## Exam C Practice Problem 15 – Examples of Frequency-Severity Models

Problem 15-A

You are given the following:

• The annual number of claims generated for a single risk follows a Poisson distribution with mean $\theta$.
• The claim size follows a uniform distribution on $(0,10 \theta)$.
• The risk parameter $\theta$ is a random variable with density function $\displaystyle \pi(\theta)=\frac{1}{2} \ \theta$ where $0<\theta<2$.
• Given $\theta$, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 4 claims in the amounts 5, 5, 15 and 20 within the last 5 years.

Determine the Buhlmann credibility estimate for total claim costs in the next year. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.39$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.61$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.69$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.25$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.69$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 15-B

You are given the following:

• The annual number of claims generated for a single risk follows a Poisson distribution with mean $\theta$.
• The claim size follows a uniform distribution on $(0,100 \theta)$.
• The prior distribution of $\theta$ has a uniform distribution on $(0,5)$.
• Given $\theta$, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 5 claims in the total amount of 360 within the last 3 years.

Determine the Buhlmann credibility estimate for total claim costs in the next year. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 97.50$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 164.33$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 179.33$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 212.50$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 257.33$

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

## Exam C Practice Problem 14 – Examples of Limited Fluctuation Credibility

Problem 14-A

You are given the following:

• The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
• The claim size follows a uniform distribution on $(0,t)$ where $t$ is unknown.
• The number of claims and the claim sizes are independent.

Using limited fluctuation credibility, how many expected claims are required to be 95% certain that actual claim costs will be within 5% of the expected claim costs? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1443$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1579$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1936$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1945$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2050$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 14-B

You are given the following:

• The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
• The claim size follows a distribution with the following moment generating function.
• $\displaystyle M(t)=\frac{1}{(1-10t)^4}$
• The number of claims and the claim sizes are independent.

What is the least number of expected claims that are required to be 90% certain that actual claim costs will be within 5% of the expected claim costs? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 820$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1230$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1353$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1376$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1396$

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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. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.50$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.70$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.97$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{57}$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{3}$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{3}$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2$

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