First-to-Default Baskets

An FTD is used when a buyer wants to protect its credit exposure to several companies at once and a seller takes on the loss on the first name; the two set the terms of the trade. In return, the seller receives a higher spread than on a single-name CDS.

A trade that ends at the first default

A first-to-default (FTD) swap bundles several reference entities into one basket and trades it as one contract. The protection buyer pays a spread at regular intervals, and the seller bears the loss if any one name in the basket defaults. The trade ends at the first default¹. A 2004 Federal Reserve Board paper describes small baskets of around five names as common. This is one way an FTD differs from a CDO tranche¹. How a credit default swap (CDS) on a single name works and how its payments flow is covered in Credit Default Swaps (CDS).

Consider a hypothetical example in which Company S has lent 2 billion yen to each of five reference entities (entities 1 to 5). It buys FTD protection on the five names with a notional amount of 2 billion yen and a term of five years. The notional covers one name, not 2 billion yen for each of the five. The spread is 7.8% a year (780bp), so Company S pays 156 million yen a year (2 billion yen × 7.8%), once a year. Accrued amounts and loan interest are omitted.

When a credit event such as a default occurs on the first name, the seller pays the loss. With a recovery rate of 40%, the payment is 2 billion yen × (1 − 40%) = 1.2 billion yen²³. Some trades instead have the seller take delivery of the debt and pay its face value in cash (physical settlement)¹.

Payments in three cases

Company S's payments depend on how many names default during the five years. The table below totals them for three cases: no defaults, one name defaulting just after the year 3 payment, and one name defaulting at year 3 and another at year 4. The loss on each loan is 60% of face value (a 40% recovery rate), or 1.2 billion yen.

With no defaults, Company S pays five years of spread, 780 million yen, and receives no compensation. The seller keeps that amount. If only one name defaults, Company S receives 1.2 billion yen, the same as the loan loss, so its net cost is the 468 million yen of spread paid. The seller loses 732 million yen, which is more than the spread it received. If two names default, the FTD compensates only the first. The 1.2 billion yen loss on the second loan stays with Company S. The seller's loss stops at one name⁴.

Figure description

The flows of Company S, the protection buyer, when only one name defaults. The spread payments continue until the default, after which one upward arrow shows the compensation for the loss, and the trade ends. The compensation covers one name only, and the amounts are the same as in the table in the text. Arrow lengths indicate relative size.

The range of the FTD spread

The FTD spread falls in a range set by the CDS spreads of the five names. The lower bound is the largest spread among the five. The upper bound is the sum of the five spreads⁵.

In this example, the FTD spread is at least 450bp and at most 1,200bp. The 780bp in Company S's example falls inside this range.

The two ends of the range correspond to the two extremes of how defaults overlap. If the defaults of the five names are unrelated, a payout can happen whenever any of the five defaults, so the spread equals the sum. If the five default at the same time, the basket is effectively one name, and the spread equals that of the largest⁵. Buying CDS on each name to cover all five costs the sum of the spreads. An FTD covers only one name's loss, and in exchange it costs less than that sum⁵.

The higher the correlation, the lower the spread

The FTD spread depends on the default correlation among the reference entities. The lower the correlation, the more likely it is that at least one of the five defaults, and the spread moves closer to the sum. The higher the correlation, the closer the probability that at least one defaults comes to that of a single name, and the spread moves closer to that of the largest name. For the seller, the trade gets riskier as correlation falls⁵.

The table below estimates the spread in this example at different correlation values. It assumes a 40% recovery rate, a constant spread for each name until maturity, zero interest rates, and continuous spread payments. The correlation is the parameter of a one-factor Gaussian copula (explained in the next section)⁶.

At a correlation of 0, the spread is the sum, 1,200bp. As the correlation approaches 1, it approaches 450bp, the spread of the largest name, entity 5. At a correlation of 0.5, the spread is about 780bp, the level in Company S's example.

A 2004 Federal Reserve Board paper shows the same pattern. Consider a hypothetical example in which a basket of five names has 5-year CDS spreads of 80, 67, 146, 25 and 34bp (a sum of 352bp). In the paper, the FTD spread was 352bp at a correlation of 0, 301bp at 0.3, and 168bp at 0.9⁷.

Valuation with a copula

Valuing an FTD separates two things: the probability that each name defaults, and how the defaults of different names overlap. The probability for each name is found by deriving a hazard rate from that name's CDS spreads. The overlap is set by a copula. A copula links the default times of the names and determines the probability that they happen together⁹. The mechanics are covered in Copulas for Credit.

In a 2012 paper by Fermanian and Vigneron, the one-factor Gaussian copula is described as the standard market method for valuing CDO tranches⁹. An FTD can be valued the same way¹. In this approach, a single factor common to all names is introduced, and once that factor is fixed, the defaults of the names are treated as independent of each other¹⁰¹¹. When the common factor moves in the bad direction, all five names become more likely to default together. The model takes three inputs: the term structure of each name's CDS spreads, the recovery rate, and the correlation⁹.

The valuation follows the same idea as for a CDS. The FTD spread is the spread at which the present value of the spread payments by the buyer equals the present value of the compensation paid by the seller¹². The difference from a CDS is that compensation is triggered by the first default among the five names. The copula is used to find the probability of that event.

Deep diveWhat the copula valuation assumes

The Gaussian copula does not specify how each name's spread moves over time. Its inputs are only the current term structure of spreads, the recovery rate, and the correlation⁹. Before any default occurs, the changes in the value of an FTD can be offset with individual CDS only when the spreads move in a particular way¹³.

The estimates in the text rest on simple assumptions: spreads constant until maturity, a single correlation value, and zero interest rates. Changing the correlation assumption does change the value, as in the example in the sidebar of "The higher the correlation, the lower the spread."

There are other valuation methods. One study values an FTD with a jump-diffusion model and compares the result with the copula method¹⁴. If the valuation method changes, the price of the same FTD changes too. This is an example of model risk.

Generalizing to the nth default

An FTD can be extended by changing the compensation trigger from the first default to the nth. In an nth-to-default trade, compensation is paid at the nth default and the trade ends there. The case n = 1 is the FTD¹⁵.

In Company S's five-name example, if n = 2, the first default is not compensated, and compensation is paid at the second default, after which the trade ends. The loss on the first default is the buyer's to bear.

A trade that compensates the losses on the first p defaults (first p-th to default) can be treated as a basic building block of many basket trades, including CDO tranches¹⁶.

What it is used for

Buyers use an FTD to protect their credit exposure to several reference entities at a lower cost than buying CDS on each name. Only one name's loss is covered, however. As in the table in "Payments in three cases," the losses on the second and later defaults stay with Company S¹⁷.

Sellers take on the loss on one name and in return receive a higher spread than on a single-name CDS¹⁷. Like the equity tranche of a CDO, the FTD seller bears the first losses. It is a leveraged credit risk trade whose value depends on correlation¹. As the table in "Payments in three cases" shows, if there are no defaults, the seller keeps the spread received as a gain, and if one name defaults, the seller's loss exceeds the spread received.

An FTD is a bespoke product, with the basket composed trade by trade. It is traded as a swap or embedded in a note¹⁵.

Risks and the difficulty of hedging

The main risks for both the buyer and the seller of an FTD are defaults of the reference entities and a wrong correlation assumption. The losses from default are shown in the table in "Payments in three cases." Correlation cannot be observed directly and market data on it is thin, so it has to be estimated for valuation, and a wrong assumption becomes a valuation error¹⁸⁸. An example of a wrong assumption is in the sidebar in "The higher the correlation, the lower the spread."

In the analysis by Fermanian and Vigneron, when CDS on each name are used as the hedging instruments, the profit and loss is flat before any default only if the spreads move in a particular way under the Gaussian copula¹³. Traders usually prepare for both spread moves and defaults, and to do so they use two CDS of different maturities on each name. In that case the profit and loss is not flat, and a carry cost is paid for the protection¹⁹.

How it differs from similar products

A CDS index compensates the loss on a constituent in proportion to its weight, and the trade continues on the remaining names²⁰. An FTD compensates the full loss on the first name (the amount net of recovery) and ends there.

Notes

1. Michael S. Gibson, "Understanding the Risk of Synthetic CDOs" (Federal Reserve Board, FEDS 2004-36, 2004), section 6.2 (page 21). Covers the payments of the party that bears the FTD loss (the seller in this text; the investor in the source), the end of the trade after one default, the size of the basket, physical settlement, and the dependence on correlation. ↩ ↩² ↩³ ↩⁴ ↩⁵

2. Joro, Na and Niu, "A Simulation-Based First-to-Default (FTD) Credit Default Swap (CDS) Pricing Approach Under Jump-Diffusion" (Proceedings of the 2004 Winter Simulation Conference), section 1 (page 1). In an FTD, when any name in the basket defaults, the seller pays the loss to the buyer, and the contract ends afterward. ↩

3. Fermanian and Vigneron, "On break-even correlation: the way to price structured credit derivatives by replication" (arXiv:1204.2251), section 5 (pages 14–15). With recovery rate R and δ indicating whether each of the n names has defaulted, the FTD payment is (1 − R) × (1 − Π(1 − δ_j)), so (1 − R) is paid if even one name defaults. ↩

4. Joro et al., section 3 (page 2). Selling an FTD limits the seller's credit risk to one name, and the seller typically earns more income than on a single-name CDS. ↩

5. Joro et al., section 3 (page 2). At a correlation of 1 the FTD spread equals that of the largest name, and at a correlation of 0 it equals the sum over the n names, which gives a range from the largest single name up to the sum. The sum of the individual CDS is an upper bound on the FTD price, and the FTD price is lower than buying CDS on all names. The risk of the FTD rises as correlation falls. ↩ ↩² ↩³ ↩⁴

6. This site's estimate. The assumptions are those in the text (recovery rate 40%, interest rate 0, spreads constant until maturity), computed under a one-factor Gaussian copula and rounded to the nearest 10bp. ↩

7. Gibson, tables 8 and 9 of section 6.2 (page 22). The 5-year CDS spreads of the five names are 80, 67, 146, 25 and 34bp. The par spread of the FTD is 352bp at a correlation of 0, 301bp at 0.3, 217bp at 0.7, and 168bp at 0.9. The names in the table are omitted here. ↩ ↩²

8. Gibson, section 6 (page 18). Correlation cannot be observed, so an error in the correlation assumption is a risk. The same section also treats the FTD as a correlation product. ↩ ↩²

9. Fermanian and Vigneron, section 1 (page 2). The Gaussian copula directly specifies the joint distribution of default times and expresses the dependence among the names' default times with a correlation parameter. Its inputs are the current term structure of spreads, the recovery rate, and the correlation, and it does not specify how spreads or intensities move. ↩ ↩² ↩³ ↩⁴

10. Fermanian and Vigneron, abstract (page 1). They treat a static framework in which default times are independent conditional on a common factor, and cite its application to valuing nth-to-default baskets. ↩

11. Gibson, section 3 (page 4). The correlation of defaults within the portfolio is assumed to be driven by a common factor. The distribution of losses is found conditional on the common factor and then integrated over the common factor. ↩

12. Joro et al., section 3 (page 2). As with a CDS, the FTD price is set by equating the sum of the present values of the spread payments with the sum of the present values of the default compensation. ↩

13. Fermanian and Vigneron, sections 1 and 2 (pages 2–7). Using individual CDS as hedging instruments, the profit and loss before any default can be made flat only when the spread moves satisfy a particular relation. ↩ ↩²

14. Joro et al., abstract (page 1). They present a method for valuing an FTD based on jump diffusion and compare it with the copula method. ↩

15. IHS Markit, "CDS Indices Primer" (2021), page 13. FTD and nth-to-default (First or nth to Default) are bespoke products traded in swap or note form. The contract ends at the first (nth) default, and compensation is paid. ↩ ↩²

16. Fermanian and Vigneron, section 5 (page 15). A trade that compensates losses up to the first p defaults can be seen as a basic building block of many basket trades, including CDO tranches. The payment is min(Σδ_i, p) × (1 − R). ↩

17. Joro et al., section 3 (page 2). In hedging credit exposure, buying CDS on all names is expensive, so an FTD protects against the first default at a lower cost than buying CDS on each name. The seller earns more income than on a single-name CDS. ↩ ↩²

18. Joro et al., section 3 (page 2). Default correlation is hard to model because market data are scarce. ↩

19. Fermanian and Vigneron, section 2 (page 7). When preparing for both spread moves and defaults, two CDS of different maturities on each name are usually used. In that case the profit and loss is not flat, and a cost is paid for the protection against default. ↩

20. IHS Markit, "CDS Indices Primer" (2021), page 22. After a credit event on a constituent, a new version is published with that name's weight set to 0, and the notional of the trade falls by that name's weight. In the example, when one name in a 100-name index defaults, a notional of USD 10 million becomes USD 9.9 million. ↩