# 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.

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.39$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.61$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.69$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.25$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.69$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

___________________________________________________________________________________

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.

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 97.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 164.33$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 179.33$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 212.50$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 257.33$

___________________________________________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

___________________________________________________________________________________

___________________________________________________________________________________

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