## Exam C Practice Problem 12 – Buhlmann Credibility Estimates

Both Problems 12-A and 12-B use the following information.

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

• For each risk in Class 1, the number of claims in a year follows a binomial distribution with mean 0.4 and variance 0.32.
• For each risk in Class 1, the size of a claim is 5 with probability 0.6 and 10 with probability 0.4.
• For each risk in Class 1, the number of claims and the claim sizes are independent.

• For each risk in Class 2, the number of claims in a year follows a binomial distribution with mean 1.6 and variance 0.32.
• For each risk in Class 2, the size of a claim is 5 with probability 0.4 and 10 with probability 0.6.
• For each risk in Class 2, the number of claims and the claim sizes are independent.

A randomly selected risk from this portfolio is observed for 3 years. Four claims are incurred in this period (the individual amounts are 5, 5, 5 and 10).

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Problem 12-A

Determine the Buhlmann credibility estimate of the next claim amount of this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.80$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.24$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.75$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11.33$

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Problem 12-B

Determine the Buhlmann credibility estimate of the aggregate claims in the next year from this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.80$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.57$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.96$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.04$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.33$

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

## Exam C Practice Problem 11 – Estimating Claim Frequency

Both Problems 11-A and 11-B use the following information.

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

• For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean 1.
• For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean 2.5.

A randomly selected risk from this portfolio has 2 claims in year 1 and 2 claims in year 2.

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Problem 11-A

What is the Bayesian estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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Problem 11-B

What is the Buhlmann estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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

## Exam C Practice Problem 10 – Examples of Claim Frequency Models

Problem 10-A

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 1.6 and variance 1.28 (80% weight) and a Gamma distribution with mean 2.5 and variance 3.125 (20% weight).

A risk is randomly selected from this portfolio and observed for 3 years and is found to have incurred 4 claims. What is the probability that this risk will incur exactly 1 claim in the upcoming year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.285$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.303$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.319$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.357$

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Problem 10-B

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 2.4 and variance $\displaystyle \frac{48}{25}$ (60% weight) and a Gamma distribution with mean 3.75 and variance $\displaystyle \frac{75}{16}$ (40% weight).

A risk is randomly selected from this portfolio and observed for 2 years and is found to have incurred 3 claims.

If this risk incurs exactly 2 claims in the upcoming year, what is the probability that the given risk is from Class 2?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.205$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.214$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.263$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.275$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.300$

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Revised: May 1, 2016.
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$\copyright \ 2013-2016 \ \ \text{Dan Ma}$

## 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.

• 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.

• 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 8 – Bayesian Estimates of Claim Frequency

Problem 8-A

A portfolio consists of ten independent risks divided into two classes. Class 1 contains 6 risks and Class 2 contains 4 risks.

The risks in each class are assumed to follow identical annual claim frequency distribution. The following table shows the distributions of the number of claims in a calendar year.

$\displaystyle \begin{bmatrix} \text{ }&\text{ }&\text{Class 1} &\text{ }&\text{Class 2} \\X=x&\text{ }&P(X=x) &\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&0.50 &\text{ }&0.20 \\ 1&\text{ }&0.25&\text{ }&0.25 \\ 2&\text{ }&0.12&\text{ }&0.30 \\ 3&\text{ }&0.08&\text{ }&0.15 \\ 4&\text{ }&0.05&\text{ }&0.10 \end{bmatrix}$

The risks in the portfolio are observed for one calendar year. The following table shows the observed results.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Number of Risks} \\\text{ }&\text{ }&\text{ } &\text{ } \\ 0&\text{ }&4 \\ 1&\text{ }&2 \\ 2&\text{ }&2 \\ 3&\text{ }&1 \\ 4&\text{ }&1 \end{bmatrix}$

Determine the Bayesian expected number of claims per risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.10$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.15$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.22$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.29$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.315$

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Problem 8-B

A portfolio consists of ten independent risks divided into two classes. Both classes contain the same number of risks.

The risks in each class are assumed to follow identical annual claim frequency distribution. The following table shows the distributions of the number of claims in a calendar year.

$\displaystyle \begin{bmatrix} \text{ }&\text{ }&\text{Class 1} &\text{ }&\text{Class 2} \\X=x&\text{ }&P(X=x) &\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&0.5 &\text{ }&0.1 \\ 1&\text{ }&0.3&\text{ }&0.4 \\ 2&\text{ }&0.1&\text{ }&0.3 \\ 3&\text{ }&0.05&\text{ }&0.1 \\ 4&\text{ }&0.05&\text{ }&0.1 \end{bmatrix}$

The risks in the portfolio are observed for one calendar year. The following table shows the observed results.

$\displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Number of Risks} \\\text{ }&\text{ }&\text{ } &\text{ } \\ 0&\text{ }&2 \\ 1&\text{ }&2 \\ 2&\text{ }&3 \\ 3&\text{ }&1 \\ 4&\text{ }&2 \end{bmatrix}$

What is the probability that a randomly selected risk in this portfolio will incur exactly 2 claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.30$

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