Low default modelling: a comparison of techniques based on a real Brazilian corporate portfolio

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1 Low default modellng: a comparson of technques based on a real Brazlan corporate portfolo MSc Gulherme Fernandes and MSc Carlos Rocha Credt Scorng and Credt Control Conference XII August 2011 Analytcs Brazl 2011 Experan Lmted. All rghts reserved. Experan & the marks used heren are servce marks or regstered trademarks of Experan Lmted. Other product & company names mentoned heren may be the trademarks of ther respectve owners. No part of ths copyrghted work may be reproduced, modfed, or dstrbuted n any form or manner wthout the pror wrtten permsson of Experan Lmted.

2 Topcs The problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 2

3 Prevous work Basel II EL = PD x EAD x LGD IRB Approach Statstcal model Hand and Henley (1997): o Statstcal classfcaton methods o Consumer credt scorng o Large data Pluto and Tasche (2006): o Low default portfolo o PD estmaton 3

4 The problem Common stuaton Vehcle fnancng Retal ndvduals Rsk drver: Percentage of depost LDP Truck fnancng Mddle companes Rsk drver: Percentage of depost Percentage of depost Default rate Rsk ncrease Default rate Rsk ncrease 0% 1%-30% 31%-50% 51%-80% 81%-99% 30% 20% 10% 5% 1% 1.5x 2x 2x 5x 2.5% 0.9% 1.1% 0.3% 0.0% 2.9x 0.75x 4.7x NA 4

5 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 5

6 Classcal logstc regresson Model statement Y ~ Bernoull ( ) p Target-covarates lnk p exp = 1+ exp ( x ) β ( x β ) Lkelhood functon ln n ( L( β y) ) = ln( 1+ exp( ( 1 2 ) β )) = 1 y x Where: = observaton n = sample sze y = response varable (1: default; 0 : non default) x = covarate's vetor β = parameter vector p = probablty of default ( - th observaton) 6

7 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 7

8 Bayesan logstc regresson Model statement Y ~ Bernoull ( ) p Target-covarates lnk p exp = 1+ exp ( x ) β ( x β ) 2 ( µ, ) βk ~ Normal k σ k Posteror dstrbuton posteror( β y) L( y β ) pror( β ) Onde: = observaton n = sample sze y = response varable (1: default; 0 : non default) pror 2 ( β ) = Normal( µ, σ ) k k k x = covarates vector β = parameters vector p = probablty of default L ( β y) = exp ln 1+ exp( ( 1 2y ) x β ) n = 1 ( ) 8

9 Bayesan logstc regresson Model statement Y ~ Bernoull ( ) p Target-covarates lnk p exp = 1+ exp ( x ) β ( x β ) 2 ( µ, ) βk ~ Normal k σ k Posteror dstrbuton posteror( β y) Pror nformaton of rsk drvers Data nformaton = L ( y β ) pror( β ) ( y β ) pror( β ) dβ PDF of parameters for the model Solved va Monte Carlo smulaton (MCMC) L 9

10 Bayesan logstc regresson Posteror dstrbuton posteror( β y) L( y β ) pror( β ) Pror dstrbutons Pror dstrbuton pror Non Informatve ( β ) = Normal( µ = 0, σ 2 = 1000) pror Informatve ( β ) = Normal( µ = 1.5, σ 2 = 1.2) Knowledge about the parameters pror to data nformaton

11 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 11

12 Lmted logstc regresson Model statement Y ~ Bernoull ( ) p Target-covarates lnk p exp = ω 1+ exp ( x ) β ( x β ) PD Classcal logstc regresson Lmted logstc regresson (w = 0.2) Lnear predtor PD Lnear predtor 12

13 Lmted logstc regresson Model statement Y ~ Bernoull ( ) p Target-covarates lnk p exp = ω 1+ exp ( x ) β ( x β ) Log-Lkelhood functon ln ( L( β y) ) = n y exp ln ω 1+ exp ( x β ) ( x β ) + ( 1 y ) exp ln 1 ω 1+ exp ( x β ) ( x β ) = 1 I (0,1) ( ) ω Onde: = observaton n = sample sze y = response varable (1: default; 0 : non default) ω = upper boundng parameter (0 < ω < 1) x = covarates vector β = parameters vector p = probablty of default 13

14 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 14

15 Oversamplng and state-dependent correcton PD: 1% PD: 15% Estmated model State-dependent correcton f(x) Orgnal LDP Default observatons artfcally created Based estmators due to oversamplng Unbased estmators (McCullagh e Nelder ) 15

16 Oversamplng technque: SMOTE* X1 X1 Artfcal observatons 1. Select defaulted observatons X2 X2 2. Randomly choose 2 observatons 3. Randomze one pont wthn the space defned X1 X1 4. Estmate model wth the nflated default rate database X2 X2 16 *SMOTE: Chawla et al. 2002

17 State-dependent sample Weghted Log-lkelhood functon ln ( L ( β y )) = ω ln ( p ) + ω ln ( 1 p ) w = 1 n = 1 [ Y ] 0 = 1 = 0 ω ln [ Y ] ' ( 1 + exp ( 1 2 y ) x β ) Parameters (β) estmated by WMLE are based, even on large data. McCullagh e Nelder (1989) present the correcton so the model wll be unbased. Onde: = observaton n = sample sze ξ x = covarates vector =.5Q 1 [( + ω ) π ] 0 ω Q = elemento 1 1 [ X ( X ' WX ) '] 1 X p = probablty of default ω = weght of τ = default rate n populaton y = vés th observaton = ω Y + ω ( 1 Y ) response varable (1: default; 0 :non default) ( ˆ 1 β ) ( X ' WX ) X ' W 1 0 β = parameters vector ω = τ y 1 ω = 0 1 y = n ( 1 τ ) ( 1 y) n y = 1 ~ = ξ β = ˆ β vés( ˆ β ) 17

18 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 18

19 Model valdaton KS Gn 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, Non default Default A B Acumulated defaulted KS = Max( NDAcum Dacum ) Acumulated non-defaulted Gn = A / (A+B) 19

20 Model valdaton Alternatve method for out-of-tme sample 1 KS Tranng sample N resamples wth replacement 2 N... N estmates for KS / Gn Gn

21 Topcs Scenaro and the problem Modellng technques: o Classcal logstc regresson o Bayesan logstc regresson o Lmted logstc regresson o Oversamplng combned wth correcton Model valdaton Results Concluson 21

22 Results Data set Corporate portfolo Revenue > US$ 130 per year obs Perod: defaulted observaton 7% 6% 5% 4% 3% 2% 1% 0% Experan Lmted. All rghts reserved. 22 set/03 Observatons nov/03 jan/04 mar/04 ma/04 jul/04 set/04 nov/04 jan/05 mar/05 ma/05 jul/05 set/05 nov/05 jan/06 mar/06 ma/06 jul/06 set/06 nov/06 jan/07 mar/07 ma/07 jul/07 set/07 nov/07 jan/08 mar/08 ma/08 jul/08 Default rate Non defaulted Defaulted Default Rate Movng Average Average Default Rate

23 Results Varables avalable and selecton Partal AUROC Balance sheet Credt demand Negatve statement Payment behavour Supplers hstory 120 varables Top partal AUROC (29 varables) Balance sheet Credt demand Negatve statement Payment behavour Supplers hstory Var 1 Var 1 Var 2 Var 3 Var 4 Var 5 Var 6 Var 7 Var 8 Var 9 Var 10 Var 11 Var 12 Var 1 Var 2 Var 3 Var 4 Var 5 Var 6 Var 7 Var 8 Var 1 Var 2 Var 3 Var 4 Var 5 Var 1 Var 2 Var 3 23 * Spearman correlaton ndex < 0.5

24 Results Parameters per model Varable Doman Classcal model Lmted logstc Bayesan model SMOTE + state dep Intercept w (upper bound lmt) Rato of short term debt over current assets hgher than 25 Total of negatve bureau statements ncluded over the past 30 days hgher than 8 Total number of dstnct companes that enqured over the past 15 days was hgher than 35 (0;1); Dummy for values hgher than 25 (0;1); Dummy for values hgher than 8 (0;1); Dummy for values hgher than Number of bankng past due contracts (Max of 4) Integer; lmted n Days snce last negatve statement pad (Max of 200 days) - square root tranformaton Maxmum of negatve statements actve at the same tme (Max of 120) - square root transformaton Real; lmted n Real; lmted n Total number of enqures over the past 15 days (Max of 50) Integer; lmted n Total number of negatve statement pad over the past 6 months (max of 45) - square root transformaton Real; lmted n * Spearman correlaton ndex < 0.5

25 Results Importance of varables Classcal logstc regresson Lmted logstc regresson Bayesan logstc regresson SMOTE + state dependent Rato of short term debt over current assets hgher than 25 Total of negatve bureau statements ncluded over the past 30 days hgher than 8 Number of bankng past due contracts (Max of 4) Days snce last negatve statement pad (Max of 200 days) - square root tranformaton Maxmum of negatve statements actve at the same tme (Max of 120) - square root transformaton Total number of enqures over the past 15 days (Max of 50) Total number of negatve statement pad over the past 6 months (max of 45) - square root transformaton Total number of dstnct companes that enqured over the past 15 days was hgher than 35 25

26 Results Performance measures bootstrap samples Box-plot Box-plot of KS KS from das bootstrap reamostras samples Max Q3 Med Q1 Mn Box-plot Box-plot of KS Gn from das bootstrap reamostras samples Classc Modelo regresson Classco Logístca Lmted Lmtada logstc Modelo Bayesan Bayesano model SMOTE State Dependent + state dep Classc Modelo regresson Classco Logístca Lmted Lmtada logstc Modelo Bayesan Bayesano model SMOTE State Dependent + state dep Bayesan model: Hgh performance and low varablty 26

27 Results Default rate assortment 25% Bayesan model - Default rate assortment % 90 20% % 70 Default rate 15% 10% 5% 0% 3.8% 4.0% 4.3% 4.6% 4.9% 5.3% 5.7% 1.1% 1.1% 1.1% 0.0% 0.0% 2.2% 0.0% 6.4% 6.8% 3.4% 2.3% 7.5% 8.4% 3.4% 3.4% 9.6% 1.1% 12.5% 9.1% 14.1% 7.9% % 0.42% 0.56% # observatons 0.71% 0.85% 1.05% 1.32% 1.61% 2.00% 2.48% 3.08% 4.17% 5.65% 8.47% 20.32% Mean pont per score class # observatons Default rate Inverted accumulated default rate 27

28 Concluson Pluto & Tasche: estmates PD for a pre-exstng ratng grade Approaches presented: o Lmted logstc regresson: Best KS, worst Gn; o SMOTE + State dependent: ncorporates dfferent varable; o Classc logstc regresson: Hgh KS and Gn, but hgher varablty on bootstrap; o Bayesan logstc regresson: Hgh KS and Gn. Bayesan model gves reasonably assorted default rate even on LDP Future research: how to ncorporate nformatve pror usng specalst s knowledge and bureau nformaton 28

29 Questons Gulherme B Fernandes gulherme.fernandes@br.experan.com Carlos A. Rocha carlos.rocha@br.experan.com 2011 Experan Lmted. All rghts reserved. Experan & the marks used heren are servce marks or regstered trademarks of Experan Lmted. Other product & company names mentoned heren may be the trademarks of ther respectve owners. No part of ths copyrghted work may be reproduced, modfed, or dstrbuted n any form or manner wthout the pror wrtten permsson of Experan Lmted.

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