How to calibrate a PD when your book has never defaulted
Zero defaults is not a zero probability of default, and it is not a reason to pick a number. It is data — and there is a standard, supervisory answer to what it is worth. On eight hundred obligors with nothing lost, the probability of default is 0.287%.
A lender or a group treasury has a book that has never lost anything. No defaults, often no internal rating, and nothing an agency has ever looked at. IFRS 9 still requires a probability-weighted expected loss, and an auditor still asks where the number came from.
What usually happens next is that somebody picks one. The honest objection to a figure chosen that way is not that it is too high or too low — it is that nothing determines it. It cannot be defended, it cannot be reproduced next period, and it moves whenever a different person is asked.
Zero cannot be used either, and not merely because it looks imprudent. A normal confidence interval on zero defaults returns zero — which is not a probability of default, it is an absence of arithmetic.
The largest probability still consistent with having seen nothing.
Observing no defaults in n obligors rules out high probabilities and says nothing about low ones. The question with an answer is: what is the largest PD that would still, reasonably often, have produced no defaults at all? That is an upper confidence bound, and at zero defaults it has a closed form — one minus the n-th root of one minus the confidence level.
It is the Clopper–Pearson exact binomial limit, which at zero defaults reduces to the closed form Pluto and Tasche give for low-default portfolios. Exact rather than approximated, because the approximations fail at precisely the default counts that motivate the question.
| Obligors, none defaulted | PD at 90% confidence |
|---|---|
| 25 | 8.80% |
| 50 | 4.50% |
| 100 | 2.28% |
| 200 | 1.14% |
| 400 | 0.574% |
| 800 | 0.287% |
| 2,000 | 0.115% |
| 5,000 | 0.046% |
The shape is the useful part. It falls fast and then flattens, so a small book cannot reach a low number by waiting — and a book of a few hundred is already most of the way to what five thousand would give you.
The confidence level is chosen, not derived, and it moves the answer by a factor of six across the range supervisors actually use. That is published here rather than buried, because a reviewer who finds it themselves stops trusting everything above it. Eight hundred obligors, none defaulted:
| Confidence | PD |
|---|---|
| 50% | 0.087% |
| 75% | 0.173% |
| 90%shipped default | 0.287% |
| 95% | 0.374% |
| 99% | 0.574% |
Whatever is chosen has to be recorded and applied consistently across periods. A confidence level quietly lowered between two reporting dates is a release to profit that the movement analysis will attribute to the book improving.
A grade with no defaults cannot be estimated from its own obligors.
But the scale carries information the grade does not. If grades are ordered by credit quality then the best grade’s PD is no higher than the next one’s, so the most prudent value for it is the one that also has to accommodate every worse grade’s experience. Each grade is estimated on itself pooled with all the grades below it.
That is what makes the result monotone by construction. Estimate each grade in isolation and on a small book the top three all come out at zero and the scale is flat — which is not a conservative estimate, it is a broken rating scale, and it is what a bound applied naively produces.
| Grade | Observed | Pooled with worse grades | PD |
|---|---|---|---|
| Internal 1 | 0 / 40 | 2 / 660 | 0.804% |
| Internal 2 | 0 / 120 | 2 / 620 | 0.856% |
| Internal 3 | 0 / 260 | 2 / 500 | 1.06% |
| Internal 4 | 1 / 180 | 2 / 240 | 2.20% |
| Internal 5 | 1 / 60 | 1 / 60 | 6.33% |
Two defaults in the whole book, both in the worst grade. Observed rates would read 0, 0, 0, 0.6%, 1.7% — flat and then a jump. Pooled, the scale rises at every step because the order of the grades is the model.
It is an upper bound, not a best estimate, and it should be described that way wherever it reaches a disclosure. It is deliberately conservative: the whole construction asks what is still consistent with the evidence rather than what is most likely given it.
It is also not a substitute for default history. It is what a defensible number looks like until there is some — derived from the absence of evidence rather than in spite of it — and it should fall as the book ages and the count grows. A bound that never moves is a sign nobody is re-running it.
Pluto and Tasche (2005), Estimating Probabilities of Default for Low Default Portfolios — the most prudent estimation principle, and the ordering across grades. Clopper and Pearson (1934) for the exact binomial interval, whose upper limit is the quantile of Beta(k+1, n−k) and which reduces at k = 0 to the closed form above.