Tranche loss and structured credit

Definition Let a reference portfolio of credit names have original notional $N > 0$ currency units, and let $L$ be its cumulative loss at a fixed horizon as a fraction of $N$, after recovery, so $0 \le L \le 1$. A name with notional $w_i$ and recovery fraction $R_i$ adds $w_i(1 - R_i)/N$ to $L$ when it defaults. The law of $L$ is taken under one measure, a pricing measure $\mathbb{Q}$ for prices or the physical measure $\mathbb{P}$ for risk statements, and $P$ and $E$ denote probability and expectation under it. A tranche with attachment point $a$ and detachment point $d$, where $0 \le a < d \le 1$, is a position that absorbs the portfolio losses lying between $a$ and $d$ and no others. Its width is $d - a$ and its original notional is $(d - a)N$. Losses below the attachment point fall on the tranches beneath it, and losses above the detachment point fall on the tranches above it. A tranche with $a = 0$ is the equity tranche, which takes the first losses, and a tranche with a higher attachment point is more senior. The names equity, mezzanine and senior say where a slice sits, not that its payoff has a different form. In a synthetic tranche the protection seller pays the tranche’s losses to the protection buyer in exchange for a premium. Real contracts add conventions that this lesson leaves out, such as how recovered amounts reduce the notional of the most senior tranche.

Definition Let $0 = k_0 < k_1 < \dots < k_m = 1$ be the boundaries of a stack of adjacent tranches, with tranche $j$ covering $[k_{j-1}, k_j]$. The loss waterfall allocates a portfolio loss from the bottom up. A tranche absorbs loss only after every tranche below it is exhausted, and it absorbs at most its width, so the tranche $[a, d]$ receives

per unit of portfolio notional. If $k_{r-1} \le L \le k_r$, the tranches below tranche $r$ are exhausted and absorb $k_{r-1}$ in total, tranche $r$ absorbs $L - k_{r-1}$, and the tranches above absorb nothing. This proves the partition identity

for every $L$ in $[0, 1]$. Taking expectations, the expected losses of the tranches add to $E[L]$ for every loss distribution, so a change in the distribution that keeps $E[L]$ fixed only moves expected loss between tranches.

Definition The tranche loss of $[a, d]$ per unit of portfolio notional is the waterfall amount $\ell_{a,d}(L)$, and it has the call-spread form

The proof is by cases. If $L \le a$, both positive parts are $0$. If $a < L \le d$, only the first is positive and the difference is $L - a$. If $L > d$, the difference is $(L - a) - (L - d) = d - a$. In each case the result is $\min\bigl((L - a)^{+}, d - a\bigr)$. As a function of $L$ the tranche loss is continuous and nondecreasing, equal to $0$ on $[0, a]$, rising with slope $1$ on $[a, d]$ and equal to $d - a$ on $[d, 1]$. The fractional tranche loss, measured against the tranche’s own notional, is

and the loss in currency units is $N\, \ell_{a,d}(L)$. So the tranche loss is the payoff of a long call on the portfolio loss struck at $a$ and a short call struck at $d$. Telescoping this form gives a second proof of the partition identity, since the interior terms cancel and leave $L^{+} - (L - 1)^{+} = L$.

Definition The expected tranche loss is $E[\ell_{a,d}(L)]$ per unit of portfolio notional, under the measure in use. The algebra is the same under $\mathbb{Q}$ and $\mathbb{P}$, while the probabilities need not be. For a discrete law with losses $L_1, \dots, L_K$ and probabilities $\pi_1, \dots, \pi_K$,

so the waterfall is applied in each state before averaging. Because $\ell_{a,d}$ is not linear, $E[\ell_{a,d}(L)] \ne \ell_{a,d}(E[L])$ in general. For an equity tranche the function $\min(L, d)$ is concave, and Jensen’s inequality gives $E[\ell_{0,d}(L)] \le \ell_{0,d}(E[L])$. For a tranche with $d = 1$ the function $(L - a)^{+}$ is convex and the inequality reverses. For a tranche with $0 < a < d < 1$ the error has no fixed sign, and a mean loss below $a$ is compatible with a positive expected tranche loss.

A contract that pays $N\, \ell_{a,d}(L_T)$ at one date $T$ has value $N\, D(0, T)\, \mathbb{E}^{\mathbb{Q}}[\ell_{a,d}(L_T)]$ when the discount factor $D(0, T)$ is deterministic, and $N\, \mathbb{E}^{\mathbb{Q}}[\ell_{a,d}(L_T)/B_T]$ with the money-market account $B_t$, $B_0 = 1$, in general. Most tranches pay over time, and the loss $L_t$ is nondecreasing in $t$. Write $\bar\ell(t) = \mathbb{E}^{\mathbb{Q}}[\ell_{a,d}(L_t)]$ for the expected tranche loss curve, and take dates $0 = t_0 < \dots < t_m$ with accrual fractions $\Delta_k$. Assume deterministic discount factors, protection that pays each period’s increase in tranche loss at the end of the period, a premium at the running spread $s$ paid at each period end on the outstanding tranche notional, and no upfront payment. Then, per unit of portfolio notional,

and when the sum multiplying $s$ is positive, the par spread is the value of $s$ that makes the two legs equal. This is an outline under the stated conventions, not a full tranche model. The loss distribution at every date still has to come from a joint default model, such as the latent factor model of Default dependence and credit portfolios, and contracts specify accrual and settlement details that change the formulas.

Definition Let $0 \le L \le 1$ and $0 \le a < d \le 1$. The expected tranche loss is the integral of the tail function of $L$ between the attachment and detachment points,

For a fixed number $L$ in $[0, 1]$, the quantity $(L - a)^{+}$ is the length of the part of $[a, 1]$ that lies below $L$, so

The integrand is nonnegative and jointly measurable, so Tonelli’s theorem allows the expectation and the integral to be exchanged, and $E[\mathbf{1}_{\{L > x\}}] = P(L > x)$. This gives $E[(L - a)^{+}] = \int_a^1 P(L > x)\, dx$. The same holds with $d$ in place of $a$, both expectations are at most $1$, and subtracting them in the call-spread form proves the identity. For a discrete law the exchange is an ordinary interchange of a finite sum and an integral. Replacing $P(L > x)$ by $P(L \ge x)$ changes nothing, because the two differ only at the atoms of $L$, which are at most countably many points and have total length zero. The case $a = 0$ and $d = 1$ gives $E[L] = \int_0^1 P(L > x)\, dx$. The expected tranche loss depends on the loss law only through its tail on $[a, d]$, so moving probability between two losses that are both at most $a$, or both at least $d$, leaves it unchanged.