# Credibility Guided Tutorials

Here are a set of exercises that guide the viewer through some of the theoretical foundations of Loss Data Analytics. Each tutorial is based on one or more questions from the professional actuarial examinations – typically the Society of Actuaries Exam C.

Tutorial Structure. Each guided tutorial has a strategy set that describes the context. When you hit the “Start quiz” button, you begin the tutorial that is comprised of a series of mini-questions designed to lead you to the target question. At each stage, hints are provided as well as feedback on the correct solution of each mini-question.

Your Assignment. In reviewing these exercises, ideally the viewer will:

• Work the problem posed referring only to basic theory
• Even if you get the answer correct, review the strategy for this type of problem by clicking (revealing) the Strategy for … header
• If you feel comfortable with the strategy and got the problem correct, then you may choose to move on. However, you might also decide to follow the step-by-step process for solving the problem by clicking on the “Start Quiz” button. It is not really a quiz — it is a guided tutorial.

Strategy for Limited Fluctuation Credibility Problems

Credibility-Poisson Frequency SOA #2
You are given:
(i) The number of claims has a Poisson distribution.
(ii) Claim sizes have a Pareto distribution with parameters $$\theta=0.5$$ and $$\alpha=6$$.
(iii) The number of claims and claim sizes are independent.
(iv) The number of claims in the first year was 1200.
(v) The aggregate loss in the first year was 6.75 million.
(vi) The manual premium for the first year was 5.00 million.
(vii) The exposure in the second year is identical to the exposure in the first year.
(vii) The observed pure premium should be within $$2\%$$ of the expected pure premium $$90\%$$ of the time.

(a) Calculate the expected number of claims needed for full credibility.
(b) Determine the limited fluctuation credibility net premium(in millions) for the second year.

Try to solve the problem on your own. (The answer is (a) 16,912.65 and (b) 5.466 million)

Then, regardless of whether you get the right answer or not, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 2.

Credibility-Non Poisson Frequency SOA #65
You are given the following information about a general liability book of business comprised of 2500 insureds:
(i) $$X_i=\sum_{j=1}^{N_i}Y_{ij}$$ is a random variable representing the annual loss of the $$i$$th insured.
(ii) $$N_1,N_2,…,N_{2500}$$ are independent and identically distributed random variables following a negative binomial distribution with parameters $$r=2$$ and $$\beta=0.2$$.
(iii) $$Y_{i1},Y_{i2},…,Y_{iN_i}$$, are independent and identically distributed random variables following a Pareto distribution with $$\alpha=3$$ and $$\theta=1000$$.
(iv) The full credibility standard is to be within $$5\%$$ of the expected aggregate losses $$90\%$$ of the time.
Using limited fluctuation credibility theory, calculate the partial credibility of the annual loss experience for this book of business.

Try to solve the problem on your own. (The answer is 0.47.)

Then, regardless of whether you get the right answer or not, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 65.

Strategy for Bühlmann Credibility

Bühlmann Credibility SOA #219

For a portfolio of policies, you are given:
(i) The annual claim amount on a policy has probability density function:
$$f(x|\theta)=\frac{2x}{{\theta}^2}, \quad 0\lt x \lt \theta$$
(ii) The prior distribution of $$\theta$$ has density function:
$$\pi(\theta)=4{\theta}^3, \quad \quad 0\lt\theta\lt 1$$
(iii) A randomly selected policy had claim amount 0.1 in Year 1.

Calculate the Bühlmann credibility estimate of the claim amount for the selected policy in Year 2

Try to solve the problem on your own. (The answer is 0.43.)

Then, regardless of whether you get the right answer or not, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 219.

Bühlmann Credibility- Aggregate losses SOA #8
You are given:
(i) Claim counts follow a Poisson distribution with mean $$\theta$$.

(ii) Claim sizes follow an exponential distribution with mean $$10\theta$$.

(iii) Claim counts and claim sizes are independent, given $$\theta$$.

(iv) The prior distribution has probability density function:
$$\pi(\theta)=\frac{5}{{\theta}^6}, \quad\quad \theta\gt 1$$

Calculate Bühlmann’s $$k$$ for aggregate losses.

Try to solve the problem on your own. (The answer is 2.25.)

Then, regardless of whether you get the right answer or not, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 8.

Strategy for Bühlmann-Straub Credibility

Bühlmann-Straub Credibility SOA #21

You are given:
(i) The number of claims incurred in a month by any insured has a Poisson distribution with mean $$\lambda$$
(ii) The claim frequencies of different insureds are independent.
(iii) The prior distribution is gamma with probability density function:

$$f(\lambda)=\frac{(100\lambda)^6 e^{-100\lambda}}{120\lambda}$$

(iv)

$${\scriptsize \begin{matrix} \begin{array}{ccc} \hline \text{Month} & \text{Number of Insureds} & \text{Number of Claims} \\ \hline 1 & 100 & 6 \\ 2 & 150 & 8 \\ 3 & 200 & 11 \\ 4 & 300 & ? \\ \hline \\ \end{array} \end{matrix} }$$

Calculate the Bühlmann-Straub credibility estimate of the number of claims in Month 4.

Try to solve the problem on your own. (The answer is 16.9)

Then, regardless of whether you get the right answer, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 21.

You are given:
(i) The amount of a claim, $$X$$, is uniformly distributed on the interval $$[0,\theta]$$.
(ii) The prior density of $$\theta$$ is $$\pi(\theta)=\frac{500}{\theta^2}, \quad \quad \theta \gt 500$$.

Two claims, $$x_1=400$$ and $$x_2=600$$ are observed. You calculate the posterior distribution as:

$$f(\theta|x_1, x_2)=3\left(\frac{600^3}{\theta^4}\right), \theta \gt 600$$

Calculate the Bayesian premium , $$E(X_3|x_1,x_2)$$.

Try to solve the problem on your own. (The answer is 450)

Then, regardless of whether you get the right answer, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution to this problem from the Society of Actuaries Exam C Question 45.

Bayesian Estimation SOA #64/11
For a group of insureds, you are given;
(i) The amount of a claim in uniformly distributed but will not exceed a certain unknown limit $$\theta$$.
(ii) The prior distribution of $$\theta$$ is $$\pi(\theta)=\frac{500}{{\theta}^2}, \theta>500$$.
(iii) Two claims of 400 and 600 are observed.

(a) Calculate the posterior probability that $$700 \lt \theta \lt 900$$ .
(b) Calculate the probability that the next claim will exceed 550.
(c) Calculate the Bayesian premium, that is, the expected value of the next claim $$y$$ given two claims of 400 and 600 are observed, $$E(y|x_1=400,x_2=600)$$.

Try to solve the problem on your own. (The answers are (a) 1/3, (b) 0.3125, and (c) 450

Then, regardless of whether you get the right answer, you can check out this “quiz.” It is not really a quiz — it is a guided tutorial. The following provides a series of questions to lead you to the solution of this problem based on the Society of Actuaries Exam C Questions 64 and 11.