Applications of the Chain Ladder Method: Practice Questions and Solutions

This section of the study guide is intended to provide practice problems and solutions to accompany the pages of Estimating Unpaid Claims Using Basic Techniques, 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. This is meant to give the student practice in answering questions of the format that will appear on Exam 6. 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.

Sources:
Friedland, Jacqueline F. Estimating Unpaid Claims Using Basic Techniques. Casualty Actuarial Society. July 2009. Chapter 7, pp. 91-97.

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

The following information applies to Problems S6-19-1 through S6-19-3:

As of December 31, 2100, you are aware of the following reported loss amounts for each accident year (AY):

For AY 2097: $315,130
For AY 2098: $310,120
For AY 2099: $200,430
For AY 2100: $180,540

You also know the following age-to-age factors:

12-24 months: 1.120
24-36 months: 1.085
36-48 months: 1.030
48 months to ultimate: 1.014

Problem S6-19-1. Use the chain ladder method to estimate the ultimate losses for each accident year.

Solution S6-19-1. The farther back in time we go, the fewer age-to-age factors we need to apply. For instance, AY 2097 data are already developed to 48 months, so only the 48-months-to-ultimate factor need apply. AY 2098 data are developed to 36 months, so we only apply the product of the 36-48-months and the 48-months-to-ultimate factors.

Ultimate losses for AY 2097: 315130*1.014 = $319,541.82
Ultimate losses for AY 2098: 310120*1.030*1.014 = $323,895.53
Ultimate losses for AY 2099: 200430*1.085*1.030*1.014 = $227,126.41
Ultimate losses for AY 2100: 180540*1.120*1.085*1.030*1.014 = $229,137.61

Problem S6-19-2. Use the chain ladder method to estimate the IBNR for each accident year as of December 31, 2100.

Solution S6-19-2. IBNR = Ultimate Claims - Reported Claims. We are given the reported claims, and, from Solution S6-19-1, we know the ultimate claims.

IBNR for AY 2097: 319541.82 - 315130 = $4411.82
IBNR for AY 2098: 323895.53 - 310120 = $13,775.53
IBNR for AY 2099: 227126.41 - 200430 = $26,696.41
IBNR for AY 2100: 229137.61- 180540 = $48,597.61

Problem S6-19-3. Based on theclaim development factors, calculate (a) the incremental percent of claims reported during each of the given age-to-age intervals, and (b) the cumulative percent of claims reported at each endpoint of the given intervals.

Solution S6-19-3. Given the age-to-age factors we have, it is actually easier to work backward from ultimate and to calculate the cumulative percent of claims reported first.

(b) For each number of months given, the cumulative percent of claims reported is 100*1/(cumulative claim development factor from that time to ultimate).

At ultimate: 100% (by definition).
At 48 months: 100*1/1.014 = 98.61932939%.
At 36 months: 100*1/(1.014*1.030) = 95.74692174%.
At 24 months: 100*1/(1.014*1.030*1.085) = 88.24601082%.
At 12 months: 100*1/(1.014*1.030*1.085*1.120) = 78.79108109%.

(a) The incremental percent claims reported between X and Y months is
(% reported at Y months) - (% reported at X months).

For 0-12 months: 78.79108109% - 0% = 78.79108109%.
For 12-24 months: 88.24601082% -78.79108109% = 9.4549297%.
For 24-36 months: 95.74692174% - 88.24601082% = 7.5009109%.
For 36-48 months: 98.61932939% - 95.74692174% = 2.8724077%.
For 48 months to ultimate: 100% - 98.61932939% = 1.38067061%.

Problem S6-19-4. Name two internal insurer changes and two changes external to the insurer that could invalidate the applicability of the chain ladder technique. (See Friedland, pp. 95-96.)

Solution S6-19-4. The following are possible choices. Any two from each category suffice as an answer. Many other valid choices are conceivable.

Internal changes:
1. Faster or slower claim settlement
2. Case outstanding increases or decreases
3. New claim-processing systems
4. Changes in claim management philosophy
5. Policy revisions (changes in offered limits, deductibles, etc.)

External changes:
1. Tort reforms
2. Changes in patterns regarding judicial rulings or jury awards
3. Changes in prevalent policyholder choices regarding limits and/or deductibles

Problem S6-19-5. (a) Fill in the blanks: "The chain ladder technique works best with lines of business where frequency is _____ (high or low?), severity is ______ (high or low?), and claims are reported _______ (quickly or slowly?) and _________ (concentrated toward a particular segment of the year or spread evenly throughout the year?)." (See Friedland, p. 96.)

(b) Justify your last choice in part (a).

Solution S6-19-5.

(a) The chain ladder technique works best with lines of business where frequency is high, severity is low, and claims are reported quickly and spread evenly throughout the year.

(b) For the chain ladder technique to work, it is desirable for claims to be spread evenly throughout the year, because an uneven claim spread implies that the average accident date will not be in the middle of the year, and this might render data for the given year non-comparable to historical data with an even spread of claims. (See Friedland, pp. 96-97.)

See other sections of The Actuary's Free Study Guide for Exam 6.