Adjusting Historical Premium to Current Rate Levels: The Extension of Exposures Method and an Introduction to the Parallelogram Method: 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 5, pp. 70-72.

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

Problem S5-33-1. Actuary X is analyzing rates for Δ Insurance Company. During the historical period for which Actuary X has data, rates were $890 per policy. Since then, rates were changed twice: there was a 5.6% increase, followed by a 4.3% decrease. Nothing else has changed. The best estimate of Actuary X is that the indicated rate - the rate the company ought to adopt - is $895 per policy. How much of a change should this be from the current rate?

Solution S5-33-1. This problem addresses the principle of adjusting historical premium to current rate levels. The $890 per policy is a historical figure and needs to be adjusted (multiplied) by the factors corresponding to changes in rates since that time. These factors are, respectively, 1 + 0.056 = 1.056 and 1 - 0.043 = 0.957. Thus, the current rate level is 890*1.056*0.957 = 899.42688. The change from the current rate is thus 895/899.42688 - 1 =
-0.0049218898 = -0.49218898%. Note that this is a rate decrease, even though the indicated rate is greater than the historical rate before the adjustments.

Problem S5-33-2. Ω Insurance Company uses a simple rating algorithm:
Premium = Exposure*Rate per Exposure*Class Factor + Policy Fee.

On November 1, 2902, the company increased its Class Factor for Class Q from 0.67 to 0.78. It also decreased its policy fee from 89 Golden Hexagons (GH) to 34 GH. Meanwhile, the rate per exposure changed from 1500 GH to 1429 GH.

Policyholder Y paid a premium of 670 GH on September 1, 2902. The company wishes to adjust this premium to the current rate level to assist in ratemaking calculations. What is the equivalent premium at the post-November 1, 2902, rate level?

Solution S5-33-2. We know that 670 = Exposure*Rate per Exposure*Class Factor + Policy Fee.

Pre-November 1, we are given that Rate per Exposure = 1500, Class Factor = 0.67, and Policy Fee = 89. We first want to find the number of exposures:

670 = Exposure*1500*0.67 + 89

Thus, Exposure = (670 - 89)/(1500*0.67) = Exposure = 0.5781094527.

The new Rate per Exposure = 1429, the new Class Factor = 0.78, and the new Policy Fee = 34. Thus, the equivalent premium at the new rate level is

0.5781094527*1429*0.78 + 34 = 678.3723582 GH.

Problem S5-33-3. Which of the following statements about the extension of exposures method of adjusting historical premium are true? More than one answer may be correct.

(a) The extension of exposures method is less practical when used in conjunction with rating guidelines that involve some subjectivity of judgment by underwriters.
(b) The extension of exposures method is one of the less accurate current rate level methods.
(c) The extension of exposures method deals with aggregate premium quantities and does not delve into individual policy data.
(d) The main limitation of the extension of exposures method today is the significant number of calculations needed to rerate each policy.
(e) The main limitation of the extension of exposures method today is the gathering of data needed to rerate each policy.
(f) It is possible to group policy data when using the extension of exposures method, but only if the rating algorithm is so complex that using individual policy data would be impractical.

Solution S5-33-3. This question is based on the discussion in Werner and Modlin, pp. 71-72.

The following statements are true:
(a) The extension of exposures method is less practical when used in conjunction with rating guidelines that involve some subjectivity of judgment by underwriters.
(e) The main limitation of the extension of exposures method today is the gathering of data needed to rerate each policy.

Choice (b) is incorrect; the extension of exposures method is the most accurate current rate level method.

Choice (c) is incorrect; the extension of exposures method will virtually always involve examination of individual policy data.

Choice (d) is incorrect; the number of calculations limitation has been overcome by advances in computing power.

Choice (f) is incorrect; the simpler the rating algorithm, the more feasible it is to group policy data with the same rating characteristics for use with the extension of exposures method.

Problem S5-33-4. Werner and Modlin (72) give six major steps for the parallelogram method of adjusting historical rate levels to current rate levels. What are these six steps?

Solution S5-33-4. The following answer is cited from Werner and Modlin (72):

"1. Determine the timing and amount of the rate changes during and after the experience period and group the policies into rate level groups according to the timing of each rate change.

"2. Calculate the portion of the year's earned premium corresponding to each rate level group.

"3. Calculate the cumulative rate level index for each rate level group.

"4. Calculate the weighted average cumulative rate level index for each year.

"5. Calculate the on-level factor as the ratio of the current cumulative rate level index and the average cumulative rate level index for the appropriate year.

"6. Apply the on-level factor to the earned premium for the appropriate year."

Problem S5-33-5. Give two ways in which the parallelogram method of adjusting historical rate levels to current rate levels is less accurate than the extension of exposures method.

Solution S5-33-5. There are more than two possible answers. The following are some of the aspects discussed by Werner and Modlin (72):
1. The parallelogram method is undertaken on a group of policies, whereas the extension of exposures method analyzes individual policy data.

2. The parallelogram method assumes that premium is written evenly throughout the time period in question, whereas the extension of exposures method considers when premium was actually written.

3. The parallelogram method involves adjusting an aggregated historical premium by an average factor, whereas the extension of exposures method involves adjusting the premium for each policy by the rate changes that would apply to that particular policy and only then aggregating the results.

4. The parallelogram method does not calculate exact rates, whereas the extension of exposures method does.

Other answers are possible.

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