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This will depend on the contractual specification. In general, you do not need to pay any premium between $\tau$ and $T$. For a CDO with attachment and detachment levels $A$ and $D$. Let $L(t)$ be the cumulative loss of the basket, that is, \begin{align*} L(t) = \sum_{i=1}^n N_i(1-R_i) 1_{\tau_i \le t}, \end{align*} where $\tau_i$ is the default time, ...

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