CDO Tranche Premiums After Portfolio Defaults

# STCDO upper tranche still paying coupons even after all default


# STCDO upper tranche still paying coupons even after all default












We assume :

- that a CDO on $n$ names, with a maturity $T$

- that at a time $\tau<T$ before the maturity of the CDO, these $n$ names have defaulted,

- that we are the protection buyer of the 22-100 tranche,

- that the full loss of this portfolio is 60,

- all the preceding quantities are expressed in percentage of the nominal of the underlying debt portfolio of this CDO.

This is where the question comes :

Will we still continue to pay a premium even between $\tau$ and $T$ ?

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/25430

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, $N_i$ is the notional amount, and $R_i$ is the recovery rate, for the $i^{th}$ entity. Moreover, let $L_{[A,\, D]}(t)$ be the tranche loss amount, that is, \begin{align*} L_{[A,\, D]}(t) &= \min\big(\max(L(t)-A, \,0), \, D-A \big)\\ &=\max\big(L(t)-A, \,0 \big) - \max\big(L(t)-D, \,0 \big). \end{align*} Then, the premium payment at time $t_j$ is based on the notional amount given by \begin{align*} \min\bigg(\big(D-A\big) - L_{[A,\, D]}(t_j),\, \sum_{i=1}^n N_i 1_{\tau_i > t_j}\bigg),\tag{1} \end{align*} which turns to zero on any premium payment date after $\tau = \max_{i=1}^n \tau_i$.

> EDIT after comments.

The second term $\sum_{i=1}^n N_i 1_{\tau_i > t_j}$ in $(1)$ is added so that, for a CDS index, the premium notional is not more than the underlying basket notional.

Note that, for an index (i.e, whole tranche $[0, 100\,\%]$), $L_{[A,\, D]}(t)=L(t)$. Assuming that the first $i_0$ entities, where $1\le i_0 < n$, have defaulted before the premium payment date $t_j$, while the remaining $n-i_0$ entities have not defaulted yet, then the notional for the premium payment on date $t_j$ is given by \begin{align*} \min\bigg(\sum_{i=1}^n N_i-\sum_{i=1}^{i_0}N_i(1-R_i),\, \sum_{i=i_0+1}^n N_i\bigg) &= \min\bigg(\sum_{i=i_0+1}^n N_i +\sum_{i=1}^{i_0} N_i R_i,\, \sum_{i=i_0+1}^n N_i\bigg)\\ &=\sum_{i=i_0+1}^n N_i. \end{align*}