Exam C Practice Problem 13 – Working with Aggregate Claims

Both Problems 13-A and 13-B use the following information.

A portfolio if independent risks is divided into two classes. Sixty five percent of the risks are in Class 1 and thirty five percent are in Class 2.

The risks in each class are assumed to follow identical annual aggregate claim distribution. The following shows the aggregate claim distributions for the two classes.

$\displaystyle \begin{bmatrix} X=x&\text{ }&P(X=x \lvert \text{Class 1}) &\text{ }&P(X=x \lvert \text{Class 2}) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{24}{40} &\text{ }&\displaystyle \frac{4}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\displaystyle \frac{6}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 10&\text{ }&\displaystyle \frac{7}{40}&\text{ }&\displaystyle \frac{3}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 15&\text{ }&\displaystyle \frac{2}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 20&\text{ }&\displaystyle \frac{1}{40}&\text{ }&\displaystyle \frac{1}{12} \end{bmatrix}$

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Problem 13-A

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year.

What is the probability that the chosen risk will have 15 in aggregate claims in the second year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

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Problem 13-B

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year and 10 in aggregate claims in the second year.

What is the probability that the chosen risk will have 15 in aggregate claims in the third year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

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

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