Policy-Year Calculations Using 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. 75-77.

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

Problem S5-35-1. Total earned premium in the policy year 4067 was 8000 Golden Hexagons (GH). Total earned premium in the policy year 4068 was 13400 GH. Total earned premium in the policy year 4069 was 6640 GH. There were three rate changes during this time period:
On August 1, 4067, the average rate level was increased by 12%.

On March 1, 4068, the average rate level was decreased by 46%.

On January 1, 4069, the average rate level was increased by 6.5%.

Assume that one-year policies were written at an even rate throughout each year. Use the parallelogram method to find the fraction of earned premium in policy year 4068 that corresponds to policies which were written before the rate change of March 1, 4068, took effect.

Solution S5-35-1. Applying the parallelogram method to policy years is easier than applying it to calendar years. All policies written in 4068 are assigned to policy year 4068, and all policies written in other years are not. Thus, we only need to concern ourselves with policies written from January 1, 4068 and before March 1, 4068. If one considers policy year 4068 to be represented by a parallelogram, then each "policy month" can be represented as a 1/12 "slice" of that parallelogram - a parallelogram of 1/12 the length of each of the horizontal sides but otherwise the same. Thus, each policy month corresponds to 1/12 the area of the policy year, and the fraction of earned premium corresponding to pre-March 1, 4068, policies is 2/12 = 1/6.

Problem S5-35-2. Total earned premium in the policy year 4067 was 8000 Golden Hexagons (GH). Total earned premium in the policy year 4068 was 13400 GH. Total earned premium in the policy year 4069 was 6640 GH. There were three rate changes during this time period:
On August 1, 4067, the average rate level was increased by 12%.

On March 1, 4068, the average rate level was decreased by 46%.

On January 1, 4069, the average rate level was increased by 6.5%.

Assume that one-year policies were written at an even rate throughout each year. Assuming that the initial rate level index (pre August 1, 4067) is 1, what is the average rate level index for policy year 4068? Use the parallelogram method.

Solution S5-35-2. We want to calculate the cumulative rate level indices after each rate change pertinent to PY 4068 data.

Post-August 1, 4067, and pre-March 1, 4068, the cumulative rate level index is 1*1.12 = 1.12.

Post-March 1, 4068 and pre-January 1, 4069, the cumulative rate level index 1.12*(1-0.46) = 0.6048.

These are the only two rate level indices relevant to PY 4068. From Solution S5-35-1, the fraction of the policy year's earned premium corresponding to pre-March 1, 4068, policies is 1/6, and therefore, the fraction corresponding to post-March 1, 4068, policies is 5/6.

Hence, the average rate level index for PY 4068 is (1/6)*1.12 + (5/6)*0.6048 = 0.69066666667.

Problem S5-35-3. Total earned premium in the policy year 4067 was 8000 Golden Hexagons (GH). Total earned premium in the policy year 4068 was 13400 GH. Total earned premium in the policy year 4069 was 6640 GH. There were three rate changes during this time period:
On August 1, 4067, the average rate level was increased by 12%.

On March 1, 4068, the average rate level was decreased by 46%.

On January 1, 4069, the average rate level was increased by 6.5%.

Assume that one-year policies were written at an even rate throughout each year. What is the policy-year 4068 earned premium, brought to the 4069 rate level? Use the parallelogram method.

Solution S5-35-3. The factor by which the policy-year 4068 earned premium of 13400 would need to be multiplied to be brought to the 4069 rate level is equal to

(Current Cumulative Rate Level Index)/(Average Rate Level Index for Historical Period).

From Solution S5-34-2, we know that Average Rate Level Index for Historical Period = 0.69066666667.

Post-March 1, 4068 and pre-January 1, 4069, the cumulative rate level index is 0.6048.

The current cumulative rate index is the post-January 1, 4069, cumulative rate index: 0.6048*1.065 = 0.644112 = Current Cumulative Rate Level Index.

Thus, the factor needed is 0.644112/0.69066666667 = 0.9325945946, and the PY 4068 earned premium, adjusted to current rate levels, is 13400*0.9325945946 = 12496.76757 GH.

Problem S5-35-4. Total earned premium in the policy year 4067 was 8000 Golden Hexagons (GH). Total earned premium in the policy year 4068 was 13400 GH. Total earned premium in the policy year 4069 was 6640 GH. There were three rate changes during this time period:
On August 1, 4067, the average rate level was increased by 12%.

On March 1, 4068, the average rate level was decreased by 46%.

On January 1, 4069, the average rate level was increased by 6.5%.

Assume that one-year policies were written at an even rate throughout each year. What is the total earned premium during policy years 4067, 4068, and 4069, brought to current (post-January 1, 4069) rate levels? Use the parallelogram method.

Solution S5-35-4. We already know from Solution S5-35-4 that the PY 4068 on-level earned premium is 12496.76757 GH. In PY 4069, all the policies written are at current rate levels, since no further rate changes were implemented after January 1, 4069. Thus, the PY 4069 on-level earned premium is exactly as given: 6640 GH. What remains is to find the PY 4067 on-level earned premium.

There are two rate level indices for PY 4067 - the original index of 1, which is in effect until August 1, or 7/12 of the policy year. The post-August 1, 4067, index of 1.12 is in effect for the remaining 5/12 of the policy year. Thus, the average historical rate level index for PY 4067 is 1*(7/12) + 1.12*(5/12) = 1.05.

The on-level factor needed is

(Current Cumulative Rate Level Index)/(Average Rate Level Index for Historical Period) =

0.644112/1.05 = 0.61344.

Thus, the PY 4067 on-level earned premium is 0.61344*8000 = 4907.52.

The answer is the sum of the on-level earned premiums for the three policy years:
4907.52 + 12496.76757 + 6640 = 24044.28757 GH.

Problem S5-35-5. Consider six-month insurance policies that are written at an even rate throughout the year. There is a rate increase of 7% on July 1, 3356. Assuming that 1 is the original rate level index, what is the average rate level index for policy year 3356?

Solution S5-35-5. Policy year X incorporates all policies written in year X. The shape of the parallelogram representing a policy year for six-month policies will be different from the shape of a parallelogram representing a policy year for one-year policies. The diagonal sides of the parallelogram will be twice as steep, as policies expire sooner, and the last policy of the policy year will expire 6 months after the end of the corresponding calendar year. What does not change, however, is that policies written during n months of the year correspond to n/12 of the parallelogram. Therefore, policies written after July 1, 3356, comprise half of the earned premium for PY 3356. Therefore, the average rate level index for PY 3356 is simply

(1/2)*1 + (1/2)*1.07 = 1.035.

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