# Free Tools for Rational Education

# Free Study Materials for the Casualty Actuarial Society (CAS) Exam 5B

# (Old Exam 6)

# Concepts Involved in the Chain Ladder Method: Practice Questions and Solutions

**This section is part of Mr. Stolyarov's Free Study Materials for the CAS Exam 5B.**

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 based on the reading. This is meant to give the student practice in answering questions of the format that will appear on Exam 5B (Old 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.

**Source:**

Friedland, Jacqueline F. *Estimating Unpaid Claims Using Basic Techniques.* Casualty Actuarial Society. July 2009. Chapter 7, pp. 84-90.

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

**Problem S6-17-1.** List the seven steps of the *development method,* i.e., the *chain ladder method*. (See Friedland, p. 85.)

**Solution S6-17-1.** The following are the seven steps of the chain ladder method (Friedland, p. 85):

1. "Compile data in a development triangle."

2. "Calculate age-to-age factors."

3. "Calculate averages of the age-to-age factors."

4. "Select claim development factors."

5. "Select tail factor."

6. "Calculate cumulative claim development factors."

7. "Project ultimate claims."

**Problem S6-17-2.** For 36 to 48 months, the age-to-age factors calculated from various accident years (AYs) of data are as follows:

AY 2112:1.315

AY 2113: 1.125

AY 2114:1.128

AY 2115: 1.269

AY 2116: 1.120

AY 2117: 1.006

Calculate the following statistics:

**(a)** Simple average for the latest six years

**(b)** Simple average for the latest five years

**(c)** Simple average for the latest three years

**(d)** Medial average for the latest six years (excluding the high and low value)

**(e)** Geometric average for the latest six years

**(f)**
Volume-weighted average for the latest six years, given that exposures
in each subsequent year are 5% higher than in the previous year

**Solution S6-17-2. (a)** Simple average for the latest six years:

(1.315 + 1.125 + 1.128 + 1.269 + 1.120 + 1.006)/6 = **1.1605.**

**(b)** Simple average for the latest five years:

(1.125 + 1.128 + 1.269 + 1.120 + 1.006)/5 = **1.1296.**

**(c)** Simple average for the latest three years:

(1.269 + 1.120 + 1.006)/3 = **1.131666667**.

**(d)** Medial average for the latest six years:

(1.125 + 1.128 + 1.269 + 1.120)/4 = **1.1605.**

**(e)** Geometric average for the latest six years:

(1.315*1.125*1.128*1.269*1.120*1.006)^(1/6) = **1.155963621**.

**(f)**
Volume-weighted average for the latest six years, given that exposures
in each subsequent year are 5% higher than in the previous year:

Assume, for convenience, that there is 1 exposure unit in AY 2112. Then, each year, the number of exposure units increases by a factor of 1.05. The volume-weighted average is thus

(1*1.315 + 1.05*1.125 +
(1.05^2)*1.128 + (1.05^3)*1.269 + (1.05^4)*1.120 + (1.05^5)*1.006)/(1 +
1.05 + 1.05^2 + 1.05^3 + 1.05^4 + 1.05^5) = **1.154704947**.

Note that I have used MS Excel notation here to facilitate ease of computerized computation. Problems of this sort are not, in most real-world applications, done by hand. On the exam, the volume of computations required will, I expect, be lower per problem than it is here.

**Problem S6-17-3.** Friedland, p. 88, discusses five
characteristics that actuaries examine when reviewing claim development
factors and the experience on the basis of which they are derived. State
the name of each characteristic and state one possible question related
to it.

**Solution S6-17-3.** The following are five
characteristics that actuaries examine when reviewing claim development
factors and the experience on the basis of which they are derived:

1. **Smooth progression of individual age-to-age factors and average factors across development periods:** Do the age-to-age factors steadily decrease toward 1 as later time periods from the accident date are considered?

2. **Stability of age-to-age factors for the same development period:**
What is the variance in the age-to-age factors for a particular time
period (X months to Y months from the accident date) among the accident
years considered?

3. **Credibility of the experience:** What
is the volume of the underlying experience, and is it sufficiently large
for the experience to be used on a stand-alone basis, or must external
data be considered?

4. **Changes in patterns:** Do any systematic differences from one time period to the next suggest a different external or internal operating environment?

5. **Applicability of the historical experience:**
Is it appropriate to assume that claims will develop in the future much
as they have developed in the past, or must one take account of new
circumstances that have not yet impacted historical claim data?

**Problem S6-17-4.** Briefly describe three approaches commonly used by actuaries to estimate tail development factors. (See Friedland, p. 90.)

**Solution S6-17-4.** The following are three approaches commonly used by actuaries to estimate tail development factors:

1. Use industry benchmark factors.

2. Extrapolate tail factors by fitting a curve - often an exponential curve - to known development factors.

3. Assume that reported development is already at ultimate and use the
ratio of reported claims to paid claims as the estimate of the tail
factor.

**Problem S6-17-5.** You know the following age-to-age factors for accident year 2033:

12-24 months: 1.134

24-36 months: 1.130

36-48 months: 1.102

48-60 months: 1.055

12 months to ultimate: 1.500

What is the tail (60-months-to-ultimate) development factor?

**Solution S6-17-5.** The 12-months-to-ultimate factor is the product of the given age-to-age factors *and* the tail factor. Thus, the tail factor is 1.500/(1.134*1.130*1.102*1.055) = **1.006852162.**

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