ \ \ \ \ \ \ t<2

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{15} \ \ \ \ \ \ \ \ t<7

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{14}{14-t}\biggr)^{9} \ \ \ \ \ \ \ \ t<14

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{15} \ \ \ \ \ \ \ \ t<2

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{16} \ \ \ \ \ \ \ \ t<7

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Problem 6-B

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the Gamma distribution with mean 4 and variance \frac{1}{2}.

After observing this risk for eight calendar years, a total of 32 claims are observed.

Determine the coefficient of variation of the posterior distribution of \theta.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{64}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{32}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{16}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{8}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{4}

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

Exam C Practice Problem 5 – Bayesian Estimate of Claim Frequency

Problem 5-A

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{2}{\theta^3}, \ \ \ \ \ \ 1<\theta<\infty

After observing for one calendar year, this risk is found to have incurred 4 claims.

Determine the Bayesian expected claim frequency for the given risk in the next year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{2}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{7}{2}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4

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Problem 5-B

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{2}{\theta^3}, \ \ \ \ \ \ 1<\theta<\infty

After observing for two calendar years, this risk is found to have incurred 2 claims in each year.

Determine the Bayesian expected claim frequency for the given risk in the next year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{3}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{5}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{3}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{5}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

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

Exam C Practice Problem 4 – Buhlmann Credibility Examples

Problem 4-A

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has a uniform distribution on (0.5,2.5).

After observing for three calendar years, this risk is found to have incurred 1 claim in year 1, 2 claims in year 2 and 3 claims in year 3.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk in year 4.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.70

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.97

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00

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Problem 4-B

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{1}{2} \ (2-\theta), \ \ \ \ \ \ 0<\theta<2

The given risk is observed for 6 calendar years and is found to have incurred a total of 10 claims.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk for the next calendar year.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{57}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{3}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{3}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

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