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Credit Models and the Crisis: A Journey into CDOs, Copulas, Correlations and Dynamic Models by Roberto Torresetti, Andrea Pallavicini, Damiano Brigo

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5
Consistency across Capital Structure and Maturities: Expected Tranche Loss
In this chapter we approach the issue of consistently representing the information embedded in tranches across attachments and maturities in a model-free way.
Let us again consider equation (2.5).
187
The numerator and denominator of the above formula depend linearly on the expected tranche loss 188 at different times (ETL).9
If a term structure of tranche upfronts, and spreads 189 for different maturities are given, then it is possible to strip back the expectations 190 in a model independent way. We will address this issue in this chapter.
We first notice the nested structure of the expected tranche losses in time. Indeed, the NPV calculations of two tranches with the same attachment and detachment points (A, B), but different maturities T1 and T2, both depend on the expected tranche loss 191 for each Ti ≤ min{T1, T2}. For example, in the NPV calculations of a 5-year maturity 0-3% tranche and a 10-year maturity 0-3% tranche, ...

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