Structured Finance

Study guides: Exam 9, Exam INV-101

Structured Finance is the pooling of credit-sensitive assets — loans, bonds, mortgages — in a special purpose vehicle, followed by the issue against that pool of a prioritised capital structure of claims (tranches), so that losses hit the junior tranches first and the senior tranches can be far safer than the average asset in the pool.

$L_{[a,\,d]} = \min\big(\max(L - a,\ 0),\ d - a\big)$

$L$ is the pool's loss, $a$ the tranche's attachment point and $d$ its detachment point. The tranche loses nothing until the claims beneath it are exhausted, then absorbs everything up to its size $d - a$ . Summed over all tranches the losses equal $L$ : tranching redistributes the pool's risk, it does not remove it.
Pooling alone achieves nothing. A pass-through (a pro-rata share of the pool) simply inherits the pool's average rating. The credit enhancement comes from prioritisation: the junior claims beneath a senior tranche set how large a pool loss it survives. The same arithmetic prices an excess-of-loss layer — a reinsurance layer is a tranche of the cedant's loss distribution.
A tranche's rating depends on the joint distribution. A single bond's rating needs only its own default probability; a senior tranche's needs the probability that many assets default together, so it is acutely sensitive to default correlation and to errors in default probabilities and recoveries. Coval, Jurek & Stafford's first point is that tranching amplifies those errors, and re-tranching tranches (the CDO-squared) amplifies them again.
Senior tranches are economic catastrophe bonds. Idiosyncratic defaults diversify away in a large pool, so a senior tranche is impaired only in a severe, economy-wide downturn. A rating measures default probability or expected loss, not when the loss occurs, so a AAA tranche deserves a far larger spread than a AAA corporate bond — their second point. A cat bond, whose losses are unrelated to the economy, is the opposite case.
See Securitization for the process and its incentive problems, and Credit Risk for default modelling.

Two-Bond Pool: The Senior Tranche Lives on Correlation Two $\$1$ bonds each default with probability $p = 10\%$ and recover nothing. They are pooled and split into a $\$1$ junior tranche (first loss) and a $\$1$ senior tranche. Find each tranche's default probability for default correlation $\rho = 0$ , $0.2$ and $1$ .
Answer The senior tranche defaults only if both bonds default; the junior defaults if either does. For two identical default indicators, $\rho = \dfrac{p_{DD} - p^2}{p(1-p)}$ , so
 $\begin{align*} p_{DD} &= p^2 + \rho\,p(1-p) \\ &= 0.01 + 0.09\rho \\[4pt] P(\text{junior defaults}) &= 2p - p_{DD} \\ &= 0.19 - 0.09\rho \end{align*}$ 
$\rho = 0$ : senior $1.0\%$ , junior $19.0\%$

$\rho = 0.2$ : senior $2.8\%$ , junior $17.2\%$

$\rho = 1$ : senior $10.0\%$ , junior $10.0\%$

A modest correlation of $0.2$ nearly triples the senior tranche's default probability while moving the junior's by under two points. At $\rho = 1$ the structure achieves no enhancement at all. The senior investor is really buying an assumption about $\rho$ .

Ten-Bond Pool: Seniority Amplifies Parameter Error Ten $\$1$ bonds with zero recovery default independently. Tranches: equity $[0, 1]$ , mezzanine $[1, 3]$ , senior $[3, 10]$ . The rating model uses $p = 5\%$ , but the true default probability is $10\%$ . How does each tranche's probability of impairment change?
Answer With $N \sim \text{Bin}(10, p)$ defaults, equity is impaired if $N \geq 1$ , mezzanine if $N \geq 2$ , senior if $N \geq 4$ .

At $p = 0.05$ the probabilities of $0, 1, 2, 3$ defaults are $0.59874, 0.31512, 0.07463, 0.01048$ ; at $p = 0.10$ they are $0.34868, 0.38742, 0.19371, 0.05740$ (Binomial Distribution).
 $\begin{align*} P_{0.05}(N \geq 4) &= 1 - 0.99897 \\ &= 0.00103 \\[4pt] P_{0.10}(N \geq 4) &= 1 - 0.98721 \\ &= 0.01279 \end{align*}$ 
Equity: $40.1\% \to 65.1\%$ (about $1.6\times$ )

Mezzanine: $8.6\% \to 26.4\%$ (about $3.1\times$ )

Senior: $0.10\% \to 1.28\%$ (about $12.4\times$ )

The more senior the tranche, the larger the relative error. A doubling of $p$ that would move a single bond a notch or two moves the senior tranche across many rating categories — and adding correlation would make it worse.