Synthetic CDOs and Tranches

Banks use synthetic CDOs to transfer the credit risk on their loans, investors to take the layer of risk and return they prefer, and dealers to hedge the tranches they sell.

The reference portfolio and its tranches

A synthetic CDO uses a reference portfolio built from credit default swaps (CDS) on several reference entities. It divides the losses on that portfolio into tranches and trades them. Losses hit the equity (first-loss) tranche first, then the mezzanine, and last the senior and super senior tranches. Unlike a cash CDO, which bundles bonds or loans, the reference portfolio is made of CDS¹.

Consider a hypothetical example in which Company T's portfolio consists of CDS on 100 reference entities bundled with equal weights. The notional per entity is 1 billion yen, for a total of 100 billion yen. Every reference entity has a recovery rate of 40%, so when one entity defaults, the portfolio loses 1 billion yen × (1 − 40%) = 600 million yen, or 0.6% of the portfolio. Each credit event, such as a default, is simply assumed to cause a loss of 600 million yen.

Divide the portfolio's losses into the following three tranches. The tranche notionals add up to the portfolio's 100 billion yen.

The side that takes on a tranche is in the position of selling protection on that tranche's losses¹. When a loss hits its tranche, the seller pays the buyer for the loss, and in return receives a spread on the tranche's remaining notional².

A tranche can be funded or unfunded. In a funded tranche, the investor pays the tranche's notional up front, and the notional shrinks by the amount of the losses. An unfunded tranche works like a swap, with no money moving at the start. The investor pays only when a loss hits, so there is counterparty credit risk: whether the investor can pay in the future³.

Loss allocation and attachment points

The level of portfolio losses at which a tranche starts to absorb losses is its attachment point. The level at which the tranche loses its entire notional once losses reach it is its detachment point⁴⁵. In the example above, the mezzanine tranche has an attachment point of 5% and a detachment point of 15%. The size of the tranches below a tranche's attachment point equals the size of the cushion that protects it⁴.

Let the portfolio loss be $L$, the attachment point $A$, and the detachment point $D$ (all converted to amounts of money). The loss on the tranche is then given by the following formula⁶.

$$\text{Tranche loss} = \min\bigl(\max(L - A,\ 0),\ D - A\bigr)$$

Using Company T's example, calculate how losses are allocated when 5, 10, 25, and 30 entities default. Each default causes a loss of 600 million yen.

With 5 defaults, the portfolio loss is only 3%, but 60% of the equity tranche's notional is lost. At 9 defaults, the loss is 5.4 billion yen, the equity tranche is wiped out, and the mezzanine tranche starts to lose (400 million yen). At 10 defaults, the mezzanine loss is 1 billion yen. The senior tranche takes a loss only once 26 or more entities have defaulted.

This formula means a tranche's loss can be read as a combination of options on the portfolio loss. It has the same form as the payoff of a call spread: buy a call with a strike at loss $A$ and sell a call with a strike at loss $D$⁷. The figure below shows the amount that the protection buyer on the mezzanine tranche receives.

Figure description

The amount that the protection buyer on the mezzanine tranche in Company T's example (losses from 5 billion yen to 15 billion yen) receives from the seller. The horizontal axis is the portfolio loss and the vertical axis is the amount the buyer receives. When the loss is 5 billion yen or less, the buyer receives nothing. Above 5 billion yen, each additional 100 million yen of loss adds 100 million yen to the amount received, and when the loss reaches 15 billion yen the amount received hits its cap of 10 billion yen. Further losses do not increase it.

The spread is paid on the notional that remains after losses⁸. Consider a hypothetical example in which the equity tranche's spread is 15% a year. At the start, the buyer pays 5 billion yen × 15% = 750 million yen a year (simplified to one payment per year). If 5 entities default, the buyer receives 3 billion yen and the remaining notional falls to 2 billion yen, so later payments drop to 2 billion yen × 15% = 300 million yen. If 10 entities default, the entire 5 billion yen notional is lost and the exchange of spread payments stops.

How correlation affects each tranche

Defaults tending to cluster in time (high default correlation) affects each tranche in a different direction. From the viewpoint of the side that takes on a tranche (the protection seller), the higher the correlation, the more the value of the equity tranche rises and the more the value of the senior tranche falls. The mezzanine tranche feels both effects, and they offset each other, so its value moves less with correlation than that of the equity or senior tranche⁹.

The reason is that higher correlation makes both cases more likely: that almost no entity defaults, and that many default at once. The equity tranche bears only the first few defaults, so what it gains when almost no one defaults outweighs what it loses when many default, and its value rises⁹. The senior tranche takes losses only when many entities default, so its value falls as that probability rises⁹. In Company T's example, 26 or more entities must default before the senior tranche takes a loss, and the higher the correlation, the more likely that becomes.

From the protection buyer's viewpoint, the directions are reversed. A buyer of senior protection gains when correlation rises, because the chance of receiving a payment increases, and a buyer of equity protection loses.

The price of a tranche reflects the default correlation the market expects. Because correlation cannot be observed directly, tranches are products that carry correlation risk⁹. For index tranches, the implied correlation among the constituents is used to value the tranche, and estimating it requires a complex model⁵.

A first-to-default basket, which bears only the first default, is a product on a small number of reference entities that is close to an equity tranche. They are similar in that they bear losses first, take a large risk on a small notional, and have a value that depends on correlation¹⁰.

Standardized index tranches

Some CDS indices are also traded as tranches that divide the index's losses into fixed ranges⁵. How the index itself works is covered in CDS Indices (iTraxx, CDX). The standard tranche ranges and coupons in the IHS Markit publication (November 2021) are as in the table below¹¹.

Standard maturities are 3, 5, 7, and 10 years. Tranches roll to a new series once a year, in September¹¹.

Index tranches differ from the index itself in that they measure losses after taking recovery into account. The index itself reduces its notional by the weight of each defaulted name. A tranche counts the name's weight multiplied by the loss rate at default (1 − recovery rate) as its loss⁵.

Treat the portfolio in Company T's example as an index of 100 equally weighted names. Now consider an equity tranche with an attachment point of 0% and a detachment point of 5%. When one entity with a 40% recovery rate defaults, the portfolio loss is 1% × 60% = 0.6%. The protection buyer on the equity tranche, with a notional of 5 billion yen, receives 5 billion yen × 0.6% ÷ 5% = 600 million yen. The equity tranche's notional falls to 4.4 billion yen¹².

After a default, the tranches trade on a new range based on the remaining 99 names. The equity tranche's new detachment point is (5% − 0.6%) ÷ 0.99 = about 4.44%, and the attachment and detachment points of the mezzanine and senior tranches change as well¹². A loss on the equity tranche does not reduce the notional of the other tranches, but it shrinks the cushion that protects them. The super senior tranche's notional falls by the amount recovered¹².

Spreads and tranche quotes

The spread of a tranche at the time of the trade is set at the value (the par spread) at which the present value of the spread payments equals the present value of the compensation for losses. The present value of the compensation is calculated from the tranche's expected loss⁶. The tranche's expected loss depends on the distribution of the portfolio's losses, and that distribution is determined by each reference entity's default probability and by correlation.

The formulas and numerical example below come from the Gibson paper (2004). In the formula, let $EL_i$ be the expected loss up to the $i$-th payment date, $P_i$ the discount factor for that date, and $\Delta_i$ the length of the payment period. The par spread $s$ is then given by the following formula. The numerator is the present value of the compensation for losses, and the denominator is the present value of the notional on which the spread is paid (the tranche width minus the expected loss)⁶.

$$s = \frac{\sum_i P_i\,(EL_i - EL_{i-1})}{\sum_i P_i\,\Delta_i\,\bigl((D - A) - EL_i\bigr)}$$

The expected loss $EL_i$ is the sum, over the number of defaults, of each probability multiplied by the tranche's loss at that number of defaults (the formula in the previous section). The model that produces these probabilities rests on the following assumptions¹³.

• Each company's asset value is expressed with a single factor common to all companies and a factor specific to each company. The strength of the common factor determines the correlation. In the numerical example, both factors are normally distributed.
• Each company's default probability is derived from its CDS spread. The recovery rate is constant and known in advance.
• In the numerical example, the propensity to default (the hazard rate) is a constant 1% a year and the risk-free rate is a constant 5% a year.

If these assumptions fail, the values change. If the CDS spreads of the reference entities move, the expected default probabilities change, and the dealer also wants to be prepared for a sudden default (jump-to-default). If views on correlation differ, the same tranche gets different values.

Consider a hypothetical example in which a portfolio of 100 names has a maturity of 5 years. Every reference entity has a CDS spread of 60bp, a recovery rate of 40%, and an asset correlation of 30%, and the model gives the par spreads below¹⁴.

The equity tranche, which bears losses first, has a high spread, and the senior tranche, which losses rarely reach, has a low one. These figures are the result of a model with assumed values (the same notionals, recovery rate, and spreads) in which a single common factor expresses correlation, and the values change if the assumptions change¹⁴.

When the CDS spreads of the reference entities change, the value of the tranches moves too. The increase in value when the spreads of all reference entities fall by 10bp is one measure of sensitivity¹⁵. When spreads fall, the value to the protection seller (the tranche investor) rises, so when spreads widen, the seller loses and the buyer gains. Even for the same spread move, the change in value relative to notional differs by tranche, and in the example's table it is larger for the equity and mezzanine tranches than for the senior tranche¹⁵.

In trades of standardized tranches, the coupon is fixed at 100bp or 500bp for each index (see the table in the previous section). For tranches from June 22, 2009, as with the indices, an upfront payment is exchanged on the trade date, the seller pays the buyer the accrued amount up to the trade date, and the buyer pays the full coupon on the next payment date⁸. With this form, the starting point for the coupon calculation is the same whatever the trade date, which makes termination, novation, and offsetting easier⁸.

What they are used for, and gains and losses

The first synthetic CDO trade (in 1997) was made for the purpose of banks hedging credit risk, reducing regulatory capital, or both¹⁷. Later, trades that create tranches with the risk and return that investors prefer became more common¹⁷. Because each tranche has a different risk and return, the paper gives an example in which different investors take them: asset managers or hedge funds for the equity tranche, regional banks for the mezzanine, and reinsurers for the senior¹⁸.

Using Company T's example, consider using tranches to reduce the credit risk on loans. A bank has lent 1 billion yen to each of the 100 companies in Company T's portfolio (100 billion yen in total), and the loan losses are the same as the reference portfolio's (600 million yen per default). The bank buys protection on the equity tranche (5 billion yen) and pays a spread of 15% a year. The results are as follows.

With up to 5 defaults, the loan loss is covered by the amount received from the equity tranche. With 10 defaults, the bank bears the 1 billion yen that exceeds the equity tranche's range. With no defaults, the bank only pays the spread and receives nothing. Adding the first year's spread of 750 million yen, the bank's combined burden of loan loss and spread is 750 million yen with 0 defaults, 750 million yen with 5 defaults, and 1.75 billion yen with 10 defaults (the first year's spread is counted as paid even when defaults occur).

From the viewpoint of the investor that sold the protection, the gains and losses are reversed. With no defaults, the investor receives 750 million yen a year. With 5 defaults, it pays 3 billion yen, and with 10 defaults, it pays 5 billion yen.

When a bank transfers loan risk through a synthetic CDO, it often keeps the equity tranche itself, and in that case the risk transferred is smaller than the share of notional would suggest¹⁹. This is because the equity tranche has a small share of notional but a large share of the risk.

Tranche risk and dealer risk

This section is based on the explanations and examples in the 2004 paper.

The risk of a tranche is not proportional to its share of notional. In the paper's example, the equity and mezzanine tranches made up 10% of the portfolio's notional but accounted for 70% to 90% of the risk (the range depends on how risk is measured)²⁰. The equity tranche carries about 13 to 17 times the risk of investing directly in the whole portfolio, and the mezzanine tranche about 5 to 7 times²¹.

In a trade in which a dealer sells just one tranche to an investor (a single-tranche CDO), the dealer is in the position of buying protection on that tranche. According to market participants' observations, around 2004, single-tranche trades were more common among new deals than the traditional form of selling all the tranches at once²². The dealer hedges by selling protection through single-name CDS on the reference entities. The amount required changes every time spreads move, so rebalancing continues²².

Consider a hypothetical example in which the dealer sells an equity tranche with a notional of USD 30 million (5 years, 100 names, USD 10 million per name, CDS spread of 60bp). The dealer hedges by selling USD 483 million of protection across the 100 names²³. That amount is about 16 times the tranche's notional.

Hedging a single tranche involves four risks²⁴.

• The hedge amount for each reference entity (delta) depends on the model, so the dealer bears model risk.
• Each rebalancing of the hedge costs money in buying and selling CDS, and if the CDS market lacks enough liquidity, the dealer cannot rebalance as it wants.
• The dealer wants to be prepared both for small moves in spreads and for sudden defaults (jump-to-default).
• There is a risk that the correlation assumed in the price and the hedge differs from the actual correlation.

Because correlation cannot be observed, correlation risk remains even after the dealer hedges²⁵. When market participants hold different views on correlation, they put different values on the same tranche, and trades that buy and sell correlation arise²⁵.

Gaps remain even when a dealer sells all the tranches and hedges with CDS on the whole portfolio. In the paper's example, the weighted average of the three tranches' spreads is 74bp, above the portfolio's 60bp¹⁴. The dealer starts out paying more than it receives (negative carry). As defaults erode the equity tranche's notional, the payments shrink, and the carry turns positive at about half of the maturity. In this example, early defaults work in the dealer's favor²⁶.

The paper's simple model does not include the order of payments (the waterfall), mechanisms that redirect payments to senior tranches under certain conditions, or restrictions on the manager. These are said to be more common in cash CDOs²⁷.

Notes

1. Michael S. Gibson, "Understanding the Risk of Synthetic CDOs" (Federal Reserve Board, revised July 2004; hereafter "the Gibson paper"), page 1, "1 Introduction" and "2 Some background on synthetic CDOs". Super senior is defined in footnote 1 on page 1. ↩ ↩²

2. The Gibson paper, pages 6–8, "4 Pricing a synthetic CDO tranche". The formula for paying the spread on the remaining notional is equation (9) on page 8. ↩

3. The Gibson paper, page 3, the explanation of "funded" and "unfunded". ↩

4. The Gibson paper, page 6, "4 Pricing a synthetic CDO tranche". The term "detachment point" comes from the Primer in the next note. ↩ ↩²

5. IHS Markit, "CDS Indices Primer" (November 2021; hereafter "the Primer"), page 25, "Overview". ↩ ↩² ↩³ ↩⁴

6. The Gibson paper, page 8, equations (7) to (9) and (11). The formulas in the text combine these and rewrite the paper's symbols L and H as A and D. ↩ ↩² ↩³

7. This site's own arithmetic rewriting the loss formula as a difference of call payoffs, not a claim made by the sources: max(L − A, 0) − max(L − D, 0) = min(max(L − A, 0), D − A). ↩

8. The Primer, page 26, "Tranche trade mechanics". ↩ ↩² ↩³

9. The Gibson paper, page 18, "6 CDO tranches are exposed to correlation risk", and Figure 4 on page 19. ↩ ↩² ↩³ ↩⁴

10. The Gibson paper, page 21, "6.2 First-to-default baskets and correlation trading". ↩

11. The Primer, page 34, "Index Tranches Availability" in "Appendix 3: Roadmap to Tranches". ↩ ↩²

12. The Primer, pages 26–27, "Tranche trading example" and "Tranche Adjustment". The source's formulas are applied here to the numbers in Company T's example. ↩ ↩² ↩³ ↩⁴ ↩⁵

13. The Gibson paper, pages 4–6 (PDF pages 6–8), equations (1) and (2) in "3 Modeling the loss distribution", and the assumptions of the numerical example on pages 9–10. ↩

14. The Gibson paper, pages 9–10, "5 The risk of CDO tranches", and Table 2. The assumptions include USD 10 million per name, quarterly payments, a hazard rate of 1% a year, and a risk-free rate of 5%. ↩ ↩² ↩³

15. The Gibson paper, page 11, "5.1 Risk measures for the hypothetical CDO tranches", and the sensitivity column of Table 5 on page 14. ↩ ↩²

16. The Primer, page 28. ↩

17. The Gibson paper, pages 2–3, "2 Some background on synthetic CDOs". The first trade was in 1997 (page 2). ↩ ↩²

18. The Gibson paper, page 9, "5 The risk of CDO tranches". These are examples of buyers in a hypothetical example. ↩

19. The Gibson paper, page 17, "5.3 Risk transfer is limited when the equity tranche is retained". ↩

20. The Gibson paper, page 12 and Table 4 on page 13. The hypothetical example is a USD 1 billion CDO (equity 0% to 3%, mezzanine 3% to 10%, senior 10% to 100%). ↩

21. The Gibson paper, page 12 and Table 5 on page 14. The ranges of about 13 to 17 times and 5 to 7 times come from three measures: sensitivity, expected loss, and unexpected loss. ↩

22. The Gibson paper, page 3 (the explanation of single-tranche CDOs) and pages 19–20, "6.1 Single-tranche CDOs". ↩ ↩²

23. The Gibson paper, page 20, Table 7. Against the equity tranche (0% to 3%, USD 30 million), the dealer sells USD 4.83 million of protection on each reference entity, USD 483 million in total. ↩

24. The Gibson paper, pages 20–21, "6.1 Single-tranche CDOs" (the four risk management challenges). ↩

25. The Gibson paper, page 21, "6.2 First-to-default baskets and correlation trading". ↩ ↩²

26. The Gibson paper, page 10, "5 The risk of CDO tranches" (the explanation after Table 2). ↩

27. The Gibson paper, pages 17–18, "5.4 CDO structuring". ↩