Consider a stylized economy with three financial instruments available:

- A single piece of (subprime) mortgage;

- Mortgage-backed security (MBS) of this single mortgage. The MBS tranching only contains a junior tranche and a senior one;

- A collateralized debt obligation (CDO) with the simplest junior-senior tranches, which is based on purchasing the senior tranche of the MBS.

The economy lives for two dates, \(t=0,1\) :

- In \(t=0\) the lender has to finance the mortgage which has a face value of \(W\). The mortgage is financed by securitization, and the senior tranche of the MBS is purchased by the CDO. The details are explained below;

- In \(t=1\) the mortgage needs to be refinanced-perhaps due to an unexpected increase in interest rate. If it is not refinanced,

it will default and the lender will be able to recover a value of \(R\). In this case, the lender incurs a loss \(L_{M}=W-R\). If it is refinanced, the renewed mortgage has a new value of \(M\). Obviously, the mortgage will not be refinanced if \(MDuring the securitization in \(t=0\), the lender sells the mortgage to the security holders at its face value, \(W\), and the aggregate return to the security holders in \(t=1\) will be either \(W\) if the mortgage is refinanced or \(R\) if it defaults. The junior tranche in the securitization has a face value of \(N\), and the senior tranche has a face value of \(W-N\). This means that the first loss worth \(N\) will be borne by the junior tranche, and the senior tranche starts to incur the cost only if the loss exceeds \(N\). Thus, if the loss to the securities is below \(N\), the senior tranche will stay intact; otherwise the senior tranche should take the remaining loss once the entire junior tranche is wiped out.

(a) Show that, for the senior tranche of MBS, the loss in \(t=1\) can be characterized by \(L_{S}=\left[L_{M}-N, 0\right]\) and the payoff in \(t=1\) is \(V_{S}=\min \left[W-N, W-L_{S}\right]\).
Note that the senior tranche of MBS is sold to a CDO, which also has two tranches, one junior and one senior. Suppose that the junior tranche in the CDO has a face value of \(N_{C}\) and the senior tranche has a face value of \(W-N-N_{C}\). The senior tranche stays intact if the loss to the CDO, or the senior tranche of MBS, is fully absorbed by the junior tranche of the CDO; otherwise the senior tranche has to take the remaining loss.

(b) Show that, for the senior tranche of CDO, the payoff in \(t=1\) is \(V_{C}=W-N-N_{C}-\max \left[\max \left(L_{M}-N, 0\right)-N_{C}, 0\right]\).