## Exam C Practice Problem 24 – Bayesian Credibility Example

Both Problems 24-A and 24-B use the following information.

You are given the following information:

• The claim size of an insured has an exponential distribution with mean $\displaystyle \frac{1}{\theta}$.
• The parameter $\theta$ has a Gamma distribution with mean 6 and variance 12.

A randomly selected insured has one claim of size 10.

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Problem 24-A

What is the Bayesian estimate of the expected amount of the next claim for this insured?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.2$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.1$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.5$

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Problem 24-B

What is the posterior probability that the size of the next claim for this insured will be greater than 5?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.19$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

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

## 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.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$

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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.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.16$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.20$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.30$

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

## Exam C Practice Problem 19 – Buhlmann Credibility Estimates

Problem 19-A

The number of claims in a year for an insurance policy in a large pool of insurance policies has a distribution with mean $\theta$ and variance $\lambda$.

• For half of the insurance policies in the large pool $\theta=1$, while for the other half $\theta=0.5$.
• For three-quarters of the insurance policies in the large pool $\lambda=0.5$, while for the other one-quarter $\lambda=0.375$.

An insurance policy is randomly selected from the large pool. Insurance company records indicate that there are 6 claims in last 5 years.

Determine the Buhlmann credibility estimate of the number of claims for the selected insurance policy in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.82$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.93$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.02$

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Problem 19-B

The number of claims in a year for an insurance policy in a large pool of insurance policies has a distribution with mean $\theta$ and variance $\lambda$.

• For three-quarters of the insurance policies in the large pool $\theta=1$, while for the other one-quarter $\theta=0.5$.
• For one-quarter of the insurance policies in the large pool $\lambda=0.5$, while for the other three-quarters $\lambda=0.375$.

An insurance policy is randomly selected from the large pool.

Determine the Buhlmann credibility factor assigned to 5 years of claim data from the selected insurance policy.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{15}{41}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{49}{133}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{19}{41}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{133}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{130}{133}$

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

## Exam C Practice Problem 16 – Another Poisson-Gamma Problem

Both Problems 16-A and 16-B use the following information.

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and forty percent are in Class 2.

• The annual number of claims for a single risk in Class 1 follows a Poisson distribution with mean 1.
• The claim size follows a Gamma distribution with mean 1.6 and variance 1.28.
• The number of claims and the claim sizes are independent

• The annual number of claims for a single risk in Class 2 follows a Poisson distribution with mean 2.6.
• The claim size follows a Gamma distribution with mean 2.5 and variance 3.125.
• The number of claims and the claim sizes are independent

A risk is randomly selected from this portfolio. According to the records of the insurer, there are 4 claims for this risk in the amounts 2, 3, 5 and 5 within the last 3 years.

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Problem 16-A

Determine the Buhlmann credibility estimate for total claim costs for this risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.7$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.7$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.4$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.5$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.0$

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Problem 16-B

Determine the Buhlmann credibility estimate for the number of claims for this risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.33$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.35$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.46$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.48$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.64$

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

## 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.

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

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

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

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

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

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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.

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

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

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

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

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

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

## Exam C Practice Problem 12 – Buhlmann Credibility Estimates

Both Problems 12-A and 12-B use the following information.

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and forty percent are in Class 2.

• For each risk in Class 1, the number of claims in a year follows a binomial distribution with mean 0.4 and variance 0.32.
• For each risk in Class 1, the size of a claim is 5 with probability 0.6 and 10 with probability 0.4.
• For each risk in Class 1, the number of claims and the claim sizes are independent.

• For each risk in Class 2, the number of claims in a year follows a binomial distribution with mean 1.6 and variance 0.32.
• For each risk in Class 2, the size of a claim is 5 with probability 0.4 and 10 with probability 0.6.
• For each risk in Class 2, the number of claims and the claim sizes are independent.

A randomly selected risk from this portfolio is observed for 3 years. Four claims are incurred in this period (the individual amounts are 5, 5, 5 and 10).

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Problem 12-A

Determine the Buhlmann credibility estimate of the next claim amount of this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.80$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.24$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.75$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11.33$

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Problem 12-B

Determine the Buhlmann credibility estimate of the aggregate claims in the next year from this risk.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6.80$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.57$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.96$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.04$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.33$

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

## Exam C Practice Problem 11 – Estimating Claim Frequency

Both Problems 11-A and 11-B use the following information.

A portfolio consists of independent risks divided into two classes. Seventy percent of the risks are in Class 1 and thirty percent are in Class 2.

• For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean 1.
• For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean 2.5.

A randomly selected risk from this portfolio has 2 claims in year 1 and 2 claims in year 2.

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Problem 11-A

What is the Bayesian estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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Problem 11-B

What is the Buhlmann estimate of the number of claims in the next year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.68$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.75$

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