Study On the Determination of - - PowerPoint PPT Presentation

신용카드 신용카드 신용카드 신용카드 적정한도에 적정한도에 적정한도에 적정한도에 관한 관한 관한 관한 연구 연구 연구 연구

KCB 연구소 연구소 연구소 연구소

• 2011. 2

NOTICE : Proprietary and Confidential This information was prepared by Korea Credit Bureau. Ltd. solely for the use of KCB personnel. No part of it may be circulated, quoted or reproduced for distribution outside the KCB organization without prior written approval from Korea Credit Bureau. Ltd

Study On the Determination of the Optimum Credit Card Limit

Yonho Song, Ph.D. Korea Credit Bureau yhsong@koreacb.com

1

I. Optimum Limit Model

• II. Determination of Optimum Purchase Limit
• III. Summary and Conclusion

> Content

2

• I. Optimum Limit Model

To determine the optimum limit that maximizes the profit,

Solve following equation: Solve following equation: Solve following equation: Solve following equation:

( ) ( ) ( ) ( ) ( ) * ( ) . . .

G G B B G G B

u x N g u x N l P x N w N g u x u x N w Max P x w r t x × × − × × =   = × − ×    

NG , NB : Number of good and bad customers : Number of good and bad customers : Number of good and bad customers : Number of good and bad customers uG(x), uB(x) : Usage function w.r.t the limit for good and bad customers : Usage function w.r.t the limit for good and bad customers : Usage function w.r.t the limit for good and bad customers : Usage function w.r.t the limit for good and bad customers w0 = = = = l/g : (losses from bad customers)/(revenue from good customers) : (losses from bad customers)/(revenue from good customers) : (losses from bad customers)/(revenue from good customers) : (losses from bad customers)/(revenue from good customers) w* = = = = NG/NB : Population Odds : Population Odds : Population Odds : Population Odds

3

• I. Optimum Limit Model

Common sense on the optimum limit

Allow higher limits to Allow higher limits to Allow higher limits to Allow higher limits to “Good Good Good Good” customers with high usage amount, lower limits customers with high usage amount, lower limits customers with high usage amount, lower limits customers with high usage amount, lower limits to to to to “Bad Bad Bad Bad” customers who are delinquent customers who are delinquent customers who are delinquent customers who are delinquent The optimum limit is the limit that maximizes the profit for a The optimum limit is the limit that maximizes the profit for a The optimum limit is the limit that maximizes the profit for a The optimum limit is the limit that maximizes the profit for a homogeneous homogeneous homogeneous homogeneous customer group customer group customer group customer group with a similar usage amount and credit rating. with a similar usage amount and credit rating. with a similar usage amount and credit rating. with a similar usage amount and credit rating.

When limits are randomly given to customers in homogeneous customer group Ai, profit function will be maximized at the optimum limit value. The optimum limit value is determined by the marginal utilization rate of good and bad customers.

* * *

( ) ( )

G B B i i i i i

du x w du x w u dx dx w w = × = ×

4

• I. Optimum Limit Model

Usage and Revenue/Cost function of Good and Bad customers

Usage functions for good and bad customers are different Usage functions for good and bad customers are different Usage functions for good and bad customers are different Usage functions for good and bad customers are different

Usage function for good customers is S-shaped Usage function for bad customers is linear with respect to the limit

The optimum limit can be obtained from Revenue and Cost function The optimum limit can be obtained from Revenue and Cost function The optimum limit can be obtained from Revenue and Cost function The optimum limit can be obtained from Revenue and Cost functions s s s

Exists where the slope of cost function is the same as the tangent of revenue function

Revenue Revenue Revenue Revenue from Good from Good from Good from Good Customers Customers Customers Customers Cost from Cost from Cost from Cost from Bad Bad Bad Bad Customers Customers Customers Customers Good Good Good Good Customer Customer Customer Customer Bad Bad Bad Bad Customer Customer Customer Customer Usage Usage Usage Usage Amount Amount Amount Amount Revenue Revenue Revenue Revenue /Cost /Cost /Cost /Cost

Average Usage Amount per person Average Usage Amount per person Revenue/Cost function per person Revenue/Cost function per person

Limit Limit Limit Limit Limit Limit Limit Limit Optimum Limit Optimum Limit Optimum Limit Optimum Limit

5

• I. Optimum Limit Model

The optimum limit depends on the customer group

Optimum limit is higher for customer groups with higher usage am Optimum limit is higher for customer groups with higher usage am Optimum limit is higher for customer groups with higher usage am Optimum limit is higher for customer groups with higher usage amount and

• unt and
• unt and
• unt and

better credit better credit better credit better credit

For a group with higher usage amount, the revenue curve moves up and to the right The optimum limit becomes higher For a group with worse credit rating, the cost curve becomes steeper The optimum limit becomes lower

A group with higher usage amount A group with higher usage amount A group with worse credit rating A group with worse credit rating

Limit Limit Limit Limit Limit Limit Limit Limit Revenue Revenue Revenue Revenue Cost Cost Cost Cost Revenue Revenue Revenue Revenue /Cost /Cost /Cost /Cost Revenue Revenue Revenue Revenue Cost Cost Cost Cost Revenue Revenue Revenue Revenue /Cost /Cost /Cost /Cost B B B B A A A A B B B B A A A A

6

• I. Optimum Limit Model

Delinquency Rate according to the Optimum Limit Model

Account Account Account Account-

• based delinquency rate

based delinquency rate based delinquency rate based delinquency rate

Defined as (# of delinquent customers)/(# of customers with balance) Property of the customer group, independent of the limit

Balance Balance Balance Balance-

• based delinquency rate

based delinquency rate based delinquency rate based delinquency rate

Defined as (delinquent amount)/(total balance) U-shaped and has a minimum value near the optimum limit

1 ( ) 1 *

B c B G

N D x w N N = = + + ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) 1 ( ) / ( )

B B b B G B G G B

b x u x D x b x b x u x u x u x u x = ≈ = + + +

7

• I. Optimum Limit Model

Can the Optimum Limit Model be applied to the actual data?

The situation described by the Optimum Limit model The situation described by the Optimum Limit model The situation described by the Optimum Limit model The situation described by the Optimum Limit model

Limits are randomly given in a homogeneous customer group Customers at different limit values are of same properties Usage amount and delinquency rate are only dependent on the limit, not the customer properties

The situation described by actual data The situation described by actual data The situation described by actual data The situation described by actual data

Higher limits are given to the customers with greater usage scale and better credit ratings Customers at different limit values have different properties Usage amount and delinquency rate are dependent on not only the limit but also the customer properties

To apply the optimum limit model to actual data, effects of the To apply the optimum limit model to actual data, effects of the To apply the optimum limit model to actual data, effects of the To apply the optimum limit model to actual data, effects of the customer properties on the customer properties on the customer properties on the customer properties on the limit need to be controlled limit need to be controlled limit need to be controlled limit need to be controlled

Customers are segmented into homogeneous customer groups In the customer groups, it must be verified that usage and delinquency depend only on the limit

8

• II. Determination of the Optimum Purchase Limit

Analysis Data

The data used for the study consisted of 21 million newly opened The data used for the study consisted of 21 million newly opened The data used for the study consisted of 21 million newly opened The data used for the study consisted of 21 million newly opened credit card credit card credit card credit card accounts with detailed customer properties accounts with detailed customer properties accounts with detailed customer properties accounts with detailed customer properties

The data was collected from 11 card companies by Korea Credit Bureau since Jan. 2006 Customers of the same properties were given different limits by different card companies due to differing limit policies

Frequently used variables in the study include: Frequently used variables in the study include: Frequently used variables in the study include: Frequently used variables in the study include:

UL : pre-signup monthly total usage amount of a customer customer customer customer (average over 3 months before signup) UN : post-signup total usage amount of a customer customer customer customer (average over 3 months after signup) uN : post-signup usage amount of a new account account account account (average over 3 months after signup) SL : pre-signup credit rating of a customer customer customer

• customer. The credit ratings lie on a scale of 1 to

10, with smaller numbers representing more creditworthy customers.

9

• II. Determination of the Optimum Purchase Limit

How customer properties affect the usage?

When pre When pre When pre When pre-

• signup usage amount is fixed, customer properties positively

signup usage amount is fixed, customer properties positively signup usage amount is fixed, customer properties positively signup usage amount is fixed, customer properties positively correlated with the limit are negatively correlated with the pos correlated with the limit are negatively correlated with the pos correlated with the limit are negatively correlated with the pos correlated with the limit are negatively correlated with the post t t t-

• signup usage

signup usage signup usage signup usage amount amount amount amount

Variable With no variables fixed With UL, SL fixed (for SL, only fix UL) UN Usage Rate SOW uN Limit UN Usage Rate SOW uN Limit UL +++

• +

++ Credit Rating(SL) 0+

• +++

0-

• ++

# of Cards Held +

• 0-

++ 0+

• +

Annual Income ++

• +

++ 0+

• 0-

0- +

( | ) ( ) ( | )

N N L

u f x X U U SOW x X = = ×

• SOW : Share Of Wallet, : Customer Properties

: Share Of Wallet, : Customer Properties : Share Of Wallet, : Customer Properties : Share Of Wallet, : Customer Properties Usage Rate = (# of customers who used the new card)/(# of new customers) = (# of customers who used the new card)/(# of new customers) = (# of customers who used the new card)/(# of new customers) = (# of customers who used the new card)/(# of new customers)

X

10

• II. Determination of the Optimum Purchase Limit

How customer properties affect the usage?

Customers with a large number of credit cards have good credit r Customers with a large number of credit cards have good credit r Customers with a large number of credit cards have good credit r Customers with a large number of credit cards have good credit ratings and atings and atings and atings and large pre large pre large pre large pre-

• signup usage amounts, so are given higher limits

signup usage amounts, so are given higher limits signup usage amounts, so are given higher limits signup usage amounts, so are given higher limits

But, usage rate and SOW decreases since their usage is distributed across many cards, yielding negative correlation between limit and post-signup usage amount

Giving higher limits to customers with more cards makes correlat Giving higher limits to customers with more cards makes correlat Giving higher limits to customers with more cards makes correlat Giving higher limits to customers with more cards makes correlation between ion between ion between ion between the usage amount and limits less strong the usage amount and limits less strong the usage amount and limits less strong the usage amount and limits less strong

But, its effect is negligible as shown by the dotted lines.

200 400 600 800 500 1000 1500 2000 2500 3000 4000 5000 500 1000 1500 2000 2500 3000 4000 5000 0% 20% 40% 60% 80% 1-2 3-5 6+ Total Post-Signup Usage Amount (USD) Percentage of New Accounts # of Cards Held=1-2 3-5 Purchase Limit (USD) 6+ 1-2 3-5 6+

11

• II. Determination of the Optimum Purchase Limit

Relation between the Limit and Usage Amount

If pre If pre If pre If pre-

• signup usage amount and credit rating are fixed, post

signup usage amount and credit rating are fixed, post signup usage amount and credit rating are fixed, post signup usage amount and credit rating are fixed, post-

• signup usage

signup usage signup usage signup usage amount is a function of the limit only amount is a function of the limit only amount is a function of the limit only amount is a function of the limit only The relation between the limit and usage amount from actual data The relation between the limit and usage amount from actual data The relation between the limit and usage amount from actual data The relation between the limit and usage amount from actual data agrees with agrees with agrees with agrees with the assumption of the optimum limit model the assumption of the optimum limit model the assumption of the optimum limit model the assumption of the optimum limit model

( | , )

N L L

u f x U S =

100 200 300 400 500 600 700 1 1000 2000 3000 1 1000 2000 3000 1 1000 2000 3000 Post-Signup Usage Amount (USD) Purchase Limit (USD) SL=6 SL=5 SL=7 Good Bad Good Bad Good Bad

12

• II. Determination of the Optimum Purchase Limit

Relation between the limit and delinquency rate

Even after fixing pre Even after fixing pre Even after fixing pre Even after fixing pre-

• signup usage amount and credit rating, account

signup usage amount and credit rating, account signup usage amount and credit rating, account signup usage amount and credit rating, account-

• based

based based based delinquency rate decreases as the limit increases. delinquency rate decreases as the limit increases. delinquency rate decreases as the limit increases. delinquency rate decreases as the limit increases.

It is possible to subdivide customers with the same credit rating according to creditworthiness. The dependence of the limit on creditworthiness cannot be completely removed by Credit Bureau data.

0% 1% 2% 3% 4% 500 1500 2500 4000 500 1500 2500 4000 500 1500 2500 4000 Account-Based Delinquency Rate 1 3 5 6 7 Total Purchase Limit (USD) U L=1-500 USD U L=1500-2000 USD Total SL=7 SL=6 SL=3 SL=5

13

• II. Determination of the Optimum Purchase Limit

Removing the dependence of the limit on creditworthiness

Account Account Account Account-

• based delinquency rate depends on the limit because the limit

based delinquency rate depends on the limit because the limit based delinquency rate depends on the limit because the limit based delinquency rate depends on the limit because the limit depends on creditworthiness depends on creditworthiness depends on creditworthiness depends on creditworthiness

where represents customer properties except pre where represents customer properties except pre where represents customer properties except pre where represents customer properties except pre-

• signup usage amount and credit rating

signup usage amount and credit rating signup usage amount and credit rating signup usage amount and credit rating

The dependence of the limit on creditworthiness can be removed b The dependence of the limit on creditworthiness can be removed b The dependence of the limit on creditworthiness can be removed b The dependence of the limit on creditworthiness can be removed by readjusting y readjusting y readjusting y readjusting

• dds at each limit value
• dds at each limit value
• dds at each limit value
• dds at each limit value

Resample customers of subgroup Ai so that sample odds( ) becomes the same as population odds( )

( )

2

( , ) ( )

c L L

D f S U x X ε = +

• (

, ) ( , )

G B G B i i i i i i i

SO A N N A N N PO ′ = → ×

/

G B i i i

SO N N = /

G B

PO N N =

2

X

14

• II. Determination of the Optimum Purchase Limit

After resampling, delinquency rate agrees with the assumption of the

• ptimum limit model

Account Account Account Account-

• based delinquency rate is independent of the limit

based delinquency rate is independent of the limit based delinquency rate is independent of the limit based delinquency rate is independent of the limit Balance Balance Balance Balance-

• based delinquency rate is a U

based delinquency rate is a U based delinquency rate is a U based delinquency rate is a U-

• shaped function of the limit

shaped function of the limit shaped function of the limit shaped function of the limit

0. 0% 0. 5% 1. 0% 1. 5% 2. 0% 2. 5% 50 100 150 200 250 300 400 500 50 100 150 200 250 300 400 500 0. 0% 0. 5% 1. 0% 1. 5% 2. 0% 2. 5% 1 3 5 6 7 Total Purchar Limit (USD) Before After SL=7 SL=6 SL=3 SL=5

15

• II. Determination of the Optimum Purchase Limit

After resampling for homogeneous customer group A, net profit per person for limit x is given by

NG , NB : Number of Good and Bad Customers : Number of Good and Bad Customers : Number of Good and Bad Customers : Number of Good and Bad Customers : Average usage amount per person for the first 12 months after : Average usage amount per person for the first 12 months after : Average usage amount per person for the first 12 months after : Average usage amount per person for the first 12 months after

• pening the new account
• pening the new account
• pening the new account
• pening the new account

r1: Merchant fee (=2.45%) : Merchant fee (=2.45%) : Merchant fee (=2.45%) : Merchant fee (=2.45%) r2: Financing interest rate (=0.46%) : Financing interest rate (=0.46%) : Financing interest rate (=0.46%) : Financing interest rate (=0.46%) AD: Average 2 : Average 2 : Average 2 : Average 2-

• period(30

period(30 period(30 period(30-

• 60 DPD) delinquent amount per person

60 DPD) delinquent amount per person 60 DPD) delinquent amount per person 60 DPD) delinquent amount per person R(2

• 7) : The roll

: The roll : The roll : The roll-

• rate at which the 2

rate at which the 2 rate at which the 2 rate at which the 2-

• period delinquent amount becomes a

period delinquent amount becomes a period delinquent amount becomes a period delinquent amount becomes a charge charge charge charge-

• off (=58%)
• ff (=58%)
• ff (=58%)
• ff (=58%)

12 12 1 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (2 7) ( ) ( ) ( ) ( ) ( )

G G B B B N M N M D B B

SO x SO x u x N x u x N x r r A x R N x PO PO P x SO x N x N x PO   × + × × × − − × → × ×     = + ×

12 12

,

G B N M N M

u u

16

• II. Determination of the Optimum Purchase Limit

Net profit function agrees with the result derived from the optimum limit model

The optimum limit that maximizes net profit is higher for custom The optimum limit that maximizes net profit is higher for custom The optimum limit that maximizes net profit is higher for custom The optimum limit that maximizes net profit is higher for customer groups with a er groups with a er groups with a er groups with a better credit rating and a larger pre better credit rating and a larger pre better credit rating and a larger pre better credit rating and a larger pre-

• signup usage amount

signup usage amount signup usage amount signup usage amount

• 40
• 20

20 40 60 80 1 3 5 7 1 3 5 7 1 3 5 7 1 3 5 7 1 3 5 7 1 3 5 7 Net Profit Per Person (USD) 4 5 6 7 Purchase Limit 1-500 500-1000 1000-1500 1500-2000 2000-2500 UL=0 6 5 S L=4 SL=7

17

• II. Determination of the Optimum Purchase Limit

Net profit function before and after resampling

Before resampling, net profit monotonically increases with the l Before resampling, net profit monotonically increases with the l Before resampling, net profit monotonically increases with the l Before resampling, net profit monotonically increases with the limit imit imit imit

Customers with different limit values are of different creditworthiness The monotonic increase is a result of the dependence of the limit on creditworthiness

After resampling, net profit decreases beyond the optimum limit After resampling, net profit decreases beyond the optimum limit After resampling, net profit decreases beyond the optimum limit After resampling, net profit decreases beyond the optimum limit

the dependence of the limit on creditworthiness has been removed

• 40
• 20

20 40 60 1 1000 2000 3000 5000 500 1500 2500 4000

• 40
• 20

20 40 60 4 5 6 7 Purchase Limit(USD) Before After 6 5 SL=4 SL=7

18

• II. Determination of the Optimum Purchase Limit

Subdividing customer groups using number of cards held increases efficiency of the optimum limit

For customer groups with small pre For customer groups with small pre For customer groups with small pre For customer groups with small pre-

• signup usage amount, the post

signup usage amount, the post signup usage amount, the post signup usage amount, the post-

• signup

signup signup signup usage amount increases as number of cards held increases, yieldi usage amount increases as number of cards held increases, yieldi usage amount increases as number of cards held increases, yieldi usage amount increases as number of cards held increases, yielding higher ng higher ng higher ng higher

• ptimum limit values
• ptimum limit values
• ptimum limit values
• ptimum limit values

For customer groups with small pre For customer groups with small pre For customer groups with small pre For customer groups with small pre-

• signup usage amount, SOW decreases as

signup usage amount, SOW decreases as signup usage amount, SOW decreases as signup usage amount, SOW decreases as number of cards held increases, yielding lower optimum limit val number of cards held increases, yielding lower optimum limit val number of cards held increases, yielding lower optimum limit val number of cards held increases, yielding lower optimum limit values ues ues ues

• 40
• 20

20 40 60 80 100 1 100 200 300 1 100 200 300 1 100 200 300 1 100 200 300 Net Profit Per Person (USD) 1-2 3-5 6+ Total Purchase Limit (USD) UL=1-500 S L=7 UL=1000-1500 SL=7 U L=1-500 SL=5 UL=1000-1500 S L=5 # of Cards=3-5 1-2 6+ 3-5 1-2 6+ 1-2 6+ 1-2 6+

19

• III. Summary and Conclusion

Establishment of the optimum limit model

Good and Bad customers respond to the limit in different manners Good and Bad customers respond to the limit in different manners Good and Bad customers respond to the limit in different manners Good and Bad customers respond to the limit in different manners

Usage functions are different

To apply the model to actual data, To apply the model to actual data, To apply the model to actual data, To apply the model to actual data,

Divide customers using credit rating and pre-signup usage amount Remove the remaining dependence of limit on the creditworthiness by resampling

Determination of the optimum limit from the net profit function

The optimum limit is higher for customers with good credit ratin The optimum limit is higher for customers with good credit ratin The optimum limit is higher for customers with good credit ratin The optimum limit is higher for customers with good credit ratings and large gs and large gs and large gs and large usage amounts usage amounts usage amounts usage amounts The optimum limit model can be used effectively to determine the The optimum limit model can be used effectively to determine the The optimum limit model can be used effectively to determine the The optimum limit model can be used effectively to determine the optimum limit

• ptimum limit
• ptimum limit
• ptimum limit

policy for mid policy for mid policy for mid policy for mid-

• range rated customers with a reasonable monthly usage amount

range rated customers with a reasonable monthly usage amount range rated customers with a reasonable monthly usage amount range rated customers with a reasonable monthly usage amount Dividing customers into more homogeneous groups yields a more ef Dividing customers into more homogeneous groups yields a more ef Dividing customers into more homogeneous groups yields a more ef Dividing customers into more homogeneous groups yields a more efficient ficient ficient ficient

• ptimum limit
• ptimum limit
• ptimum limit
• ptimum limit