Measuring the Quality of Credit Scoring Models
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1 Measur the Qualty of Credt cor Models Mart Řezáč Dept. of Matheatcs ad tatstcs, Faculty of cece, Masaryk Uversty CCC XI, Edurh Auust 009
2 Cotet. Itroducto 3. Good/ad clet defto 4 3. Measur the qualty 6 4. Idexes ased o dstruto fucto 7 5. Idexes ased o desty fucto 7 6. oe results for orally dstruted scores 4 7. Coclusos 30 /30
3 Itroducto It s possle to use scor odel effectvely wthout kow how ood t s. Usually oe has several scor odels ad eeds to select just oe. The est oe. Before easur the qualty of odels oe should kow (ao other ths): ood/ad defto expected reject rate 3/30
4 Good/ad clet defto Good defto s the asc codto of effectve scor odel. The defto usually depeds o: days past due (DPD) aout past due te horzo Geerally we cosder follow types of clet: Good Bad Ideterate Isuffcet Excluded Rejected. 4/30
5 Good/ad clet defto BAD Custoer Accepted Rejected Default (60 or 90 DPD) Fraud (frst delayed payet, 90 DPD) Early default (-4 delayed payet, 60 DPD) Late default (5 delayed payet, 60 DPD) GOOD Not default Isuffcet INDETERMINATE 5/30
6 Measur the qualty Oce the defto of ood / ad clet ad clet's score s avalale, t s possle to evaluate the qualty of ths score. If the score s a output of a predctve odel (scor fucto), the we evaluate the qualty of ths odel. We ca cosder two asc types of qualty dexes. Frst, dexes ased o cuulatve dstruto fucto lke Koloorov-rov statstcs (K) G dex C-statstcs Lft. The secod, dexes ased o lkelhood desty fucto lke Mea dfferece (Mahalaos dstace) Iforatoal statstcs/value (I Val ). 6/30
7 Idexes ased o dstruto fucto D K, 0, clet s ood otherwse. Nuer of ood clets: Nuer of ad clets: Proportos of ood/ad clets: p G, p B Eprcal dstruto fuctos: F ( a) I ( s a D ). GOOD K Koloorov-rov statstcs (K) K ax F, BAD ( a) F, a [ L, H ] GOOD ( a) F F ( a) I ( s a D 0). BAD K N. ALL ( a) N I N ( s a ) a, [ L H ] I ( A) 0 A s true otherwse 7/30
8 Idexes ased o dstruto Lorez curve (LC) fucto x y G dex F F. BAD. GOOD ( a) ( a), a [ L, H ]. A G A A B G k ( F F ) ( F F ). BAD k. BAD k. GOOD k. GOOD k F. BAD k F. GOOD where ( ) s k-th vector value of eprcal dstruto fucto of ad (ood) clets k 8/30
9 Idexes ased o dstruto fucto C-statstcs: c stat A C G c It represets the lkelhood that radoly selected ood clet has hher score tha radoly selected ad clet,.e. c stat P ( s s D 0) K D K 9/30
10 0/30 Aother possle dcator of the qualty of scor odel ca e cuulatve Lft, whch says, how ay tes, at a ve level of rejecto, s the scor odel etter tha rado selecto (rado odel). More precsely, the rato dcates the proporto of ad clets wth less tha a score a,, to the proporto of ad clets the eeral populato. Forally, t ca e expressed y: [ ] H L a, ( ) ( ) ( ) ( ) ( ) ( ) N a s I Y a s I Y Y I Y I a s I Y a s I BadRate a CuBadRate a Lft ) ( ) ( Idexes ased o dstruto fucto BadRate a BadRate a aslft ) ( ) (
11 Idexes ased o dstruto fucto Usually t s coputed us tale wth uers of all ad ad clets soe ads (decles). asolutely cuulatvely decle # clets # ad clets Bad rate as. Lft # ad clets Bad rate cu. Lft ,0% 3,0 6 6,0% 3,0 00,0%,40 8 4,0%, ,0%,60 36,0%, ,0%,00 4 0,3%, ,0% 0, ,8%, ,0% 0, ,7%, ,0% 0,0 47 6,7%, ,0% 0,0 48 6,0%,0 9 00,0% 0,0 49 5,4%, ,0% 0,0 50 5,0%,00 All ,0% Lft value 0,8 0,6 0,4 3,50 as. Lft 3,00 cu. Lft,50,00,50,00 0, decle G0,55 0, Lorz curve Base le 0 0 0, 0,4 0,6 0,8 /30
12 Idexes ased o dstruto fucto Whe ad rates are ot ootoe: LC looks fe G s slhtly lowered Lft looks strae asolutely cuulatvely decle # clets # ad clets Bad rate as. Lft # ad clets Bad rate cu. Lft ,0%,60 8 8,0%,60 00,0%,40 0 0,0%, ,0% 3,0 36,0%, ,0%,00 4 0,3%, ,0% 0, ,8%, ,0% 0, ,7%, ,0% 0,0 47 6,7%, ,0% 0,0 48 6,0%,0 9 00,0% 0,0 49 5,4%, ,0% 0,0 50 5,0%,00 All ,0% 0,8 G0,48 3,50 3,00,50 as. Lft cu. Lft 0,6 0,4 Lft value,00,50,00 0, Lorz curve Base le 0 0 0, 0,4 0,6 0,8 0, decle /30
13 Idexes ased o dstruto Whe score s reversed, we ota reversed fures. fucto asolutely cuulatvely decle # clets # ad clets Bad rate as. Lft # ad clets Bad rate cu. Lft ,0% 3,0 6 6,0% 3,0 00,0%,40 8 4,0%, ,0%,60 36,0%, ,0%,00 4 0,3%, ,0% 0, ,8%, ,0% 0, ,7%, ,0% 0,0 47 6,7%, ,0% 0,0 48 6,0%,0 9 00,0% 0,0 49 5,4%, ,0% 0,0 50 5,0%,00 All ,0% asolutely cuulatvely decle # clets # ad clets Bad rate as. Lft # ad clets Bad rate cu. Lft 00,0% 0,0,0% 0,0 00,0% 0,0,0% 0,0 3 00,0% 0,0 3,0% 0,0 4 00,0% 0,0 4,0% 0,0 5 00,0% 0,40 6,% 0, ,0% 0,60 9,5% 0, ,0%,00 4,0% 0, ,0%,60,8% 0, ,0%, ,8% 0, ,0% 3,0 50 5,0%,00 All ,0% Lft value 3,50 as. Lft 3,00 cu. Lft,50,00,50,00 0, decle G - 0,55 0,8 0,6 0,4 0, Lorz curve Base le 0 0 0, 0,4 0,6 0,8 3/30
14 Idexes ased o dstruto The G s ot eouh!!! C : C : decle # clets # ad clets Bad rate ,0% ,0% ,0% ,0% ,0% ,0% ,0% ,0% ,0% ,0% All ,0% decle # clets # ad clets Bad rate ,0% ,0% ,0% ,0% 5 00,0% ,0% ,0% ,0% ,0% 0 00,0% All ,0% 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, 0 fucto ood ad 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, 0 ood ad K , 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 K , 0, 0,3 0,4 0,5 0,6 0,7 0,8 0,9 0,8 0,6 0,4 0, 0 0,8 0,6 0,4 0, 0 G 0.4 G 0,4 0 0, 0,4 0,6 0,8 Lorz curve Base le 0 0, 0,4 0,6 0,8 Lorz curve Base le 4/30
15 Idexes ased o dstruto fucto C : C : 4,00 3,50 3,00 as. Lft cu. Lft,50,00 as. Lft cu. Lft Lft value,50,00,50 Lft value,50,00,00 0,50 0, decle decle Lft 0%.55 > Lft 50%.48 < Lft 0%.90 Lft 50%.64 C s etter f reject rate s expected aroud 50%. C s uch ore etter f reject rate s expected y 0%. 5/30
16 Idexes ased o dstruto fucto Lft ca e expressed ad coputed y forulae: Lft( a) F N. ALL ( a). BAD a [ L, H ] F ( a) Lft q F F F N. ALL ( FN. ALL ( q)) F. BAD N. ( F ( q)) q ( F ( )). BAD q ALL N. ALL N. ALL { a [ L, H], F ( a) q} ( q) N. ALL Lft 0% ( 0 F F (0.)).. BAD N. ALL 6/30
17 Idexes ased o desty fucto Lkelhood desty fuctos: f GOOD (x) f BAD (x) Kerel estates: ~ f GOOD ( x, h), D K K h ( x s ) ~ f BAD ( x, h) D 0 k K h ( x s ) Optal adwdth (axal sooth): h O, k 3 k ( )! ( 5) k k k k ~ k (k 3)! where: k ~ s the order of kerel fucto (e.. for Epaechkov kerel) s uer of actual cases s a estate of stadard devato 7/30
18 Idexes ased o desty fucto Mea dfferece (Mahalaos dstace): D M M where s pooled stadard devato: M, M are eas of ood (ad) clets, are stadard devatos of ood (ad) clets 8/30
19 Idexes ased o desty fucto Iforato value (I val ) cotuous case (Dverece): f I val dff f ( x) GOOD BAD f BAD x ( ) GOOD ( f ( x) f ( x) ) l dx ( x) f ( x) f ( x) GOOD BAD f LR ( x) l f f GOOD BAD ( x) ( x) 9/30
20 Iforato value (I val ) dscretzed cotuous case: Replace desty fuctos y ther kerel estates ad copute teral uercally (e.. y coposte trapezodal rule). Us Epaechkov kerel, ve y ad optal adwdth we have ~ f ( x) For ve M pots Idexes ased o desty IV h O, k we ota ( x ) I [,] 3 K( x) x 4 ( ) ~ ~ ( ) fgood( x, ho,) fgood ( x, ho,) fbad( x, ho,) l ~ fbad( x, ho,) x 0, K, x M fucto ~ I val M xm x0 ~ ~ ~ fiv ( x0) fiv ( x ) fiv ( xm ) M x0 xm 0/30
21 Idexes ased o desty fucto Iforato statstcs/value (I val ) dscrete case: Create tervals of score typcally decles. Nuer of oods (ads) -th terval s arked y ( ). It ust holds > 0, > 0 The we have I val l score t. # ad clets #ood clets % ad [] % ood [] [3] [] - [] [4] [] / [] [5] l[4] [6] [3] * [5] 0,0%,% -0,0 0,53-0,64 0,0 5 4,0%,6% -0,0 0,39-0,93 0, ,0% 5,5% -0, 0,34 -,07 0, ,0% 9,8% -0,8 0,35 -,05 0, ,0% 5,4% -0,05 0,77-0,6 0, ,0% 6,0% 0,4,7 0,77 0, ,0% 4,4% 0,06,80 0,59 0, ,0%,% 0,05,84 0,6 0, ,0% 0,% 0,08 5,,63 0,3 0 48,0% 5,% 0,03,53 0,93 0,03 All Ifo. Value 0,68 /30
22 Iforato value for our exaple of two scorecards: C : C : Idexes ased o desty fucto decle # clets # ad clets #ood % ad [] % ood [] [3] [] - [] [4] [] / [] [5] l[4] [6] [3] * [5] cu. [6] ,0% 7,% -0,8 0, -,58 0,44 0, ,0% 9,3% -0,07 0,58-0,54 0,04 0, ,0% 0,% 0,0,8 0,5 0,0 0, ,0% 0,% 0,0,8 0,5 0,0 0, ,0% 0,3% 0,03,48 0,39 0,0 0, ,0% 0,4% 0,04,74 0,55 0,0 0, ,0% 0,4% 0,04,74 0,55 0,0 0, ,0% 0,6% 0,06, 0,75 0,04 0, ,0% 0,6% 0,06, 0,75 0,04 0, ,0% 0,7% 0,07,67 0,98 0,07 0,70 All Ifo. Value 0,70 decle # clets # ad clets #ood % ad [] % ood [] [3] [] - [] [4] [] / [] [5] l[4] [6] [3] * [5] cu. [6] ,0% 8,9% -0, 0,44-0,8 0,09 0, ,0% 9,% -0,09 0,5-0,68 0,06 0, ,0% 9,% -0,08 0,54-0,6 0,05 0, ,0% 9,4% -0,06 0,63-0,46 0,03 0, ,0% 9,8% -0,0 0,8-0,0 0,00 0, ,0% 0,4% 0,04,74 0,55 0,0 0, ,0% 0,7% 0,07,67 0,98 0,07 0, ,0% 0,8% 0,08 3,59,8 0,0 0, ,0% 0,8% 0,08 3,59,8 0,0 0, ,0% 0,9% 0,09 5,44,69 0,5 0,67 All Ifo. Value 0,67 /30
23 Idexes ased o desty Us arks C : C : 0,0 0,05 0,00-0,05-0,0-0,5-0,0-0,5-0,30 0,0 0,05 0,00-0,05-0,0-0,5 I_dff I_LR I dff I_dff I_LR ,50,00 0,50 0,00-0,50 -,00 -,50 -,00,00,50,00 0,50 0,00 I -0,50 -,00 LR fucto l we have: 0,50 I_df * I_LR 0,45 cu. I_dff * I_LR 0,40 0,35 0,30 0,5 0,0 0,5 0,0 0,05 0, ,6 I_df * I_LR 0,4 cu. I_dff * I_LR 0, 0,0 0,08 0,06 0,04 0,0 0, ,80 0,70 0,60 0,50 0,40 0,30 0,0 0,0 0,00 0,80 0,70 0,60 0,50 0,40 0,30 0,0 0,0 0,00 K G 0.4 Lft 0%.55 Lft 50%.48 I val 0.70 I val0% 0.47 I val50% 0.50 K G 0.4 Lft 0%.90 Lft 50%.64 I val 0.67 I val0% 0.5 I val50% 0.3 3/30
24 4/30 Assue that the scores of ood ad ad clets are orally dstruted,.e. we ca wrte ther destes as Estates of paraeters ad : Pooled stadard devato: Estates of ea ad stadard dev. of scores for all clets : oe results for orally dstruted scores ( ) ) ( x GOOD e x f µ π ( ) ) ( x BAD e x f µ π µ µ,,,. M M, are stadard devatos of ood (ad) clets, are eas of ood (ad) clets M M M M ALL ( ) ALL ALL ALL µ,
25 µ µ D D D K Φ Φ D G Φ Lft q I val D oe results for orally Φ q ALL dstruted scores Assue that stadard devatos are equal to a coo value : ( ) Φ D Φ ( q) p D G Where Φ s the stadardzed oral dstruto fucto, Φ the oral dstruto fucto wth paraeters, ad Φ µ ( ), ( ) s the stadard quatle fucto. µ Lft q D M Φ q ALL Φ M ( q) p D G 5/30
26 6/30 Geerally (.e. wthout assupto of equalty of stadard devatos): Φ Φ c D a D a c D a D a K * * * * oe results for orally dstruted scores, a * D µ µ * M M D where c l, ( ) ( ) ( ) ( ) Φ Φ D D D D K l l * * * *
27 7/30 Geerally (.e. wthout assupto of equalty of stadard devatos): ( ) * Φ D G oe results for orally dstruted scores ( ) ( ) ( ) Φ Φ Φ Φ ALL ALL ALL ALL q q q q q Lft µ µ µ µ, *, ) ( val A A D A I *, ) ( val A A D A I ( ) Φ Φ ALL q M M q q Lft
28 oe results for orally dstruted scores K: µ 0, G µ 0, K ad the G react uch ore to chae of µ ad are alost uchaed the drecto of. G > K 8/30
29 oe results for orally dstruted scores Lft 0% : µ 0, I val : µ 0, I case of Lft 0% t s evdet stro depedece o µ ad sfcatly hher depedece o tha case of K ad G. Aa stro depedece o µ. Furtherore value of I val rses very quckly to fty whe teds to zero. 9/30
30 Coclusos It s possle to use scor odel effectvely wthout kow how ood t s. It s ecessary to jude scor odels accord to ther streth score rae where cutoff s expected. The G s ot eouh! Results cocer Lft ad Iforato value ca e used to ota the est avalale scor odel. Results for orally dstruted scores ca help wth coputato of referred dexes. Furtherore they ca help to uderstad how those dexes ehave. 30/30
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