Constant hazard, a transition matrix, and what the choice does to an IFRS 9 lifetime provision.
Chain the 12-month PD through survival, never multiply it by the years: with a constant hazard, lifetime PD = 1 − (1 − PD12m)T, so a 1.00 per cent PD becomes 4.90 per cent over five years. That assumes the borrower's risk never changes. Run the same borrower through an illustrative rating transition matrix and the five-year PD is 8.44 per cent, 1.72 times the constant-hazard figure, while a high-risk borrower's falls from 34.09 to 27.10 per cent.
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 scorecard or a PD model is usually calibrated to a twelve-month default window. IFRS 9 needs more: stage 1 loans carry twelve-month expected credit loss, but once credit risk has increased significantly the provision moves to lifetime loss, and lifetime loss needs a PD for every remaining year. The conversion looks like a formula; it is really an assumption about how risk evolves.
| From \ to | Low risk (L) | High risk (H) | Default |
|---|---|---|---|
| Low risk (L) | 92% | 7% | 1% |
| High risk (H) | 15% | 77% | 8% |
| Default | 0% | 0% | 100% |
The default column is each grade's 12-month PD: 1.00 per cent for L and 8.00 per cent for H. The loan is a five-year amortising loan of 100,000, repaid in five equal annual principal payments, with LGD of 45 per cent and an effective interest rate of 6 per cent.
Survival to year t = (1 − PD)t
Marginal PD in year t = (1 − PD)t−1 × PD
Cumulative PD to year T = 1 − (1 − PD)T
Grade L: 1 − 0.995 = 4.90%. Grade H: 1 − 0.925 = 34.09%.
Excel: =1-(1-B2)^B3 for the cumulative PD; =(1-B2)^(t-1)*B2 for the marginal PD of year t.
The marginal PD falls each year, 1.00, 0.99, 0.98, 0.97 and 0.96 per cent for L, because only survivors can default. Multiplying 1.00 per cent by five gives 5.00 per cent and overstates; for H, eight times five gives 40.00 per cent against 34.09. The error grows with the PD and the term: a 10.0 per cent PD over ten years is 65.13 per cent, not 100.
State vector after t years = starting vector × Mt. The default column of the result is the cumulative PD; the year-on-year change is the marginal PD.
Excel: =MMULT(previous_row, M) copied down five rows, starting from {1, 0, 0} for a grade L borrower.
| Year | Still in L | Migrated to H | Marginal PD | Cumulative PD | Constant hazard |
|---|---|---|---|---|---|
| 1 | 92.0% | 7.0% | 1.00% | 1.00% | 1.00% |
| 2 | 85.7% | 11.8% | 1.48% | 2.48% | 1.99% |
| 3 | 80.6% | 15.1% | 1.80% | 4.28% | 2.97% |
| 4 | 76.4% | 17.3% | 2.01% | 6.30% | 3.94% |
| 5 | 72.9% | 18.7% | 2.15% | 8.44% | 4.90% |
Year one is identical by construction. After that the grade L borrower's marginal PD rises, because a growing share of the cohort has migrated to H, where it defaults at 8 per cent a year. By year five 18.7 per cent of the original L borrowers are high risk, and the cumulative PD is 8.44 per cent: 3.54 points above the constant-hazard answer, an implied average annual PD of 1.75 per cent rather than 1.00.
Grade H runs the other way. Its marginal PD falls from 8.00 to 3.52 per cent by year five, because 40.0 per cent of the cohort has cured to L. Five-year PD is 27.10 per cent, 7.00 points below constant hazard. The matrix pulls both grades towards the portfolio average, the mean reversion that rating agency cohort studies typically report.
Lifetime ECL = Σ marginal PDt × LGD × EADt × (1 + EIR)−t, with EAD the balance outstanding at the start of each year.
| Year | Marginal PD | EAD | Discount factor | ECL |
|---|---|---|---|---|
| 1 | 1.00% | 100,000 | 0.9434 | 425 |
| 2 | 1.48% | 80,000 | 0.8900 | 474 |
| 3 | 1.80% | 60,000 | 0.8396 | 409 |
| 4 | 2.01% | 40,000 | 0.7921 | 287 |
| 5 | 2.15% | 20,000 | 0.7473 | 144 |
| Lifetime | 1,739 |
The 12-month provision is 425; the lifetime provision is 1,739, 4.1 times as much, which is the jump a stage 2 transfer produces on this loan. With constant-hazard PDs the lifetime figure would be 1,167, so the method choice alone moves this loan's provision by 572, or 49.0 per cent. For grade H the matrix gives 7,403 against 8,563 under constant hazard, 13.5 per cent less.
| 12-month PD | 3 years | 5 years | 10 years | PD × 10 |
|---|---|---|---|---|
| 0.5% | 1.49% | 2.48% | 4.89% | 5.0% |
| 1.0% | 2.97% | 4.90% | 9.56% | 10.0% |
| 2.0% | 5.88% | 9.61% | 18.29% | 20.0% |
| 5.0% | 14.26% | 22.62% | 40.13% | 50.0% |
| 10.0% | 27.10% | 40.95% | 65.13% | 100.0% |
For good grades over short terms, multiplication is close enough. For weak grades or long terms it is not, and it can exceed 100 per cent, which no probability can.
Treating constant hazard as neutral. It is a strong assumption that a borrower's grade never changes, and it errs in a predictable direction: too low for good grades, too high for weak ones. If the bank has rating histories, estimate the matrix from them, check that its first-year default column reproduces the calibrated 12-month PDs, and only then raise it to the power of the term. If the PD model itself has drifted, fix its level first, as in recalibrating a PD model to a new default rate; a lifetime curve inherits every error in year one.
A lifetime PD is a 12-month PD plus a view on migration. State the view, show both curves, and expect the gap to be largest exactly where the provision is largest. The one-period version of the loss formula is in expected loss from PD, LGD and EAD, and the calibrated scorecard these PDs start from is built in the free workbooks for Machine Learning for Finance.
Only as a rough approximation for low PDs and short terms. Multiplication ignores that a borrower who has already defaulted cannot default again. A 2.0 per cent PD over five years is 9.61 per cent with survival chained, against 10.0 per cent multiplied; a 10.0 per cent PD over ten years is 65.13 per cent, while multiplication gives 100.0 per cent.
It is the probability, seen from today, of defaulting in a given future year, the increase in cumulative PD over that year. For a 1.00 per cent constant hazard the marginal PDs are 1.00, 0.99, 0.98, 0.97 and 0.96 per cent. Lifetime ECL sums each year's marginal PD times LGD times exposure, discounted at the effective interest rate.
Because some good borrowers migrate to weaker grades and then default at the weaker grade's rate. In the illustrative matrix, 18.7 per cent of low-risk borrowers are high risk by year five, so the five-year PD is 8.44 per cent instead of the 4.90 per cent a constant 1.00 per cent hazard implies.
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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