Analysis of Insurance Losses Using the Parallelogram Method and Age-to-Age Development Factors: 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 6, pp. 101-106.

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

Problem S5-42-1. Law X requires all workers' compensation insurance benefits to increase by 7% above what their levels would have been in the absence of the law. The law takes effect on May 1, 2060, and applies only to insurance policies written after that date. Assume that all insurance policies have annual terms and that losses are spread evenly throughout any given year. What is the adjustment factor by which losses for the second quarter of accident year 2060 must be multiplied in order to be brought to the post-Law-X level? Use the parallelogram method.

Solution S5-42-1. We can consider each calendar or accident year to be a rectangle of horizontal length t = 12 months and height p = 100 percent - representing the proportion of a given policy that has expired. Each accident quarter is one-fourth of this rectangle, having length 3 months and height 100 percent. We can draw a diagonal line from point (t = May 1, 2060, and p = 0%) to point (t = May 1, 2061, and p = 100%). This line represents policies that were written on May 1, 2060. These policies and all policies thereafter are post-Law-X. The second quarter of AY 2060 is concluded on July 1, 2060, which on our diagonal line corresponds to p = 2/12 = 1/6 = 16.66666667%. Thus, the area of the triangle representing the proportion of the second quarter of AY 2060 (AQ2 2060) that is affected by the law change is (1/2)*2*16.66666667 = 16.66666667, out of a total area of 3*100 = 300, representing the entirety of AQ2 2060. The corresponding proportion is 16.66666667/300 = 1/18, meaning that 17/18 of the policies in effect during AQ2 2060 are not affected by the law change.

If 1 is the loss level index before the law change, then 1.07 is the loss level index after the law change. The average loss level index during AQ2 2060 is therefore 1.07*(1/18) + 1*(17/18) = 1.003888889. The adjustment factor we desire can be obtained via the following formula:
Adjustment = (Current Loss Level)/(Average Loss Level of Historical Period). The current loss level is the post-Law-X level of 1.07, so the adjustment factor is 1.07/1.003888889 = 1.065855008.

Problem S5-42-2. Law X requires all workers' compensation insurance benefits to increase by 7% above what their levels would have been in the absence of the law. The law takes effect on May 1, 2060, and applies only to insurance policies written after that date. Assume that all insurance policies have annual terms and that losses are spread evenly throughout any given year. What is the adjustment factor by which losses for the second quarter of policy year 2060 must be multiplied in order to be brought to the post-Law-X level? Use the parallelogram method.

Solution S5-42-2. The parallelogram method is easier to use on policy year data because policy year is defined to consider the time at which policies are written, and so the lines separating policy years and quarters are (in the case of annual policies) parallel to the lines representing policies written at certain specified dates. The diagonal line representing policies written on May 1, 2060, splits the parallelogram corresponding to the second quarter of policy year 2060 (PQ2 2060) into two sections. The first section is 1/3 of the PQ2 2060 parallelogram and represents the policies written between April 1 and May 1, 2060. The rest of the policies, 2/3 of PQ2 2060, were written between May 1 and July 1, 2060.

If 1 is the loss level index before the law change, then 1.07 is the loss level index after the law change. The average loss level index during PQ2 2060 is therefore 1*(1/3) + 1.07*(2/3) = 1.0466666667. The adjustment factor we desire can be obtained via the following formula:
Adjustment = (Current Loss Level)/(Average Loss Level of Historical Period). The current loss level is the post-Law-X level of 1.07, so the adjustment factor is 1.07/1.0466666667 = 1.022292994.

Problem S5-42-3. Law Y requires all workers' compensation insurance benefits to increase by 7% above what their levels would have been in the absence of the law. The law takes effect on May 1, 2060, and applies to all losses thereafter, irrespective of when the corresponding insurance policies were written. Assume that all insurance policies have annual terms and that losses are spread evenly throughout any given year. What is the adjustment factor by which losses for the second quarter of accident year 2060 must be multiplied in order to be brought to the post-Law-Y level? Use the parallelogram method.

Solution S5-42-3. Even though the parallelogram method is being used, no actual parallelograms apply to this situation. The law change applies to all losses after May 1, 2060, irrespective of when policies were written. This can be diagrammatically represented as a vertical line at t = May 1, 2060. Since we are analyzing losses on an accident year basis, the vertical line simply divides the rectangle representing AQ2 2060 into two areas; the smaller area, comprising 1/3 of AQ2060, denotes losses occurring before the law change. 2/3 of the losses in AQ2 2060 will occur after the law change.

If 1 is the loss level index before the law change, then 1.07 is the loss level index after the law change. The average loss level index during PQ2 2060 is therefore 1*(1/3) + 1.07*(2/3) = 1.0466666667. The adjustment factor we desire can be obtained via the following formula:
Adjustment = (Current Loss Level)/(Average Loss Level of Historical Period). The current loss level is the post-Law-Y level of 1.07, so the adjustment factor is 1.07/1.0466666667 = 1.022292994.

Note that policy-year analysis of policy-based loss level changes will result in the same answer as accident-year analysis of accident-based loss level changes.

Problem S5-42-4. Law Y requires all workers' compensation insurance benefits to increase by 7% above what their levels would have been in the absence of the law. The law takes effect on May 1, 2060, and applies to all losses thereafter, irrespective of when the corresponding insurance policies were written. Assume that all insurance policies have annual terms and that losses are spread evenly throughout any given year. What is the adjustment factor by which losses for the second quarter of policy year 2060 must be multiplied in order to be brought to the post-Law-Y level? Use the parallelogram method.

Solution S5-42-4. In the diagram for this problem, the parallelograms represent policy years (and policy quarters) of data, and a vertical line at t = May 1, 2060, represents the law change. The diagonal line representing the beginning of PQ2 2060 (beginning at t = April 1, 2060, and p = 0%, ending at t = April 1, 2061, and p = 100%) intersects the vertical line at p = 1/12 = 8.333333333%. The vertical line and the diagonal line form a triangle pertaining to losses in PQ2 2060 that took place before the effective date of Law Y. The area of this triangle is (1/2)*1*8.33333333 = 8.33333333, out of the area of 300 corresponding to the parallelogram representing PQ2 2060 (a policy quarter parallelogram has the same area as an accident quarter rectangle). The corresponding fraction is 8.33333333/300 = 1/36 - meaning that 1/36 of losses in PQ2 2060 that took place before the law change, and 35/36 of losses took place after the law change.

If 1 is the loss level index before the law change, then 1.07 is the loss level index after the law change. The average loss level index during PQ2 2060 is therefore (1/36)*1 + (35/36)*1.07 = 1.06805555556. The adjustment factor we desire can be obtained via the following formula:
Adjustment = (Current Loss Level)/(Average Loss Level of Historical Period). The current loss level is the post-Law-Y level of 1.07, so the adjustment factor is 1.07/1.06805555556 = 1.001820546.

Problem S5-42-5. For a certain kind of "long-tailed" loss that occurred in Accident Year 2052, you have the following information:
As of 12 months, total losses were $56,000.

As of 24 months, total losses were $65,700.

As of 36 months, total losses were $87,000.

As of 48 months, total losses were $90,000.

As of 60 months, total losses were $102,000.

From the information above, it is possible to calculate four age-to-age development factors. Find the factors and specify the time periods to which each factor pertains.

Solution S5-42-5.

An age-to-age development factor for the time period X through Y is found via the expression

(Losses at time Y)/(Losses at time X).

The factor for months 12-24 is thus 65700/56000 = 1.173214286.

The factor for months 24-36 is 87000/65700 = 1.324200913.

The factor for months 36-48 is 90000/87000 = 1.034482759.

The factor for months 48-60 is 102000/90000 = 1.1333333333.

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