Basics of Classical Credibility, Bühlmann Credibility, and Bayesian Analysis of Credibility: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Basic Ratemaking, cited below. Students are encouraged to read these pages before attempting the problems. This study guide is entirely an independent effort by Mr. Stolyarov and is not affiliated with any organization(s) to whose textbooks it refers, nor does it represent such organization(s).

Some of the questions here ask for short written answers based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5. Students are encouraged to type their own answers first and then to compare these answers with the solutions given here. Please note that the solutions provided here are not necessarily the only possible ones.

Source:
Werner, Geoff and Claudine Modlin. Basic Ratemaking. Casualty Actuarial Society. 2009. Chapter 12, pp. 213-220.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S5-92-1. You know the following information:

An actuary establishes a full credibility standard such that it is desired that the observed value be within ± 2% of the true value 85% of the time.

Claim occurrence follows a Poisson distribution, all exposures are homogeneous, and no variation exists in the claim costs.

Based on 346 claims, the observed pure premium is $533.

The complement of credibility - based on data external to the situation - is $316 for pure premium.

What is the number of claims needed to attain full credibility?

Use the Normal Distribution Table in the set of Exam 4 / C Tables as necessary.

Solution S5-92-1. We use the formula E(Y) = ((z_p/2)/k)². Here, k = 0.02, and p = 1 - 0.85 = 0.15, meaning that p/2 = 0.075. Thus, E(Y) = ((z_0.075)/0.02)². We find z_0.075 using the Normal Distribution Table. This is the z value for which the entry in the table is closest to 1 - 0.075 = 0.925. The closest value to this is 1.44 (with standard normal cumulative distribution function Φ(z) = 0.9251). Thus,

E(Y) = (1.44/0.02)² = 5184 claims needed for full credibility.

Problem S5-92-2. You know the following information:

An actuary establishes a full credibility standard such that it is desired that the observed value be within ± 2% of the true value 85% of the time.

Claim occurrence follows a Poisson distribution, all exposures are homogeneous, and no variation exists in the claim costs.

Based on 346 claims, the observed pure premium is $533.

The complement of credibility - based on data external to the situation - is $316 for pure premium.

What is the credibility-weighted pure premium, according to classical (limited fluctuation) credibility theory?

Solution S5-92-2. To find the credibility of the observed data, we use the square root rule, where Z = Credibility = √((Number of claims observed)/(Full credibility standard)) = √(346/5184) = 0.2583482672.

Thus, we can apply the formula

(Credibility-weighted pure premium) = Z*(Observed Pure Premium) + (1-Z)*(Complement of Credibility) = 0.2583482672*533 + (1-0.2583482672)*316 = 372.061574 = $372.06.

Problem S5-92-3. You know the following information:

Based on 260 claims, the observed pure premium is $376.

The prior mean pure premium estimated by the actuary is $400.

The expected value of the process variance (EVPV) is 146.

The variance of the hypothetical means (VHM) is 3.

What is the credibility-weighted pure premium, according to Bühlmann (least squares) credibility theory?

Solution S5-92-3.

First, we find Bühlmann's K = EVPV/VHM = 146/3 = 48.6666667.

To find the credibility of the data, we use the formula Z = N/(N + K), where N is the number of observations (here, 260):

Z = (260)/(260 + 48.6666667) = 0.8423326134.

Thus, we can apply the formula

(Credibility-weighted pure premium) = Z*(Observed Pure Premium) + (1-Z)*(Complement of Credibility) = 0.8423326134*376 + (1-0.8423326134)*400 = 379.7840173 = $379.78.

Problem S5-92-4. Which of the following statements are true regarding the Bühlmann credibility formula? More than one answer may be correct.

(a) It is assumed that the risk process and risk parameters do not shift over time.
(b) There is a number of observations that can result in full credibility being granted to the observed data.
(c) It is assumed that as the number of observations N increases, the variance of hypothetical means (VHM) of the sum of these observations also increases.
(d) It is assumed that as the number of observations N increases, the variance of expected value of the process variance (EVPV) of the sum of these observations decreases.
(e) The major challenge of the Bühlmann approach is the determination of EVPV and VHM.
(f) The major challenge of the Bühlmann approach is the determination of the complement of credibility.
(g) The "square root rule" applies to Bühlmann credibility.

Solution S5-92-4. This question is based on the discussion by Werner and Modlin, pp. 213-219.

The following answers are correct:

(a) It is assumed that the risk process and risk parameters do not shift over time.
(c) It is assumed that as the number of observations N increases, the variance of hypothetical means (VHM) of the sum of these observations also increases.
(e) The major challenge of the Bühlmann approach is the determination of EVPV and VHM.

Choice (f) cannot be correct if choice (e) is correct. Choices (b) and (g) apply to classical credibility but not to Bühlmann credibility. Choice (d) is incorrect: it is assumed that as the number of observations N increases, the variance of expected value of the process variance (EVPV) of the sum of these observations increases.

Problem S5-92-5. Which of the following statements about Bayesian analysis of credibility are true? More than one answer may be correct.

(a) Bayesian analysis is typically more complex to apply in practice than classical credibility analysis.
(b) Like classical credibility and Bühlmann credibility, Bayesian analysis of credibility requires a calculation of a credibility factor Z, which is then applied to the observed value of the quantity in question.
(c) In Bayesian analysis, a crucial assumption is that the prior estimate of the quantity in question remains constant.
(d) In Bayesian analysis, the prior estimate of the quantity in question changes to reflect new information.
(e) In Bayesian analysis, new information is added to the prior estimate via further observations of empirical data, beyond the observed data originally given.
(f) In Bayesian analysis, new information is added to the prior estimate via Bayes's Theorem.
(g) The Bühlmann credibility estimate corresponds to the weighted least-squares line associated with the Bayesian estimate.

Solution S5-92-5. This question is based on the discussion of Bayesian analysis by Werner and Modlin, p. 220.

The following answers are correct:

(a) Bayesian analysis is typically more complex to apply in practice than classical credibility analysis.
(d) In Bayesian analysis, the prior estimate of the quantity in question changes to reflect new information.
(f) In Bayesian analysis, new information is added to the prior estimate via Bayes's Theorem.
(g) The Bühlmann credibility estimate corresponds to the weighted least-squares line associated with the Bayesian estimate.

Choice (b) is not correct; Bayesian analysis does not require an explicit calculation of Z; rather, the prior estimate is adjusted according to new information via the use of Bayes's theorem. This means that choice (c) is also not correct, as the whole point of Bayesian analysis is to adjust the prior estimate. Choice (e) is not correct; Bayes's theorem does not depend on the injection of new empirical data in order to be applied to existing data.

See other sections of The Actuary's Free Study Guide for Exam 5.