Annals of Operations Research 74(1997)159 – 171

159

A two-stage least cost credit scoring model ★ William V. Gehrlein and Bret J. Wagner Department of Business Administration, University of Delaware, Newark, DE 19716, USA E-mail: [email protected]

A least cost two-stage credit scoring model is developed and evaluated. The model is two-stage in the sense reflected by existing practice. That is, based on preliminary information, a decision is made to grant credit, deny credit, or to seek additional information before making a decision. If additional information is sought, the decision goes to the second stage, where a final decision to grant or deny credit is made. The model is based on an integer programming formulation that attempts to minimize the total combined cost of granting credit to likely defaulters, denying credit to likely payers, and for obtaining all of the information that is required to make the decisions. Keywords: credit scoring, integer programming, discriminant analysis, two stage, cost minimization.

1.

Introduction

Credit scoring models are mechanisms for determining whether or not credit should be extended to applicants. These decisions are based on information that is obtained about applicants. Chandler and Parker [5] list typical applicant information as: • applicant’s age, • time at currentyprevious residence, • time at currentyprevious job, • housing status, • occupation group, • income, • number of dependants, • phone at residence, • banking relationship, • debt ratio, • co-applicant information, • credit references. ★

This research was supported by a grant from the Financial Institutions Research and Education Center of the University of Deleware.

© J.C. Baltzer AG, Science Publishers

160

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

A simple version of a credit scoring model will assign weights to some numerical measures of the relevant items of applicant information to obtain a numerical weighted sum. A prespecified cutoff value determines whether or not the assessed value of the weighted sum for a particular loan applicant warrants the approval of the loan being applied for. Most procedures for determining the weights to be used for credit scoring models result from relying on statistical discriminant analysis. More sophisticated statistically-based procedures have also been developed for evaluating simple credit scoring models. For example, Boyes et al. [3] use probit analysis to evaluate the likelihood that an applicant will default if the loan is approved. Models used for credit scoring can also have other applications in finance. Wynn [15] describes how these models can be used to set appropriate credit limits for individuals applying for a revolving line of credit. Also, Wynn [16] describes how analysis of this type can be used to establish the rate of interest that should be charged on small business loans to reflect the degree of risk of default for applicants. There is debate concerning the wisdom of using statistically derived credit scoring models, as opposed to using the judgment of loan officers, to analyze loan applications. Chalos [4] performed an empirical study to find that a statistical credit scoring model would be expected to outperform an individual loan officer, but that a loan review committee would be expected to outperform the scoring model. However, an excessive amount of time would be required if a committee must evaluate a large number of applications [4]. Alexander [1] examined applicants for loans that were approved by a credit scoring model, and compared them to applicants who received loans when a loan officer had overridden a recommendation of rejection by the credit scoring model. Results showed delinquency rates that were seven times greater for overrides than for recommendations by the credit scoring model. Edmister [6] presents empirical results to indicate that both judgmental decisions and the use of credit scoring models can both be very accurate, and showed that the incorporation of both approaches into making a decision can be effective in improving loan loss accuracy. There is increasing evidence to suggest that the use of credit scoring models can be a very effective tool in evaluating loan applicants. There are a number of reasons for implementing credit scoring models beyond the issue of their accuracy. Customer service is an issue; auto loan applications can typically be analyzed in five to six minutes by a credit scoring model [1]. Overstreet and Kemp [13] suggest the use of credit scoring models to evaluate and compare the consistency of loan officers’ credit evaluations. The consistency of evaluation by loan officers is an important issue, with potential legal implications. 2.

Two-stage procedures and cost implications

Mehta [12] suggested a sequential decision process in granting credit, where there is a cost for each piece of information obtained on the firm requesting credit. A decision tree was developed based on the expected cost, which was based on the

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

161

probability of firm default. Two-stage procedures for evaluating loan applications have been suggested, where in the first stage a credit scoring model uses basic application information to divide applicants into three groups: loan approval, loan rejection, and undecided. The undecided group is then examined more thoroughly in a second stage in which additional information, in the form of credit bureau reports, is used to make a final decision [13]. Credit bureau reports have been shown to substantially increase the accuracy of credit scoring models [5]. There is, of course, additional cost incurred with obtaining the credit bureau information. The studies mentioned to this point have primarily been concerned with the predictive ability of credit scoring models in terms of their propensity to accurately detect potential defaulters. There is an obvious cost involved with granting a loan to an applicant who will ultimately default. However, there is also an opportunity cost associated with denying loans to applicants who would have repaid them. And, there are the costs involved with obtaining information to make the determination as to whether or not the loan should be made. Our ultimate goal should be to find a credit scoring model that will minimize the sum of all of these costs. A number of researchers mention the notion of striving for a credit scoring model that will maximize profit, or minimize cost [1, 3, 5]. Gehrlein and Gempesaw [8] develop an integer programming formulation to obtain a simple credit scoring model to minimize total cost. Our purpose in this paper is to obtain a model to minimize total cost for a two-stage credit scoring model. The single-stage model will be presented before the two-stage model, since it is used for comparison. 3.

The single-stage model

The single-stage integer programming formulation is based on a set of n observations on previous applicants. These observations would include a list of the numerical values for each of the k attributes recorded on the initial application. We also require the knowledge of whether or not the applicant defaulted. For applicant i, the Aij denote the numerical value of attribute j for 1 ≤ j ≤ k. Let wj denote the weight to be assigned to attribute j and F(i) be the resulting credit score, where F (i ) =

k

∑ w j Aij .

(1)

j =1

A cutoff value Xc is used to determine whether the applicant should be accepted or rejected. The applicant is accepted if F(i) > Xc and rejected if F(i) < Xc . Defining D as the set of applicants who will default and P as the set of applicants who will pay, the single-stage model is Minimize

Cdp

∑ Ii + C pd ∑ Ii

i ∈D

i ∈P

(2)

162

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

subject to

k

F (i ) =

∑ w j Aij ,

(3)

j =1

F (i ) + MI i ≥ X c + e, F (i ) − MI i ≤ X c − e,

∀i ∈ P,

∀i ∈ D,

(4) (5)

where Cdp is the cost associated with classifying a defaulter as a payer and Cpd is the cost of classifying a payer as a defaulter, M is a large number and e is a small number. Banks and Abad [2] discuss considerations that must be given to the relative sizes of M and e in formulations of this type. The binary variable Ii takes on the value zero if the classification is made correctly and one if the classification is made incorrectly. Since Cpd and Cdp are constant for all applicants, the model would seem to be most appropriate for a situation like determining whether or not credit card applications, for a fixed line of credit, should be approved. Since defaulters would likely default after reaching the approved line of credit, the loss Cdp would be fairly consistent for all defaulters. Similarly, Cpd would be associated with the discounted foregone profits over the lifetime of the credit card, and similar arguments could be made that Cpd is relatively consistent over applicants. 4.

A least cost two-stage model

For the two-stage model, there are k attributes denoted by Aij , with 1 ≤ i ≤ n and 1 ≤ j ≤ k, for the n applicants at the first stage of classification and k* attributes denoted by Bij that would be obtained at the second stage of the decision process. Two credit scores are calculated. The first score, F1 (i), is used in the first stage to determine whether to grant credit, deny credit or obtain additional information, and is defined as

F1 (i ) =

k

∑ w j Aij .

(6)

j =1

If F1 (i) ≥ Xp , then credit is granted. If F1 (i) ≤ Xd , credit is denied and if Xd < F1 (i) < Xp , additional information is requested and a second credit score, F2 (i), is calculated as

F2 (i ) =

k

∑ u j Aij

j =1

+

k*

∑ υ j Bij .

( 7)

j =1

If F2 (i) ≥ Xss , credit is granted, otherwise it is denied. Thus, the information from the first stage is retained, but the weights that are associated with the various factors are allowed to change in the second stage. To determine the weights wj , uj and υ j , both Aij and Bij are required for all applicants in the initial data set, even if the decision is made in the first stage of the process.

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

163

The integer programming formulation begins with a set of four constraints for applicant i: (8) F1 (i ) + MI1i ≥ X p , F1 (i ) − MI 2i ≤ X d ,

with

(9)

F1 (i ) − M ( 2 − I1i − I 2i ) ≤ X p − e,

(10)

F1 (i ) + M ( 2 − I1i − I 2i ) ≥ X d + e,

(11)

X p − X d ≥ 2 e,

(12 )

where I 1i and I 2i are binary variables with value 0 or 1. Given the set of constraints in (8) through (11), applicant i will be approved in this first stage if I 1i = 0 and I 2i = 1. Applicant i will be denied credit in this first stage if I 1i = 1 and I 2i = 0, and the applicant will go on to the second stage for evaluation if I 1i = I 2i = 1. It is not possible to have I 1i = 0 and I 2i = 0 simultaneously with these constraints. Costs of improperly classifying applicants in the first stage can be determined in the following manner. Suppose applicant i was a defaulter. The cost Cdp of classifying a defaulter as a payer in the first stage is only incurred if I 1i = 0. If applicant i is a payer, then a cost Cpd of classifying a payer as a defaulter in the first stage is only incurred when I 2i = 0. The cost of misclassifying applicants in the first stage is therefore given by First Stage Misclassification Cost = Cdp

∑ (1 − I1i ) + Cpd ∑ (1 − I 2i ).

i ∈D

(13)

i ∈P

All applicants will be evaluated in the first stage, so the cost of evaluation at that stage is fixed, and is therefore not significant in finding a least cost credit scoring model. There is an incremental cost Css associated with obtaining the information to be used in the second stage. The cost of evaluation for applicants going to the second stage is given by n

Second Stage Evaluation Cost = Css ∑ ( I1i + I 2i − 1).

(14 )

i =1

This cost function works because constraints (8) through (11) make it impossible to have I1i + I2i = 0, a decision was made in the first stage when I1i + I2i = 1 and the second stage will be used only when I1i + I2i = 2. It remains to establish the cutoff value Xss to be used in the second stage of the evaluation. Applicant i will be categorized as a defaulter during the second stage of evaluation if I1i + I2i = 2 (so that a second stage evaluation takes place) and F2 (i) ≤ Xss . For each applicant i ∈D, we add a constraint of the form

F2 (i ) − MI 3i − M ( 2 − I1i − I 2i ) ≤ X ss .

(15)

164

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

As above, I 3i is a binary variable with values 0 or 1. This constraint is met trivially if I 1i + I 2i = 1, which indicates that a classification as a payer or defaulter was made in the first stage. A decision will be made correctly in the second stage if I 1i + I 2i = 2 and I 3i = 0. For each applicant i ∈P, we write a constraint of the form F2 (i ) + MI 3i + M ( 2 − I1i − I 2i ) ≥ X ss + e.

(16 )

As above, applicant i will be properly categorized as a payer in the second stage if I 1i + I 2i = 2 and I 3i = 0, and a solution can always be found with I 3i = 0 if a classification as a payer or defaulter is made in the first stage. The cost of misclassifications in the second stage is given by Second Stage Misclassification Cost = Cdp

∑ I3i

i ∈D

+ C pd

∑ I3i .

(17)

i ∈P

Again, the formulation for the second stage misclassification cost will work, because a solution with I 3i = 0 can always be found trivially if a classification is made for observation i in the first stage of classification. The cost terms given in (13), (14) and (17) can be combined and simplified. The complete model is Minimize

∑ [Cdp I3i + Css I2i + (Css − Cdp ) I1i ]

i ∈D

+

∑ [Cpd I3i + Css I1i + (Css − Cpd ) I2i ]

(18)

i ∈P

subject to

F1 (i ) =

k

∑ wj Aij ,

(19)

j =1

F2 (i ) =

k

k*

j =1

j =1

∑ uj Aij + ∑ υ j Bij ,

(20)

F1 (i ) + MI1i ≥ X p ,

(21)

F1 (i ) + MI2i ≤ Xd ,

(22)

F1 (i ) − M (2 − I1i − I2i ) ≤ X p − e,

(23)

F1 (i ) + M (2 − I1i − I2i ) ≥ Xd + e,

(24)

X p − Xd ≥ 2e,

(25)

F2 (i ) − MI3i − M (2 − I1i − I2i ) ≤ Xss ,

∀i ∈ D,

(26)

F2 (i ) + MI3i + M (2 − I1i − I2i ) ≥ Xss + e,

∀i ∈ P.

(27)

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

5.

165

Evaluation of the model

The two-stage model was evaluated by comparing it to the single-stage model, using the data in the appendix, which comes from Johnson and Wickern [9]. The data set represents financial information for 46 firms on four indexes from Moody’s Industrial Manuals regarding firms that went bankrupt or remained solvent during a two-year interval following the measurement of the attributes. Two attributes – cash flowytotal debt and net incomeytotal assets – were arbitrarily selected as the attributes that were available during the first-stage evaluation. Current assetsycurrent liabilities and current assetsynet sales were assumed to be available for the second-stage evaluation at additional cost. Leonard and Banks [10] suggest that five “good” accounts are needed to offset the losses from one “bad” account, so costs were assigned in this ratio (Cpd = $100 and Cdp = $500), as well as two other ratios (Cpd = $300 and Cdp = $300; Cpd = $500 and Cdp = $100) for each of three costs for obtaining the second-stage attributes: Css = $10, $25 and $50. Three models were evaluated: the two-stage model, the single-stage model using the two first-stage attributes, and the single-stage model with all four attributes. The models were solved using IBM’s Optimization Subroutine Library (OSL) release 2 on a Sun Microsystems SPARC-center 2000 consisting of eight 50 MHz T1 SuperSPARC CPUs with 2 MB of Supercache. Problems were submitted to OSL using GAMS version 2.25.073 requiring 30 MB of core memory. Problem solution times ranged from 246 to 2231 CPU seconds. Table 1 presents the results for Css = $10. When Cpd = $100 and Cdp = $500, that is, when the costs are biased against granting credit to those that will end up defaulting, the two-stage model makes no mistakes in granting credit (7 approved in the first stage, 14 in the second stage). When the costs are biased against denying credit to those who will pay (Cpd = $500 and Cdp = $100), the two-stage model makes no mistakes in denying credit (4 denied in the first stage, 14 denied in the second stage). When the costs are equal (C pd = C dp = $300), the two-stage model makes only four decisions in the first stage and waits to the second stage to decide on the remaining 42 applicants. It appears that when the costs are biased in one direction, then more decisions can be made in the first stage, with a slight increase in errors to save the costs of obtaining information for the second-stage analysis. For example, when Cpd = $100 and Cdp = $500, only 17 applicants are evaluated in the second stage versus 42 applicants when Cpd = Cdp = $300. The number of correct classifications with the two-stage model when Css = $10 is always greater than the number of correct classifications for either single-stage model. Even when all four attributes are used in the single-stage model, the two-stage model makes more correct classifications using the same information. As would be expected, the two-stage model can make some classifications in the first stage, remove those applicants from the pool and then calculate a new set of weights (ui and υ i ) that perform better at categorizing the remaining pool of applicants than can be done when all applicants are considered in a single stage. The single-stage model with four

166

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

Table 1 Comparison of single- and two-stage models, Css = 10. Cpd = 100 Cdp = 500 Two stage

Initially denied Correctly denied

1 1

4 4

7 7

3 3

25 22

17 17 3 14

42 41 19 23

17 17 14 3

4

1

3

Denied Correctly denied

30 20

21 18

11 11

Approved Correctly approved

16 15

25 22

35 25

Total incorrectly classified

11

6

10

Denied Correctly denied

26 21

20 19

15 15

Approved Correctly approved

20 20

26 24

31 25

5

3

6

Additional information used Correct in second stage Denied in second stage Approved in second stage Total incorrectly classified

Single stage, 4 attributes

Cpd = 500 Cdp = 100

22 18

Initially approved Correctly approved

Single stage, 2 attributes

Cpd = 300 Cdp = 300

Total incorrectly classified

attributes makes no mistakes in approving applicants when Cpd = $100 and Cdp = $500 (biased against granting credit incorrectly) and no mistakes in denying credit when Cpd = $500 and Cdp = $100 (biased against denying credit incorrectly). Table 2 shows the results for the case when Css = $25. For the case when Cpd = $100 and Cdp = $500, the results are identical to the two-stage model for Css = $10 for the same misclassification costs. In fact, the results are not changed for Cpd = $100 and Cdp = $500 when Css = $50. Obviously, there is no group of applicants that are evaluated in the second stage that can be moved to a first stage such that the classification accuracy is sufficient to preserve the information cost savings that would result. When Cpd = Cdp = $300, and when Cpd = $500 and Cdp = $100, the number of applicant decisions deferred until the second stage drops by 28 and 3, respectively, from the case when Css = $10. For Cpd = Cdp = $300, two additional applicants are approved incorrectly (all in the first stage) which increases the classification costs by $600, but this is offset by

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

167

Table 2 Comparison of single- and two-stage models, Css = 25. Cpd = 100 Cdp = 500 Two stage

Cpd = 300 Cdp = 300

Cpd = 500 Cdp = 100

22 18

11 11

11 11

7 7

21 18

21 18

17 17 3 14

14 14 7 7

14 14 7 7

4

3

3

Denied Correctly denied

30 20

21 18

11 11

Approved Correctly approved

16 15

25 22

35 25

Total incorrectly classified

11

6

10

Denied Correctly denied

26 21

20 19

15 15

Approved Correctly approved

20 20

26 24

31 25

5

3

6

Initially denied Correctly denied Initially approved Correctly approved Additional information used Correct in second stage Denied in second stage Approved in second stage Total incorrectly classified

Single stage, 2 attributes

Single stage, 4 attributes

Total incorrectly classified

the information cost savings of $700 that results from the reduced information requirements. The single-stage model results are identical to those in table 1 (and table 3), since neither model considers information costs. In all cases, with Css = $25, the two-stage model has the same (one case) or fewer incorrect classifications than either single-stage model. As with the case when Css = $10, the two-stage model with four attributes makes no mistakes in approving applicants when Cpd = $100 and Cdp = $500 (biased against granting credit incorrectly) and no mistakes in denying credit when Cpd = $500 and Cdp = $100 (biased against denying credit incorrectly). Table 3 presents the results for Css = $50. The decisions are identical to the case when Css = $25 when Cpd = Cdp = $300. For the case where Cpd = $500 and Cdp = $100, three fewer applicant decisions are made in the second stage than with Css = $25. Even so, all denials are made correctly. Again, the two-stage model makes the same or fewer incorrect decisions than the single-stage model.

168

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

Table 3 Comparison of single- and two-stage models, Css = 50. Cpd = 100 Cdp = 500 Two stage

Cpd = 300 Cdp = 300

Cpd = 500 Cdp = 100

22 18

11 11

11 11

7 7

21 18

24 20

17 17 3 14

14 14 7 7

11 11 6 5

4

3

4

Denied Correctly denied

30 20

21 18

11 11

Approved Correctly approved

16 15

25 22

35 25

Total incorrectly classified

11

6

10

Denied Correctly denied

26 21

20 19

15 15

Approved Correctly approved

20 20

26 24

31 25

5

3

6

Initially denied Correctly denied Initially approved Correctly approved Additional information used Correct in second stage Denied in second stage Approved in second stage Total incorrectly classified

Single stage, 2 attributes

Single stage, 4 attributes

Total incorrectly classified

Table 4 presents a cost comparison for the single- and two-stage models. For the single-stage model with two attributes, the cost of additional information has no impact on costs. When four attributes are used, the additional information cost Css is expended for all applicants, thus it varies with Css . When C ss = $25 or $50, the additional information cost for the single-stage model with four attributes makes it significantly more costly than the single-stage model with two attributes. When Css = $10, the four-attribute single-stage model produces lower costs than the singlestage model with two attributes, except when Cpd = $500 and Cdp = $100. The twostage model is superior in all cases to either single-stage model. When Css = $10, the two-stage model has costs that range from 40% to 69% lower than the best singlestage model. When Css = $25, the two-stage model is 30 – 45% lower in cost than the best single-stage model, which in this case is the two-attribute model. When Css = $50, the cost reductions are less significant, ranging from 5 to 16%.

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

169

Table 4 Cost comparison of single- and two-stage models. Cost of additional information

Model

10

Two stage

570

420

440

Single stage, 4 attributes

960

1,360

1,060

Two stage

825

1,250

650

Single stage, 4 attributes

1,650

2,050

1,750

Two stage

1,250

1,600

950

Single stage, 4 attributes

2,800

3,100

2,900

Single stage, 2 attributes

1,500

1,800

1,000

25

50

NyA

6.

Cpd = 100 Cdp = 500

Cpd = 300 Cdp = 300

Cpd = 500 Cdp = 100

Conclusions

The two-stage least cost credit scoring model provided clearly superior performance to the single-stage credit scoring models in the experiments. When some credit information is obtained at additional cost, the two-stage model can reduce total costs significantly by making the easy decisions without obtaining the additional information. Future research on this model could focus on evaluating its performance with actual credit application data sets. Most likely, in practice, the data sets that would be used to fit the model would be larger and would contain a smaller proportion of defaulters. Appendix: Financial data for firm bancruptcy

Obs. 1 2 3

Cash flowy total debt 0.4485 – 0.5633 0.0643

Net incomey total assets

Current assetsy current liabilities

Current assetsy net sales

Outcome after two years

– 0.4106 – 0.3114 0.0156

1.0865 1.5134 1.0077

0.4526 0.1642 0.3978

Bankrupt Bankrupt Bankrupt . . . continues

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

170

Financial data for firm bankruptcy (continued)

Obs.

Cash flowy total debt

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

– 0.0721 – 0.1002 – 0.1421 0.0351 – 0.0653 0.0724 – 0.1353 – 0.2298 0.0713 0.0109 – 0.2777 0.1454 0.3703 – 0.0757 0.0451 0.0115 0.1227 – 0.2843 0.5135 0.0769 0.3776 0.1933 0.3248 0.3132 0.1184 – 0.0173 0.2169 0.1703 0.1460 – 0.0985 0.1398 0.1379 0.1486 0.1633 0.2907 0.5383 – 0.3330 0.4785 0.5603 0.2029 0.4746 0.1661 0.5808

Net incomey total assets

Current assetsy current liabilities

Current assetsy net sales

Outcome after two years

– 0.0930 – 0.0917 – 0.0651 0.0147 – 0.0566 – 0.0076 – 0.1433 – 0.2961 0.0205 0.0011 – 0.2316 0.0500 0.1098 – 0.0821 0.0263 – 0.0032 0.1055 – 0.2703 0.1001 0.0195 0.1075 0.0473 0.0718 0.0511 0.0499 0.0233 0.0779 0.0695 0.0518 – 0.0123 – 0.0312 0.0728 0.0564 0.0486 0.0597 0.1064 – 0.0854 0.0910 0.1112 0.0792 0.1380 0.0351 0.0371

1.4544 1.5644 0.7066 1.5046 1.3737 1.3723 1.4196 0.3310 1.3124 2.1495 1.1918 1.8762 1.9941 1.5077 1.6756 1.2602 1.1434 1.2722 2.4871 2.0069 3.2651 2.2506 4.2401 4.4500 2.5210 2.0538 2.3489 1.7973 2.1692 2.5029 0.4611 2.6123 2.2347 2.3080 1.8381 2.3293 3.0124 1.2444 4.2918 1.9936 2.9166 2.4527 5.0594

0.2589 0.6683 0.2794 0.7080 0.4032 0.3361 0.4347 0.1824 0.2497 0.6969 0.6601 0.2723 0.3828 0.4215 0.9494 0.6038 0.1655 0.5128 0.5368 0.5304 0.3548 0.3309 0.6279 0.6862 0.6925 0.3484 0.3970 0.5174 0.5500 0.5778 0.2643 0.5151 0.5563 0.1978 0.3786 0.4835 0.4730 0.1847 0.4443 0.3018 0.4487 0.1370 0.1268

Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Bankrupt Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent Solvent

From R.A. Johnson and D.W. Wickern, Applied Multivariate Statistical Analysis, PrenticeHall, Englewood Cliffs, NJ, 1988, pp. 525 – 526.

W.V. Gehrlein, B.J. Wagner y A two-stage least cost credit scoring model

171

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

W. Alexander, What’s the score?, ABA Banking Journal (August 1989)58 – 63. W.J. Banks and P.L. Abad, An efficient optimal solution algorithm for the classification problem, Decision Sciences 22(1991)1008 – 1023. W.J. Boyes, D.L. Hoffman and S.A. Low, An econometric analysis of the bank credit scoring problem, Journal of Econometrics 40(1989)3 – 14. P. Chalos, The superior performance of loan review committees, The Journal of Commercial Bank Lending (December 1985)60 – 66. G.G. Chandler and L.E. Parker, The predictive value of credit bureau reports, Journal of Retail Banking 11(Winter 1989)47 – 54). R.O. Edmister, Combining human credit analysis and numerical credit scoring for business failure prediction, Akron Business and Economic Review 19(Fall 1988)6 –14. W.V. Gehrlein, General mathematical programming formulations for the statistical classification problem, Operations Research Letters 5(1986)299– 303. W.V. Gehrlein and V. Gempesaw, A cost minimization approach to classification, Proceedings of the National Decision Sciences Institute Meeting, November, 1991, pp. 1162 – 1163. R.A. Johnson and D.W. Wickern, Applied Multivariate Statistical Analysis, Prentice-Hall, 1988, pp. 525 – 526. K.J. Leonard and W.J. Banks, Automating the credit decision process, Journal of Retail Banking 16(1994)39 – 44. C. Loucopoulos and R. Pavur, A modified form of a general mathematical programming formulation for the statistical classification problem, Mimeograph, 1991. D. Mehta, The formulation of credit policy models, Management Science 15(1968)B30 – B-50. G.A. Overstreet and R.S. Kemp, Managing control in credit scoring systems, Journal of Retail Banking 8(1986)79 – 86. R.C. Soltysik and P.R. Yarnold, Determining a lower bound for M in multivariable optimal discriminant analysis, Mimeograph, 1991. A. Wynn, Credit scoring for mobile communications, Credit Management (1991)32. A. Wynn, A fresh look at the old duo, Credit Management (1991)36 – 37.