Structured Credit Products (CDOs and Securitizations) › Collateralized Debt Obligations

The key parameters defining each tranche are:

• Attachment point: The cumulative loss level at which the tranche begins absorbing losses
• Detachment point: The loss level at which the tranche is completely wiped out
• Thickness: The difference between detachment and attachment points

To understand how these parameters work together, consider a mezzanine tranche with a 3% attachment point and 10% detachment point, giving it a 7% thickness. This tranche sits in the middle of the capital structure, protected by the equity tranche below it but subordinate to the senior tranche above. The tranche absorbs no losses whatsoever until the total pool losses exceed 3%. Once pool losses cross that 3% threshold, the mezzanine tranche begins absorbing all marginal losses. It continues absorbing every additional dollar of loss until pool losses reach 10%, at which point the mezzanine tranche is completely exhausted and the senior tranche begins to suffer.

Mathematically, we can express the tranche loss percentage as:

$$L_{\text{tranche}} = \frac{\min(\max(L_{\text{pool}} - A, 0), D - A)}{D - A}$$

where:

• $L_{\text{tranche}}$: percentage loss allocated to the tranche
• $L_{\text{pool}}$: cumulative percentage loss of the underlying collateral pool
• $A$: attachment point (e.g., 3%), the threshold where the tranche begins to lose value
• $D$: detachment point (e.g., 10%), the threshold where the tranche is wiped out
• $D - A$: tranche thickness

This formula captures the layered loss allocation through nested minimum and maximum functions. Let's break down each component to understand the mechanics:

The innermost expression, $L_{\text{pool}} - A$, calculates the amount by which pool losses exceed the attachment point. If the pool has lost 7% and the attachment point is 3%, this difference equals 4%, representing the losses that have "reached" this tranche.

The maximum function, $\max(L_{\text{pool}} - A, 0)$, ensures the tranche loses nothing if pool losses remain below the attachment point $A$. When pool losses are only 2% against a 3% attachment point, the difference is negative, but the maximum function returns zero because the tranche is still fully protected by the subordination below it.

The minimum function, $\min(..., D - A)$, caps the loss at the tranche thickness ($D - A$), so the tranche cannot lose more than its total notional value. Once pool losses exceed the detachment point, this tranche has nothing more to lose; the excess losses pass through to the next senior tranche.

Finally, the division by $D - A$ converts the dollar loss into a percentage of the tranche's notional, normalizing the result to a value between 0 and 1.

This "call spread" payoff structure means the tranche behaves like an option on the pool losses. Specifically, owning a tranche is economically equivalent to being long a call option on pool losses struck at the attachment point and short a call option struck at the detachment point. This options-based perspective proves valuable when we consider hedging strategies and the relationship between tranche pricing and correlation.