Why your base scenario should not sit at Z = 0
Anchoring the central case at the average state of the economy is the obvious thing to do and it biases the allowance downwards on every book. How far down depends on asset correlation — up to 27.4% on a sovereign and public sector book, and in the opposite direction on the one sector where correlation is lowest.
The conditional PD at the average economy is not the through-the-cycle PD. It is lower.
A through-the-cycle PD is an unconditional probability — the average over every state of the economy. Conditioning it on a particular state, which is what the Vasicek transformation does, gives a different quantity. The mapping is convex in the region where credit PDs actually live, so by Jensen’s inequality the value at the average input sits below the average of the values. Credit loss distributions are right-skewed; the mean is above the central case.
Which is exactly why IFRS 9.5.5.17(a) asks for an unbiased probability-weighted amount rather than a best-estimate scenario. Run the model on the base case alone, or on a set whose base case is anchored at zero, and the allowance is systematically light — not because any single input is wrong, but because of where the scenarios were placed.
Below: the conventional arrangement — base at Z = 0, downside at −1, upside at +0.7, weighted 50/30/20 — measured against the through-the-cycle PD it is supposed to reproduce. A negative figure means the scenario set understates the allowance. Beside it, the set this tool ships with, which is calibrated to reproduce it.
| Sector | Asset correlation | Base at Z = 0 | Calibrated |
|---|---|---|---|
| Commercial real estate | 0.25 | −19.5% | +1.2% |
| Financial institutions | 0.24 | −27.0% | 0.0% |
| Energy and resources | 0.22 | −16.9% | +0.7% |
| Sovereign and public sector | 0.20 | −27.4% | −1.3% |
| Manufacturing | 0.19 | −15.0% | +0.2% |
| Corporate | 0.18 | −15.9% | 0.0% |
| Retail mortgage | 0.15 | −14.3% | −0.3% |
| Services | 0.15 | −9.4% | −0.3% |
| Agriculture | 0.14 | −5.8% | −0.5% |
| SME | 0.12 | −3.0% | −0.7% |
| Retail unsecured | 0.04 | +2.3% | −0.9% |
Sorted by asset correlation, because that is the finding. Computed on this page by the same function that blocks a model review when a scenario set fails it.
It is not one error. It scales with asset correlation.
At ρ = 0.20 on a sovereign and public sector book the conventional anchor takes 27.4% off the probability of default. At ρ = 0.04 on a retail unsecured book it goes the other way and +2.3% — it over-provides.
That matters commercially, not just technically. The sectors where the error is largest are the high-correlation ones — sovereign, financial institutions, commercial real estate — which is to say a bank’s largest and most concentrated exposures. A single quoted percentage would have been easier to publish and would have been wrong in both directions.
Place the scenarios so that, weighted, the conditional PDs reproduce the unconditional PD they came from. That is a testable property rather than a matter of judgement, and it means the base case sits above zero — this tool ships it at Z = 0.4 and not at the average economy. The downside then carries the weight the skew requires.
Then check it, on every set, including your own. Most implementations never do — it is cheap to check, and it is the difference between a scenario set that is defensible in front of a model-risk committee and one that merely looks reasonable. In this tool the check runs on save and refuses to let a biased set go to review.
The transformation is the single-factor Vasicek model, the same one underneath the Basel IRB risk-weight formulas. The requirement for an unbiased, probability-weighted estimate is IFRS 9.5.5.17(a). The sector parameters used above are the values this tool ships as a starting point, and it flags every one of them as uncalibrated and refuses to report on them until they are replaced with your own — the arithmetic on this page holds whatever they are.