**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 5, authored by Mr. Stolyarov. This is Section 93 of the Study Guide. See an index of all sections by following the link in this paragraph.**

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. 220-226.

**Original Problems and Solutions from The Actuary’s Free Study Guide**

**Problem S5-93-1.** The following characteristics of a desirable complement of credibility are mentioned by Werner and Modlin, p. 232:

1. Accuracy;

2. Lack of bias;

3. Statistical independence from the base statistic;

4. Availability;

5. Ease of computation;

6. Logical relationship to the base statistic.

In each of the following scenarios, one of the above qualities is absent. Identify the quality that is clearly missing.

(a) The experience to which the credibility factor Z is being applied constitutes 60% of the experience considered in the complement of credibility.

(b) The complement of credibility is consistently close to the observed experience to which the credibility factor Z is being applied, but it is also consistently greater than that observed experience.

(c) Experience from aviation insurance is used to develop a complement of credibility to experience from private passenger automobile insurance.

(d) The expected value of the complement of credibility is the expected value of the observed experience to which the credibility factor Z is being applied, but there is a large variance to the data pertaining to the complement of credibility.

(e) The calculation of the complement of credibility requires extremely complex formulas that have been developed internally to an insurance company, are deemed by the company to be proprietary and confidential, and are only permitted to be used by one highly trained actuary/statistician.

(f) A complement of credibility is used to evaluate loss data for homeowners’ insurance. The data pertaining to the complement of credibility apply to an exotic class of insurance for which no losses occur during most years, and for which loss data are irregularly compiled and inadequately retained.

**Solution S5-93-1.**

Scenario **(a)** is an instance of the absence of characteristic **3. Statistical independence from the base statistic**.

Scenario **(b)** is an instance of the absence of characteristic **2. Lack of bias**.

Scenario **(c)** is an instance of the absence of characteristic **6. Logical relationship to the base statistic**.

Scenario **(d)** is an instance of the absence of characteristic **1. Accuracy**.

Scenario **(e)** is an instance of the absence of characteristic **5. Ease of computation** and possibly of characteristic **4. Availability,** particularly with regard to parties external to the company.

Scenario **(f)** is an instance of the absence of characteristic **4. Availability**.

**Problem S5-93-2.** Actuaries sometimes use the experience of a larger group (here, X) that includes the group being considered (here, Y) to develop a complement of credibility to the experience of Y.

(a) Briefly describe two advantages of this approach.

(b) Briefly describe two problems that might arise with this approach.

**Solution S5-93-2.** This problem is based on the discussion by Werner and Modlin, p. 222-223.

**(a)** The following are some advantages of this approach:

1. The experience of X is likely to have a lower process variance than the experience of Y.

2. Data for X will typically be readily available, if data for Y is available.

3. There is, in most cases, a logical relationship between the data for Y and the data for X.

4. A complement of credibility based on X will typically be easy to compute.

5. The problem of the dependence of the experience of X on the experience of Y might be addressed by considering only the experience in X *except* the experience of Y in developing the complement of credibility; this is only possible if Y does not constitute a major fraction of X.

Any two of the above suffice as answers. Other valid answers may be possible.

**(b)** The following are some problems that might arise with this approach:

1. The experience of X is likely to be a biased estimator of the experience of Y.

2. If the experience of Y comprises a large fraction of the experience of X, then X would not be statistically independent from Y.

Other valid answers may be possible.

**Problem S5-93-3.** To find a complement of credibility to the loss cost experience of a particular group (here, A), an actuary can use the loss costs of a larger related group (here, B) (call this Approach 2) instead of using the loss costs of a larger group that includes the group being rated (here, C) (call this Approach 1).

**(a)** What is a major advantage of Approach 2 over Approach 1?

**(b)** What is a major weakness of Approach 2? How might that weakness be mitigated?

**Solution S5-93-3.** This problem is based on the discussion by Werner and Modlin, p. 223-224.

**(a)** Since, unlike C, B does not include A, the experience of C is most likely independent from the experience of A. Approach 2 is thus more likely than Approach 1 to produce a complement of credibility that is independent to the data to which the credibility factor (Z) is being applied.

**(b)** A major weakness of Approach 2 is that the complement of credibility is likely to be biased. The bias arises in that the experience of B is likely to consistently diverge from the experience of A in some way. To mitigate this bias, it might be possible to adjust the experience of B to match the exposure to loss in group A. This would require a comparison of exposures between group A and group B.

**Problem S5-93-4.** To find a complement of credibility to the loss cost experience of a particular group (here, A), an actuary can use the loss costs of a larger related group (here, B). Consider a situation in which the actuary uses an indicated rate change for B and applies it to present rates. It is known that the current loss cost of group A is $360, the current average loss cost of group B is $376, and the indicated loss cost for group B is $463. What should be the complement of credibility using this approach?

**Solution S5-93-4.** This problem is based on the discussion by Werner and Modlin, p. 224. We use the formula

C = (Current Loss Cost of Subject Experience)*((Larger Group Indicated Loss Cost)/(Larger Group Current Average Loss Cost)) = 360*(463/376) = 443.2978723 = **C = $443.30.**

**Problem S5-93-5.**

(a) In what kinds of situations is Harwayne’s Method for determining a complement of credibility most often used?

(b) What are two possible disadvantages of Harwayne’s Method?

**Solution S5-93-5.** This problem is based on the discussion by Werner and Modlin, p. 225-226.

**(a)** Harwayne’s Method is most often used when the subject experience (to which the credibility factor Z is being applied) has a significantly different distribution from that of the related experience (which is used for determining the complement of credibility.

**(b)** Two possible disadvantages of Harwayne’s Method are that (1) the necessary computations can be complicated and time-consuming and (2) it may be more difficult to explain the relationship of the complement of credibility to the subject experience, as a result of the computational complexity.

Quantitative applications of Harwayne’s Method will be covered in Section 94 of the study guide.

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