## Exam C Practice Problem 22 – Estimating Aggregate Claims

Both Problems 22-A and 22-B use the following information.

You are given the following:

• The annual number of claims for a policyholder follows a binomial distribution with mean 0.5 and variance 0.375.
• The following is the probability function of the claim size $X$.
• $\displaystyle \begin{bmatrix} X=x&\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } \\ 10&\text{ }&0.35 \\ 20&\text{ }&0.35 \\ 30&\text{ }&0.25 \\ 40&\text{ }&0.05 \end{bmatrix}$

• The number of claims and the claim sizes are independent.

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Problem 22-A

Calculate the variance of the annual aggregate claim amount for the policyholder.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 71.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 180$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 190$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 230$

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Problem 22-B

In a certain year, the policyholder has incurred at least one claim and the aggregate claim amount is below 45. Given this information, what is the mean aggregate claim amount for the policyholder?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.0$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.0$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12.5$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 20.0$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 21.3$

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

## Exam C Practice Problem 21 – Working with Aggregate Claims

Problem 21-A

You are given the following:

• The annual number of claims follows a Poisson distribution with mean 800.
• The claim size follows a Gamma distribution with $\alpha$ = 5 and $\theta$ = 2.
• The number of claims and the claim sizes are independent.

Using the normal approximation for the distribution of aggregate claim costs, calculate the probability that the aggregate claim costs will exceed 8350?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.17$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.20$

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Problem 21-B

You are given the following:

• The annual number of claims follows a geometric distribution with mean 8.
• The claim size follows an exponential distribution with mean 5.
• The number of claims and the claim sizes are independent.

Using the normal approximation for the distribution of aggregate claim costs, calculate the 75th percentile of aggregate claim costs?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 64$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 68$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 70$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 75$

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

## Exam C Practice Problem 2 – Variance of Pure Premium

Problem 2-A

You are given:

• For a given risk, the number of claims in a calendar year is 0 (with probability 0.4), 1 (with probability 0.5) and 2 (with probability 0.1).
• In case of only 1 claim, the claim size will be 100 (with probability 0.75) and 200 (with probability 0.25).
• In case of 2 claims, the claim size will be 100 (with probability 0.5) and 200 (with probability 0.5). The two claim sizes are independent.

Calculate the variance of the pure premium for this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 135$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8,650$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9,694$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10,296$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 18,250$

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Problem 2-B

You are given:

• For a given risk, the number of claims in a calendar year is 0 (with probability 0.4), 1 (with probability 0.5) and 2 (with probability 0.1).
• The claim size will be 100 (with probability 0.75) and 200 (with probability 0.25).
• The number of claims and the claim size are independent.
• In case of 2 claims, the two claim sizes are independent.

Calculate the variance of the pure premium for this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 310$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7,189$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7,236$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7,719$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 96,100$

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