Methods for Finding Complements of Credibility: Harwayne's Method and the Use of Competitors' Rates: 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. 225-228.

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

Problem S5-94-1. You are given the following information:

There are three classes used for rating: X, Y, and Z.
There are three states: J, K, and L.

In State J:
There are 420 exposures for class X, and the pure premium for class X is $333.
There are 550 exposures for class Y, and the pure premium for class Y is $215.
There are 150 exposures for class Z, and the pure premium for class Z is $761.

In State K:
There are 120 exposures for class X, and the pure premium for class X is $412.
There are 163 exposures for class Y, and the pure premium for class Y is $125.
There are 600 exposures for class Z, and the pure premium for class Z is $800.

In State L:
There are 636 exposures for class X, and the pure premium for class X is $312.
There are 642 exposures for class Y, and the pure premium for class Y is $200.
There are 120 exposures for class Z, and the pure premium for class Z is $620.

Apply the first steps of Harwayne's method for finding the complement of credibility for the experience of class X of State J as follows:

(a) Find the average pure premium for State J.

(b) Find the average pure premium for State K, based on the exposure distribution for State J.

(c) Find the average pure premium for State L, based on the exposure distribution for State J.

Solution S5-94-1.

(a) The average pure premium for State J is the exposure-weighted average of the pure premiums by class: (420*333 + 550*215 + 150*761)/(420 + 550 + 150) = 332.375 = $332.38.

(b) We retain the exposure weights from State J but apply them to State K pure premium data:
(420*412 + 550*125 + 150*800)/(420 + 550 + 150) = 323.0267857 = $323.07.

(c) We retain the exposure weights from State J but apply them to State L pure premium data:
(420*312 + 550*200 + 150*620)/(420 + 550 + 150) = $298.25.

Problem S5-94-2. You are given the following information:

There are three classes used for rating: X, Y, and Z.
There are three states: J, K, and L.

In State J:
There are 420 exposures for class X, and the pure premium for class X is $333.
There are 550 exposures for class Y, and the pure premium for class Y is $215.
There are 150 exposures for class Z, and the pure premium for class Z is $761.

In State K:
There are 120 exposures for class X, and the pure premium for class X is $412.
There are 163 exposures for class Y, and the pure premium for class Y is $125.
There are 600 exposures for class Z, and the pure premium for class Z is $800.

In State L:
There are 636 exposures for class X, and the pure premium for class X is $312.
There are 642 exposures for class Y, and the pure premium for class Y is $200.
There are 120 exposures for class Z, and the pure premium for class Z is $620.

Continue the application of Harwayne's method for finding the complement of credibility for the experience of class X of State J as follows:

(a) Develop an adjustment factor F_K for the experience from State K.

(b) Develop an adjustment factor F_L for the experience from State L.

Solution S5-94-2.

(a) We find F_K by dividing the average pure premium for State J by the average pure premium for State K, based on the exposure distribution for State J. We found the former figure to be 332.375 in Solution S5-94-1(a), and we found the latter figure to be 323.0267857 in Solution S5-94-1(b). Thus, F_K = 332.375/323.0267857 = 1.02893944.

(b) We find F_L by dividing the average pure premium for State J by the average pure premium for State L, based on the exposure distribution for State J. We found the former figure to be 332.375 in Solution S5-94-1(a), and we found the latter figure to be 298.25 in Solution S5-94-1(c). Thus, F_L = 332.375/298.25 = 1.114417435.

Problem S5-94-3. You are given the following information:

There are three classes used for rating: X, Y, and Z.
There are three states: J, K, and L.

In State J:
There are 420 exposures for class X, and the pure premium for class X is $333.
There are 550 exposures for class Y, and the pure premium for class Y is $215.
There are 150 exposures for class Z, and the pure premium for class Z is $761.

In State K:
There are 120 exposures for class X, and the pure premium for class X is $412.
There are 163 exposures for class Y, and the pure premium for class Y is $125.
There are 600 exposures for class Z, and the pure premium for class Z is $800.

In State L:
There are 636 exposures for class X, and the pure premium for class X is $312.
There are 642 exposures for class Y, and the pure premium for class Y is $200.
There are 120 exposures for class Z, and the pure premium for class Z is $620.

Continue the application of Harwayne's method for finding the complement of credibility for the experience of class X of State J as follows:

(a)Find the adjusted pure premium for class X of State K, based on the adjustment factor F_K, which adjusts for differences in experience from State J.

(b)Find the adjusted pure premium for class X of State L, based on the adjustment factor F_L, which adjusts for differences in experience from State J.

Solution S5-94-3.

(a) From Solution S5-94-2(a), F_K = 1.02893944.For class X in State K, the pure premium is $412. The adjusted pure premium is thus 412*1.02893944 = 423.9230493 = $423.92.

(b) From Solution S5-94-2(b), F_L = 1.114417435.For class X in State L, the pure premium is $412. The adjusted pure premium is thus 312*1.114417435 = 347.6982397 = $347.70.

Problem S5-94-4. You are given the following information:

There are three classes used for rating: X, Y, and Z.
There are three states: J, K, and L.

In State J:
There are 420 exposures for class X, and the pure premium for class X is $333.
There are 550 exposures for class Y, and the pure premium for class Y is $215.
There are 150 exposures for class Z, and the pure premium for class Z is $761.

In State K:
There are 120 exposures for class X, and the pure premium for class X is $412.
There are 163 exposures for class Y, and the pure premium for class Y is $125.
There are 600 exposures for class Z, and the pure premium for class Z is $800.

In State L:
There are 636 exposures for class X, and the pure premium for class X is $312.
There are 642 exposures for class Y, and the pure premium for class Y is $200.
There are 120 exposures for class Z, and the pure premium for class Z is $620.

Complete the application of Harwayne's method for finding the complement of credibility for the experience of class X of State J by developing a combined adjusted pure premium for State K and State L, weighted by the class X exposures in each state. This will be the complement of credibility in this situation.

Solution S5-94-4. In State K, there are 120 exposures in class X, and, according to Solution S5-94-3(a), the adjusted pure premium is 423.9230493. In State L, there are 636 exposures in class X, and, according to Solution S5-94-3(b), the adjusted pure premium is 347.6982397. Thus, the exposure-weighted adjusted pure premium that will serve as the complement of credibility is (120*423.9230493 + 636*347.6982397)/(120 + 636) = 359.7974158 = $359.80.

Problem S5-94-5. Some insurers use complements of credibility based on competitors' rates.

(a) Briefly discuss two likely advantages of this approach.

(b) Briefly discuss two likely disadvantages of this approach.

Solution S5-94-5. This question is based on the discussion in Werner and Modlin, pp. 227-228.

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

1. Competitors' rates are likely to produce a complement of credibility that is independent of the subject experience analyzed by the company.

2. If the competitors have more exposures than a small and/or new insurer that is using this method, then the complement of credibility is likely to have less process error.

3. The calculations for this approach are often straightforward.

4. This approach is relatively easy communicate with regard to the complement of credibility having a logical relationship to the subject experience.

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

(b) The following are some possible disadvantages of this approach:

1. Competitors' rates may reflect marketing considerations and/or regulatory constraints that are specific to the competitors and may not apply to the company that is using the competitors' data. This may reduce the accuracy of the complement of credibility.

2. Competitors may have different underwriting and claim practices than the company that is using the competitors' data. This may introduce bias to the complement of credibility.

3. Data from competitors may be time-consuming and difficult to obtain.

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

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