1 Garland Actuarial LLC Estimating Gift Certificate Liability: An Actuarial Approach CAS 2003 Annual...

1Garland Actuarial LLC

Estimating Gift Certificate Liability:

An Actuarial Approach

CAS 2003 Annual MeetingNew Orleans, LouisianaNovember 9-12, 2003

Harry T. Garland, [email protected]


Terminology

Liability* – Expected value of all future gift certificate redemptions from those already issued.

Breakage – Expected value of issued gift certificates that will never be redeemed.

Breakage Factor – Percentage of total value of issued gift certificates contributing to Breakage.

Escheat Law – Legal basis in some states to claim from the issuer the dollar value of unredeemed gift certificates.

* Auditors may take other factors into account when determining liability for accounting purposes


The Problem

Let TI = Total value of certificates issued

Let TR = Total value of certificates redeemed

Let L = Liability

Let B = Breakage

Then,

TI – TR = L + B

For unredeemed certificates how do you estimate those that will be redeemed at some future time (L) vs. those that will never be redeemed (B)? And what if the company does not have accurate records to be able to determine TI and TR?


Actuary

Attorney

Company

Accountant

AnalysisAnalysis

Data

Escheat Law Compliance

Balance Sheet Liability

The Process


Sample Company Data

Gift Certificate History

0

1

2

3

4

5

6

7

8

9

Oct-00

Nov-0

0

Dec-0

0

Jan-

01

Feb-0

1

Mar

-01

Apr-0

1

May

-01

Jun-

01

Jul-0

1

Aug-0

1

Sep-0

1

Oct-01

Nov-0

1

Dec-0

1

Jan-

02

Feb-0

2

Mar

-02

Apr-0

2

May

-02

Jun-

02

Jul-0

2

Aug-0

2

Sep-0

2

Oct-02

Nov-0

2

Dec-0

2

Jan-

03

Feb-0

3

Mar

-03

Apr-0

3

May

-03

Jun-

03

Jul-0

3

Mill

ion

s

ISSUED

REDEEMED


Analytical Approach

1) Build a model that predicts monthly certificate redemptions from historical monthly certificate sales.

2) Determine breakage from the model.

3) Determine liability from the model


Building the Model

1) Assume that certificate redemptions in any month can expressed as a linear combination of certificate sales for that month and all past months.

2) Express the model as a redemption vector with elements RV(0), RV(1), RV(2)… where

RV(0) is the percentage of dollar value issued in the current month contributing to redemptions in the current month.

RV(1) is the percentage of dollar value issued in the immediately preceding month contributing to redemptions in the current month.

And so on. The length of the vector should be determined by when the contribution of certificates issued in more distant months becomes immaterial to current redemptions.


Techniques for Calculating the Redemption Vector

1) If transaction data on redemptions by issue date is available, build an actuarial triangle of redemptions by age of month of issue.

2) If only cumulative sales and redemption data is available consider multiple regression analysis and/or exponential decay model.


Redeemed

rm-11 rm-3 rm-2 rm-1 rm rm

Oct-00 428,574 … 517,356 523,851 496,936 577,415 466,258

Nov-00 5,629,747 … 523,851 496,936 577,415 888,371 524,951

Dec-00 415,249 … 496,936 577,415 888,371 6,891,057 2,730,390

Jan-01 392,317 … 577,415 888,371 6,891,057 553,241 2,330,785

Feb-01 431,581 … 888,371 6,891,057 553,241 539,711 1,107,643

Mar-01 597,022 … 6,891,057 553,241 539,711 555,744 869,286

Apr-01 599,215 … 553,241 539,711 555,744 645,149 942,968

May-01 835,034 … 539,711 555,744 645,149 745,722 722,195

Jun-01 517,356 … 555,744 645,149 745,722 1,030,556 869,404

Jul-01 523,851 … 645,149 745,722 1,030,556 581,295 757,378

Aug-01 496,936 … 745,722 1,030,556 581,295 616,551 658,365

Sep-01 577,415 … 1,030,556 581,295 616,551 567,971 543,792. . . . . . .. . . . . . .. . . . . . .

Monthly Dollar Value of Certificates Issued

Example of Regression Matrix

Dollar amount redeemed each month is modeled as a linear combination of the dollar amounts issued in preceding months. The regression coefficients form the elements of the redemption vector. Assumption for this example is that there are no significant redemptions from certificates more than one year old.


Regression Vector

for Sample Company Data

Gift Certificate History

0

1

2

3

4

5

6

7

8

9

Oct-00

Nov-0

0

Dec-0

0

Jan-

01

Feb-0

1

Mar

-01

Apr-0

1

May

-01

Jun-

01

Jul-0

1

Aug-0

1

Sep-0

1

Oct-01

Nov-0

1

Dec-0

1

Jan-

02

Feb-0

2

Mar

-02

Apr-0

2

May

-02

Jun-

02

Jul-0

2

Aug-0

2

Sep-0

2

Oct-02

Nov-0

2

Dec-0

2

Jan-

03

Feb-0

3

Mar

-03

Apr-0

3

May

-03

Jun-

03

Jul-0

3

Mill

ion

s

ISSUED

REDEEMED

RV = (33.2%, 27.9%, 8.9%, 6.1%, 4.3%, 2.9%, 2.4%, 2.3%, 1.8%, 0.2%. 0.1%, .06%)

For this example, the elements of the redemption vector are the coefficients of the regression.


Actual Redemptions vs. Model

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Oct-00

Dec-00

Feb-0

1

Apr-0

1

Jun-

01

Aug-0

1

Oct-01

Dec-01

Feb-0

2

Apr-0

2

Jun-

02

Aug-0

2

Oct-02

Dec-02

Feb-0

3

Apr-0

3

Jun-

03

Mill

ion

s

ActualModel

Comparison of Actual Data with Model


Calculating Breakage Factor

Since each element of the redemption vector, RV(n), represents the percentage contribution to monthly redemption of those certificates issued n months prior to the redemption month, then the sum of all the elements of the redemption vector represents the percentage of certificate sales that is estimated to be redeemed. The breakage factor then is given by:

n

k

kRVctorBreakageFa0

)(1


Calculating Liability

1) Convolution of the redemption vector with the vector of monthly certificate sales yields the vector of estimated monthly certificate redemptions.

2) Convolution of the liability vector with the vector of monthly certificate sales yields the vector of estimated liabilities (as of the end of each month.)

3) The liability vector models what has not yet been redeemed. It can be derived from the redemption vector as follows:

For a redemption vector with n+1 elements RV(0), RV(1), … RV(n), the liability vector, LV, can be formulated as an n-element vector where each element (x ranging from 0 to n-1) is given by:

n

xk

kRVxLV)1(

)()(


Summary

We have shown how actuarial techniques can be used to help a company estimate future gift certificate liabilities. This information is needed to help the company determine its balance sheet liability for gift certificates and its escheat obligations (which vary form state to state.)

Companies may have very poor records regarding their paper gift certificate transactions (e.g. no redemption data by issue date.) Companies that have merged or acquired other companies may have incomplete historic data. As a result, a variety of techniques may be needed to accurately model redemptions.

Many companies are replacing their gift certificates with magnetically-striped gift cards. Use of gift cards will facilitate the acquisition of data for actuarial analysis.

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