An empirical study for credit card approvals in the Greek banking … · 2014-12-18 · An empirical study for credit card approvals in the Greek banking sector Maria Mavri George

An empirical study for credit card approvals in the Greek banking sector

Maria Mavri

George Ioannou

Maria MavriMaria Mavri

George George IoannouIoannou

Management Sciences Laboratory

Department of Management Science & Technology

Athens University of Economics & Business

Bergamo, Italy 17-21 May 2004

• Introduction - Motivation

• Literature

• Problem Description

• Model Definition & Methodology

• Results

• Model Evaluation

• Validation Test

• Conclusions

Contents

Introduction

The new economy in the information society creates a new business environment. What is different is that more and more business gets transacted in a computer-mediated environment. Transactions, bill payments, purchasing, reservations (hotels, travel or cinema tickets) are done electronically. A Credit Card is an electronic payment systemthat is used more than two decades.

During the last two decades, credit cards have become one of the main ways for executing financial transactions. Credit cards constitute the largest part of individual financing. Their success is owed to the fact thatafter the issue of the card and the determination of the credit limit, the owner is free to charge and sometimes to overcharge whenever he wants the predefined limit

Introduction

Although there are few credit card networks - Visa, MasterCard and AmericanExpress are the three largest ones - the number of commercial banksoffering credit cards affiliated with these networks has been reported as roughly 4000 during the 1980s (Ausubel, 1991) and 6,000 during the 1990s (GAO, 1994). Thus, a wide range of credit cards are present in themarket, issued by a number of banks or by different financial institutions.

Managing credit cards is a complex business. A factor that contributes to this complexity is that credit card customers use their cards for a numberof non credit reasons, namely: payment convenience, smoothing of finances,paying regular bills, emergencies and spontaneous spending.

The cost of using a credit card instead of money relates to a high interestrate that must be paid at the end of each period on balances debited to the card within previous period.

Motivation

The subject of this study is the investigation of the reasoning behind theacceptance or rejection of a customer for the allowance of a credit card. The applicant may be reliable and the card will be issued to him/her or unreliable and his/her application will be rejected. Several studies have examined the criteria according to which a bank characterizes an individualas a proper cardholder while another one as inappropriate.

On the other hand, a series of studies have been undertaken in order to evaluate factors that play a crucial role in someone’s decision to adopt or ignore this electronic payment system, while some others examine thephenomenon “credit cards” from a financial perspective.

Literature Review

� Meidan and Davos (1994) used factor analysis to identify the issues thatinfluence someone to apply for a credit card. Convenience, security, scales of economy by credit card usage and status of users are reasons that are recognized as the most popular within this study.

� Min Qi and Sha Yang (2003) used neural networks in order to predict customers’ behavior with respect to credit cards. Factors such as convenience,security, reliability were examined and the correlation among them wasdetermined.

� Nash and Sinkey (1997) assumed that the market for credit cards has beenthe subject of attention because of high profits earned on credit cards. Theytried to estimate risk-return profiles for banks credit-card offering and toexplore the role of intangible assets in determining resale premiums on credit card receivables

Literature Review

� Zopounidis and Doumpos (2002) used the multi-group discrimination approach embedded in multi-criteria analysis. This method is based on aniterative binary segmentation procedure. In their two-stage procedure, first they discriminated the accepted credit card applications from all others, while in the second they discriminated the applications requiring furtherinvestigation from the rejected ones

Approach

� The purpose of this study is to develop a procedure for determining the factors which affect a bank’s decision for issuing or not a credit card.

� The approach, through its mathematical model, aims at estimating or forecasting the bank’s management team decision of approval or rejection of an application form.

� The analysis is based on real data of application-forms from a leadingEuropean bank, whose recent strategy has focused on business growth through sales and the expansion of credit cards issuance.

� The proposed approach differs from previous researches since it uses a generalized linear model through which factors that influence more or less bank’s decision to accept or reject a credit card application, are identified.

� The approach is also applicable to any other financial products that follow the same scheme of application – examination – evaluation –

Problem Description

Consider a sample of m customers of a bank or of another financial institution that request a credit card and have filed-out an application form.Information about demographic data and each individual’s banking activity(debts, number and types of cards they hold, etc) are provided within theapplication.

Demographical and market data are known and can be considered in orderto forecast the potential of each individual’s application form.

The goal is to find the important factors according to which an applicantis judged as appropriate for holding a credit card and to predict the percentage of customers that fulfill the bank’s criteria for the issuanceof the credit card.

.

Problem Definition (I)Predicted Variable

Suppose that we have n customers. We introduce a binary variable Yi whichfails into one of two categories such as “yes” or “no”

As Yi has binomial distribution we specify probabilities

P (Yi=1)=pi if the application form is approved and

P (Yi =0)=1-pi if not.

Thus

i

1 if the applicant is judged as appropriateY =

0if the applicant is judged as inappropriate

i

i

i

if p >=0.51Y =

if p <0.5 0

Problem Definition (II)Predicted Variable

Y is the number of customers who were judged as appropriate in our sample

Y has binomial distributionbinomial distribution Y~ binomial (n,p) with probability density function

y n-ynf(y;p) = p (1-p)

y

+−+−−=

y

nlogp)log(1 np)log(1y logpy expp)f(y;

The probability function can be rewritten in the form of exponential family

∑i

nΥ

i=1Y=

Problem Definition (III)Factorial Variables

Our dependent variable Y is influenced by quantitative and qualitative variables

Demographic dataDemographic data(age, education, monthly income)

IndividualIndividual’’s Banking activity s Banking activity (bank’s accounts,financial credibility, own property)We model them by

Xji where j∈ J, J={1,2… k} is the set of factors

with cardinality to k for the i-th customer i∈I where I={1,2… n}

Some of them are continuous variables, some others are binary and finally Some of them are ordinal (we model them by using dummy variables too

Problem Definition (III)Variable Definition

Variable

Choice (1=yes, 0 otherwise)

Demographic Characteristic

x1i = Gender (1=male, 0= female) x2i = Age (ordinal) x3i = Education (ordinal)

x4i = Married or not (1= married, 0=otherwise)

Economic Data

x5i = Monthly income (ordinal) x6i = How long is working in the same work (1= a year or more, 0=otherwise)

x7i = bad financial credibility (1=bankruptcy, 0= otherwise)

x8i = property (1=has property, 0= otherwise)

Problem Definition (IV)The Logistic Regression Model

The probability pi for the i-th customer to use online banking services is

(1)

The correlation between the k –explanatory variables x1i, x2i, …,xki and decision of using or not online services of the i-th customer is modeled by the linearlogistic model.

logit (pi)= log (pi/(1-pi))= β0 +β1* x1i +β2,* x2i+...+ βκ* xki

1i ki0 1 k

1i ki0 1 k

i

exp( +...+ )β β βx x=p

1+exp( +...+ )β β βx x

++

∑i jij j= βn x

i

i

n

i n

ep =

1+e

Problem Definition (V)The Logistic Regression Model

In order to fit the linear logistic model to our given set of data the m +1 unknown parameters βο, β1,...,βm have first to be estimated.

These parameters are estimated using the method of maximum likelihoodmethod of maximum likelihood.

The likelihood functionlikelihood function is given by L(β)

∏ i i in

y n -yi

i iii=1

n= p (1-p )

y

The likelihood function depends on the unknown success probabilities that in turn depend on the βs and so likelihood function can be regarded as afunction of β

Problem Definition (VI)The Logistic Regression Model

The problem now is to obtain those values which maximize L (β) but is usually more convenient to maximize the logarithm of likelihood log L(logarithm of likelihood log L(ββ))

log L(β)=

=

=

where and xoi=1 for all values of i

∑i

i i ii i

i i

n log + y log logp +(n y ) (1-p )

y-

(1 )

−

∑i i

i ii i i

i

n p log + y log log+n (1-p )

y p

)

∑ ii n

i ii i

i

n log + y n - log (1 +n ey

∑k

i jij=0 j= βn x

Demographical data of our sample

% %

Gender Education

Male 68.4 Less than high school 18.8

Female 31.6 High School 53.4

Age University 27.7

<27 4.2

27-40 59.1 Monthly Income (in euros)

41-50 21.5 <590 31.4

>50 15.2 590-1110 48.2

Percentage employed at the

Same work within last year 91 >1100 20.4

Family Status Holding Other Cards

Married 65.4 Yes 70.6

Not Married 34.6 No 29.4

Property Ownership Bad Financial Credibility

Yes 63.8 Yes 20.4

No 36.2 No 79.6

Results of the Logistic Analysis

We used the Likelihood Foreword method for calculating the beta coefficients βjAccording to the Likelihood Foreword Method the analysis begins with a model which includes only a constant and then adds single factors into the model basedon the criterion of the score statistic (the variable with the most significant score statistic is added to the model.

The Uj –score statistic for the xji variable is given by

where E(yi) is the mean value of the yi variable.

The analysis proceeds until none of the remaining variables have a significant score statistic. The cut-off point for significance is 0.05.

( )( )( )

( )ni i i

j jii=1 i i

y - E y E yU = x

Var y n

∂ ∂

∑

�Examination of the variables’ score statistic leads to the conclusion that 2 steps are necessary in order to enter all variables that significantly improve the model.

�The score statistic of the variables which are not included in the equation in steps 1 and 2 respectively are shown in Table.

�Note that “df” is the degrees of freedom of the variables and “Sig” is the rateof significance of the variables



Step 1Variables Score df Sig.

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 Education ,038 1 ,845

Family Status ,018 1 ,893 Income(<=590) 3,456 1 ,063

Income(590-1100) ,575 1 ,448 Income(>1100) 1,622 1 ,203

Duration in the Same work ,306 1 ,580 Property 4,605 1 ,032

Education (less than high school)

,319 1 ,572

Education (High School) 1,125 1 ,289 Education (University) 2,253 1 ,133

Overall Statistics 15,307 14 ,357


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

Family Status ,018 1 ,893 Income (<=590) 3,456 1 ,063

Income(590-1100) ,575 1 ,448 Income(>1100) 1,622 1 ,203

Duration at the Same work ,425 1 ,515 Education (less than high

school)

,319 1 ,572

Education (High School) 1,125 1 ,289

Education (University) 2,253 1 ,133

Overall Statistics 10,855 13 ,668


In order to evaluate the βj coefficients for the variables that are determined as significant from the above test we used the method of the maximum likelihood.

L depends on the unknown success probabilities which in turn depend on the βjcoefficients, so the likelihood function can be regarded as a function of βj. The problem now is to obtain the values which maximize L or equivalently log L

The derivatives of this log-likelihood function with respect to the unknown Parameters are

where j=0,7,8.

∂ −∑ ∑∂

-1

i ii ij

m m

jii=1 i=1i

n nlogL= (1+ )ny x e exβ

j

o , 7 , 8β β β


Table presents βj and the changes of the βj coefficients, due to the stepwise method we used. The values of the βj coefficient that we will use in equation (1) are those resulting from step 2 of the Likelihood forward method. βj represents the change in the logit of yi (logit(yi)=log(p/1-p)) (approve or rejection of an application form) associated with a one unit change in factorial variable xji (j=1,2).

βj Standard Errorj

Step 1 Bad Financial Credibility ,866 ,386 Constant -,571 ,347

Step 2 Bad Financial Credibility ,887 ,391

Property -,702 ,321 Constant -,334 ,365


Consequently the proposed logistic regression model is:logit (pi)=

log (pi/(1-pi))= -0,702* x7,i + 0,887* x8, i -0,334 (2)

7,i 8, i

7,i 8, ii

exp(-0,702 * x + 0,887 * x - 0,334)=p

1+ exp(-0,702 * x + 0,887 * x - 0,334)

pi represents the probability of the individual i to be judged as appropriate for financing to him a credit card, while the explanatory variables x7i and x8i

represent overall measures of factors regarding property and his financial position, respectively. Since our study did not explicitly introduce monthlyincome and duration in the same work, the effects of these factors are parameterized in terms of the logistic constant.


1. The logistic result for β7 indicates that individual’s property is a significant determinant of bank’s management final decision to approve or to reject an application for credit card. An increase in this factorial coefficient would increasethe willingness of the bank to approve the application.

2.According to the logistic result for β8 the credibility of an individual is also a significant factor. As we mentioned above someone’s bad financial position is a prohibitive factor for credit card issuance. If an individual has debts then his application form is characterized as inappropriate for approval. A decrease in this coefficient would affect radically the bank’s final decision.

3.The rate of significance of variables x1i = Gender and x4i = Family status are0.690 and 0.893 respectively (cut value is 0.05). These large values of significance denote that these two variables won’t have any contribution in the estimation of probability pi for i-th customer.


4. Concerning the variables x2i (age) conveys different information associate with the four score statistic values. The rate of significance for the second category(i.e. individuals who aged between 27 and 40 years old ) is 0.117. This rate of significance is good, indicating that the target group for credit card applicants is between individuals who aged from 27 to 40 years old, somethingthat is confirmed in practice as well. The significance rates of levels “41-50”and “more than 50 years old” which are 0.237 and 0.290 respectively are also well. On the contrary, the rate of significance of the first category (individuals aged until 27 years old) is small, something which is very common, as many people in this age are still students and they don’t work, they don’t have any property.


5. The variable “education” has also 3 levels. The rate of significance ofindividuals who fulfilled university is 0.133 which signifies that this variable

could also be significant if we take into account additional variables such as

profession.

6. The variable “monthly income” has also 3 levels. The rate of significance oflevel 3 (monthly income more than 1100 euros) is 0.203 which signifies that monthly income is an important factor for the bank’s final decision.

7. The constant coefficient βο is consistent with our construction, under which monthly income and applicants, who aged until to 40 years old, areparameterized in the logistic constant.

Model’s evaluation

A measure for the significance of the coefficients β7 and β8 of the variables x7iand x8i that represent “property” and “Bad Financial Credibility” is given by Waldstatistic test.

(Waldj=).

High values of Waldj in combination with low number of the degrees of freedom(df) indicate high significance. The Wald test, the significance of the test (Sig) and the degrees of freedom of each variable are presented in next Table.

2β

j

St.Errorj

Waldj df Sig.

Step 1 Bad Financial Credibility 5,038 1 ,025 Constant 2,703 1 ,100

Step 2 Bad Financial Credibility 5,134 1 ,023

Property 4,778 1 ,029 Constant ,837 1 ,360


Constant Bad Financial

Credibility

Property

Step 1 Constant 1,000 -,806

Bad Financial Credibility

X7i -,899 1,000

Step 2 Constant 1,000 -,849 -,273Bad Financial

Credibility X7i

-,849 1,000 -,052

Property x8i -,273 -,052 1,000

We examined the multicollinearity issue. Table provides the correlation matrix of the variables in the logistic equation. At each step we determine the significant variables and we define the correlation between them. From the final results of the Step 2 we conclude that the value of the correlation betweenthe two final explanatory variables x7i and x8i. is low (-0,52).


Deviance is denoted by D and is given by

where Lp is the value of equation (2) when we used the estimated coefficients(-0.334,-0.702, 0.887 respectively) and La is the value of equation (8) when we used the estimated coefficients of all variables x1i-x8i (significant andno significant) that are included in the model.

Deviance measures the extent to which our model fits the data deviates fromthe model which include all variablesSmaller values mean that the model fits the data better. The value of deviance decreases from Step1 to Step2, indicating that the variables that entered in thelogistic equation in each step explained the probability of the predicted variablebetter (From 242,237 in step 1 to 237,383 in Step 2).

p

p aa

LD = -2log = -2 logL - logLL

Step -2 Log likelihood

1 242,237 2 237,383


The most common approach for determining the accuracy of our model is Hosmer and Lemeshow rate ( )

( )∑d i

i

mi

d

i i

2

2 i

i=1

y - n pX =

n p 1 - p

Pi <0.5 (Rejection) Pi>=0.5 (Approval) Total

Observed Expected Observed Expected

Step 2 1 10 9,596 3 3,404 13 2 13 13,403 10 9,597 23

3 27 27,403 24 23,597 51 4 34 33,598 58 58,402 92

Table indicates the value of Hosmer and Lemeshow test. The significance of thetest is 0.944.

Step X2HL df Sig.

2 ,114 2 ,944

Validation TestIn order to evaluate the accuracy of the logistic regression model we used a sample of 100 applicants of the same bank. Information about these applicants is presented below

Demographic Data

% %

Gender Education

Male 68.4 Less than high school 11.1

Female 31.6 High School 60

Age University 13.3

<27 24.4

27-40 44.4 Monthly Income (in euros)

41-50 15.5 <590 33.3

>50 17.7 590-1110 37.8

Percentage employed at the

Same work within last year 91 >1100 31.1

Family Status Holding Other Cards

Married 54.3 Yes 75

Not Married 45.6 No 25

Property Ownership Bad Financial Credibility

Yes 66.7 Yes 22.2

No 33.3 No 77.7

Validation Test

Calculating the estimated probability pi of each applicant we conclude that theestimated percentage of the applicants who will receive a credit card is 46.6%while the estimated percentage of the rejected application forms is 53.3%.

The observed percentages of approvals and rejections are 55% and 44% respectively. Table 11 presents the absolute error of these two predictions.

Predicted Percentage

Observed Percentage

Absolute

Error of prediction

Accepted application forms 46.6% 55% 8,4%

Rejected application forms 53.3% 44% 9.3%

Validation Test

Comparing the actual data and the predicted probabilities for the accepted application forms, we calculated the percentage of the correctness of theestimation of approvals (46.6%) which is 71,4%. The corresponding percentageof correctness of the rejected application forms is 72.72%. As a consequence the correctness of our proposed model in the validated sampleis 72.07%. Table 12 summarizes the information about the correctness of our predictions


Correctness of the

predicted percentage

Accepted application form 46.6% 71.4% Rejected application form 53.3% 72.72%

Model’s overall correctness of estimated percentages

72.07%

Comparing with LP

In the linear model the dependent variable that presents the approval or the rejection of an application form for the i-th applicant is yi*. yi* is a continuous variable that takes values between 0 and 1 only. We consider that if

[ )[ ]

∈

∈

0,0.5 the application form is rejected

0.5,1 the application form is approved

*

i

*

i

y

y

The independents variables are x1i-x8i , as they are determined in Table 2. Through linear regression technique, all variables x1i-x8i. are examined for theirsignificance (their contribution in the explanation of yi*).

The same variables x8i =“Property’ and x7i =“Bad Financial Credibility” areidentified as significant. The coefficients of x7i and x8i are -0,184 and 0,186 respectively.

Comparing with LP

R 2 adjusted R2

Model Accuracy ,052 ,042

Simple regression model is:

yiyi*= *= --0,172 x7i +0,202 x8i +0,5330,172 x7i +0,202 x8i +0,533

Calculating yi* for each of the 100 applicants, we estimate the percentage of approvals and rejections of application forms. Table 14 presents the predicted and the actual percentages of approvals and rejections and the absolute error ofthese two predictions.


Observed Percentage

Absolute

Error of prediction

Accepted application forms 93.3% 55% 38,3%

Rejected application forms 6.6% 44% 37.4%

Comparing with LP

Comparing the actual data and the predicted values of yi for the accepted application forms, we calculated the percentage of the correctness of the estimation of approvals (93.3%) which is 48%.

Table summarizes the information about the correctness of our predictions


Correctness of the

predicted percentage

Accepted application form 93.3% 48% Rejected application form 6.6% 33.3%

Model’s overall correctness of estimated

percentages

40.65%

Conclusions

•We have provided a choice model that estimates probabilities of approval or rejection of an application form of a credit card. We used logistic regression analysis and we determined the contribution of a number of factors –demographical and financial- in the calculation of the estimated probabilities.

•In order to validate the accuracy of the model we looked for the Hosmer-Lemeshow rate and the Deviance value. We estimated all the essential coefficients of the factorial variables in the equation of the probability and we checked the correctness of the proposed technique by calculating the probabilitiesof a new sample of applicants where we estimated the percentage of theforecasting approves and the forecasting rejects

•According to the results of a validation test we claim that the model is accurate. Comparing the results by logistic analysis with those by simple regression analysis we led to the conclusion that the logistic model is more precise

Documents

An empirical study for credit card approvals in the Greek banking … · 2014-12-18 · An empirical study for credit card approvals in the Greek banking sector Maria Mavri George