Tim Olson
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93 lines
4.5 KiB
Markdown
93 lines
4.5 KiB
Markdown
---
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description: "Reference overview of Collateralized Debt Obligation (CDO) mechanics, tranche structure, mark-to-market valuation, spread pricing, and risky duration — the foundational concepts for all CDO trading strategies."
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tags: [structured-assets, cdo, credit, tranche, abs, cds]
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---
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# CDO Generalities: Collateralized Debt Obligations
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**Section**: 11.1 | **Asset Class**: Structured Assets | **Type**: Reference / Foundational
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## Overview
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A CDO is an asset-backed security consisting of a basket of assets (bonds, loans, credit default swaps, etc.) divided into tranches with different credit ratings and interest rates. Each tranche has an attachment point a and a detachment point d. When cumulative portfolio losses exceed a, the tranche begins to lose value; when losses exceed d, the tranche is completely wiped out. Understanding CDO valuation is the foundation for all CDO carry and curve trading strategies (Sections 11.2–11.7).
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## Construction / Mechanics
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### Tranche Structure
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- **Attachment point a**: portfolio loss level at which the tranche begins to absorb losses
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- **Detachment point d**: portfolio loss level at which the tranche is fully wiped out
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- Example: a 3–8% tranche loses value when portfolio losses exceed 3% and is fully wiped at 8%
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- Typical tranche hierarchy (decreasing default risk, decreasing premium rate):
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- Equity: 0–3%
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- Junior mezzanine: 3–7%
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- Senior mezzanine: 7–10%
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- Senior: 10–15%
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- Super senior: 15–30%
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### Buyer/Seller Roles
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- **Buyer (long tranche, protection seller)**: receives periodic premium payments; obligated to cover defaults up to the tranche size in the event of a default.
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- **Seller (short tranche, protection buyer)**: makes periodic premium payments; receives a payment in the event of a default.
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### Synthetic CDOs
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Synthetic CDOs are constructed using credit default swaps (CDS) on a reference pool. Exchange-traded single-tranche CDOs reference CDS indexes such as CDX or iTraxx.
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### Expected Loss Computation
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Let t_i (i = 1,...,n) be the periodic premium payment dates. Let H(t) be the set of possible default scenarios ℓ_α (α = 1,...,K) with probabilities p_α(t). The expected loss is:
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```
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L(t) = Σ p_α(t) × max(min(ℓ_α, L_d) - L_a, 0) (481)
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```
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where L_a = a × M_CDO and L_d = d × M_CDO, and M_CDO is the CDO notional in dollars.
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### Mark-to-Market (MTM) Valuation
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The MTM value M of the tranche (from the long investor's perspective) is:
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```
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M = P - C (482)
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```
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The premium leg (what the long investor receives):
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```
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P = S × Σ D_i Δ_i [M_tr - L(t_i)] (483)
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```
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The contingent (default) leg (what the long investor pays):
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```
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C = Σ D_i [L(t_i) - L(t_{i-1})] (484)
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```
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where:
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- S = spread (annual premium rate)
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- D_i = risk-free discount factor for payment date t_i
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- Δ_i = t_i - t_{i-1} (time between payment dates)
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- M_tr = L_d - L_a (tranche notional)
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- L(t_0) = 0
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### Fair Spread and Risky Duration
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Setting M = 0 fixes the fair spread S = S*. The risky duration D of the tranche is the first derivative of M w.r.t. the spread:
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```
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M(S) = (S - S*) × Σ D_i Δ_i [M_tr - L(t_i)] (485)
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D = ∂M/∂S = Σ D_i Δ_i [M_tr - L(t_i)] (486)
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```
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The risky duration D_ix can also be defined analogously for a CDS index.
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## Return Profile
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The long tranche position earns carry (spread income) when no defaults occur. The contingent leg represents the tail-risk cost: large losses materialise only when portfolio losses exceed the attachment point. Junior tranches have higher spreads but greater default exposure; senior tranches have lower spreads but protection from defaults up to the attachment point.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| a, d | Attachment and detachment points (%) |
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| M_CDO | CDO notional in dollars |
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| S | Tranche spread (annual premium rate) |
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| D (risky duration) | Sensitivity of MTM to spread change; Eq. (486) |
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| L(t) | Expected loss at time t; Eq. (481) |
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## Notes
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- The expected loss L(t) and the probabilities p_α(t) are model-dependent; different models (Gaussian copula, etc.) give different valuations.
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- Risky duration is the primary hedging metric for CDO tranche positions (used in Sections 11.2–11.5).
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- CDS indexes (CDX, iTraxx) provide liquid reference points for both outright hedging and relative-value trades.
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- Convexity of tranche value w.r.t. spread means that hedging ratios change as spreads move; dynamic rehedging is required.
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