Reproduce CDS Index Default Probability via Tranche [0,100] Probability

Question

The tranche survival probability up to time $t$ between attachment $K_1$ and detachment $K_2$ is defined as $$Q(t,K_1,K_2) \quad=\quad 1 - \mathbb{E}[L(t,K_1,K_2)]$$ with tranche loss function $$L(t,K_1,K_2) \quad=\quad \frac{\min(L(t),K_2) - \min(L(t),K_1)}{K_2 - K_1}$$ and index loss function $$L(t) \quad=\quad \frac{1}{N} \cdot \sum_{i=1}^N (1-R_i)\cdot 1_{\{\tau_i<t\}}$$

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Now, if we set $K_1=0\%$ and $K_2=100\%$ we get $$Q(t,0,1) \quad =\quad 1 - \frac{\mathbb{E}[\min(L(t),1)] - 0}{1-0} \quad =\quad 1 - (1-R)\cdot \mathbb{P}(\tau<t)\tag{1}$$ (assuming $R_i\equiv R$ and $\mathbb{P}(\tau_i<t)\equiv \mathbb{P}(\tau<t)$ for simplicity)

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However, if $K_1=0\%$ and $K_2=100\%$, should we not recover the pure index probabilities? That is $$Q(t,0,1) \quad=\quad 1 -\mathbb{P}(\tau<t)\tag{2}$$ It seems $\color{red}{(1-R)}$ is somehow incorrectly showing up in formula $(1)$.
How does one reconcile formula $(1)$ and $(2)$?

Answer 1

It is actually that you forgot your $1 - R$ in formula (2) :) The index survival curve is defined similarly to the tranche's : $Q\left(t\right) = 1 - \mathbb{E} \left[L\left(t\right)\right] = 1 - \left(1 - R\right)\mathbb{P}\left(\tau < t\right)$. Hence, your formula for the 0-100 tranche survival curve does coincide with the index'.

That history of loss consistency between index and tranche is handled the following ways :

• On the index, the protection seller pays the loss and will receive coupons on $1-w_i$ of their notional, $w_i$ being the weight of the defaulted entity. $1 - \sum_{i = 1}^N{w_i1_{\tau_i < t}}$ is often called the index factor.
• On the tranche, the equity protection seller pays the loss and receives coupons on $1 - L \left(t, 0, K\right)$ of their notional. The super senior tranche holder, though they do not take any loss, receive coupons on $1 - \sum_{i = 1}^N{w_iR_i1_{\tau_i < t}}$, i.e. on a diminished notional. Practitionners say they are attacked by the top.

This is to ensure that when all names have defaulted, a protection buyer would not pay remaining coupons on $R$ of the notional, which would not make any sense. A good reference on the subject is O'Kane's textbook on credit derivatives (2008).