Credit Models and the Crisis: A Journey into CDOs, Copulas, Correlations and Dynamic Models

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).

The numerator and denominator of the above formula depend linearly on the expected tranche loss at different times (ETL).⁹

If a term structure of tranche upfronts, and spreads for different maturities are given, then it is possible to strip back the expectations 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 for each T_i ≤ 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, ...