Base correlation

Given implied correlations for the 0–3%, 3–7% and 7–10% tranches, you can obtain base correlations for 0–3%, 0–7% and 0–10%¹. The loss of tranche $A$ to $B$ is the loss of the 0 to $B$ equity tranche minus that of the 0 to $A$ tranche, and a separate base correlation is used for $A$ and for $B$².

Consider a hypothetical example in which the 0–3% base correlation is 0.1 with an expected loss of 89.7%, and the 0–7% base correlation is 0.3 with an expected loss of 53.5%. The expected loss of the 3–7% tranche is about 26.3%, using unrounded values (0.53484 and 0.89737): $(7 \times 0.53484 - 3 \times 0.89737)/4 \approx 0.2629$.

Unlike compound correlation, which finds one correlation per tranche, base correlation is smoother across upper ends and makes it easier to value non-standard tranches by interpolation². On the other hand, because the two ends of one tranche are valued with different correlations, the expected loss can turn negative².

Notes

1. Christoffersen, Ericsson, Jacobs and Jin, presentation at the Bank of Canada Fixed Income Conference (September 13, 2008), slide 7. ↩

2. Damiano Brigo, Andrea Pallavicini and Roberto Torresetti, "Credit Models and the Crisis, or: How I learned to stop worrying and love the CDOs" (arXiv:0912.5427v3, February 2010), sections 3.2–3.3 (pages 23–28). ↩ ↩² ↩³