How To Model A Credit Card From A Bank Card To A Creditcard

Transcription

1 An emprcal study for credt card approvals n the Greek bankng sector Mara Mavr George Ioannou Bergamo, Italy May 2004 Management Scences Laboratory Department of Management Scence & Technology Athens Unversty of Economcs & Busness

2 Contents Introducton - Motvaton Lterature Problem Descrpton Model Defnton & Methodology Results Model Evaluaton Valdaton Test Conclusons

3 Introducton The new economy n the nformaton socety creates a new busness envronment. What s dfferent s that more and more busness gets transacted n a computer-medated envronment. Transactons, bll payments, purchasng, reservatons (hotels, travel or cnema tckets) are done electroncally. A Credt Card s an electronc payment system that s used more than two decades. Durng the last two decades, credt cards have become one of the man ways for executng fnancal transactons. Credt cards consttute the largest part of ndvdual fnancng. Ther success s owed to the fact that after the ssue of the card and the determnaton of the credt lmt, the owner s free to charge and sometmes to overcharge whenever he wants the predefned lmt

4 Introducton Although there are few credt card networks - Vsa, MasterCard and Amerca Express are the three largest ones - the number of commercal banks offerng credt cards afflated wth these networks has been reported as roughly 4000 durng the 1980s (Ausubel, 1991) and 6,000 durng the 1990s (GAO, 1994). Thus, a wde range of credt cards are present n the market, ssued by a number of banks or by dfferent fnancal nsttutons. Managng credt cards s a complex busness. A factor that contrbutes to ths complexty s that credt card customers use ther cards for a number of non credt reasons, namely: payment convenence, smoothng of fnances payng regular blls, emergences and spontaneous spendng. The cost of usng a credt card nstead of money relates to a hgh nterest rate that must be pad at the end of each perod on balances debted to the card wthn prevous perod.

5 Motvaton The subject of ths study s the nvestgaton of the reasonng behnd the acceptance or rejecton of a customer for the allowance of a credt card. The applcant may be relable and the card wll be ssued to hm/her or unrelable and hs/her applcaton wll be rejected. Several studes have examned the crtera accordng to whch a bank characterzes an ndvdual as a proper cardholder whle another one as napproprate. On the other hand, a seres of studes have been undertaken n order to evaluate factors that play a crucal role n someone s decson to adopt or gnore ths electronc payment system, whle some others examne the phenomenon credt cards from a fnancal perspectve.

6 Lterature Revew Medan and Davos (1994) used factor analyss to dentfy the ssues that nfluence someone to apply for a credt card. Convenence, securty, scales of economy by credt card usage and status of users are reasons that are recognzed as the most popular wthn ths study. Mn Q and Sha Yang (2003) used neural networks n order to predct customers behavor wth respect to credt cards. Factors such as convenence securty, relablty were examned and the correlaton among them was determned. Nash and Snkey (1997) assumed that the market for credt cards has been the subject of attenton because of hgh profts earned on credt cards. They tred to estmate rsk-return profles for banks credt-card offerng and to explore the role of ntangble assets n determnng resale premums on credt card recevables

7 Lterature Revew Zopounds and Doumpos (2002) used the mult-group dscrmnaton approach embedded n mult-crtera analyss. Ths method s based on an teratve bnary segmentaton procedure. In ther two-stage procedure, frst they dscrmnated the accepted credt card applcatons from all others, whle n the second they dscrmnated the applcatons requrng further nvestgaton from the rejected ones

8 Approach The purpose of ths study s to develop a procedure for determnng the factors whch affect a bank s decson for ssung or not a credt card. The approach, through ts mathematcal model, ams at estmatng or forecastng the bank s management team decson of approval or rejecton of an applcaton form. The analyss s based on real data of applcaton-forms from a leadng European bank, whose recent strategy has focused on busness growth through sales and the expanson of credt cards ssuance. The proposed approach dffers from prevous researches snce t uses a generalzed lnear model through whch factors that nfluence more or less bank s decson to accept or reject a credt card applcaton, are dentfed. The approach s also applcable to any other fnancal products that follow the same scheme of applcaton examnaton evaluaton

9 Problem Descrpton Consder a sample of m customers of a bank or of another fnancal nsttuton that request a credt card and have fled-out an applcaton form. Informaton about demographc data and each ndvdual s bankng actvty (debts, number and types of cards they hold, etc) are provded wthn the applcaton. Demographcal and market data are known and can be consdered n order to forecast the potental of each ndvdual s applcaton form. The goal s to fnd the mportant factors accordng to whch an applcant s judged as approprate for holdng a credt card and to predct the percentage of customers that fulfll the bank s crtera for the ssuance of the credt card..

10 Problem Defnton (I) Predcted Varable Suppose that we have n customers. We ntroduce a bnary varable Y whch fals nto one of two categores such as yes or no 1 f the applcant s judged as approprate Y = 0f the applcant s judged as napproprate As Y has bnomal dstrbuton we specfy probabltes P (Y =1)=p P (Y =0)=1-p f the applcaton form s approved and f not. Thus 1 Y = 0 f p >=0.5 f p <0.5

11 Problem Defnton (II) Predcted Varable Y s the number of customers who were judged as approprate n our sample ΥY=Y has bnomal dstrbuton Y~ bnomal (n,p) wth probablty densty functon n y f(y;p) = p (1-p) y The probablty functon can be rewrtten n the form of exponental famly n-y f(y; p) = exp y logp y log(1 p) + n log(1 p) + n log y

12 Problem Defnton (III) Factoral Varables Our dependent varable Y s nfluenced by quanttatve and qualtatve varables Demographc data (age, educaton, monthly ncome) Indvdual s s Bankng actvty (bank s accounts,fnancal credblty, own property) We model them by X j where j J, J={1,2 k} s the set of factors wth cardnalty to k for the -th customer I where I={1,2 n} Some of them are contnuous varables, some others are bnary and fnally Some of them are ordnal (we model them by usng dummy varables too

13 Problem Defnton (III) Varable Defnton Varable Choce (1=yes, 0 otherwse) Demographc Characterstc x 1 = Gender (1=male, 0= female) x 2 = Age (ordnal) x 3 = Educaton (ordnal) x 4 = Marred or not (1= marred, 0=otherwse) Economc Data x 5 = Monthly ncome (ordnal) x 6 = How long s workng n the same work (1= a year or more, 0=otherwse) x 7 = bad fnancal credblty (1=bankruptcy, 0= otherwse) x 8 = property (1=has property, 0= otherwse)

14 Problem Defnton (IV) The Logstc Regresson Model The probablty p for the -th customer to use onlne bankng servces s logt (p )= log (p /(1-p ))= β 0 +β 1 * x 1 +β 2,* x β κ * x k (1) The correlaton between the k explanatory varables x 1, x 2,,x k and decson of usng or not onlne servces of the -th customer s modeled by the lnear logstc model. p exp( β + 0 β1x ) 1 βkx k = 1+exp( β + β x β x ) k n k = β j j x j e n p = 1+ e n

15 Problem Defnton (V) The Logstc Regresson Model In order to ft the lnear logstc model to our gven set of data the m +1 unknown parameters β ο, β 1,...,β m have frst to be estmated. These parameters are estmated usng the method of maxmum lkelhood. The lkelhood functon s gven by L(β) n n = p (1-p ) =1 y y n -y The lkelhood functon depends on the unknown success probabltes that n turn depend on the βs and so lkelhood functon can be regarded as a functon of β

16 Problem Defnton (VI) The Logstc Regresson Model The problem now s to obtan those values whch maxmze L (β) but s usually more convenent to maxmze the logarthm of lkelhood log L(β) log L(β)= n log + ylogp +(n -y ) log(1-p ) y = n p log + ylog +n log(1-p ) y (1 p ) = n n log + y n - n log(1 + ) y e where k j=0 j n = β x j and x o =1 for all values of

17 Demographcal data of our sample Gender Educaton % % Male 68.4 Less than hgh school 18.8 Female 31.6 Hgh School 53.4 Age Unversty 27.7 < Monthly Income (n euros) < > Percentage employed at the Same work wthn last year 91 > Famly Status Holdng Other Cards Marred 65.4 Yes 70.6 Not Marred 34.6 No 29.4 Property Ownershp Bad Fnancal Credblty Yes 63.8 Yes 20.4 No 36.2 No 79.6

18 Results of the Logstc Analyss We used the Lkelhood Foreword method for calculatng the beta coeffcents βj Accordng to the Lkelhood Foreword Method the analyss begns wth a model whch ncludes only a constant and then adds sngle factors nto the model based on the crteron of the score statstc (the varable wth the most sgnfcant score statstc s added to the model. The Uj score statstc for the xj varable s gven by U = j n ( y - E ( y )) Var ( y ) where E(y) s the mean value of the y varable. The analyss proceeds untl none of the remanng varables have a sgnfcant score statstc. The cut-off pont for sgnfcance s ( ) E y x j =1 n

19 Results of the Logstc Analyss Examnaton of the varables score statstc leads to the concluson that 2 steps are necessary n order to enter all varables that sgnfcantly mprove the model. The score statstc of the varables whch are not ncluded n the equaton n steps 1 and 2 respectvely are shown n Table. Note that df s the degrees of freedom of the varables and Sg s the rate of sgnfcance of the varables

20 Results of the Logstc Analyss Step 1Varables Score df Sg. Gender,003 1,959 Age( <=27),001 1,971 Age (27-40)) 3,121 1,077 Age(41-50) 2,069 1,150 Age (>=51)),378 1,539 Educaton,038 1,845 Famly Status,018 1,893 Income(<=590) 3,456 1,063 Income( ),575 1,448 Income(>1100) 1,622 1,203 Duraton n the Same work,306 1,580 Property 4,605 1,032 Educaton (less than hgh,319 1,572 school) Educaton (Hgh School) 1,125 1,289 Educaton (Unversty) 2,253 1,133 Overall Statstcs 15,307 14,357

21 Results of the Logstc Analyss Step 2Gender,160 1,690 Age( <=27),208 1,648 Age (27-40)) 2,457 1,117 Age(41-50) 1,398 1,237 Age (>=51)) 1,118 1,290 Famly Status,018 1,893 Income (<=590) 3,456 1,063 Income( ),575 1,448 Income(>1100) 1,622 1,203 Duraton at the Same work,425 1,515 Educaton (less than hgh,319 1,572 school) Educaton (Hgh School) 1,125 1,289 Educaton (Unversty) 2,253 1,133 Overall Statstcs 10,855 13,668

22 Results of the Logstc Analyss In order to evaluate the βj coeffcents for the varables that are determned as sgnfcant from the above test we used the method of the maxmum lkelhood. L depends on the unknown success probabltes whch n turn depend on the βj coeffcents, so the lkelhood functon can be regarded as a functon of βj. The problem now s to obtan the values whch maxmze L or equvalently log L The dervatves of ths log-lkelhood functon wth respect to the unknown Parameters β are o, β7, β8 where j=0,7,8. logl m m = n n y x n (1+ ) =1 j =1 x j e e β j -1

23 Results of the Logstc Analyss Table presents βj and the changes of the βj coeffcents, due to the stepwse method we used. The values of the βj coeffcent that we wll use n equaton (1) are those resultng from step 2 of the Lkelhood forward method. βj represents the change n the logt of y (logt(y)=log(p/1-p)) (approve or rejecton of an applcaton form) assocated wth a one unt change n factoral varable xj (j=1,2). β j Standard Error j Step 1 Bad Fnancal Credblty,866,386 Constant -,571,347 Step 2 Bad Fnancal Credblty,887,391 Property -,702,321 Constant -,334,365

24 Results of the Logstc Analyss Consequently the proposed logstc regresson model s: logt (p)= log (p/(1-p))= -0,702* x7, + 0,887* x8, -0,334 (2) p = exp(-0,702 * x + 0,887 * x - 0,334) 7, 8, 1+ exp(-0,702 * x + 0,887 * x - 0,334) 7, 8, p represents the probablty of the ndvdual to be judged as approprate for fnancng to hm a credt card, whle the explanatory varables x 7 and x 8 represent overall measures of factors regardng property and hs fnancal poston, respectvely. Snce our study dd not explctly ntroduce monthly ncome and duraton n the same work, the effects of these factors are parameterzed n terms of the logstc constant.

25 Results of the Logstc Analyss 1. The logstc result for β7 ndcates that ndvdual s property s a sgnfcant determnant of bank s management fnal decson to approve or to reject an applcaton for credt card. An ncrease n ths factoral coeffcent would ncreas the wllngness of the bank to approve the applcaton. 2.Accordng to the logstc result for β8 the credblty of an ndvdual s also a sgnfcant factor. As we mentoned above someone s bad fnancal poston s a prohbtve factor for credt card ssuance. If an ndvdual has debts then hs applcaton form s characterzed as napproprate for approval. A decrease n ths coeffcent would affect radcally the bank s fnal decson. 3.The rate of sgnfcance of varables x1 = Gender and x4 = Famly status are and respectvely (cut value s 0.05). These large values of sgnfcance denote that these two varables won t have any contrbuton n the estmaton of probablty p for -th customer.

26 Results of the Logstc Analyss 4. Concernng the varables x2 (age) conveys dfferent nformaton assocate wth the four score statstc values. The rate of sgnfcance for the second category (.e. ndvduals who aged between 27 and 40 years old ) s Ths rate of sgnfcance s good, ndcatng that the target group for credt card applcants s between ndvduals who aged from 27 to 40 years old, somethng that s confrmed n practce as well. The sgnfcance rates of levels and more than 50 years old whch are and respectvely are also well. On the contrary, the rate of sgnfcance of the frst category (ndvduals aged untl 27 years old) s small, somethng whch s very common, as many people n ths age are stll students and they don t work, they don t have any property.

27 Results of the Logstc Analyss 5. The varable educaton has also 3 levels. The rate of sgnfcance of ndvduals who fulflled unversty s whch sgnfes that ths varable could also be sgnfcant f we take nto account addtonal varables such as professon. 6. The varable monthly ncome has also 3 levels. The rate of sgnfcance of level 3 (monthly ncome more than 1100 euros) s whch sgnfes that monthly ncome s an mportant factor for the bank s fnal decson. 7. The constant coeffcent βο s consstent wth our constructon, under whch monthly ncome and applcants, who aged untl to 40 years old, are parameterzed n the logstc constant.

28 Model s evaluaton A measure for the sgnfcance of the coeffcents β7 and β8 of the varables x7 and x8 that represent property and Bad Fnancal Credblty s gven by Wald statstc test. 2 β (Waldj=). j St.Error j Hgh values of Waldj n combnaton wth low number of the degrees of freedom (df) ndcate hgh sgnfcance. The Wald test, the sgnfcance of the test (Sg) and the degrees of freedom of each varable are presented n next Table. Wald j df Sg. Step 1 Bad Fnancal Credblty 5,038 1,025 Constant 2,703 1,100 Step 2 Bad Fnancal Credblty 5,134 1,023 Property 4,778 1,029 Constant,837 1,360

29 Model s evaluaton We examned the multcollnearty ssue. Table provdes the correlaton matrx of the varables n the logstc equaton. At each step we determne the sgnfcant varables and we defne the correlaton between them. From the fnal results of the Step 2 we conclude that the value of the correlaton between the two fnal explanatory varables x7 and x8. s low (-0,52). Constant Bad Fnancal Credblty Property Step 1 Constant 1,000 -,806 Bad Fnancal -,899 1,000 X Credblty 7 Step 2 Constant 1,000 -,849 -,273 Bad Fnancal -,849 1,000 -,052 X 7 Credblty Property x 8 -,273 -,052 1,000

30 Model s evaluaton Devance s denoted by D and s gven by Lp D = -2log p a L = -2 logl - logl a where Lp s the value of equaton (2) when we used the estmated coeffcents (-0.334,-0.702, respectvely) and La s the value of equaton (8) when we used the estmated coeffcents of all varables x1-x8 (sgnfcant and no sgnfcant) that are ncluded n the model. Devance measures the extent to whch our model fts the data devates from the model whch nclude all varables Smaller values mean that the model fts the data better. The value of devance decreases from Step1 to Step2, ndcatng that the varables that entered n the logstc equaton n each step explaned the probablty of the predcted varable better (From 242,237 n step 1 to 237,383 n Step 2). Step -2 Log lkelhood 1 242, ,383

31 Model s evaluaton The most common approach for determnng the accuracy of our model s Hosmer and Lemeshow rate m ( y - n p ) d 2 X = d =1 n p ( 1 - p ) P <0.5 (Rejecton) P >=0.5 (Approval) Total Observed Expected Observed Expected Step , , , , , , , , Table ndcates the value of Hosmer and Lemeshow test. The sgnfcance of the test s Step X 2 HL df Sg. 2,114 2,944 2

32 Valdaton Test In order to evaluate the accuracy of the logstc regresson model we used a sample of 100 applcants of the same bank. Informaton about these applcants s presented below Gender Educaton Demographc Data % % Male 68.4 Less than hgh school 11.1 Female 31.6 Hgh School 60 Age Unversty 13.3 < Monthly Income (n euros) < > Percentage employed at the Same work wthn last year 91 > Famly Status Holdng Other Cards Marred 54.3 Yes 75 Not Marred 45.6 No 25 Property Ownershp Bad Fnancal Credblty Yes 66.7 Yes 22.2 No 33.3 No 77.7

33 Valdaton Test Calculatng the estmated probablty p of each applcant we conclude that the estmated percentage of the applcants who wll receve a credt card s 46.6% whle the estmated percentage of the rejected applcaton forms s 53.3%. The observed percentages of approvals and rejectons are 55% and 44% respectvely. Table 11 presents the absolute error of these two predctons. Predcted Percentage Observed Percentage Absolute Error predcton Accepted applcaton forms 46.6% 55% 8,4% Rejected applcaton forms 53.3% 44% 9.3% of

34 Valdaton Test Comparng the actual data and the predcted probabltes for the accepted applcaton forms, we calculated the percentage of the correctness of the estmaton of approvals (46.6%) whch s 71,4%. The correspondng percentage of correctness of the rejected applcaton forms s 72.72%. As a consequence the correctness of our proposed model n the valdated sample s 72.07%. Table 12 summarzes the nformaton about the correctness of our predctons Predcted Percentage Correctness of the predcted percentage Accepted applcaton form 46.6% 71.4% Rejected applcaton form 53.3% 72.72% Model s overall correctness of estmated percentages 72.07%

35 Comparng wth LP In the lnear model the dependent varable that presents the approval or the rejecton of an applcaton form for the -th applcant s y*. y* s a contnuous varable that takes values between 0 and 1 only. We consder that f y y * * [ ) [ ] 0, 0.5 the applcaton form s rejected 0.5,1 the applcaton form s approved The ndependents varables are x1-x8, as they are determned n Table 2. Through lnear regresson technque, all varables x1-x8. are examned for ther sgnfcance (ther contrbuton n the explanaton of y*). The same varables x8 = Property and x7 = Bad Fnancal Credblty are dentfed as sgnfcant. The coeffcents of x7 and x8 are -0,184 and 0,186 respectvely.

36 Comparng wth LP Smple regresson model s: y*= -0,172 x7 +0,202 x8 +0,533 R 2 adjusted R 2 Model Accuracy,052,042 Calculatng y* for each of the 100 applcants, we estmate the percentage of approvals and rejectons of applcaton forms. Table 14 presents the predcted and the actual percentages of approvals and rejectons and the absolute error of these two predctons. Predcted Observed Absolute Percentage Percentage Error of predcton Accepted applcaton forms 93.3% 55% 38,3% Rejected applcaton forms 6.6% 44% 37.4%

37 Comparng wth LP Comparng the actual data and the predcted values of y for the accepted applcaton forms, we calculated the percentage of the correctness of the estmaton of approvals (93.3%) whch s 48%. Table summarzes the nformaton about the correctness of our predctons Predcted Percentage Correctness of the predcted percentage Accepted applcaton form 93.3% 48% Rejected applcaton form 6.6% 33.3% Model s overall correctness of estmated percentages 40.65%

38 Conclusons We have provded a choce model that estmates probabltes of approval or rejecton of an applcaton form of a credt card. We used logstc regresson analyss and we determned the contrbuton of a number of factors demographcal and fnancal- n the calculaton of the estmated probabltes. In order to valdate the accuracy of the model we looked for the Hosmer- Lemeshow rate and the Devance value. We estmated all the essental coeffcents of the factoral varables n the equaton of the probablty and we checked the correctness of the proposed technque by calculatng the probablte of a new sample of applcants where we estmated the percentage of the forecastng approves and the forecastng rejects Accordng to the results of a valdaton test we clam that the model s accurate. Comparng the results by logstc analyss wth those by smple regresson analys we led to the concluson that the logstc model s more precse