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---
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description: "CDO carry strategy that buys a low-quality tranche and delta-hedges with a single-name CDS, using the CDS risky duration as the hedge ratio denominator rather than the index."
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tags: [structured-assets, cdo, carry, cds-hedge, single-name, delta-hedge]
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---
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# Carry — CDS Hedging
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**Section**: 11.5 | **Asset Class**: Structured Assets | **Type**: Carry / Delta-Hedged
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## Overview
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Instead of hedging a long low-quality tranche position with the CDS index (Section 11.2) or a high-quality tranche (Section 11.4), this strategy uses a single-name CDS as the hedging instrument. The single-name CDS typically pays a lower premium than the long tranche, generating positive carry. The hedge ratio uses the risky duration of the CDS as the denominator.
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## Construction / Mechanics
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**Position:**
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- Long a low-quality tranche: receive high spread S_tranche
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- Short a single-name CDS (protection buyer): pay lower spread S_CDS
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**Hedge ratio:**
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```
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Δ_CDS = D / D_CDS (489)
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```
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where:
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- D = risky duration of the low-quality tranche (Eq. 486)
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- D_CDS = risky duration of the single-name CDS
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Note: Eq. (489) is Eq. (487) with D_ix replaced by D_CDS.
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**Economics:**
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- Premium received on the long tranche > premium paid on the short single-name CDS
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- Net carry = S_tranche - Δ_CDS × S_CDS
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## Return Profile
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Earns positive carry from the spread differential between the tranche and the CDS. The hedge approximately neutralises single-name credit spread sensitivity. Residual risks include basis risk between the tranche and the CDS (they do not move in perfect lockstep), default event risk (the CDS may not pay out at the same time or magnitude as tranche losses), and gamma risk.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| Δ_CDS = D / D_CDS | Hedge ratio: tranche risky duration / CDS risky duration |
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| S_tranche | Spread received on long tranche |
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| S_CDS | Spread paid on short single-name CDS |
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| Net carry | S_tranche - Δ_CDS × S_CDS |
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| D_CDS | Risky duration of the CDS instrument |
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## Variations
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- Use multiple single-name CDS (one per reference entity or a basket) for more precise hedging across the reference pool.
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- Combine with an index hedge (Section 11.2) to address both systematic and idiosyncratic credit spread risk.
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## Notes
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- Single-name CDS has more precise default coverage for specific reference entities than the index, but may not be available for all names in the pool.
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- Basis risk between the tranche and the single-name CDS is a key risk: the tranche spread reflects pool-wide correlation dynamics, while the CDS spread reflects individual credit risk.
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- Liquidity of single-name CDS varies widely; less-liquid names may have wide bid-ask spreads that erode carry.
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- The tranche remains exposed to correlated defaults (systemic events) even when individual names are hedged.
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---
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description: "CDO carry strategy that buys the equity (lowest quality) tranche and delta-hedges credit spread risk by selling the CDS index, earning the spread differential between tranche premium and index cost."
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tags: [structured-assets, cdo, carry, equity-tranche, delta-hedge, cds-index]
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---
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# Carry — Equity Tranche with Index Hedging
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**Section**: 11.2 | **Asset Class**: Structured Assets | **Type**: Carry / Delta-Hedged
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## Overview
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The equity tranche (e.g., 0–3%) of a CDO pays the highest periodic premium of all tranches because it absorbs the first losses in the reference portfolio. By buying this tranche and delta-hedging the credit spread exposure through a short position in the CDS index, the trader earns the premium differential between the equity tranche and the index hedge cost while neutralising systematic spread movements.
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## Construction / Mechanics
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**Position:**
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- Long the equity (0–3%) tranche (protection seller): receive spread S_equity
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- Short the CDS index (protection buyer): pay spread S_index
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**Delta (hedge ratio):** The number of index units to short per unit of equity tranche is:
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```
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Δ_ix = D / D_ix (487)
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```
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where:
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- D = risky duration of the equity tranche (Eq. 486 from Section 11.1)
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- D_ix = risky duration of the CDS index
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**Economics:**
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- Premium received from the equity tranche > premium paid on the short index position
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- The net carry = S_equity × M_tr - S_index × Δ_ix × M_index (per unit time, before defaults)
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## Return Profile
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The trade earns positive carry (net spread income) as long as defaults do not erode the equity tranche beyond its detachment point. The hedged position is approximately spread-neutral to small parallel moves in credit spreads. The primary remaining risk is convexity: large spread moves change Δ_ix and require rehedging.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| Δ_ix = D / D_ix | Hedge ratio: equity tranche risky duration / index risky duration |
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| S_equity | Spread on the equity tranche (higher premium) |
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| S_index | Spread on the CDS index (lower premium) |
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| Net carry | S_equity - Δ_ix × S_index (approximately) |
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| Rehedging frequency | Required as spreads move (risky durations change) |
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## Variations
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- Leave the position partially unhedged to maintain more credit spread exposure (higher carry but more market risk).
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- Use individual CDS names in the reference pool instead of the index for more precise hedging.
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## Notes
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- The equity tranche has the highest spread income but is the first to lose principal if defaults occur in the reference pool; this is the primary tail risk.
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- Δ_ix changes as spreads widen or tighten, requiring dynamic rehedging; gamma (convexity) cost reduces net carry.
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- During credit crises, equity tranche spreads can blow out dramatically; the hedge may not keep pace if the index and tranche spreads move non-proportionally.
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- The strategy is exposed to correlation risk: if default correlations in the reference pool increase (systemic stress), equity tranche losses occur faster than priced.
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---
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description: "CDO carry strategy that sells a senior or mezzanine tranche and delta-hedges by buying the CDS index, earning the index spread premium over the tranche cost."
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tags: [structured-assets, cdo, carry, senior-tranche, mezzanine, delta-hedge, cds-index]
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---
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# Carry — Senior/Mezzanine Tranche with Index Hedging
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**Section**: 11.3 | **Asset Class**: Structured Assets | **Type**: Carry / Delta-Hedged
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## Overview
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This is the mirror image of the equity tranche carry trade (Section 11.2). Instead of buying a low-quality tranche and hedging with a short index position, the trader sells a high-quality (senior or mezzanine) tranche and delta-hedges by buying the CDS index. The premiums received from the index exceed the premiums paid on the short senior/mezzanine tranche, generating positive net carry.
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## Construction / Mechanics
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**Position:**
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- Short the senior/mezzanine tranche (protection buyer): pay spread S_senior
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- Long the CDS index (protection seller): receive spread S_index
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**Delta (hedge ratio):** Same formula as Eq. (487):
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```
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Δ_ix = D / D_ix (487)
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```
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where D is the risky duration of the senior/mezzanine tranche and D_ix is the risky duration of the CDS index.
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**Economics:**
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- Premium received from the long index position > premium paid on the short senior/mezzanine tranche
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- This trade is the opposite of the equity tranche strategy: the index premiums fund the tranche premium cost
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## Return Profile
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Earns positive carry as long as defaults do not trigger payouts on the index position that exceed the spread income. The hedged position is approximately spread-neutral to small parallel credit spread moves. The long index position compensates for portfolio-level default losses; the short senior tranche position profits from non-extreme loss scenarios.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| Δ_ix = D / D_ix | Hedge ratio: senior/mezzanine tranche risky duration / index risky duration |
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| S_index | Spread received on long CDS index position |
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| S_senior | Spread paid on short senior/mezzanine tranche |
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| Net carry | Δ_ix × S_index - S_senior (approximately) |
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## Variations
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- Adjust the quality of the tranche sold (from mezzanine to super-senior) to tune the risk/carry trade-off.
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- Use multiple tranches at different quality levels to diversify the credit curve exposure.
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## Notes
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- This strategy is "opposite" to the equity tranche carry trade: the long CDS index position exposes the trader to portfolio-level defaults, while the short senior tranche benefits from low-to-moderate loss scenarios.
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- The net carry is typically lower than the equity tranche trade because senior tranche spreads are lower.
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- Correlation risk applies here as well: if correlation decreases (idiosyncratic defaults increase), the index position is hit harder than a diversified senior tranche.
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- The CDS index is a liquid instrument; basis risk between the index and individual tranche spreads is a key residual risk.
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---
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description: "CDO carry strategy that buys a low-quality tranche and delta-hedges by selling a high-quality tranche, earning the spread differential between the two tranches while hedging credit spread exposure."
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tags: [structured-assets, cdo, carry, tranche-hedging, delta-hedge, spread-differential]
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---
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# Carry — Tranche Hedging
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**Section**: 11.4 | **Asset Class**: Structured Assets | **Type**: Carry / Delta-Hedged
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## Overview
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Rather than using the CDS index as the hedge vehicle, this strategy hedges a long position in a low-quality tranche by selling a high-quality tranche. The hedge ratio is calibrated to equate the risky durations of the two positions, and the trader earns the spread differential between the high-yield low-quality tranche and the low-yield high-quality tranche.
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## Construction / Mechanics
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**Position:**
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- Long a low-quality tranche (e.g., equity 0–3%): receive high spread S_low
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- Short a high-quality tranche (e.g., senior): pay lower spread S_high
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**Hedge ratio:** The number of high-quality tranche units to short per unit of low-quality tranche is:
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```
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Δ_high = D_low / D_high (488)
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```
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where:
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- D_low = risky duration of the low-quality tranche
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- D_high = risky duration of the high-quality tranche
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**Economics:**
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- Net carry = S_low - Δ_high × S_high (per unit time, before defaults)
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- Since S_low >> S_high, the trade generates positive net carry when spread-neutral
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## Return Profile
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Profits from the spread differential between the low and high-quality tranches. The delta-hedge neutralises small parallel credit spread moves. Residual exposure includes the correlation between the two tranche spreads, curvature (gamma) as spreads shift, and the risk that defaults breach the low-quality tranche while leaving the high-quality tranche intact.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| Δ_high = D_low / D_high | Hedge ratio: risky durations of low vs. high quality tranches |
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| S_low | Spread on low-quality (long) tranche |
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| S_high | Spread on high-quality (short) tranche |
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| Net carry | S_low - Δ_high × S_high |
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| Rehedging | Required as risky durations change with spread moves |
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## Variations
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- Use mezzanine as the hedge instead of senior to change the risk profile.
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- Stack multiple tranche pairs (e.g., equity vs. mezzanine and mezzanine vs. senior) in a ladder structure.
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## Notes
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- The correlation between the two tranche spreads is the key residual risk: if the low-quality tranche widens while the high-quality tranche tightens (de-correlation of credit curves), the hedge becomes less effective.
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- Unlike the index-hedged strategies (Sections 11.2 and 11.3), there is no single liquid instrument representing the hedge; both legs may have limited secondary market liquidity.
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- The hedge ratio Δ_high changes dynamically; frequent rebalancing is required in volatile credit markets.
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- This trade is sensitive to the shape of the loss distribution: a bi-modal loss distribution (either very few or very many defaults) affects the two tranches differently.
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---
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description: "CDO curve flattener/steepener strategy that takes simultaneous long and short positions in CDO tranches of different maturities, betting on changes in the shape of the CDO spread curve."
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tags: [structured-assets, cdo, curve-trade, flattener, steepener, term-structure]
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---
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# CDO Curve Trades
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**Section**: 11.6 | **Asset Class**: Structured Assets | **Type**: Relative Value / Curve
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## Overview
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Analogous to curve trades in bonds (Section 5.13), CDO curve trades involve simultaneous long and short positions in CDO tranches of different maturities. A flattener (steepener) bets that the spread curve will flatten (steepen), meaning the differential between long-term and short-term tranche spreads will decrease (increase). The trade can be structured to be dollar-neutral, risky-duration-neutral, or carry-neutral.
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## Construction / Mechanics
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**Flattener**: Sell (buy protection on) a short-term tranche + Buy (sell protection on) a long-term tranche.
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- Profits if long-term spreads fall relative to short-term spreads (curve flattening).
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- In a flattener with S_long > S_short: the trade has positive carry.
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**Steepener**: Buy (sell protection on) a short-term tranche + Sell (buy protection on) a long-term tranche.
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- Profits if long-term spreads rise relative to short-term spreads (curve steepening).
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**Carry over period [t, t+Δt]:**
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```
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C(t, t+Δt) = (M_long × S_long - M_short × S_short) × Δt (490)
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```
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where M_long, M_short are the long and short tranche notionals, and S_long, S_short are their spreads.
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**P&L:** The MTM P&L of the strategy is:
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```
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P&L = M_long - M_short (491)
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```
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where M_long and M_short are the long and short tranche MTM values (Eq. 485 from Section 11.1).
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**Structuring options:**
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- **Dollar-neutral**: M_long = M_short (equal notionals)
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- **Risky-duration-neutral**: D_long = D_short (Eq. 486); equal risky duration exposure
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- **Carry-neutral**: M_long × S_long = M_short × S_short (zero net carry at inception)
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## Return Profile
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Profits from the anticipated change in the shape of the CDO spread curve (flattening or steepening). The trade eliminates much of the outright credit spread level exposure when structured as duration-neutral. For an upward-sloping curve (S_long > S_short), a flattener has positive carry (favourable in stable conditions).
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| S_long, S_short | Spreads on long and short maturity tranches |
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| M_long, M_short | Tranche notionals |
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| C(t, t+Δt) | Carry over the holding period (Eq. 490) |
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| Structuring method | Dollar-neutral / duration-neutral / carry-neutral |
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| P&L = M_long - M_short | Mark-to-market gain/loss (Eq. 491) |
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## Variations
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- **Index curve trade**: implement with CDS index tranches of different tenors (e.g., 5-year vs. 10-year CDX).
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- **Cross-tranche curve trade**: combine different tranche quality (e.g., equity 5yr vs. senior 10yr) for a combined curve and quality bet.
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## Notes
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- Upward-sloping CDO spread curves give positive carry on flattener trades (sell short-term protection, buy long-term protection when S_long > S_short).
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- Duration-neutral structuring ensures that small parallel shifts in the spread curve do not generate P&L; the trade is a pure curve bet.
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- The risky duration changes as spreads move, requiring periodic rebalancing to maintain duration neutrality.
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- Liquidity at the long end of the CDO curve is typically lower than at the standard 5-year tenor; transaction costs may be significant.
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@@ -0,0 +1,92 @@
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---
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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) |
|
||||
| L(t) | Expected loss at time t; Eq. (481) |
|
||||
|
||||
## Notes
|
||||
- The expected loss L(t) and the probabilities p_α(t) are model-dependent; different models (Gaussian copula, etc.) give different valuations.
|
||||
- Risky duration is the primary hedging metric for CDO tranche positions (used in Sections 11.2–11.5).
|
||||
- CDS indexes (CDX, iTraxx) provide liquid reference points for both outright hedging and relative-value trades.
|
||||
- Convexity of tranche value w.r.t. spread means that hedging ratios change as spreads move; dynamic rehedging is required.
|
||||
@@ -0,0 +1,60 @@
|
||||
---
|
||||
description: "Mortgage-backed security (MBS) passthrough trading strategy that buys MBS and hedges interest rate duration with interest rate swaps, leaving prepayment risk as the primary exposure and profit driver."
|
||||
tags: [structured-assets, mbs, mortgage, prepayment, duration-hedge, interest-rate-swap]
|
||||
---
|
||||
|
||||
# Mortgage-Backed Security (MBS) Trading
|
||||
|
||||
**Section**: 11.7 | **Asset Class**: Structured Assets | **Type**: Carry / Relative Value / Duration-Hedged
|
||||
|
||||
## Overview
|
||||
MBS passthrough securities pool mortgage cash flows and pass them through to investors. The primary risk specific to MBS (beyond general interest rate risk) is prepayment risk: homeowners can prepay their mortgages when interest rates fall, refinancing at lower rates. This creates negative convexity in MBS prices. The core trading strategy buys MBS passthroughs and hedges the interest rate duration exposure with interest rate swaps, isolating prepayment risk as the residual P&L driver.
|
||||
|
||||
## Construction / Mechanics
|
||||
**Position:**
|
||||
- Long MBS passthrough (receive mortgage cash flows)
|
||||
- Short interest rate swap notional calibrated to hedge the MBS duration (pay fixed, receive floating; or receive fixed, pay floating depending on direction)
|
||||
|
||||
**Prepayment mechanics:**
|
||||
- When interest rates fall, homeowners refinance (prepay mortgages early)
|
||||
- The MBS investor receives principal early (at par) but loses the above-market coupon stream
|
||||
- This creates **negative convexity**: the MBS price rises less than a comparable bond when rates fall, and falls similarly when rates rise
|
||||
- The 5-year swap rate is the key reference benchmark for prepayment behaviour
|
||||
|
||||
**Hedge ratio determination:**
|
||||
The hedge ratios are model-dependent and require prepayment models. Two approaches:
|
||||
|
||||
1. **Model-based**: use a prepayment model to compute duration, then size the swap position to match the MBS dollar duration.
|
||||
|
||||
2. **Non-parametric / empirical**: using historical data, estimate the first derivative of MBS price P with respect to the 5-year swap rate R:
|
||||
- Constrain P to be a non-increasing function of R (reflecting negative convexity)
|
||||
- Estimate via constrained regression (e.g., Ait-Sahalia and Duarte, 2003; Duarte, Longstaff and Yu, 2006)
|
||||
|
||||
## Return Profile
|
||||
Profits from:
|
||||
1. **Carry**: MBS passthroughs typically offer a spread over Treasuries and swaps as compensation for prepayment risk.
|
||||
2. **Prepayment mispricing**: if the market overprices prepayment risk (i.e., the MBS is cheap relative to the modelled value), the hedged position profits as spreads tighten.
|
||||
|
||||
The primary risk is prepayment model error and the negative convexity of MBS when rates decline sharply.
|
||||
|
||||
## Key Parameters / Signals
|
||||
| Parameter | Description |
|
||||
|-----------|-------------|
|
||||
| Prepayment rate (CPR) | Conditional prepayment rate; key driver of MBS cash flows |
|
||||
| Duration | Interest rate sensitivity of MBS; hedge-determining quantity |
|
||||
| Hedge instrument | Interest rate swaps (typically 5-year or 10-year) |
|
||||
| OAS (option-adjusted spread) | Spread over swap curve after adjusting for the embedded prepayment option; buy high OAS |
|
||||
| Negative convexity | MBS price appreciation is capped when rates fall (prepayments accelerate) |
|
||||
| Hedge ratio ∂P/∂R | First derivative of MBS price w.r.t. 5-year swap rate; model or regression estimated |
|
||||
|
||||
## Variations
|
||||
- **TBA (to-be-announced) trading**: most liquid MBS market; trade generic pools before specific pools are specified.
|
||||
- **Specified pool trading**: target pools with lower prepayment characteristics (e.g., low-loan-balance, geographic concentrations) that command a premium.
|
||||
- **IO/PO strips**: trade interest-only or principal-only strips to isolate specific prepayment exposures.
|
||||
- **Agency vs. non-agency MBS**: agency MBS (Fannie/Freddie/Ginnie) carry implicit government guarantee; non-agency MBS add credit risk.
|
||||
|
||||
## Notes
|
||||
- Prepayment modelling is complex; the standard PSA (Public Securities Association) benchmark model and its descendants require significant calibration.
|
||||
- Negative convexity is the key distinguishing risk vs. standard fixed income: as rates fall, the effective duration of MBS shortens (prepayments accelerate), creating re-hedging costs.
|
||||
- The non-parametric hedge approach avoids model specification risk but requires substantial historical data and may be slow to adapt to regime changes in prepayment behaviour.
|
||||
- The 2008 financial crisis demonstrated that non-agency MBS carries substantial credit risk in addition to prepayment risk; agency MBS largely avoided credit losses but still suffered spread widening.
|
||||
Reference in New Issue
Block a user