Redesigning Ratings: Assessing the Discriminatory Power of Credit Scores under Censoring

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1 Redeigning Rating: Aeing the Dicriminatory Power of Credit Score under Cenoring Holger Kraft, Gerald Kroiandt, Marlene Müller Fraunhofer Intitut für Techno- und Wirtchaftmathematik (ITWM) Thi verion: June 29, 2004 Abtract: In the light of Bael II, redeigning rating ytem ha been becoming an important iue for bank and other financial intitution. The available data bae for thi tak typically contain only the accepted credit applicant and i thu cenored. To evaluate exiting and alternative rating ytem, we would actually need the full data bae of all pat credit applicant. In thi paper we dicu how to ae the performance of credit rating under the aumption that for credit data only a part of the default and non-default i oberved. The paper invetigate criteria that are baed on the difference of the core ditribution under default and non-default uch a the accuracy ratio. We how how to etimate bound for thee criteria in the uual ituation that the bank torage only data of the accepted credit applicant. Keyword: credit rating, credit core, dicriminatory power, ample election, Gini coefficient, accuracy ratio JEL Claification: G21 Correponding author: Marlene Müller, Fraunhofer ITWM, Potfach 3049, D Kaierlautern, Germany, Phone: , marlene.mueller@itwm.fraunhofer.de

2 1 Introduction A bank which wihe to decide whether a credit applicant will obtain a credit or not ha to ae if the applicant will be able to redeem the credit. Among other criteria, the bank need an etimate of the probability that the applicant will default prior to the maturity of the credit. At thi tep, a rating of the applicant i a valuable deciion upport. The idea of a rating ytem i to identify criteria which eparate the good from the bad creditor, uch a equity and liquidity ratio or factor concerning the capital tructure of a firm. In a more formal ene a rating correpond to a gue of the default probability of the credit. Obviouly, the quetion arie how a bank can identify a ufficient number of elective criteria and, epecially, what electivity and dicriminatory power mean in thi context. A particular problem of credit coring i that default and non-default are only oberved for a ubample of applicant. In the following ection we try to make a firt tep to a rigorou treatment of thi ubject which i rarely addreed in literature. Apart from the theoretical attractivene thi iue i of highly practical importance. Thi i due to the fact that the Bael Committee on Banking Superviion i working on a New Capital Accord (Bael II) where default rik adjuted capital requirement hall be etablihed. In thi context rating and the deign of rating play an important role. Clearly, the committee want the bank to identify factor which have an ability to differentiate rik [and] have predictive and dicriminatory power (Banking Committee on Banking Superviion; 2001, p. 50). Conequently, bank are forced to ae the quality of their rating ytem and to optimize them with repect to the above-mentioned requirement. The available data bae for thi tak typically contain only the accepted credit applicant (the debitor of the bank). Data entrie for the rejected credit applicant do often not exit. Thi lead to a non-repreentative data bae which give biaed etimate of all relevant parameter if thi cenoring i not appropriately handled. To evaluate a new rating ytem, e.g. by comparing it with the exiting rating ytem, we would actually need the full data bae of all pat credit applicant. One poibility would be to introduce a model which allow u to extrapolate on the data for the rejected applicant (Greene; 1998; Feelder; 2000). For intance, Ah and Meeter (2002) and Crook and Banaik (2004) report that uch bia correction typically have a maller effect than neceary. Therefore, we ue an approach which avoid any pecification of a model for the rejected applicant. We perform a wort cae analyi to derive lower and upper bound for criteria ued to evaluate rating ytem. More preciely, we conider meaure for the dicriminatory power of a rating ytem and epecially it correponding credit core (numerical value that reflect rating of the credit applicant). We introduce a criterion that i baed on the dicrepancy between the core ditribution of defaulted and non-defaulted credit applicant. A another criterion of interet we tudy the accuracy ratio (computed from Gini coefficient, ee for example Keenan and Sobehart; 1999) which compare the core ditribution of defaulted applicant with that of all credit applicant. In a different context, Horowitz and Manki (1998) conider a imilar cenoring problem, namely urvey nonrepone. They derive bound for the regreion function, which in our cae would correpond to default probabilitie (PD). In contrat to their analyi, our focu lie on performance meaure for credit core. For thee meaure we can exploit the fact 1

3 that the probabilitie of default and non-default cannot vary independently. To ummarize, the main contribution of the paper are the following: We dicu how to evaluate the dicriminatory power of credit core given that only a part of the default and non-default i oberved. It i trongly emphaized that cenoring lead to biaed etimate for any kind of performance meaure for rating ytem. We derive lower and upper bound for criteria that are baed on the difference of the core ditribution under default and non-default. The paper i organized a follow: In Section 2 we dicu how to define the dicriminatory power of a credit core. We introduce two criteria that are imple to illutrate and meaure the difference between the core ditribution of defaulted and non-defaulted loan. Section 3 dicue the conequence of cenoring. In our context, cenoring mean that we aume to have default or non-default information only for a retricted et of applicant. In Section 4 we how how to find lower and upper bound for the propoed criteria under very weak aumption. Section 5 illutrate the obtained bound by a real data example. Finally, Section 6 ummarize our reult. 2 Dicriminatory Power of a Score To ae the default rik of an credit applicant a bank uually identifie everal indicator X 1,..., X p uch a debt equity ratio or return on invetment which are then aggregated to a ingle value, the rating of the credit applicant. We will call the mechanim that i ued to aggregate the factor core function. Formally, the core function S i a real valued function which map the indicator variable X 1,..., X p into the ingle value S(X 1,..., X p ). For the ake of brevity we will denote the random variable S(X 1,..., X p ) by S. To formalize the default event for debitor i, we introduce the random variable Y (i) that can take on only the two value 1 (default) and 0 (non-default). The correponding core S (i) aim to reflect the rik of thi default event. In the following we will tudy the relation between S (i) and Y (i) for a credit portfolio coniting of i = 1,..., n debitor. In practice, a bank typically calculate the core value S (i) at time t and oberve the default event Y (i) at ome future time point (for example t + 1). For the redeign of a rating ytem the bank i able to check if it hitorical rating (produced by the core function) have led to reliable prediction of the default event. An important criterion to ae the quality of the core function i it dicriminatory power, i.e. it ability to eparate the good from the bad applicant. More preciely, the bank can ue it ex-pot obervation of the default to plit the ample of all debitor into two ubample: the non-defaulted and the defaulted debitor. If we denote the core cumulative ditribution function of thee two ample by F 0 and F 1, dicriminatory power then mean that the location of the probability mae of thee ditribution are a different a poible. From now on and without lo of generality high core value tand for high default rik and low core value tand for low default rik. It i important to note that comparing rating with 2

4 repect to their dicriminatory power i only meaningful if thee rating imply a reaonable ordering of the debitor. In thi context reaonable mean that the default probability of a good rated debitor i lower than for bad rated debitor. Formally thi property mean that F 1 tochatically dominate F 0. For thi reaon we can exclude all thoe rating which do not atify thi minimal requirement. Obviouly no practitioner would take rating into conideration that do not exhibit thi form of tochatic dominance. From a formal point of view we need to define how the ditance between F 0 and F 1 hould be meaured. We will conider the following two approache: the maximum ditance meaured by T = max{f 0 () F 1 ()}, the average ditance meaured by the area under curve of the receiver operating characteritic (ROC) curve AUC = 1 F 1 () df 0 () which will hown to be equivalent to the accuracy ratio AR derived from the Lorenz curve. In the remaining part of thi ection we will decribe ome propertie of thee meaure. Intead of analyzing T one can alo conider R = 1 T which i given by R = min {F 1 () + 1 F 0 ()}. (1) If F 0 and F 1 have denitie f 0 and f 1 poeing only one point of interection, then R ha the intuitive interpretation a the overlapping region of thee denitie. Thi i illutrated in Figure 1. It i obviou that for ditribution F 0, F 1 on completely different upport (perfect eparation), the value of T i 1. If both ditribution are identical (no eparation) then T equal 0. In all other cae T will take on value between 0 and 1. In practice we have obervation S (i) for the core and Y (i) for the default event. Etimate of the cumulative ditribution function F 0, F 1 can be eaily found by the empirical ditribution function i F j () = I(S(i), Y (i) = j) i I(Y, j = 0, 1, (2) (i) = j) where I( ) denote the indicator function. We wih to remark that T = 1 R { = max F0 () F 1 ()} alo erve a the tet tatitic in the Kolmogorov-Smirnov tet which check the hypothei of tochatic dominance of F 1 over F 0. Applying a Kolmogorov-Smirnov tet, thi minimal requirement can alo be teted formally. However, our goal i to identify a core function leading to a maximal poible value of T. The econd meaure for the dicriminatory power introduced above i the the area under the ROC curve (Hand and Henley; 1997; Engelmann et al.; 2003; Sobehart and Keenan; 3

5 Overlapping of Denitie f0, f f0 f Figure 1: Overlapping area R 2001). The ROC curve i obtained by plotting cumulated percentage of the non-defaulted and defaulted debitor on the horizontal and vertical cale, repectively. If we ort the percentage of applicant from bad to good core, thi mean to graph 1 F 0 () v. 1 F 1 () for all (+, ). Figure 2 how how the area under curve (AUC) i then calculated: AUC = + {1 F 1 ()} d {1 F 0 ()} = 1 + F 1 () d F 0 (). An optimal core would exactly eparate default and non-default. The correponding optimal ROC lead to an area under curve that fill the complete unit quare o that AUC equal 1 in thi cae. The wort core i the one which doe not contain any information about default and non-default, i.e. it randomly aign core value to credit applicant. The correponding ROC curve i thu identical to the diagonal and it AUC equal 1. 2 The AUC i directly related to the accuracy ratio AR which i baed on the Lorenz curve and it Gini coefficient (Keenan and Sobehart; 1999; Engelmann et al.; 2003). In contrat to the ROC curve the Lorenz curve plot 1 F () v. 1 F 1 () for all (+, ) where F denote the cumulative ditribution function of S, the core value of all debitor. Figure 3 how the principle of the Lorenz curve. The Lorenz curve i alo known a the power curve or the cumulative accuracy profile (CAP). The optimal Lorenz curve (for a core that exactly eparate default and non-default) reache the vertical axi 100% at a horizontal percentage of P (Y = 1), the probability 4

6 1 F () 1 100% ROC curve Percentage of default AUC (area under curve) Percentage of non default Figure 2: ROC curve for credit core 100% 1 F 0 () 1 F () 1 100% PD optimal curve Lorenz curve Percentage of default 1_ 2 G Percentage of applicant Figure 3: Lorenz curve for credit core 100% 1 F() 5

7 of default. The wort Lorenz curve i identical to the diagonal ince F 1 = F in that cae. A quantitative meaure for the performance of a core i then baed on the area between the Lorenz curve and the diagonal. The Gini coefficient G equal twice thi area, i.e. G = 2 + (1 F 1 )() d(1 F )() 1 = F 1 () df (). (3) To compare different core, their accuracy ratio AR are defined by relating the Gini coefficient of each core to the Gini coefficient of the optimal Lorenz curve. The accuracy ratio i defined by AR = G G = G opt P (Y = 0). The AR i a poitive linear tranformation of AUC which can be hown a follow: By the Baye theorem we have F = P (Y = 0) F 0 + P (Y = 1) F 1. Thi provide G = 1 2 = 1 2 P (Y = 0) F 1 df = 1 2 F 1 d {P (Y = 0)F 0 + P (Y = 1)F 1 } F 1 df 0 2 P (Y = 1) F 1 df 1 = P (Y = 0) (2 AUC 1), where the relation F 1 df 0 = 1 AUC, P (Y = 1) = 1 P (Y = 0) and F 1 df 1 = 1 are 2 ued. We therefore obtain G AR = = 2 AUC 1. P (Y = 0) In practice the integral in AUC and AR are etimated by numeric integration of F 1 over F 0 or F where F denote the empirical ditribution function i F () = I(S(i) ). (4) n The AUC and hence the accuracy ratio AR are related to the Wilcoxon rank um tet and it equivalent, the Mann Whitney U tet. Both are claical nonparametric tet to check if two ditribution are identical, in our cae F 0 and F 1. In it implet form, the U tet i derived for continuou core ditribution. Denote by S 0 the core given non-default and by S 1 the core given default. Let S (i) 0 and S (k) 1 be two ample obervation from the credit portfolio. The U tet tatitic then count the number of pair where S (i) 0 < S (k) 1 hold, i.e. Û = #{S (i) 0 < S (k) 1 } over all i, k. If the core function perfectly eparate default and non-default we hould obviouly obtain Û = n 0 n 1. If there i no relation between core and default event, then S (i) 0 < S (k) 1 happen with probability 1 uch that Û 0.5(n 2 0 n 1 ). Conequently, a recaled verion of the tet tatitic, Û/(n 0 n 1 ) i an etimate for P {S 0 < S 1 } = 1 F 1 () df 0 () = AUC = AR In the cae of dicrete core ditribution thi equation involve an additional term for P (S 0 = S 1 ). The U tet i another way to formally tet our minimal requirement of tochatic dominance of F 1 over F 0. Again we wih to tre that our aim i to find a core function leading to the maximal poible value of AUC or AR. 6

8 3 Credit Scoring under Cenoring A particular problem with credit data in practice i that we uually oberve default and non-default only for a ubample of applicant. Thi i o becaue the bank compute core for N applicant but only n of them (n < N) are accepted for a loan. Hence, default and non-default obervation are preelected by a condition which we denote by A. Thi type of ample preelection can be decribed a cenoring or ample election. Note that condition A formalize the criterion for granting credit applied by the bank under conideration. Overlapping for Credit f0, f Figure 4: Truncated overlapping area for credit data Obviouly, thi data ampling will reult in biaed etimate of all relevant parameter due to the non-repreentative data bae. We wih to tre that thi bia can be poitive or negative. In the equel, we will ee that formally thi come from the fact that we are working with conditional probabilitie. The problem of bia correction in thi cae ha been mainly tudied by uing (regreion) model that extrapolate on the unoberved data and with a focu on the etimating regreion coefficient and PD. In the econometric literature, bivariate regreion model for ample election (Heckman; 1979) are well-known. For example, Greene (1998) and Boye et al. (1989) ue a bivariate probit model for credit data. In the tatitical literature, thi bia correction technique i know a reject inference. Feelder (2000) and Crook and Banaik (2004) are reference here. To illutrate the effect of cenoring (or ample election) for etimating T, aume again that we have two continuou denitie f 0, f 1. Aume further for a moment that the cenoring condition ha the intuitive form A = {S c}, (5) where c i a threhold uch that credit applicant are accepted for a loan if their core S i maller than c. Figure 4 how thi modified ituation in comparion to Figure 1. The 7

9 ditribution right to the black line (here c = 2) cannot be oberved but need in fact to be conidered for a correct aement of the performance of the core. Let S and Ỹ denote the oberved part of the core and the group indicator. Hence, we have only obervation for the cenored default core S 1 and non-default core S 0, while we are intereted in the non-cenored core S 0 and S 1. Under the aumption (5), the relation between S j and S j (j = 0, 1) i given by P ( S j ) = P ( S Ỹ = j) = P ( S, Ỹ = j) P (Ỹ = j) = P (S, Y = j A). P (Y = j A) From A = {S c} and P (S j ) = P (S, Y = j)/p (Y = j) it follow that which then how P ( S j ) = P (S, Y = j) P (S c, Y = j) = P (S j ) P (S j c) F j () = F j() F j (c) for c, for c. (6) Here F j denote the cumulative ditribution function of S j. Under the aumption that S j ha a continuou ditribution, (6) reult in an equivalent recaling of the denitie by F j (c). AkFigure 5 illutrate thi cae, note the difference to Figure 4 on the vertical cale. Overlapping of Truncated Denitie f0, f Figure 5: Oberved overlapping area R We will now analye the relation between R and R, the region of overlapping for the cenored (oberved) and the non-cenored (partially unoberved) ample. Computing R in the ame way a R and uing (6), would hence give { R = min F1 () + 1 F } 0 () = min { F1 () F 1 (c) + 1 F } 0(). (7) F 0 (c) 8

10 Thi how that the naive calculation of the overlapping region from incompletely oberved data i uually different (biaed) from the objective overlapping region R. The difference in T = 1 R and T = 1 R can be coniderably important a the following Monte Carlo imulation how. We have imulated 250 data et, each of N = 500 applicant for a credit. The core S (i) of thee applicant are generated only once and come from a normal ditribution with expectation 2.5 and variance The imulated PD are obtained from a Logit model, i.e. we define p() = exp( ) and generate the Y (i) a Bernoulli random variable with probability parameter p(s (i) ). We reject the 2% wort cored credit applicant. Thi lead to a credit portfolio coniting of n = 490 debitor. Dicriminatory Power T cenored full Figure 6: Difference in T (upper boxplot) and T (lower boxplot) In Figure 6 boxplot for the realized ditribution of the etimated T and T are diplayed. The graphic how that in our imulated example T i typically maller than T (in 235 out of the 250 cae). A imilar boxplot could be hown for ÃR and AR. So uing T and ÃR intead of T and AR can milead in both direction (over- and underetimation) if the bank wihe to ae the performance of it rating ytem. Recall that formally thi come from the fact that the acceptance condition A lead to conditional probabilitie. 9

11 4 Inequalitie for the Nonparametric Cae A we have een in Section 3, the computation of T and AR from S j require pecific aumption on the ditribution of S j and their relation to the cenoring condition A. In the general cae in which the relation between A and S i completely unknown there i no poibility to etimate thee ditribution beyond A. A pointed out, thi i a relevant problem for a bank redeigning it rating ytem ince data on rejected applicant are uually not available. A poible remedy to thi problem i the calculation of lower and upper bound on the meaure for dicriminatory power. Our approach i inpired by Horowitz and Manki (1998) who conider a imilar cenoring problem in urvey nonrepone. The general aumption throughout thi ection i that we know the percentage of rejected loan, i.e. the full number of credit applicant. Denote thi number of all credit (accepted or rejected) by N. Under the aumption that the percentage of both rejected applicant and default are mall, relatively narrow bound can be found for T and AR. We emphaize that N typically doe not contain applicant who are rejected without being rated. 4.1 Maximum Ditance Meaure T Recall that the computation of T = max {F 0 () F 1 ()} require the cumulative ditribution function F j () of S j = (S Y = j). However, we only oberve F j (), the cumulative ditribution function of S j = ( S j Ỹ = j) = (S Y = j, A). To derive upper and lower bound for T we have to relate the unobervable function F j () to the obervable function F j (). The following lemma how thi relation. For the ake of clarity, we have collected all more complex derivation in the appendix. Lemma 4.1 Uing the notation α j = P (A Y = j), we have α j Fj () F j () 1 α j {1 F j ()}. To apply thi lemma for calculating bound for T, we need bound for α j. Thee follow from P (Y = j, A) P (Y = j) P (Y = j, A) + P (A). (8) which i a conequence of P (Y = j) = P (Y = j, A) + P (Y = j, A), where A tand for the complement of A. Since α j = P (Y = j A)P (A) P (Y = j) = P (Ỹ = j)p (A) P (Y = j) 10

12 it follow by (8) that α j [α low j, 1], where α low j = P (Ỹ = j)p (A). (9) P (Ỹ = j)p (A) + P (A) Lemma 4.1 together with (9) yield upper and lower bound for T. We ummarize thi reult in the following propoition: Propoition 4.2 Bound for T are given by [ max α0 low F 0 () + α1 low {1 F ] 1 ()} 1 [ T 1 min α0 low {1 F 0 ()} + α1 low ] F 1 (). We wih to tre that in the pecial cae of no cenoring (i.e. if all credit applicant were accepted for a loan and we oberve their default or non-default) we have P (A) = 0 and α0 low = α1 low = 1. A a conequence, the inequality of Propoition 4.2 reduce to T = max {F 0 () F 1 ()} which i exactly the definition for the uncenored cae. Let u further remark that the bound in Propoition 4.2 are quite ueful but not the optimal one. In contrat to Horowitz and Manki (1998), we can exploit that the probabilitie of default and non-default are complement and cannot vary independently. Uing thi fact, we can derive improved bound which are ummarized in Propoition 4.3. It turn out, however, that in Monte-Carlo imulation the improvement by Propoition 4.3 i very modet. We refer here to the imulation example preented in Section 5. Propoition 4.3 Improved bound for T are given by [ β0 max F p up 0 () + β ] 1 1 p up {1 F 1 ()} 1 T 1 min [ β0 p low {1 F 0 ()} + β 1 1 p low ] F 1 (), where β j = P (Y = j, A) and the function p low the appendix. and p up are defined a in (14) and (17) in To apply thee bound to empirical data, we need to etimate all unknown quantitie in Propoition 4.2 or 4.3. Thi i poible becaue we know the total number of cored credit applicant N. More preciely: For the oberved core under default and non-default we know their empirical ditribution function F j which can be obtained analogouly to (2). To 11

13 etimate αj low, β j, p low 0, and p up 0 we conider the probabilitie of {Ỹ = j} = {Y = j A}, A, and A, which can be approximated by their oberved relative frequencie P (Ỹ = j) = n j n, P (A) = n N, P (A) = N n N. (10) Here n 0 denote the number of oberved non-default and imilarly n 1 denote the number of oberved default. A before, n tand for the ample ize of the oberved credit (i.e. n = n 0 + n 1 ). Thi provide the etimate α low j = n j n j + N n, βj = n j N. (11) Etimate for p low 0 and p up 0 can be found by ubtituting β j, P (A) and F j () into (14) (15) and (17) (18). Dicriminatory Power T imulation Figure 7: Etimated T (mooth olid line), T (olid) and bound (dahed) The following Monte Carlo imulation illutrate the effect of the etimated bound. We ue the previouly imulated data et. Figure 7 how etimate for T, T and the etimated upper and lower bound according to Propoition 4.3 for all 250 imulated data et. To implify the comparion, all imulated value are orted by the etimated value of T. The bound according to Propoition 4.2 are only lightly wider o that we omit them here. 12

14 Note that in practice the etimation of T could not have been carried out becaue data on rejected applicant are uually not collected. However, due to our imulation experiment, we have the opportunity to etimate both T and T. The imulation analyi how in particular, that T might be maller or larger than T. A cloer inpection of the imulated data reveal that T tend to be larger than T if P (Ỹ = 1) P (Y = 1). In other word, T tend to overetimate T if the cenoring condition doe reject a too mall number of default. In thee cae uing T intead of T would have led to a too optimitic value for the dicriminatory power of the core. The upper and lower bound, however, lead to a correctly pecified range for T. We ee that the lower bound in Figure 7 eem to be quite far away from both etimated T and T. Thi i a conequence of the fact that our bound do not require any information about the tructure of the cenoring condition A. A narrower lower bound could be calculated if additional information on A i available, for example if A i determined by a different core S and we know the dependence tructure between S and S. 4.2 Accuracy Ratio AR We conider the cenored accuracy ratio ÃR = G P (Ỹ = 0), where G denote the cenored Gini coefficient and P (Ỹ = 0) = P (Y = 0 A). Thu, analogouly to T, the Gini coefficient G and the accuracy ratio ÃR are biaed. We will how how to obtain upper and lower bound for both G and AR. 1 F( Y=1,A) 100% Lorenz curve of oberved loan Percentage of oberved default _1 ~ G 2 Percentage of oberved applicant 100% 1 F( A) Figure 8: Lorenz curve under cenoring Suppoe that the Lorenz curve for the oberved loan look a in Figure 8. To obtain lower and upper bound for the Lorenz curve of all credit applicant, we conider two extreme 13

15 cae for the unoberved part: the unoberved loan poe either the lowet or the bet rating. Thee aignment lead to the Lorenz curve in Figure 9. Hence, lower and upper bound for G (and ubequently for AR) can be derived by calculating the area under the curve in Figure 9. The reulting inequality for AR i ummarized by the following propoition. A before we refer to the appendix for a detailed proof. Propoition 4.4 Bound for AR are given by (ÃR + 1 ) β 0 β ) 1 (ÃR p 0(1 p 0) 1 AR β 0 β 1 1 p 0(1 p 0) + 1 where again β j = P (Y = j, A) and β 0 if β 0 > 1, 2 p 1 0 = if β β 0 + P (A), β 0 + P (A) if β 0 + P (A) < 1. 2 Let u remark that in the pecial cae if all credit applicant are accepted, it hold p 0 = β 0 and 1 p 0 = β 1. Hence, the upper and lower bound for the Lorenz curve a well for Gini coefficient and accuracy ratio coincide with their repective value in the non-cenored cae. In practice, we ue the etimate α 1 low, F 1 (), P (A) from Section 4 and i F () = I(S(i) ). n To illutrate the reult of Propoition 4.4, we ue the data from the Monte Carlo imulation in Section 4 again. Figure 10 how the etimated AR and ÃR a well a the etimated upper and lower bound according to all 250 imulated data et (orted by the etimated AR). We find ÂR > ÃR in 237 (out of the 250) cae. A for T we can conclude that uing ÃR intead of AR would have led to too large or mall value for the dicriminatory power of the core, wherea the upper and lower bound provide a correctly pecified range for ÂR. The remark on the imulation in Section 4 apply here a well. We ee, however, that the etimated bound are wider (relative to the value of ÃR and AR) and that the lower bound may be negative. Thu, only the upper bound ha a ueful interpretation. 14

16 1 F( Y=1) 100% _ P(A Y=1) _1 P(A)P(A Y=1)G ~ 2 P(A Y=1) 1 F() 100% P(A) _ P(A,Y=0) _ P(A,Y=1) 1 F( Y=1) 100% ~ _1 P(A)P(A Y=1)G 2 P(A Y=1) 1 F() _ P(A Y=1) 100% _ P(A,Y=1) P(A) _ P(A,Y=0) Figure 9: Lorenz curve under cenoring 15

17 Accuracy Ratio AR imulation Figure 10: Etimated AR (mooth olid line), ÃR (olid) and bound (dahed) 5 Application Let u now conider a brief illutration on real data. We ue the data from Fahrmeir and Tutz (1994) which are publicly available 1. The data et comprie 1000 obervation of private loan. One of the variable i credit hitory. We will now ae dicriminatory power under the aumption that cutomer with a negative credit hitory (thoe which howed a heitant payment of previou credit ) would not have granted a loan and that their default or non-default would not have been oberved. Thi mean we ue a ample of n = 960 oberved cutomer wherea the ample ize of all applicant i equal to N = We etimate two different Logit pecification. The correponding variable are lited in Table 1. The firt pecification ue more peronal and credit information but i not a uperet of the econd pecification. We compare the core etimated by a Logit model with repect to T and AR. The reulting criteria on the oberved data a well a the etimated lower and upper bound are hown in Table 2. We recognize that a in Figure 7 and 10 the interval for AR are clearly wider. Conequently, information that we get out of the interval etimate i more precie in the cae of T. In particular, we oberve a negative lower bound for AR in pecification 2. However, in thi pecial example the interval for AR do not have an interection. Thi mean that 1 See 16

18 Variable Specification 1 Specification 2 previou loan (1 for OK, 0 for unknown) employed (1 for more than one year, 0 otherwie) duration of the loan (dicretized with dummie for 10 12, 13 18, and more than 24 month) amount of the loan (+ amount quared) age of the borrower (+ age quared) interaction term for amount and age aving (1 for more than 1000 DM, 0 otherwie) foreigner (1 if ye, 0 otherwie) purpoe (1 if loan i ued to buy a car, 0 otherwie) houe owner (1 if ye, 0 otherwie) Table 1: Variable for core etimation Etimated criterion Specification 1 Specification 2 T maximal range of T [0.222,0.349] [0.108,0.235] ÃR maximal range of AR [0.238,0.492] [-0.018,0.236] Table 2: Dicriminatory power of the core pecification 1 i definitely better than pecification 2. Here, the unoberved data cannot improve the accuracy ratio for pecification 2 over that for pecification 1. 6 Concluion The dicriminatory power of a credit core can be etimated by comparing the core ditribution of the default with that of the non-default or the full ample. We conider two poible criteria in thi paper: The maximal difference of the cumulative core ditribution function for non-default and default and the accuracy ratio T = max {F 0 () F 1 ()} AR = 1 2 F 1 () df () P (Y = 0) 17

19 A we have een, a cenored ample can lead to coniderable bia when uing the criteria to evaluate the core with repect to dicriminatory power. Our imulation how that the bia might be poitive or negative, i.e. there i no imple rule to take account for thi bia. Corrected calculation of the criteria are poible if detail about the rejected credit applicant are known. However, often no precie information about thee applicant i available. For thi cae the paper demontrate how to ae dicriminatory power by computing lower and upper bound of uch criteria. The calculation of bound i poible under the weak aumption that only the percentage of rejected credit i known. Appendix: Proof Proof of Lemma 4.1 We have F j () = P (S Y = j) = P (S, A Y = j) + P (S, A Y = j) = P (S A, Y = j)p (A Y = j) + P (S, A Y = j), hence F j () = F j () P (A Y = j) + P (S, A Y = j). (12) We find an upper bound for F j () by uing that {S } A A in the econd term of (12), i.e. F j () F j ()P (A Y = j) + P (A Y = j) = 1 P (A Y = j){1 F j ()}. A lower bound for F j () i given by omitting the econd term of (12) completely, uch that F j () F j ()P (A Y = j). Proof of Propoition 4.2 The reult follow directly by combining Lemma 4.1 and the bound in (9). Proof of Propoition 4.3 We introduce the additional abbreviation β j = P (Y = j, A) and p = P (Y = 0), uch that α 0 = β 0 /p and α 1 = β 1 /(1 p). We will firt conider bound for R and later on tranfer them into bound for T. Conider the lower bound for R firt. From the proof of Lemma 4.1 we ee F 1 () + 1 F 0 () α 1 F1 () + α 0 {1 F 0 ()} = β 1 1 p F 1 () + β 0 p {1 F 0 ()} (13) 18

20 In the lat term each of the probabilitie can be etimated from the oberved data except for p. Hence, for given the lat term ha to be minimized with repect to p. For thi minimization one ha to conider the three cae β 1 F1 () = β 0 {1 F 0 ()}, β 1 F1 () > β 0 {1 F 0 ()}, and β 1 F1 () < β 0 {1 F 0 ()}, which all lead to the ame optimum: and p low = γ = β 0 if γ < β 0, β 0 + P (A) if γ > β 0 + P (A), γ, otherwie, β 0 {1 F 0 ()} β 0 {1 F 0 ()} + β 1 F1 () (14). (15) The upper and lower threhold in (14) are conequence of the bound in (8). To derive the upper bound of R we have again from the proof of Lemma 4.1 F 1 () + 1 F 0 () 2 α 1 {1 F 1 ()} α 0 F0 () = 2 β 1 1 p {1 F 1 ()} β 0 p F 0 (). (16) Maximization of the lat term with repect to p lead to a imilar reult a before: β 0 if δ < β 0, p up = β 0 + P (A) if δ > β 0 + P (A), δ, otherwie, and δ = Combining the reult we obtain β 1 1 p low F 1 () + β 0 p low β 0 F0 () + {1 F 0 ()} β 0 F0 () β 1 {1 F 1 ()} F 1 () + 1 F 0 () 2 β 1 1 p up uch that by uing T = 1 R the tatement i proved. Proof of Propoition 4.4 (17). (18) {1 F 1 ()} β 0 p up F 0 () (19) We recall the notation β j = P (Y = j, A), which allow u to write β 0 + β 1 intead of P (A). Additionally, we introduce the notation p j = P (Y = j). Obviouly thee probabilitie are related by p 0 + p 1 = 1. We can now expre the following term uing p j and β j. P (Y = j A) = β j β 0 + β 1 19

21 ÃR = β 0 + β 1 β 0 G G = β 0 β 0 + β 1 ÃR P (A, Y = j) = p j β j P (A Y = j) = p j β j p j Conider firt the lower bound for AR. From the firt plot of Figure 9 we ee that the lower bound for G (twice the area under the curve minu 1) equal G low = P (A)P (A Y = 1) G + P (A)P (A Y = 1) + P (A, Y = 1) {1 P (A Y = 1)} 1 = (β 0 + β 1 ) β ( 1 ( p G + 1) (p 1 β 1 ) 1 + β ) p 1 Uing the relation between G and ÃR lead to G low = 1 { } p (ÃR + 1) β 0β 1 p 0 p 1. 1 Thu we obtain ) AR low = Glow β0 β 1 = (ÃR (20) p 0 p 0 p 1 We now ue the ame approach for the upper bound of AR. From the econd plot in Figure 9 we calculate a an upper bound for G Thi reult in G up = P (A Y = 1)P (A, Y = 1) + P (A)P (A Y = 1) G + P (A) {P (A Y = 1) + 1} + P (A, Y = 0) 1 = p 1 β 1 (p 1 β 1 ) + (β 0 + β 1 ) β 1 G p 1 p ( 1 ) p1 β 1 + (β 0 + β 1 ) (p 0 β 0 ) 1 p 1 = 1 { } β 0 β 1 p (ÃR 1) + p 0p 1. 1 ) AR up = Gup β0 β 1 = (ÃR (21) p 0 p 0 p 1 Hence, we obtain together with (20) ) β0 β ) 1 β0 β 1 (ÃR AR (ÃR (22) p 0 p 1 p 0 p 1 To achieve the minimal value for the lower and the maximal value for the upper bound, it i obviou that p 0 p 1 = p 0 (1 p 0 ) mut be maximal. It i important to note that p 0 cannot vary freely ince from (8) we have β 0 p 0 β 0 + P (A). A a conequence, we have to ditinguih three cae: 20

22 (1) β β 0 + P (A) In that cae, the value that maximize p 0 (1 p 0 ) i p 0 = 1 2 (a if p 0 could take on all value between 0 and 1). (2) 1 2 < β 0 Here, the optimal value i p 0 = β 0. (3) 1 2 > β 0 + P (A) Here, the optimal value i p 0 = β 0 + P (A). Reference Ah, D. and Meeter, S. (2002). Bet practice in reject inferencing, Conference preentation, Credit Rik Modelling and Deciioning Conference, Wharton Financial Intitution Center, Philadelphia. Banking Committee on Banking Superviion (2001). The New Bael Capital Accord, Bank for International Settlement. Boye, W. J., Hoffman, D. L. and Low, S. A. (1989). Meauring default accurately, Journal of Econometric 40: Crook, J. and Banaik, J. (2004). Doe reject inference really improve the performance of application coring model?, Journal of Banking and Finance 28(4): Engelmann, B., Hayden, E. and Tache, D. (2003). Teting rating accuracy, Rik 16: Fahrmeir, L. and Tutz, G. (1994). Multivariate Statitical Modelling Baed on Generalized Linear Model, Springer. Feelder, A. J. (2000). Credit coring and reject inference with mixture model, International Journal of Intelligent Sytem in Accounting, Finance & Management 9: 1 8. Greene, W. H. (1998). Sample election in credit-coring model, Japan and the World Economy 10: Hand, D. J. and Henley, W. E. (1997). Statitical claification method in conumer credit coring: a review, Journal of the Royal Statitical Society, Serie A 160: Heckman, J. (1979). Sample election bia a a pecification error, Econometrica 47: Horowitz, J. L. and Manki, C. F. (1998). Cenoring of outcome and regreor due to urvey nonrepone: Identification and etimation uing weight and imputation, Journal of Econometric 84: Keenan, S. C. and Sobehart, J. R. (1999). Performance meaure for credit rik model, Reearch report # , Rik Management Service, Moody Invetor Service. Sobehart, J. R. and Keenan, S. C. (2001). Meauring default accurately, Rik 14:

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