THE GINI COEFFICIENT

Transcription

1 THE GINI COEFFICIENT The Gii coefficiet is a statistic which measures the ability of a scorecard or a characteristic to rak order risk. A Gii value of 0% meas that the characteristic caot distiguish good from bad cases, e..g. Goods Bads Bad Rate Phoe 80% 80% 15% No Phoe 20% 20% 15% A Gii value of 100% meas that a characteristic/scorecard distiguishes perfectly. Goods Bads Bad Rate Phoe 100% 0% 0% No Phoe 0% 100% 100% A typical credit scorecard has a Gii coefficiet of 40-60%. Behaviour scorecards have values of 70-80%. A very powerful characteristic ca have a Gii coefficiet of 25%. To calculate Gii values, assume that oe has good ad bad accouts rak ordered by score with the score sufficietly fiely graded such as that there is oly oe case per score. The essetial otio is that of a flip. A flip is a traspositio of cosecutive good ad bad accouts. The Gii coefficiet is the percetage of flips required to reach the rak orderig from a radom assigmet of goods ad bads by score (i.e. with Gii = 0). The example below illustrates this, with the accouts to be flipped uderlied. EXAMPLE (b = bad, g = good, cases raked i icreasig score order) (1) b b g b g b b g g g b g g g g (the startig order, raked by score) 2) b b b g g b b g g g b g g g g 3) b b b g b g b g g g b g g g g 4) b b b b g g b g g g b g g g g 5) b b b b g b g g g g b g g g g 6) b b b b b g g g g g b g g g g 7) b b b b b g g g g b g g g g g 8) b b b b b g g g b g g g g g g 9) b b b b b g g b g g g g g g g 10) b b b b b g b g g g g g g g g b b b b b b g g g g g g g g g 5/13/2004 Gii Coefficiet 1/6

2 Therefore there are 10 flips to reach the ideal state from the actual (where the ideal state is that show after flip 10). A scorecard that had o effect would rak the goods ad bads radomly, therefore o average each bad would have to flip with half of the goods. Therefore to get from the radom state to the ideal state would take 6 x 9/2 flips (there are 6 bad ad 9 goods) = 27 flips. Therefore, to reach state (1) from the radom state would take flips = 17 flips out of a maximum of 27. The Gii coefficiet is the percetage of such flips = 17/27 = 63%. Where there is more tha oe accout per score iterval, or where oe is dealig with a discrete characteristic a refiemet to the calculatio is ecessary, but the priciple is the same. Assume we have a characteristic with attributes (or alteratively a scorecard with scores or score itervals). Let A 1,..., A be the attributes. Let G i be the probability of beig attribute i give that the accout is good, ad B i be the probability of beig attribute i give that the accout is bad. Let G be the vector of G i s ad B be the verctor of B i s, e.g.: G B Teat`.2.5 LWP.1.1 Ower.7.4 Let U be the upper triagular matrix. The the Gii coefficiet is give by the formula. = G (U T -U) B E.G., from the values above, U = & = [.2,.1,.7] = [.8,.5,-3].5.1 = 0.33 (or 33%).4 There are two alterative calculatio methods. The first uses the logic i the followig SAS program : INPUT GOODS BADS; BETWEEN + 2 X CUMGOODS X BADS; WITHIN + GOODS * BADS; CUMGOODS + GOODS; CUMBADS + BADS; GINI = 100 x 1 - BETWEEN+WITHIN CUMGOODS x CUMBADS 5/13/2004 Gii Coefficiet 2/6

3 which is a derivative of the formula below: 1 - r 2 (B r s r 1 1 G s ) + ½ r 1 = 100 ½ r 1 G r r 1 B r G r B r The other is a direct calculatio of the shaded area i the followig graph expressed as a percetage of the triagle above the diagoal. Scorecard Ascedig Cumulative Bads % Ascedig Cumulative Good% 5/13/2004 Gii Coefficiet 3/6

4 Characteristic with 3 attributes (represeted by the 3 lies) Ascedig 0 Cumulative Bads % 0 Key poits regardig the Gii are : Ascedig Cumulative Good% - it is uaffected by uiformly weightig either the goods or the bads. - it is very simple to calculate it for a biary characteristic e.g. G B Teat.3.5 Ower.7.5 The Gii is just = 0.2 (i.e. 20%) - Whe calculatig it for a cotiuous characteristic, oe usually assumes that the rak orderig is the atural oe. Whe calculatig it for a discrete characteristic, oe assumes that the rak orderig is a logical oe if oe is cocered with the fie-classed values, ad a reducig bad rate order if oe is cocered with course-classed values. - Negative Gii values are possible if the characteristic rak orders i the opposite way to expectatio. - The Gii coefficiet has certai logical qualities. For istace, if a attribute is split ito two such that each compoet has the same bad rate the the Gii value will ot chage. This desirable feature is ot mirrored by other power measures such as Divergece. 5/13/2004 Gii Coefficiet 4/6

5 Example Pascal Gii ad R-Squared Program Program GINI; Uses CRT; Var ifile : strig; f,g : text; : iteger; cumgoods, cumbads, goods, bads : real; withi, betwee, mss, tss, giy, rsq : real; Begi assig(g,'statlist.lst'); rewrite(g); Writel(g,'Program GINI'); Writel(g); ifile :='dat1.txt'; Write (g,'file : '); Writel(g,ifile); writel(g); Assig(f,ifile); Reset(f); cumgoods := 0; cumbads := 0; withi := 0; betwee := 0; := 0; mss := 0; tss := 0; While ot eof(f) do Begi Readl(f, goods, bads); := + 1; mss := mss + (goods * bads) / (goods + bads); betwee := betwee + 2 * bads * cumgoods; withi := withi + goods * bads; cumgoods := cumgoods + goods; cumbads := cumbads + bads; Ed; tss := (cumgoods * cumbads) / (cumgoods + cumbads); rsq := 100 * (1 - mss / tss); giy := 100 * (1 - (withi + betwee) / (cumgoods * cumbads)); Write (g,'cumulative Goods : '); Writel (g,cumgoods:8:2); Write (g,'cumulative Bads : '); Writel(g,cumbads:8:2); Writel(g); Write(g,'Gii % : '); Writel(g,giy:8:2); Write(g,'R-Squared % : '); Writel(g,rsq:8:2); close(f); close(g); Ed. Example Quick Basic Gii ad R-Squared Program Program to calculate GINI coefficiets COLOR 0, 7, 8 CLS PRINT Program GINI : PRINT INPUT Iput File : ; ifile$ OPEN I, 1, ifile$ cumgoods = 0 cumbads = 0 withi = 0 betwee = 0 = 0 DO WHILE NOT EOF (1) INPUT 1, goods, bads = + 1 5/13/2004 Gii Coefficiet 5/6

6 betwee = betwee + 2 * bads * cumgoods withi = withi + goods * bads cumgoods = cumgoods + goods cumbads = cumbads + bads LOOP PRINT Cumulative Goods : ; cumgoods PRINT Cumulative Bads : ; cumbads PRINT gii = 100 * (1 - (withi + betwee) / (cumgoods * cumbads)) PRINT Gii : ; : PRINT USING. ; gii END Ala Lucas Rhio Risk Ltd. Alalucas@rhiorisk.com 5/13/2004 Gii Coefficiet 6/6