From pool expected loss to tranche expected loss with the one-factor model, and why correlation matters for thin layers but hardly at all for a thick first-loss piece.
The expected loss of an SRT tranche is the average, across every possible pool loss, of the part that falls between the tranche's attachment and detachment points, divided by its thickness. On an illustrative €4,000 million corporate pool with a 1.60 per cent expected loss, a one-factor model with 15 per cent correlation gives the 0 to 7 per cent first-loss tranche an expected loss of 22.60 per cent of its notional, close to the 22.86 per cent of simple division. The split inside it is where the model earns its keep: 79.25 per cent on the first 1 per cent, 3.17 per cent on the 4 to 7 per cent layer.
Worked in full in Significant Risk Transfer by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Credit funds price SRT tranches as a margin against expected and unexpected loss. The pool's expected loss is easy: probability of default times loss given default. Turning it into a tranche's expected loss needs a view of how losses are distributed, because a tranche only feels the part of the pool loss above its attachment point, capped at its detachment point. The standard starting point is the same one-factor Gaussian model that underlies the IRB capital formula, applied to a large, granular pool.
| Input | Value |
|---|---|
| Reference portfolio | 4,000 |
| Cumulative probability of default over the weighted life | 4.0% |
| Loss given default | 40% |
| Pool expected loss, PD × LGD | 1.60% = 64.0 |
| Asset correlation | 15% |
| Weighted life, for the per-year translation | 3 years |
Large homogeneous pool: every loan has the same PD, LGD and correlation, and there are enough of them that idiosyncratic noise averages out. Real SRT pools of several hundred corporate names are close enough for a first pass; concentrated pools need a loan-level simulation.
Pool loss given the systematic factor Z: L(Z) = LGD × Φ( (Φ−1(PD) − √ρ × Z) ÷ √(1 − ρ) ).
Tranche loss as a share of its notional, attachment A, detachment D: min( max(L − A, 0), D − A ) ÷ (D − A).
Tranche expected loss = the average of that over the distribution of Z, a standard normal. Integrate numerically: a grid of Z values, each weighted by its normal density.
Any pool loss quantile in closed form, in Excel: =LGD*NORM.S.DIST((NORM.S.INV(PD)+SQRT(rho)*NORM.S.INV(q))/SQRT(1-rho),TRUE). For the tranche, put Z in one column from −6 to 6 in small steps, L(Z) beside it, the tranche loss beside that, and use =SUMPRODUCT(TrancheLoss,NORM.S.DIST(Z,FALSE))*Step.
The pool loss distribution is skewed. Its median is 1.15 per cent, below the 1.60 per cent mean; the 90th percentile is 3.47 per cent, the 99th 7.13 per cent and the 99.9th 10.96 per cent. Most outcomes are quieter than average and a few are much worse, which is exactly what a tranche structure slices up.
| Tranche | Notional, €m | Expected loss, % of notional | Expected loss, €m | Share of pool EL | Probability of total loss |
|---|---|---|---|---|---|
| 0 to 1% | 40 | 79.25% | 31.7 | 49.5% | 55.79% |
| 1 to 4% | 120 | 23.15% | 27.8 | 43.4% | 7.09% |
| 4 to 7% | 120 | 3.17% | 3.8 | 5.9% | 1.08% |
| 7 to 100%, senior | 3,720 | 0.02% | 0.7 | 1.1% | 0.00% |
| Pool | 4,000 | 1.60% | 64.0 | 100% | |
| 0 to 7%, as one first-loss tranche | 280 | 22.60% | 63.3 | 98.9% | 1.08% |
Two checks. The euro expected losses of the layers add back to the pool's 64.0, because every euro of pool loss lands in exactly one tranche. And the 0 to 7 per cent tranche is the sum of its three layers: 31.7 + 27.8 + 3.8 = 63.3.
A 0 to 7 per cent tranche absorbs 98.9 per cent of the pool's expected loss, which is why its expected loss is close to the naive 1.60 ÷ 7 = 22.86 per cent. The model matters less for the thick first-loss piece than for anything carved out of it. The equity sliver is a near-certain loss, 79.25 per cent expected and wiped out in 55.79 per cent of outcomes; the 4 to 7 per cent layer loses 3.17 per cent on average and is wiped out in 1.08 per cent of them. Spread over a three-year life, those are roughly 2,642, 772 and 106 basis points a year, before timing and amortisation, against 753 for the whole 0 to 7 per cent tranche.
| Case | 0 to 1% | 1 to 4% | 4 to 7% | 0 to 7% | Senior |
|---|---|---|---|---|---|
| Correlation 5% | 93.95% | 21.76% | 0.25% | 22.86% | 0.00% |
| Correlation 15%, base | 79.25% | 23.15% | 3.17% | 22.60% | 0.02% |
| Correlation 25% | 67.53% | 22.40% | 5.73% | 21.71% | 0.09% |
| PD 3.0%, pool EL 1.20% | 70.08% | 14.98% | 1.45% | 17.05% | 0.01% |
| PD 5.0%, pool EL 2.00% | 85.40% | 31.32% | 5.58% | 28.01% | 0.04% |
| PD 6.0%, pool EL 2.40% | 89.61% | 39.12% | 8.62% | 33.26% | 0.08% |
Correlation moves expected loss from the bottom of the structure to the top. Higher correlation means more outcomes with almost no losses and a few with many: the equity piece does better, the 4 to 7 per cent layer and the senior do worse. The whole 0 to 7 per cent tranche barely moves, from 22.86 to 21.71 per cent, because nearly all of the pool's loss stays inside it either way. A fund buying the full first-loss tranche is mostly taking a view on PD and LGD. A fund buying a mezzanine slice is taking a view on correlation, whether it models one or not.
The common mistake is dividing pool expected loss by tranche thickness for every layer. For the 0 to 7 per cent tranche it is a fair shortcut, 22.86 against 22.60 per cent. For the bottom 1 per cent it gives 100 per cent, a certain total loss, against 79.25; for the 4 to 7 per cent layer, an average pool loss of 1.60 per cent never reaches the 4 per cent attachment, so the shortcut says zero, against 3.17 per cent once the bad outcomes are counted. Thin tranches need the distribution. The second mistake is using an annual PD on a three-year deal: the PD has to match the horizon over which the tranche is exposed, including any replenishment period.
Expected loss is only the first input to price. The tranche's return, with coupons paid on a balance that losses write down, is built in how to calculate the return on an SRT tranche, and the bank's side of the same pool starts from KIRB for a securitised loan portfolio.
Tranche expected loss = average of min(max(L − A, 0), D − A) ÷ (D − A) over the pool loss distribution. For a thick first-loss tranche it lands near pool EL over thickness, 22.60 per cent here; for thin layers it does not, and correlation decides how it is split. The book's portfolio, tranche capital and investor return models are in the free workbook for this case.
Only for a tranche that absorbs almost all of the pool's losses. Here the 0 to 7 per cent tranche takes 98.9 per cent of pool expected loss, so 1.60 divided by 7 gives 22.86 per cent against a modelled 22.60. For thinner layers the shortcut fails: it gives a certain total loss for the 0 to 1 per cent piece, whose modelled expected loss is 79.25 per cent.
Higher correlation concentrates losses into fewer, worse outcomes, which helps the most junior layer and hurts more senior ones. Moving correlation from 5 to 25 per cent takes the 0 to 1 per cent layer's expected loss from 93.95 to 67.53 per cent and the 4 to 7 per cent layer's from 0.25 to 5.73 per cent, while the full 0 to 7 per cent tranche moves only from 22.86 to 21.71 per cent.
A cumulative PD over the period the tranche is exposed, not a one-year PD, including any replenishment period in which defaults still hit the tranche. The illustrative case uses 4.0 per cent over a three-year weighted life. Raising it to 6.0 per cent lifts the 0 to 7 per cent tranche's expected loss from 22.60 to 33.26 per cent.
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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