Study On the Determination of 신용카드 신용카드 신용카드 적정한도에 신용카드 적정한도에 적정한도에 적정한도에 관한 관한 관한 관한 연구 연구 연구 연구 the Optimum Credit Card Limit KCB 연구소 연구소 연구소 연구소 2011. 2 Yonho Song, Ph.D. Korea Credit Bureau yhsong@koreacb.com 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
> Content I. Optimum Limit Model II. Determination of Optimum Purchase Limit III. Summary and Conclusion 1
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 ( ) ( ) × × − × × u x N g u x N l ( ) = P x N G w N 0 G B ( ) ( ) = × − × g u x u x * N w ( ) . . . Max P x w r t x N G , N B : Number of good and bad customers : Number of good and bad customers : Number of good and bad customers : Number of good and bad customers u G ( x ), u B ( 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 w 0 = = 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) = N G /N B : Population Odds w* = = = : Population Odds : Population Odds : Population Odds 2
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 ” customers who are delinquent Bad Bad 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 with a similar usage amount and credit rating. with a similar usage amount and credit rating. customer group customer group 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 A i , 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 ( ) ( ) du x w du x w * 0 0 B i i = × = × u i * * dx dx w w i i 3
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 Average Usage Amount per person Revenue/Cost function per person Average Usage Amount per person Revenue/Cost function per person Revenue Revenue Revenue Revenue Usage Usage Usage Usage Cost from Cost from Cost from Cost from /Cost /Cost /Cost /Cost Amount Amount Amount Amount Bad Bad Bad Bad Customers Customers Customers Customers Bad Bad Bad Bad Customer Customer Customer Customer Revenue Revenue Revenue Revenue from Good from Good from Good from Good Customers Customers Customers Customers Good Good Good Good Customer Customer Customer Customer Optimum Limit Optimum Limit Optimum Limit Optimum Limit Limit Limit Limit Limit Limit Limit Limit Limit 4
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 ount and ount and ount 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 worse credit rating A group with higher usage amount A group with worse credit rating Revenue Revenue Revenue Revenue Revenue Revenue Revenue Revenue B B B B /Cost /Cost /Cost /Cost /Cost /Cost /Cost /Cost B B A A B B A A A A A A Cost Cost Cost Cost Revenue Revenue Revenue Revenue Cost Cost Cost Cost Revenue Revenue Revenue Revenue Limit Limit Limit Limit Limit Limit Limit Limit 5
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 B 1 N ( ) = = D x c B G 1 * + w + N N Balance- Balance -based delinquency rate based delinquency rate Balance Balance - - based delinquency rate based delinquency rate Defined as (delinquent amount)/(total balance) U-shaped and has a minimum value near the optimum limit B B ( ) ( ) 1 b x u x ( ) = ≈ = D x b B G B G G B ( ) ( ) ( ) ( ) 1 ( ) / ( ) + + + b x b x u x u x u x u x 6
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 7
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