Assessing the Discriminatory Power of Credit Scores

Transcription

1 Aeing the Dicriminatory Power of Credit Score Holger Kraft 1, Gerald Kroiandt 1, Marlene Müller 1,2 1 Fraunhofer Intitut für Techno- und Wirtchaftmathematik (ITWM) Gottlieb-Daimler-Str. 49, Kaierlautern, Germany 2 Humboldt-Univerität zu Berlin, Intitut für Statitik & Ökonometrie Spandauer Str. 1, Berlin, Germany Augut 15, 2002 We dicu how to ae the performance for credit core under the aumption that for credit data only a part of the default and nondefault i oberved. The paper introduce a criterion that i baed on the difference of the core ditribution under default and nondefault. We how how to etimate bound for thi criterion, the Gini coefficient and the accuracy ratio. Keyword: credit rating, credit core, dicriminatory power, ample election, Gini coefficient, accuracy ratio JEL Claification: G21

2 1 Introduction A bank which want to decide whether a credit applicant will get a credit or not ha to ae if the applicant will be able to redeem the credit. Among other criteria, the bank require 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, a for example liquidity ratio or ratio 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. 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). Unfortunately, they do not give any formal definition of predictive or dicriminatory power. The paper i organized a follow: In Section 2 we dicu how to meaure dicriminatory power of a core (a numerical value that reflect the rating of a credit applicant). We introduce a criterion that i baed on the difference between the ditribution of the core conditioned on default or non-default and i imple to compute. Section 3 dicue the conequence of the typical cenoring in credit data due to the fact that not all credit applicant are accepted. Thi implie that we do have default or non-default information only for a retricted et of applicant. To keep thing imple we firt dicu dicriminatory power uing a parametric etting. In Section 4 we conider the nonparametric cae and how how to find lower and upper bound for the propoed criterion. Finally, Section 5 extend our approach to lower and upper bound for the Gini coefficient and the accuracy ratio (AR). 2 Dicriminatory Power of a Score Let u tart with the following claification problem: Conider random variable X 1,..., X p and a group indicator Y {0, 1}. A core S (ued to rate applicant 1

3 for a loan) i an aggregation of the variable X 1,..., X p into a ingle number. Hence, we can conider any real valued function S(X 1,..., X p ) to be a core. For the ake of brevity we will ue S to denote the random variable S(X 1,..., X p ). In the following we will only tudy the relation between S and Y. There exit a variety of criteria to ae the quality of a core. A reaonable core function for credit rating hould aign higher core value to credit applicant who have higher probabilitie of default (PD). Therefore the capability to eparate the two group of obervation correponding to Y = 1 (default) and Y = 0 (non-default) i a baic feature of a credit core function. A meaure for the dicriminatory power can conequently be ued a a performance meaure for a credit core. A traightforward approach to ae dicriminatory power i the comparion of the conditional ditribution of S given default or non-default. We will firt focu on the difference of thee two conditional ditribution. The methodology that i derived here can however be ued for other meaure of performance a well. Overlapping of Normal Denitie f0, f Figure 1: Overlapping area U for two normal denitie In the cae of a normal ditribution the conditional denitie of S given Y = j, j = 0, 1 are eay to viualize and to compute. Denote f 0, f 1 the probability denitie of S Y = 0 and S Y = 1, further F 0, F 1 their cumulative ditribution function. Conider firt the pecial cae that f 0 and f 1 have exactly one point of interection, cf. Figure 1. (A condition for thi property will be given in a moment.) Let be the horizontal coordinate of thi interection. Auming a normal ditribution mean that both denitie f 0 and f 1 are determined by their expectation µ 0, µ 1 and tandard deviation σ 0, σ 1. We uppoe (w.l.o.g.) in the 2

4 following that µ 1 > µ 0. Then the region of overlapping U for the two denitie can be calculated a U = F 1 () + 1 F 0 (). (1) If in the normal cae both tandard deviation are identical (σ 0 = σ 1 ), there i exactly one point of interection which i given by = µ 0 + µ 1 2 For different tandard deviation (σ 0 σ 1 ), there may be one or two point of interection (a in quadratic dicriminance analyi) and the horizontal coordinate are determined by f 0 () = f 1 () i.e. a olution of the quadratic equation 2 (σ 2 1 σ 2 0) + 2(µ 1 σ 2 0 µ 0 σ 2 1) + µ 2 0σ 2 1 µ 2 1σ σ 2 1 log(σ 0 ) σ 2 0 log(σ 1 ) = 0. The definition of U can be eaily generalized to the nonparametric cae when no ditributional aumption for S i made: U = min{f 0 (), f 1 ()} d. (2) Thi definition allow any number of interection point of f 0 and f 1. Alternatively, auming a monotone relationhip between the core S and the default probability, a variant of the definition can be given by. U = min {F 1 () + 1 F 0 ()}. (3) Thi definition i baed on the idea that only one optimal interection point hould exit in thi cae. A for the normal cae, we aume that f 1 i right of f 0. An analogou definition could be formulated for a monotone decreaing relationhip. It i obviou that for denitie f 0, f 1 on completely different upport (perfect eparation) the region of overlapping U i zero. If both denitie are identical (no eparation) then U equal one. In all other cae U will take on value between 0 and 1. An indicator of dicriminatory power i now given by T = 1 U. (4) A U, the dicriminatory power indicator T take on value in the interval [0, 1]. In practice we have obervation S (i) for the core and Y (i) for the group (default and non-default in credit coring). Under the aumption of a normal ditribution U (and hence T ) can be computed uing the empirical moment µ 0, µ 1, σ 0, and σ 1. Under more general aumption on the ditribution, U and T can be computed for example by nonparametric etimate of the denitie (hitogram, kernel denity etimator). In the monotone cae it i ufficient to have nonparametric 3

5 etimate of the cumulative ditribution function F 0, F 1. Thoe etimate can be eaily found by the empirical ditribution function i F j () = I(S(i), Y (i) = j) i I(Y, j = 0, 1. (5) (i) = j) We remark that the ditribution of T i related to the Kolmogorov-Smirnov tet tatitic, which check the hypothei F 0 = F 1. Hence, thi tet can be applied to find out if the core influence the PD at all. 3 Credit Scoring & Unobervable Area Conider now a ample of n credit applicant, for which a et of variable i given (e.g. age of the applicant, amount and duration of the loan, income etc.). A above we aume that a real valued core S i calculated from thee variable at time t = 0 and the default (Y = 1) or non-default (Y = 1) i oberved at time t = 1. The particular problem of credit coring i that we oberve default and nondefault only for a ubample of applicant. In more detail, thi mean that 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 i uually decribed a cenoring or ample election. The problem of ample election ha been mainly tudied in the (econometric) literature with a focu on the etimating regreion coefficient and PD. Greene (1998) for example ue a Heckman two-tep procedure (ee Heckman, 1979) for etimating probit and count data model for credit data. Gourieroux and Jaiak (2001, Ch. 7) conider maximum-likelihood probit and a Bayeian approach wherea a rather general approach i introduced by Horowitz and Manki (1998). We conider the problem of etimating dicriminatory power in the cenored cae under very general ditributional aumption. Thi will lead to upper and lower bound for the performance criteria rather than claical point etimate. To illutrate the effect of cenoring (or ample election) for etimating U and T aume again that both denitie f 0, f 1 have exactly one interection point. Aume alo that the cenoring condition i A = {S c}, (6) where c i a threhold uch that no credit applicant are accepted for a loan when their core S i larger than c. Figure 2 how thi modified ituation in comparion to Figure 1. The ditribution right to the black line (here c = 2) 4

6 Overlapping for Credit f0, f Figure 2: Truncated overlapping area for credit data cannot be oberved but need in fact to be conidered for a correct aement of the performance of the core. Denote S = (S A) und Ỹ = (Y A) the oberved part of the core and the group variable. Hence, we have only obervation for S j = ( S Ỹ = j), j = 0, 1 while we are intereted in S j = (S Y = j). Under the aumption (6), the relation between S j and S j i given by P ( S j ) = P ( S, Ỹ = j) P (Ỹ = j) = P (S, Y = j) = P (S c, Y = j) P (S, Y = j A) P (Y = j A) if c. Since P (S j ) = P (S Y = j) = P (S, Y = j)/p (Y = j) it follow that P ( S j ) = P (S j ) P (Y = j) P (S c, Y = j) = P (S j ) P (S j c), which how F j () = F j() F j (c). (7) Here F j denote the cumulative ditribution function of S j. Under the aumption that S j ha a continuou ditribution, (7) reult in an equivalent recaling of the denitie by F j (c). Thee denitie and their region of overlapping Ũ for 5

7 Overlapping of Truncated Denitie f0, f Figure 3: Oberved overlapping area Ũ the normal cae are hown in Figure 3. Note the difference to Figure 2 on the vertical cale, ince f j () f j (). We will now examine the difference between Ũ and U, the region of overlapping for the cenored (oberved) and the non-cenored (partially unoberved) ample. In the following we will conider the monotone verion of the overlapping region: U = min {F 1 () + 1 F 0 ()}. Computing the overlapping region Ũ in the ame way and uing (7), would hence give { Ũ = min F1 () + 1 F } { F1 () 0 () = min F 1 (c) + 1 F } 0(). (8) F 0 (c) Thi how that the naive calculation of the overlapping from incompletely oberved data i uually different (biaed) from the objective overlapping region U. The difference in Ũ and U (or T and T ) can be coniderably important a a mall Monte Carlo imulation how. We have imulated 100 data et, each of N = 500 obervation. The core S (i) are generated only once and come from a normal ditribution with expectation 3 and variance The imulated PD are obtained from a Logit model, i.e. p() = exp( ) and the Y (i) are Bernoulli random variable with probability parameter p(s (i) ). The threhold i choen a c = 0.5, thi give here n =

8 Dicriminatory Power T T Figure 4: Boxplot for T (upper) and T (lower) Figure 4 how boxplot for the realized ditribution of the etimated T = 1 Ũ (lower boxplot) and T = 1 U (upper boxplot). The graphic how that in our imulated example T i typically maller than T. In particular, both mean and median of the 100 etimated T are a large a the upper quartile of the etimated T. A cloer inpection of the data how that in 95 cae T < T and in 5 cae T > T. So uing T at the place of T can milead in aeing the performance of the core in both direction (over- and under-etimation). Under the aumption that the type of the ditribution of S j are known a correction for Ũ can be eaily calculated. Let u outline thi for the example of normal ditribution: Here the moment of S j, j = 0, 1, can be calculated (Greene, 1993, Theorem 22.2) by E(S j S j c) = µ j + σ j λ(α j ), (9) V ar(s j S j c) = σ 2 j [1 λ(α j ){λ(α j ) α j }], (10) with µ j and σ j denoting the moment of the unconditional ditribution, α j = c µ j and λ(α) = φ(α) σ j Φ(α) denoting the invere Mill ratio. The expectation µ j = E(S j ) and variance σ 2 j = V ar(s j ) can hence be calculated from the credit data uing the empirical moment of S j and by olving the ytem of equation (9) (10). Etimate of f j and F j are then obtained by plugging µ j, σ j into the denity and cumulative ditribution function of the normal ditribution. 7

9 We remark that thi idea can be generalized to any monotone tranformation of the normal ditribution. For example, many variable ued for credit coring have a kewed ditribution. Thi typically tranfer to core which are linearly weighted um of thee variable. The log-normal ditribution, which can model uch a kewed core, ha a direct relation to the normal ditribution: Aume S j i log-normal with parameter µ j, σ j, then for the log-core Since the logarithm i monotone log(s j ) N(µ j, σ 2 j ). (11) F j () = P (S j ) = P (log(s j ) log()). (12) The computation for log-normal core i therefore completely determined by the normal cae. An even wider cla of ditribution i covered by uing any monotone ditribution a e.g a Box Cox tranformation. A correction of Ũ i alo poible if the cenoring i determined by another core function S, i.e. A = {S c}. (13) Thi i a more realitic aumption ince in practice S can be conidered a the core function from a previou credit rating ytem. If the credit rating ytem i redeigned, the performance of the new core function S need to be aeed. Under the very retrictive aumption of a joint normal ditribution of S j and S j with moment µ j, σ j, µ j, σ j and correlation ρ j it i known that E(S j S j < c) = µ j + ρ j σ j λ(α j ), (14) V ar(s j S j < c) = (σ j ) 2 [1 ρ 2 jλ(α j ){λ(α j ) α j }], (15) ee e.g. Greene (1993, Theorem 22.4). Here α j = c µ j σ j and λ denote the invere Mill ratio a before. In addition we have ( x µ F j j (x) = Φ 2, c µ ) { ( )} 1 j c µj, ρ σj j Φ. (16) σ j σ j The moment of S j could be etimated from equation analogou to (9) (10). With thee etimate for µ j, σ j, the ytem of equation (14) (16) could be ued to find etimate of the unconditional moment µ j, σ j and ρ j. Thi technique could again be generalized to monotone tranformation a the logarithm or the Box-Cox tranformation. However, apart from the retrictive ditributional aumption thi approach require that obervation for both core function S and S given A = {S c} are available. 8

10 4 Inequalitie for the Nonparametric Cae A we have een in Section 3, the computation of U from S j require pecific aumption on the ditribution of S j and their relation to the cenoring condition A. In the cae of completely unknown ditribution there i no poibility to etimate thee ditribution beyond A. Thi i a relevant problem when a bank redeign it credit rating ytem, ince data on rejected applicant are normally not available. A poible remedy to thi problem i the calculation of upper and lower bound for the dicriminatory power T. 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. We want to tre that N typically doe not contain applicant who are rejected without being rated. Recall that the computation of U require the cumulative ditribution function F j () of S j = (S Y = j). However, we only oberve F j (), the cumulative ditribution function of Sj = (S Y = j, A). Therefore we conider now the relation between F j () and F j () in thi general cae. We have hence 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), F j () = F j () P (A Y = j) + P (S, A Y = j) (17) where A denote the complement of A. We find an upper bound for F j () by uing that {S } A A in the econd term of (17), i.e. F j () F j ()P (A Y = j) + P (A Y = j) = 1 P (A Y = j){1 F j ()}. (18) A lower bound for F j () i given by omitting the econd term of (17) completely, uch that F j () F j ()P (A Y = j). (19) Both inequalitie (18) and (19) involve P (A Y = j) which can not be directly etimated, ince the ditribution of Y in A i unknown. However, we can decribe the range of P (A Y = j). We tart with a firt approximation. Let u introduce the notation α j = P (A Y = j), 9

11 uch that (18) and (19) can be written a α j Fj () F j () 1 α j + α j Fj (). (20) From P (Y = j) = P (A, Y = j) + P (A, Y = j) we conclude that P (A, Y = j) P (Y = j) P (A, Y = j) + P (A). (21) Thu from α j = P (A Y = j) = P (Y = j A)P (A) P (Y = j) = P (Ỹ = j)p (A) P (Y = j) it follow that α j α j 1, where α j = P (Ỹ = j)p (A). (22) P (Ỹ = j)p (A) + P (A) Equation (20) together with (22) yield α 1 F 1 () + α 0{1 F 0 ()} F 1 () + 1 F 0 () 2 α 1 {1 F 1 ()} α 0 F 0 (). (23) A a conequence we obtain upper and lower bound for the dicriminatory power indicator T = 1 U = 1 min{f 1 () + 1 F 0 ()} which are given by [ 1 min 2 α1 {1 F 1 ()} α F ] 0 0 () [ T 1 min α F 1 1 () + α 0{1 F ] 0 ()}. (24) We want to tre that in the pecial cae where all credit applicant are accepted we have A = Ω and α 0 = α 1 = 1. A a conequence (24) reduce to T = 1 min {F 1 () + 1 F 0 ()}, which i exactly the definition introduced in Section 2. More ophiticated bound for F 1 () + 1 F 0 () can be obtained a follow. We ue the additional abbreviation β j = P (A, Y = j), p j = P (Y = j), uch that α j = β j p j. 10

12 Conider the lower bound firt. From (18) and (19) we have F 1 () + 1 F 0 () α 1 F1 () + α 0 {1 F 0 ()} = β 1 1 p 0 F1 () + β 0 p 0 {1 F 0 ()} (25) In the lat term every probability can be etimated from the oberved data except for p 0. Hence, for given the lat term ha to be minimized with repect to p 0. 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 reult: β 0 if γ < β 0, p 0 = β 0 + P (A) if γ > β 0 + P (A), (26) γ, otherwie, and γ = β 0 {1 F 0 ()} β 0 {1 F 0 ()} + β 1 F1 (). (27) The upper and lower threhold in (26) are conequence of the bound in (21). To derive an upper bound of F 1 () + 1 F 0 () we conclude from (18) and (19) F 1 () + 1 F 0 () 2 α 1 {1 F 1 ()} α 0 F0 () = 2 β 1 1 p 0 {1 F 1 ()} β 0 p 0 F0 (). (28) Maximization of the lat term with repect to p 0 lead to a imilar reult a before: β 0 if δ < β 0, p 0 = β 0 + P (A) if δ > β 0 + P (A), (29) δ, otherwie, and δ = Combining the reult we obtain β 0 F0 () + β 0 F0 () β 1 {1 F 1 ()}. (30) β 1 1 p 0 F 1 () + β 0 {1 p F 0 ()} 0 F 1 () + 1 F 0 () 2 β 1 {1 1 p F 1 ()} β 0 0 p 0 F 0 () (31) 11

13 and a in (24) 1 min [ 2 β 1 1 p 0 {1 F 1 ()} β 0 T 1 min [ β1 1 p 0 p 0 ] F 0 () F 1 () + β ] 0 {1 p F 0 ()} 0. (32) All quantitie in the inequalitie (24) and (32) can be etimated. For the oberved core under default and non-default we have their empirical ditribution function a in (5). To etimate αj, β j, p 0 and p 0 we conider the probabilitie of the event {Ỹ = j}, A and A which can be etimated by their oberved relative frequencie P (Ỹ = j) = n j n, P (A) = n N, P (A) = N n N. (33) Here n 0 denote the number of oberved non-default (Y (i) = 0) and n 1 the number of oberved default (Y (i) = 1). A before we ue n for the ample ize of the oberved credit (i.e. n = n 0 + n 1 ), and N for the number of all the credit. Thi give the etimate α j = n j n j + N n, βj = n j N. (34) Etimate for p 0 and p 0 can be found by plugging β j, P (A) and F j () into (27) and (30). A before we ue a Monte Carlo imulation to illutrate the effect of thee etimated bound. The contruction of the imulated data et i a above with one modification: We ue core S (i) with a variance of Thi yield a value of n = 491 for the ample ize of the obervable core. We find T > T in 91 cae and T < T in 9 cae. Figure 5 how etimate for T (thick olid line), T (thin olid line) and the etimated upper and lower bound according to (32) for all 100 imulated data et (orted by the etimated T ). The bound according to (24) are wider but of very imilar ize, uch that we omit them here. Recall that in practice the etimation of T could not have been carried out, thi i only poible here for imulated data. The imulation how in particular, that in the mentioned 9 cae T a a replacement of T would have led to a too large value for the dicriminatory power of the core. The upper and lower bound however (which cover both T and T ) indicate a correctly pecified range for T. We remark that the lower bound in Figure 5 eem to be quite far away from both T and T. Thi i a conequence of the fact that thi bound doe not require 12

14 Dicriminatory Power T and Bound T Figure 5: Etimated T (thick olid), T (olid) and bound (dahed) any information about the tructure of the cenoring condition A. Thi bound could be coniderably improved if additional information a e.g. A = {S c} i ued. 5 Gini coefficient and Accuracy Ratio An alternative and frequently ued meaure for the performance of a core i the accuracy ratio AR which i baed on the Lorenz curve and it Gini coefficient. In the cae of cenored data, the accuracy ratio computed from the oberved part of the data i biaed a well. A for T we can etimate bound for the AR if the ditribution of the core i unknown. Let u firt introduce the relevant term. The Lorenz curve viualize core by mean of comparing the ditribution of S 1 and S. Figure 6 how the principle of the Lorenz curve. On the horizontal and vertical cale, the percentage of applicant are orted from high to low core. The Lorenz curve i alo known a election curve. Variant of the Lorenz curve are the receiver operating characteritic (ROC) curve (Hand and Henley, 1997) and the performance curve (Gourieroux and Jaiak, 2001, Ch. 4). 13

15 To operate with cumulative ditribution function denote the negative core by V = S. The Lorenz curve of S i then defined by the coordinate {L 1 (v), L 2 (v)} = {P (V < v), P (V < v Y = 1)}, v (, ). Since P (V < v) = 1 F ( v), thi i equivalent to {L 1 (), L 2 ()} = {1 F (), 1 F 1 ()}, (, ). A etimate of the Lorenz curve can be computed by mean of the empirical cumulative ditribution function F and F 1. 1 F( Y=1) 100% optimal curve Lorenz curve Percentage of Default Percentage of Applicant (ordered from bad to good) 100% 1 F() Figure 6: Lorenz curve for Credit Scoring Recall that core hould aign higher core value to credit applicant with higher PD. Such a credit core i obviouly good if all vertical coordinate of the Lorenz curve are large. The bet (optimal) Lorenz curve correpond to a core that exactly eparate default and non-default. Thi optimal curve reache the vertical 100% at a horizontal percentage of P (Y = 1), the probability of default. A random aignment of credit applicant to core value correpond to a Lorenz curve identical to the diagonal. Lorenz curve can alo be ued to compare different core function. Better core are more cloe to the optimal Lorenz curve. A quantitative meaure for 14

16 the performance of a core i baed on the area between the Lorenz curve and the diagonal. The Gini coefficient G denote twice thi area, i.e. G = {1 F 1 (H(z))} dz 1 = F 1 (H(z)) dz (35) where H i the invere of 1 F. In practice the latter integral i etimated by numeric integration of F 1 over the range of F. 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 hence defined a AR = G G opt = G P (Y = 0). In the cenored cae we would compute G and ÃR intead of G and AR. Note that a for T and T the Gini coefficient and accuracy ratio are biaed. We will now how how to obtain upper and lower bound for G and AR in thi cenored cae, i.e. if obervation for A are not available. A before let S 1, S denote the oberved core and F 1, F their cumulative ditribution function. We ue (18) and (19) for F 1 and derive imilar inequalitie for F uing the ame idea we ued for F j. Conider firt F () = P (S, A) P (A) P (S ) P (A) = F () P (A). Alo we have Together thi give F () = F () P (A) + P ( S A) P (A) = F () P (A) + P ({ S } A) F () P (A) + P (A) = 1 P (A) {1 F ()}. F () P (A) F () 1 P (A) {1 F ()}. (36) Uing thi together with (20) for F 1, we find lower bound [ { L 1(), L 2()} = P (A) {1 F } { (), α1 1 F }] 1 () and upper bound { L 1(), L 2()} = { 1 P (A) F } (), 1 α 1 F 1 () 15

17 Accuracy Ratio AR and Bound T Figure 7: Etimated AR (thick olid), ÃR (olid) and bound (dahed) for the Lorenz curve. In practice we ue the etimate α 1, Section 4 and i F () = I(S(i) ). n F 1 (), P (A) from The upper and lower bound for the Lorenz curve obviouly lead to upper and lower bound Ĝ and Ĝ for the Gini coefficient ince integration preerve monotonicity. For the accuracy ratio AR we need the additional etimate for P (Y = 0). A we dicued before, a point etimate of P (Y = 0) i not available. However (21) motivate upper and lower etimate n 0 N P (Y = 0) N n 1 N. Hence, bound for the etimated accuracy ratio can be found from N N n 1 Ĝ ÂR N n 0 Ĝ. (37) A we have een for T, in the pecial cae that all credit applicant are accepted, it hold A = Ω and α 0 = α 1 = 1. Hence, the upper and lower bound for the 16

18 Lorenz curve a well for Gini coefficient and accuracy ratio coincide with their repective value in thi fully oberved cae. A an illutration, we ue the data from the Monte Carlo imulation in Section 4. Figure 7 how the etimated AR (thick olid line) and ÃR (thin olid line) a well a the etimated upper and lower bound according for all 100 imulated data et (orted by the etimated AR). We find ÂR > ÃR in 97 cae and ÂR < ÃR in 3 cae. A for T we can conclude that uing ÃR a a replacement of AR would have led to too large or mall value for the dicriminatory power of the core, wherea the upper and lower bound indicate a correctly pecified range for ÂR. We alo remark that the reulting plot in Figure 7 i very imilar to that for T in Figure 5. Reference Banking Committee on Banking Superviion (2001). Accord, Bank for International Settlement. The New Bael Capital Gourieroux, C. and Jaiak, J. (2001). Financial Econometric: Problem, Model, and Method, Princeton Univerity Pre. Greene, W. H. (1993). Econometric Analyi, 2 edn, Prentice Hall. Greene, W. H. (1998). Sample election in credit-coring model, Japan an 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: