A credit scorecard is a logistic regression rewritten in points. Three choices fix the scale; the arithmetic that follows turns every coefficient and weight of evidence into a number an underwriter can add up.
Choose the points to double the odds (PDO), an anchor score and the odds at that anchor, then derive two constants: factor = PDO / ln 2 and offset = anchor − factor × ln(anchor odds). With 20 points to double the odds and 600 points at 50 to 1, the factor is 28.8539 and the offset 487.1229. Every bin then scores minus its coefficient times its weight of evidence times the factor.
Worked in full in Machine Learning for Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The card is the Brechin small-business scorecard from Machine Learning for Finance: a logistic regression on seven characteristics, each entered as the weight of evidence (WoE) of its bin, fitted on 6,000 development applications. The regression models the log-odds of bad, which is why its coefficients are negative: a bin with a positive WoE holds proportionally more goods, and should lower the log-odds of bad. All figures are illustrative.
| Input | Value |
|---|---|
| Points to double the odds (PDO) | 20 |
| Anchor score | 600 |
| Odds of good to bad at the anchor | 50 |
| Fitted intercept, log-odds of bad | −2.6983 |
A score is a linear function of the log-odds of good: score = offset + factor × ln(odds). Two conditions pin down the two constants. Adding PDO points must double the odds, so factor × ln 2 = PDO. And the anchor score must sit at the anchor odds.
Factor = PDO / ln 2 = 20 / 0.693147 = 28.8539
Offset = anchor − factor × ln(odds) = 600 − 28.8539 × 3.912023 = 487.1229
Excel: =PDO/LN(2) and =Anchor-Factor*LN(Odds)
Check: 620 points should be odds of 100 to 1, and 580 points 25 to 1. Both hold, and 600 points corresponds to a probability of default of 1.96 per cent.
The log-odds of good is minus the model's log-odds of bad: −(intercept + Σ coefficient × WoE). Multiply through by the factor and add the offset, and the score splits into a constant and one term per characteristic.
Base score = offset − factor × intercept = 487.1229 + 28.8539 × 2.6983 = 564.98
Points for a bin = −coefficient × WoE × factor
Excel: =-Coef*WoE*Factor
Score one applicant: a construction business trading for three years, with debt service cover of 1.3x, net debt at four times EBITDA, no county court judgments, a good bank conduct score and a loan at 65 per cent of value.
| Characteristic | Bin | Coefficient | WoE | Points |
|---|---|---|---|---|
| Base score | 564.98 | |||
| Age of business | 24-47 months | −1.0021 | 0.1013 | 2.93 |
| Debt service cover | 1.25-1.49 | −1.0890 | 0.1062 | 3.34 |
| Net debt / EBITDA | 3.50-4.49 | −0.9934 | −0.1452 | −4.16 |
| Sector | Construction | −1.0810 | −0.2878 | −8.98 |
| Worst CCJ | None | −1.0351 | 0.2693 | 8.04 |
| Bank conduct score | 75-84 | −1.1475 | 0.2306 | 7.64 |
| Loan to value | 0.60-0.74 | −1.1686 | −0.0146 | −0.49 |
| Score | 573.30 |
The scale is invertible, so a score is always a statement about odds.
Odds of good = e(score − offset) / factor = e(573.30 − 487.1229) / 28.8539 = 19.82
Probability of default = 1 / (1 + odds) = 4.80 per cent
The same 4.80 per cent comes straight out of the logistic regression, which is the check that the points table has been built correctly. If the card is published with points rounded to whole numbers, as most are, this applicant scores 574 rather than 573: rounding each bin separately adds up to a point of noise, which matters only for applicants sitting on the cut-off.
PDO, anchor and odds are presentation choices. Change them and every number on the card moves, but no applicant's probability of default and no ranking changes.
| PDO | Anchor | Odds | Factor | Offset | Base score | Applicant | PD | Old 550 cut-off becomes |
|---|---|---|---|---|---|---|---|---|
| 20 | 600 | 50 | 28.8539 | 487.12 | 564.98 | 573.30 | 4.80% | 550.00 |
| 40 | 600 | 50 | 57.7078 | 374.25 | 529.96 | 546.59 | 4.80% | 500.00 |
| 20 | 600 | 20 | 28.8539 | 513.56 | 591.42 | 599.73 | 4.80% | 576.44 |
| 20 | 500 | 50 | 28.8539 | 387.12 | 464.98 | 473.30 | 4.80% | 450.00 |
Doubling PDO doubles the spread of every characteristic: debt service cover goes from a range of 47.16 points across its bins to 94.31. That makes the card look more discriminating and is not. The one thing that must move with the scale is the cut-off, and the last column shows where it goes.
Policy rules written in points, such as a referral band or a manual override threshold, are tied to the scale they were written on. A rescaled card with unchanged policy rules is a different credit policy.
The most frequent error is mixing logarithms: the factor computed with the natural log and the offset with base 10, or the reverse. Using base 10 throughout, coefficients and weights of evidence included, is harmless: the factor becomes 66.4386 and the scores are identical. Mixed, the offset becomes 550.98 instead of 487.1229, and every score on the card moves up by 63.86 points while the cut-off stays put. The second is a sign error: forgetting that the regression models bad, so that the best debt service cover bin scores −29.39 points instead of +29.39 and the card ranks backwards. Both errors are caught by the same test: score one application, convert it back to a probability, and compare it with the regression's prediction.
The scale fixes how points read; where the line is drawn on it is a separate economic decision, worked in what leaving a credit score cut-off unchanged actually costs.
The bins, weights of evidence and points table are built from counts in the free workbook for this case, where changing PDO from 20 to 40 shows the same thing on all 8,400 applications.
Points to double the odds: the number of points by which the score rises when the odds of good to bad double. With a PDO of 20 and a score of 600 at 50 to 1, a score of 620 means 100 to 1 and 580 means 25 to 1. The factor that converts log-odds into points is PDO divided by ln 2, here 28.8539.
Invert the scale. Odds of good to bad equal e to the power of (score minus offset) divided by factor, and the probability of default is 1 over (1 + odds). At 573.30 on a scale with offset 487.1229 and factor 28.8539 the odds are 19.82 to 1, a probability of default of 4.80 per cent.
No, provided the cut-off is moved with it. The scale is a unit of measurement. Doubling PDO to 40 doubles every bin's points and moves the base score to 529.96, but the applicant's probability of default stays 4.80 per cent and a cut-off of 550 on the old scale becomes 500.00 on the new one.
This article is one calculation from Machine Learning for Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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