## Exam C Practice Problem 23 – Working with Credibility Estimates

Both Problems 23-A and 23-B use the following information.

You are given the following:

• A portfolio of independent risks is divided into two classes.
• Each class contains the same number of risks.
• For each risk in Class 1, the claim size follows a zero-truncated geometric distribution with mean 1.5.
• For each risk in Class 2, the claim size follows a zero-truncated geometric distribution with mean 2.5.
• See definition of zero-truncated distribution here.

A risk is selected at random from the portfolio. The first claim observed for this risk is 3.

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Problem 23-A

Calculate the Bayesian credibility estimate of the expected value of the next claim that will be observed for this risk. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 23-B

Calculate the Buhlmann credibility estimate of the expected value of the next claim that will be observed for this risk. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$

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