Rank the book by score, cut it into deciles, and read lift against the break-even bad rate rather than against 1.
Sort the book by score, riskiest first, cut it into ten equal deciles, and divide each decile's bad rate by the bad rate of the whole book: that is lift. Cumulative gains is the share of all bads caught by the first k deciles. On an illustrative book of 10,000 loans at 6.25 per cent bad, the riskiest decile has a lift of 2.69, and the riskiest 30 per cent of applicants hold 59.5 per cent of the bads. What makes the table useful is the decile where lift falls below 1.318, the point at which declining stops paying.
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 →
A fictional small-business lender has scored 10,000 loans that have run their full outcome window. 625 of them went bad, a base rate of 6.25 per cent. The loans are ranked by score and cut into deciles of 1,000 each, decile 1 being the riskiest. Every figure is illustrative.
| Decile | Bads | Bad rate | Lift | Cumulative bads | Cumulative gains | Cumulative lift |
|---|---|---|---|---|---|---|
| 1 | 168 | 16.8% | 2.69 | 168 | 26.9% | 2.69 |
| 2 | 116 | 11.6% | 1.86 | 284 | 45.4% | 2.27 |
| 3 | 88 | 8.8% | 1.41 | 372 | 59.5% | 1.98 |
| 4 | 69 | 6.9% | 1.10 | 441 | 70.6% | 1.76 |
| 5 | 55 | 5.5% | 0.88 | 496 | 79.4% | 1.59 |
| 6 | 43 | 4.3% | 0.69 | 539 | 86.2% | 1.44 |
| 7 | 33 | 3.3% | 0.53 | 572 | 91.5% | 1.31 |
| 8 | 25 | 2.5% | 0.40 | 597 | 95.5% | 1.19 |
| 9 | 17 | 1.7% | 0.27 | 614 | 98.2% | 1.09 |
| 10 | 11 | 1.1% | 0.18 | 625 | 100.0% | 1.00 |
| Book | 625 | 6.25% | 1.00 |
Decile lift = (bads in decile ÷ loans in decile) ÷ (all bads ÷ all loans)
Cumulative gains at decile k = bads in deciles 1 to k ÷ all bads
Cumulative lift at decile k = cumulative gains ÷ (k ÷ 10)
Excel, with bads in B2:B11 and 1,000 loans a decile: =(B2/1000)/(SUM($B$2:$B$11)/10000) for lift, =SUM($B$2:B2)/SUM($B$2:$B$11) for gains.
Step 1, the base rate. 625 bads over 10,000 loans is 6.25 per cent. Every lift is measured against this one number.
Step 2, decile lift. Decile 1 holds 168 bads in 1,000 loans, 16.8 per cent. Divided by 6.25 per cent, the lift is 2.69: a loan in that decile is 2.69 times as likely to go bad as a loan picked at random. By decile 5 the lift is 0.88, below average, and the safest decile reads 0.18.
Step 3, cumulative gains. Add the bads down the table and divide by 625. The first two deciles hold 284 bads, 45.4 per cent of them, in 20 per cent of the book. A random score would catch 20 per cent; a perfect one would catch all 625 in the first decile alone, because 625 is fewer than 1,000.
Step 4, cumulative lift. 45.4 per cent of bads in 20 per cent of the loans is a cumulative lift of 2.27. It always ends at 1.00, because the whole book holds all the bads.
The gains curve and the cumulative accuracy profile are the same curve. Its area, measured against a perfect model, gives the accuracy ratio, which equals the Gini. From these ten bands it comes to 43.4 per cent, slightly lower than a loan-by-loan figure would be because ties inside a decile are counted half. The method is in how to calculate Gini and KS.
Lift on its own describes ranking. To decide anything, set it against what a loan earns. Take the unit economics of the book's fictional lender: a loan that performs earns 6,982, a loan that defaults loses 77,785. One bad wipes out 11.14 good loans, so a group of applicants is worth declining only if its bad rate exceeds 6,982 ÷ (6,982 + 77,785), or 8.2367 per cent. Against a 6.25 per cent book, that is a break-even lift of 1.318.
Read down the lift column. Deciles 1 to 3 are above 1.318 (2.69, 1.86 and 1.41) and lose money; decile 4 at 1.10 earns 1,133,077. The profit-maximising rule declines the riskiest 30 per cent.
| Declined | Bads avoided | Share of bads | Goods turned away | Profit on the book |
|---|---|---|---|---|
| None | 0 | 0.0% | 0 | 16,840,625 |
| Riskiest 10% | 168 | 26.9% | 832 | 24,099,481 |
| Riskiest 20% | 284 | 45.4% | 1,716 | 26,950,453 |
| Riskiest 30% | 372 | 59.5% | 2,628 | 27,427,949 |
| Riskiest 40% | 441 | 70.6% | 3,559 | 26,294,872 |
| Riskiest 50% | 496 | 79.4% | 4,504 | 23,975,057 |
The marginal decile decides, not the average. Declining the first 40 per cent turns away 8.07 goods per bad on average, comfortably below 11.14, yet it earns less than declining 30, because decile 4 on its own turns away 13.49 goods for each bad it stops. Decile 3 turns away 10.36, which is why it stays out.
Model B sees the same 625 bads but spreads them more evenly: 130 in the riskiest decile, 15 in the safest.
| Measure | Model A | Model B |
|---|---|---|
| Top-decile lift | 2.69 | 2.08 |
| Gains at 20% | 45.4% | 37.6% |
| Gains at 30% | 59.5% | 51.2% |
| Accuracy ratio from deciles | 43.4% | 33.1% |
| Best decline rule | Riskiest 30% | Riskiest 30% |
| Profit at the best rule | 27,427,949 | 23,020,065 |
At the same 30 per cent decline rate, model B lets through 52 more bads, 8.3 points of gains, and the book earns 4,407,884 less. Same approval rate, same applicants: the whole difference is ranking.
Comparing lift across books with different bad rates. Lift has a ceiling set by the base rate. At 6.25 per cent bad, a perfect model puts every bad in the top decile, a lift of 10.00, and model A reaches 26.9 per cent of that ceiling. On a book running at 15 per cent bad, the ceiling is 6.67, and the same 16.8 per cent top-decile bad rate is a lift of only 1.12. A vendor quoting a top-decile lift of 4 on a prime card and 2 on a subprime one may be describing two equally good models. Compare lift only on the same population, and say which population it is.
The second mistake is recutting the deciles on each new sample. On a monitoring sample the bands should stay on the development score boundaries, or a population shift disappears into the cut points.
The bands, the unit economics and the cut-off grid are all live in the free workbooks for this case, and what an unchanged cut-off costs shows the same economics moving over time.
It depends on the bad rate, because the ceiling does. At a 6.25 per cent bad rate a perfect model reaches a top-decile lift of 10.00; on a book at 15 per cent bad the ceiling is 6.67. An illustrative application card with a top-decile lift of 2.69 reaches 26.9 per cent of that ceiling, with an accuracy ratio of 43.4 per cent. Compare lifts only on the same population.
They plot the same counts two ways. The gains chart shows the cumulative share of bads caught against the share of the book, for example 45.4 per cent of bads in the riskiest 20 per cent. The lift chart divides that by the share of the book, giving a cumulative lift of 2.27 at the same point. Gains rises to 100 per cent; cumulative lift falls to 1.00.
Yes, once it is scaled. The gains curve is the cumulative accuracy profile, and its area above the diagonal divided by the same area for a perfect model is the accuracy ratio, which equals the Gini. On ten illustrative deciles the area is 0.7033 and the accuracy ratio 43.4 per cent, slightly below a loan-level figure because ties within a decile count half.
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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