A worked synthetic securitisation: the p-factor term by term, the exponential formula, and the attachment point where the floor takes over.
Under SEC-IRBA a tranche's risk weight comes from four inputs: KIRB, the attachment and detachment points, and a supervisory p-factor built from the pool's granularity, LGD and the tranche's maturity. For the senior tranche of the fictional Northwall 2026-1, attaching at 7.00 per cent above a KIRB of 6.51 per cent, p is 0.47 and the risk weight is 35.34 per cent. It reaches the 15 per cent floor only if the senior attaches at 9.74 per cent.
Worked in full in Significant Risk Transfer by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Northwall 2026-1 is the synthetic securitisation worked in Significant Risk Transfer: a 4,000 million portfolio of corporate and mid-market loans. The bank buys protection on a junior tranche from 0 to 7.00 per cent and keeps the senior from 7.00 to 100 per cent. Everything below is illustrative and follows the wholesale, granular (N of 25 or more) coefficient rows of the framework as currently written (CRR Articles 259 and 260, Basel CRE44); a retail pool uses different rows. The Commission's 2025 proposal to amend the securitisation rules would recalibrate p and the floors, so check the text in force before relying on the numbers.
| Input | Value |
|---|---|
| KIRB, capital plus expected loss of the pool | 6.51% |
| Effective number of exposures, N | 166.27 |
| Exposure-weighted LGD | 40.62% |
| Tranche maturity, MT | 5 years |
| Senior attachment, A | 7.00% |
| Senior detachment, D | 100% |
| Risk-weight floor, non-STS | 15% |
Step 1: the p-factor. p sets how steeply capital falls as you move up the structure. A higher p means more capital on the senior. It is a linear formula with five coefficients, floored at 0.30:
p = max[0.30; A + B / N + C × KIRB + D × LGD + E × MT]
Senior, wholesale, granular: A = 0.00, B = 3.56, C = −1.85, D = 0.55, E = 0.07
p = 0.0214 − 0.1203 + 0.2234 + 0.3500 = 0.4745
The maturity term, 0.07 times five years, is the largest. MT is the tranche maturity, floored at one year and capped at five under the CRR, so five years is already the maximum. Granularity contributes almost nothing once N is above a hundred.
Step 2: the supervisory formula. Measure the tranche relative to KIRB and apply the exponential:
a = −1 / (p × KIRB) = −1 / (0.4745 × 6.51%) = −32.40
u = D − KIRB = 0.9349; l = max(A − KIRB, 0) = 0.0049
KSSFA = (ea·u − ea·l) / (a × (u − l)) = (6.98e−14 − 0.8518) / (−30.13) = 2.83%
Risk weight = 12.5 × KSSFA = 35.34%, above the 15% floor
Excel, for a tranche attaching at or above KIRB: =MAX(Floor,12.5*(EXP(a*(D-K))-EXP(a*MAX(A-K,0)))/(a*(D-MAX(A,K))))
The senior is 3,720 million, so it carries 1,314.5 million of risk-weighted assets. At an 8 per cent minimum that is 105.2 million of capital; at a 13.5 per cent internal target, 177.5 million. Before the trade the whole pool averaged 76.93 per cent.
For a tranche straddling KIRB the slice below KIRB is weighted at 1250 per cent and the rest by the formula, pro rata to thickness. For a tranche wholly below KIRB the weight is 1250 per cent. The junior 0 to 7.00 per cent tranche, with the non-senior p of 0.55, comes out at 1244.11 per cent: just under 1250 because its top half point sits above KIRB.
Attachment is the lever the bank controls. Each point of junior thickness moves the senior weight sharply until the floor takes over.
| Attachment | Formula result | Risk weight applied |
|---|---|---|
| 6.00% | 47.76% | 47.76% |
| 6.50% | 41.33% | 41.33% |
| 7.00% | 35.34% | 35.34% |
| 8.00% | 25.83% | 25.83% |
| 9.00% | 18.89% | 18.89% |
| 9.74% | 14.98% | 15.00% |
| 12.00% | 7.39% | 15.00% |
Below 6.51 per cent the senior itself straddles KIRB and picks up a 1250 per cent slice. Above 9.74 per cent the floor binds and every extra point of protection bought releases nothing. For a non-STS deal the economic sweet spot lies somewhere between the two: a protected tranche detaching a little above KIRB.
| MT, years | p | Risk weight |
|---|---|---|
| 1 | 0.30 | 20.35% |
| 2 | 0.30 | 20.35% |
| 3 | 0.33 | 23.29% |
| 4 | 0.40 | 29.30% |
| 5 | 0.47 | 35.34% |
Shorter maturity and STS both work through p. Qualifying as STS halves p, which here lands on the 0.30 floor, giving the same 20.35 per cent as a one-year tranche, and lowers the risk-weight floor to 10 per cent.
The most frequent error is using the wrong coefficient row. The non-senior row has A = 0.16 and a less negative coefficient on KIRB (−1.03 against −1.85); applied to the senior it gives p of 0.55 and a risk weight of 41.49 per cent instead of 35.34, 6.15 points too high. The retail rows set A and B to zero and change C, D and E. The coefficient rows for senior and non-senior are easy to transpose in a spreadsheet, and the error does not look wrong because both answers are plausible.
The second error is to treat MT as the pool's weighted average maturity. MT is the tranche maturity, and in many structures, with replenishment and a time call, it is longer than the average loan life. A senior priced with MT of 3 years at 23.29 per cent that should have used 5 years is understated by 12.05 points of risk weight, and the capital release in the investment memo is overstated accordingly.
Whether the retained junior slice should then be weighted at 1250 per cent or deducted is a separate decision, worked in should a retained 1250 per cent position be deducted or risk-weighted?
The p-factor, every detachment from 3 to 14 per cent and both thresholds are live formulas in the free workbook for this case.
p is a supervisory parameter that controls how fast capital falls as a tranche moves above KIRB. It is A + B/N + C x KIRB + D x LGD + E x MT, floored at 0.30, with coefficients that depend on wholesale or retail and senior or non-senior. For Northwall's senior it is 0.4745, of which 0.3500 comes from a five-year tranche maturity alone.
The part below KIRB is weighted at 1250 per cent and the part above by the supervisory formula, each pro rata to its thickness. Northwall's junior tranche from 0 to 7.00 per cent sits mostly below a 6.51 per cent KIRB, so it weighs 1244.11 per cent rather than the full 1250. A senior attaching at 6.00 per cent would weigh 47.76 per cent for the same reason.
For a qualifying senior position, STS halves p, subject to the 0.30 floor, and lowers the risk-weight floor from 15 to 10 per cent. On Northwall halving p takes it to the 0.30 floor and the senior weight falls from 35.34 to 20.35 per cent, the same result as a one-year tranche maturity.
This article is one calculation from Significant Risk Transfer. 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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