CDO tranche pricing
The model structured finance ran on, and the one that failed in 2007 — not because the arithmetic was wrong, but because a single correlation was treated as a constant to be calibrated rather than a variable to be understood. This prices a tranche under the one-factor Gaussian copula and shows why the equity and senior ends disagree about what correlation means.
| Tranche | Attach | Detach | Expected loss | Wipe-out |
|---|---|---|---|---|
| 0–3% | 0.0% | 3.0% | 62.767% | 34.89% |
| 3–7% | 3.0% | 7.0% | 19.469% | 9.78% |
| 7–10% | 7.0% | 10.0% | 6.488% | 4.07% |
| 10–15% | 10.0% | 15.0% | 2.174% | 0.99% |
| 15–30% | 15.0% | 30.0% | 0.230% | 0.01% |
| 30–100% | 30.0% | 100.0% | 0.000% | 0.00% |
The equity tranche is long correlation — the pool either defaults together or not at all — and the senior tranche is short it. That is why a single ρ cannot price the whole stack, and why base correlation is quoted as a curve rather than a number.
c = Φ⁻¹(PD) · The default point in the latent variable.
p(M) = Φ[(c − √ρ M)/√(1−ρ)] · Independent defaults once M is fixed.
E[min(max(L−a,0), b−a)]/(b−a) · Integrated over the normal factor.
P(L > b) · Solved from p(M*) = b/LGD.
A clear path from inputs to results.
The model structured finance ran on, and the one that failed in 2007 — not because the arithmetic was wrong, but because a single correlation was treated as a constant to be calibrated rather than a variable to be understood.
How to use
- Set the pool default probability and recovery rate.
- Choose the asset correlation.
- Set the tranche attachment and detachment points.
- Read the expected loss, the wipe-out probability and the whole tranche stack.
The formula
Reference: Vasicek (1987) loan portfolio value; Li (2000) for the Gaussian copula applied to CDOs; BCBS (2005) for the regulatory use.
Worked example
A 3–7% tranche on a 5% default pool
- Pool PD 5%, recovery 40%, correlation 20%.
- Tranche attaches at 3% and detaches at 7%.
- 1
The expected pool loss is 5% × 60% = 3.00%.
- 2
Conditional on the common factor, the default probability ranges from near zero to over 50%.
- 3
Integrating the capped and floored conditional loss over the factor gives 19.47% expected loss on the tranche.
- 4
The tranche is wiped out 9.78% of the time and never touched 65.1% of the time.
- 5
The equity tranche below it loses 62.77% and the senior tranche above it loses essentially nothing.
19.47% tranche EL · 9.78% wipe-out · 65.1% untouched.
Interpreting the result
The equity and senior tranches disagree about what correlation means, and that disagreement is the model. Equity is long correlation — the pool either defaults together or not at all — and senior is short it. A single ρ cannot price the whole stack, which is why base correlation is quoted as a curve.
Assumptions & limitations
- A large homogeneous pool: every obligor has the same PD, LGD and correlation.
- One systematic factor, with idiosyncratic risk independent across obligors.
- Recovery is constant and independent of default — the assumption that failed in 2008.
- A single-period (static) model with no path dependence.
Frequently asked questions
Why did the model fail in 2008?
Because recovery fell exactly when defaults rose, so the constant-LGD assumption understated senior losses, and because the correlation was calibrated to a benign period and then treated as stable.
Why is my mezzanine tranche not invertible?
Because expected loss for a mezzanine tranche is not monotone in correlation — it rises and then falls, so a given price can correspond to two correlations.
What is base correlation?
The correlation that reprices a tranche attaching at zero. It is quoted as a curve because each detachment point implies a different number.
What is a wipe-out probability?
The probability that the pool loss exceeds the detachment point, so the tranche loses all of its notional.
Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.