A stable score and a rising default rate
The bank, the model and every number in this exercise are fictional, made up for practice.
Background
A regional bank uses a probability-of-default model for small business loans. The quarterly monitoring report compares the distribution of model scores this quarter with the distribution when the model was developed, and compares predicted with actual defaults on loans booked a year earlier.
| Score band | At development | This quarter |
|---|---|---|
| 1 (lowest risk) | 10% | 8% |
| 2 | 20% | 18% |
| 3 | 40% | 38% |
| 4 | 20% | 22% |
| 5 (highest risk) | 10% | 14% |
| Item | Value |
|---|---|
| Loans | 5,000 |
| Average predicted 12-month probability of default | 3.0% |
| Loans that defaulted within 12 months | 210 |
Your task
- Calculate the population stability index (PSI) for the score distribution. Is the population stable by the common rule of thumb (below 0.1 stable, 0.1 to 0.25 a moderate shift, above 0.25 a significant shift)?
- Are defaults in line with the prediction? How confident are you?
- What would you tell the bank’s model owner, and what would you do next?
One way to approach it
PSI. For each band, (this quarter − development) × ln(this quarter ÷ development):
- Band 1: (0.08 − 0.10) × ln(0.8) = −0.02 × −0.2231 = 0.0045
- Band 2: (0.18 − 0.20) × ln(0.9) = −0.02 × −0.1054 = 0.0021
- Band 3: (0.38 − 0.40) × ln(0.95) = −0.02 × −0.0513 = 0.0010
- Band 4: (0.22 − 0.20) × ln(1.1) = 0.02 × 0.0953 = 0.0019
- Band 5: (0.14 − 0.10) × ln(1.4) = 0.04 × 0.3365 = 0.0135
Total PSI ≈ 0.023: stable by the rule of thumb. The only notable movement is a shift into the highest-risk band, which contributes most of the total.
Defaults. The model predicted 3.0% × 5,000 = 150 defaults; there were 210, 40% more. Treating defaults as independent, the standard deviation of the count is √(5,000 × 0.03 × 0.97) = √145.5 ≈ 12.1, so 210 is about (210 − 150) ÷ 12.1 ≈ 5 standard deviations above the prediction. Even allowing for correlated defaults, which widen the range in practice, that is far outside normal variation.
What it means. The inputs look stable but the outcomes do not: the relationship between score and default has changed. A stable PSI was never evidence that the model still works; it only shows that the population looks similar. Likely causes to check: an economic change affecting small businesses, a change in the bank’s lending policy or product, or a change in how defaults are recorded. Break the comparison down by score band and by segment to see where the gap is.
What to tell the model owner. The model is underpredicting defaults materially; that should be escalated under the bank’s model risk policy now, not at the next annual review. Interim options include an overlay on predicted defaults while the cause is investigated, closer monitoring, and a decision on whether to recalibrate or redevelop. Under SR 11-7, ongoing monitoring and outcomes analysis exist for exactly this situation, and the report should record what was found and what was decided.