Calculations Using the Parallelogram Method when Some Rate Changes Are Applied to Already Written Insurance Policies: 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. 77-78.

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

Problem S5-36-1. Earned premium in calendar year 5777 was 9000 Golden Hexagons (GH). Earned premium in calendar year 5778 was 10000 GH. The insurer in question writes one-year policies.

There were two rate changes during this time period:

On July 1, 5777, the insurer increased its rates by 10%.

On September 1, 5777, because of the passage of the Radical Rate Reduction Act of 5777, all rates - including the rates of existing policies, were reduced by 20%.

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 5777 that corresponds to policies which were written after July 1, 5777 - before the September 1 law change took effect.

Solution S5-36-1. Let calendar year 5777 be a rectangle of horizontal length t = 12 (in months) and height p = 100 (in percentage points). Then the area of this rectangle is 1200. To represent the dividing line pertaining to the July 1, 5777, rate change, we can draw a diagonal time from t = July 1, 5777, and p = 0 to t = January 1, 5778, and p = 50. To represent the dividing line pertaining to the September 1, 5777, rate change, we just draw a vertical line starting at t = September 1, 5777, and p = 0 - since this rate change applies to all policies, not just policies written after September 1.

We can divide the parallelogram into four areas:
(a) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - before the September 1 law change took effect.

(b) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - after the September 1 law change took effect.

(c) The area corresponding to earned premium on policies which were written after July 1, 5777 - before the September 1 law change took effect.

(d) The area corresponding to earned premium on policies which were written after July 1, 5777 - after the September 1 law change took effect.

To find (c), we note that at September 1, policies written on July 1 will have been in force for 2 months, meaning that they will be 2/12 = 16.6666666667% to expiration. This is represented graphically by the legs of the right triangle which forms the boundaries of area (c). This triangle has horizontal length 2 months and height 16.6666666667 percentage points, and therefore an area of (1/2)*2*16.6666666667 = 16.6666666667. The corresponding fraction of the rectangle representing CY 5777 is 16.6666666667/1200 = 0.0138888888889 = 1/72.

Problem S5-36-2. Earned premium in calendar year 5777 was 9000 Golden Hexagons (GH). Earned premium in calendar year 5778 was 10000 GH. The insurer in question writes one-year policies.

There were two rate changes during this time period:

On July 1, 5777, the insurer increased its rates by 10%.

On September 1, 5777, because of the passage of the Radical Rate Reduction Act of 5777, all rates - including the rates of existing policies, were reduced by 20%.

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 5777 that corresponds to policies which were written after July 1, 5777 - after the September 1 law change took effect.

Solution S5-36-2. From Solution S5-36-1, we recall how we structured the graphic solution to this problem:

Let calendar year 5777 be a rectangle of horizontal length t = 12 (in months) and height p = 100 (in percentage points). Then the area of this rectangle is 1200. To represent the dividing line pertaining to the July 1, 5777, rate change, we can draw a diagonal time from t = July 1, 5777, and p = 0 to t = January 1, 5778, and p = 50. To represent the dividing line pertaining to the September 1, 5777, rate change, we just draw a vertical line starting at t = September 1, 5777, and p = 0 - since this rate change applies to all policies, not just policies written after September 1.

We can divide the parallelogram into four areas:
(a) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - before the September 1 law change took effect.

(b) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - after the September 1 law change took effect.

(c) The area corresponding to earned premium on policies which were written after July 1, 5777 - before the September 1 law change took effect.

(d) The area corresponding to earned premium on policies which were written after July 1, 5777 - after the September 1 law change took effect.

In Solution S5-36-1, we found that (c) = 16.666666667.

We wish the find the fraction of the rectangle corresponding to area (d).

It is easiest to first find (c) + (d), which is simply the area corresponding to policies written after July 1, 5777. This is the area of a triangle with base 6 months and height 50 percentage points. This area is (1/2)*6*50 = 150.

To find (d), we take (c) + (d) - (c) = 150 - 16.666666667 = 133.333333333.

The fraction of the rectangle (d) represents is 133.333333333/120 = 0.11111111111 = 1/9.

Problem S5-36-3. Earned premium in calendar year 5777 was 9000 Golden Hexagons (GH). Earned premium in calendar year 5778 was 10000 GH. The insurer in question writes one-year policies.

There were two rate changes during this time period:

On July 1, 5777, the insurer increased its rates by 10%.

On September 1, 5777, because of the passage of the Radical Rate Reduction Act of 5777, all rates - including the rates of existing policies, were reduced by 20%.

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 5777 that corresponds to policies which were written prior to July 1, 5777 - before the September 1 law change took effect.

Solution S5-36-3. From Solution S5-36-1, we recall how we structured the graphic solution to this problem:

Let calendar year 5777 be a rectangle of horizontal length t = 12 (in months) and height p = 100 (in percentage points). Then the area of this rectangle is 1200. To represent the dividing line pertaining to the July 1, 5777, rate change, we can draw a diagonal time from t = July 1, 5777, and p = 0 to t = January 1, 5778, and p = 50. To represent the dividing line pertaining to the September 1, 5777, rate change, we just draw a vertical line starting at t = September 1, 5777, and p = 0 - since this rate change applies to all policies, not just policies written after September 1.

We can divide the parallelogram into four areas:
(a) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - before the September 1 law change took effect.

(b) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - after the September 1 law change took effect.

(c) The area corresponding to earned premium on policies which were written after July 1, 5777 - before the September 1 law change took effect.

(d) The area corresponding to earned premium on policies which were written after July 1, 5777 - after the September 1 law change took effect.

In Solution S5-36-1, we found that (c) = 16.666666667.

In Solution S5-36-2, we found that (d) = 133.33333333.

We are interested in finding area (a).

The relevant area that we can find most directly is area (a) + (c), which is simply the rectangle corresponding to all the calendar months prior to September 1, 5777. The area of this rectangle is 8*100 = 800.

Thus, (a) = (a) + (c) - (c) = 800 - 16.666666667 = 783.333333333.
The fraction of the CY 5777 rectangle corresponding to area (a) is 783.333333333/1200 = 0.652777777778 = 47/72.

Problem S5-36-4. Earned premium in calendar year 5777 was 9000 Golden Hexagons (GH). Earned premium in calendar year 5778 was 10000 GH. The insurer in question writes one-year policies.

There were two rate changes during this time period:

On July 1, 5777, the insurer increased its rates by 10%.

On September 1, 5777, because of the passage of the Radical Rate Reduction Act of 5777, all rates - including the rates of existing policies, were reduced by 20%.

Assume that one-year policies were written at an even rate throughout each year. What is the average historical rate level index for calendar year 5777? Use the parallelogram method.

Solution S5-36-4. In Solutions S5-36-1 through S5-36-3, we have divided the rectangle corresponding to CY 5777 into four areas:

(a) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - before the September 1 law change took effect.

(b) The area corresponding to earned premium on policies which were written prior to July 1, 5777 - after the September 1 law change took effect.

(c) The area corresponding to earned premium on policies which were written after July 1, 5777 - before the September 1 law change took effect.

(d) The area corresponding to earned premium on policies which were written after July 1, 5777 - after the September 1 law change took effect.

We found that the fraction of the rectangle corresponding to (a) is 47/72, the fraction corresponding to (c) is 1/72, and the fraction corresponding to (d) = 1/9. Thus, the fraction corresponding to (b) is 1 - 47/72 - 1/72 - 1/9 = 2/9.

The rate level associated with (a) is 1, since policies in (a) are prior to either of the rate changes.

The rate level associated with (b) is 0.80, since only the legally mandated rate reduction of 20% applies to these policies.

The rate level associated with (c) is 1.10, since only the company's rate increase of 10% applies.

The rate level associated with (d) is 1.10*0.80 = 0.88, since both of the rate changes apply.

The average historical rate level index is therefore 1*(47/72) + 0.80*(2/9) + 1.10*(1/72) + 0.88*(1/9) = 0.94361111111111.

Problem S5-36-5. Earned premium in calendar year 5777 was 9000 Golden Hexagons (GH). Earned premium in calendar year 5778 was 10000 GH. The insurer in question writes one-year policies.

There were two rate changes during this time period:

On July 1, 5777, the insurer increased its rates by 10%.

On September 1, 5777, because of the passage of the Radical Rate Reduction Act of 5777, all rates - including the rates of existing policies, were reduced by 20%.

Assume that one-year policies were written at an even rate throughout each year. An actuary in the year 5780 is studying earned premium data from calendar years 5777 and 5778. No new rate changes have occurred, besides the two mentioned above. What is the total earned premium from those two calendar years, brought to current rate levels?

Solution S5-36-5. Based on the average historical rate level index of 0.94361111111, found in Solution S5-36-4, we can find the on-level earned premium for CY 5777 by multiplying the historical earned premium of 9000 by a factor of

(Current Cumulative Rate Level Index)/(Average Rate Level Index for Historical Period), where the current cumulative rate level index is 1.10*0.80 = 0.88, taking both rate changes into account.

Thus, the on-level CY 5777 earned premium is 9000*(0.88/0.94361111111111) = 8393.288195.

We will also want to find the average rate level index for CY 5778. All policies in CY 5778 are affected by the September 1, 5777, legally mandated rate change. However, not all policies in CY 5778 were written after July 1, 5777. The upper-left-hand corner of the rectangle corresponds to policies that were written before July 1, 5777. This is a triangular area bounded by a diagonal line extending from t = January 1, 5778 and p = 50 to t = July 1, 5778 and p = 100.

This area is therefore (1/2)*50*6 = 150, and is thus 150/1200 = 1/8 of the CY 5778 rectangle. The associated rate level index is simply 0.80, since the July 1, 5777, 10% rate increase does not apply to these policies.

Hence, the average historical rate level index for CY 5778 is 0.80*(1/8) + 0.88*(7/8) = 0.87.

To bring the historical earned premium of 10000 for CY 5778 to current rate levels, we will need to multiply it by

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

0.88/0.87, producing a result of 10000*0.88/0.87 = 10114.94253.

The sum of the on-level CY 5777 and CY 5778 earned premiums is 8393.288195 + 10114.94253 = 18508.23072 GH.

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