Prob of customer default (PD): p ( Y = 1 ∣ x ) = p p(Y=1|x) = p p(Y=1∣x)=p Prob of customer not default: p ( Y = 0 ∣ x ) = 1 − p p(Y=0|x) = 1-p p(Y=0∣x)=1−p Odds, bad/good: o d d s = p 1 − p odds = \frac{p}{1-p} odds=1−pp Formula for Prob to Score can be expressed as: s c o r e = A + B ∗ l n ( o d d s ) score = A+B*ln(odds) score=A+B∗ln(odds)
set p 0 p_{0} p0 when o d d s = Θ 0 odds =\Theta _{0} odds=Θ0 when Odds double, number of points increase PDO（point of double odds） formula is: { p 0 = A + B l n ( Θ 0 ) p 0 + P D O = A + B l n ( 2 Θ 0 ) \left\{\begin{matrix} p_{0}=A+Bln(\Theta _{0}) \\ p_{0}+PDO=A+Bln(2\Theta _{0}) \end{matrix}\right. {p0=A+Bln(Θ0)p0+PDO=A+Bln(2Θ0) Calculate A, B { B = P D O l n ( 2 ) A = p 0 − B l n ( Θ 0 ) ) \left\{\begin{matrix} B=\frac{PDO}{ln(2)}\\ A=p_{0}-Bln(\Theta _{0})) \end{matrix}\right. {B=ln(2)PDOA=p0−Bln(Θ0))
Set Θ 0 = 1 / 60 ， p 0 = 600 ， P D O = − 20 \Theta _{0}=1/60，p_{0}=600，PDO=-20 Θ0=1/60，p0=600，PDO=−20 which means: when Odds = 1/60, score is 600, and when odds doubled, 1/30 in this case, score increased by -20 points (higher the odds, more the bad, lower the score) { B = − 20 l n ( 2 ) A = 600 − B l n ( 1 / 60 ) ) \left\{\begin{matrix} B=\frac{-20}{ln(2)}\\ A=600-Bln(1/60)) \end{matrix}\right. {B=ln(2)−20A=600−Bln(1/60)) s c o r e = 481.89 − 28.85 ∗ l n ( o d d s ) score = 481.89-28.85*ln(odds) score=481.89−28.85∗ln(odds)
