## 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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.38$

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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. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.10$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.15$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.22$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.29$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.315$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.30$

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