Application of the Parallelogram Method for Adjusting Historical Premium to Current Rate Levels: 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. 72-75.

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

Problem S5-34-1. Total earned premium in the year 4067 was 6700 Golden Hexagons (GH). Total earned premium in the year 4068 was 6630 GH. Total earned premium in the year 4069 was 5640 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 calendar year 4068 that corresponds to policies which were written before the rate change of March 1, 4068, took effect.

Solution S5-34-1. First, we need to find the fraction of earned premium in calendar year 4068 that corresponds to policies which were written after the rate change of March 1, 4068, took effect. To do this, we can, following Werner and Modlin (83), draw a graph with time (t) on the horizontal axis and percent of policy earned (p) on the vertical axis. The policies written on March 1, 4068, would be represented by a diagonal line corresponding to p = 0 at t = March 1, 4068, and to p = 100 at t = March 1, 4069. At t = January 1, 4069, p will be (10/12)*100 = 83.3333333%. If we consider the t axis to be denominated in months and the p axis to be denominated in percentage points of policy earned, then the area of the rectangle corresponding to calendar year 4068 is 12*100 = 1200. The area of the triangle corresponding to the policies written after March 1, 4068, is (1/2)*10*83.333333333 = 416.666666667. The fraction of the rectangle occupied by such policies is 416.666666667/1200 = 0.3472222222222, implying that the fraction of the rectangle occupied by pre-March 1, 4068, policies is 1 - 0.3472222222222 = 0.6527777777778.

Problem S5-34-2. Total earned premium in the year 4067 was 6700 Golden Hexagons (GH). Total earned premium in the year 4068 was 6630 GH. Total earned premium in the year 4069 was 5640 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 calendar year 4068 that corresponds to policies which were written before the rate change of March 1, 4068, took effect, but after the rate change of August 1, 4067, took effect.

Solution S5-34-2. From Solution S5-34-2, we know that the fraction of the rectangle corresponding to calendar year 4068 occupied by pre-March 1, 4068, policies is 0.6527777777778. We first want to find the part of this area occupied by pre-August 1, 4067, policies. To do this, we can, following Werner and Modlin (83), draw a graph with time (t) on the horizontal axis and percent of policy earned (p) on the vertical axis. The policies written on August 1, 4067 would have p = (5/12)*100 = 41.66666667 at t = January 1, 4067. They will expire on August 1, 4068. So the area in calendar year 4068 corresponding to these policies is that of an upper-left-hand-corner triangle with height of 100 - 41.66666667 = 58.333333333% and horizontal length of 8 months. This area is (1/2)*8*58.333333333 = 233.33333333. Since the area of the CY 4068 rectangle is 1200, the are corresponding to pre-August 1, 4067, policies is 233.33333333/1200 = 0.19444444444 of the rectangle. The rest of the pre-March 1, 4068, area is comprised of post-August 1, 4067, but pre-March 1, 4068, policies. This is our desired answer: 0.6527777777778 - 0.19444444444 = 0.458333333333 = 11/24.

Problem S5-34-3. Total earned premium in the year 4067 was 6700 Golden Hexagons (GH). Total earned premium in the year 4068 was 6630 GH. Total earned premium in the year 4069 was 5640 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 calendar year 4068? Use the parallelogram method.

Solution S5-34-3. We want to calculate the cumulative rate level indices after each rate change pertinent to CY 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.

From Solutions S5-34-1 and S5-34-2, we know that 0.19444444444 = 7/36 of CY 4068 policies are pre-August 1, 4067, 0.458333333333 = 11/24 of CY 4068 policies are post-August 1, 4067, and pre-March 1, 4068, and 0.3472222222222 = 25/72 of CY 4068 policies are post-March 1, 4068. These are the factors by which the corresponding cumulative rate level indices should be multiplied to get the average rate level index for CY 4068:

1*(7/36) + 1.12*(11/24) + 0.6048*(25/72) = 0.91777777778.

Problem S5-34-4. Total earned premium in the year 4067 was 6700 Golden Hexagons (GH). Total earned premium in the year 4068 was 6630 GH. Total earned premium in the year 4069 was 5640 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 calendar-year 4068 earned premium, brought to the 4069 rate level? Use the parallelogram method.

Solution S5-34-4. The factor by which the calendar-year 4068 earned premium of 6630 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 S5-34-3, we know that Average Rate Level Index for Historical Period = 0.917777777778.

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 0.644112/0.917777777778 = 0.7018169492, and the CY 4068 earned premium, adjusted to current rate levels, is 6630*0.7018169492 = 4653.046373 GH.

Problem S5-34-5. Total earned premium in the year 4067 was 6700 Golden Hexagons (GH). Total earned premium in the year 4068 was 6630 GH. Total earned premium in the year 4069 was 5640 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 calendar year 4069, brought to current (post-January 1, 4069) rate levels? Use the parallelogram method.

Solution S5-34-5. For CY 4069, half the policies during the calendar year are written after January 1, 4069, while half are written before. Of the half written before January 1, 4069, some were written before March 1, 4068. It is useful to find this fraction first. To do this, we can, following Werner and Modlin (83), draw a graph with time (t) on the horizontal axis and percent of policy earned (p) on the vertical axis. The policies written on March 1, 4068 would have p = (10/12)*100 = 83.333333333 at t = January 1, 4069. This means that in the rectangle corresponding to CY 4069, the area of the triangle corresponding to such policies written before March 1, 4068, would be in the upper-left-hand-corner of the rectangle and would have height 100 - 83.333333333 = 16.6666667 and horizontal length 2 months, for an area of

(1/2)*16.6666667*2 = 16.666666667. The corresponding fraction of the rectangle (for is 16.6666667/1200 = 0.0138888889. Thus, the fraction of the rectangle corresponding to policies written after March 1, 4068, but before January 1, 4069, would be 0.5 - 0.0138888889 = 0.48611111111.

We multiply these fractions by the cumulative rate indices corresponding to the time periods in question:

0.0138888889*1.12 + 0.48611111111*0.6048 + 0.5*0.644112 = 0.6316115556. This is the average rate level index for CY 4069. To bring the CY 4069 earned premium to current rate levels, we would need to multiply it by a factor of

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

0.644112/0.6316115556 = 1.019791349, getting a result of 5640*1.019791349 = 5751.623206 GH.

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