Hull J.C. Risk management and Financial institutions
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Credit Derivatives 311
LIBOR plus 25 basis points to buy the bond. The payer retains ownership
of the bond for the life of the swap and faces less credit risk than it would
have done if it had lent money to the receiver to finance the purchase of
the bond, with the bond being used as collateral for the loan. If the
receiver defaults, the payer does not have the legal problem of trying to
realize on the collateral.
8
The spread over LIBOR received by the payer is compensation for
bearing the risk that the receiver will default. The payer will lose money
if the receiver defaults at a time when the reference bond's price has
declined. The spread therefore depends on the credit quality of the receiver
and of the bond issuer, and on the default correlation between the two.
There are a number of variations on the standard deal we have
described. Sometimes, instead of a cash payment for the change in the
value of the bond, there is physical settlement where the payer exchanges
the underlying asset for the notional principal at the end of the life of the
swap. Sometimes the change-in-value payments are made periodically
rather than all at the end of the life of the swap.
13.6 BASKET CREDIT DEFAULT SWAPS
In what is referred to as a basket credit default swap there are a number of
reference entities. An add-up basket CDS provides a payoff when any of
the reference entities default. A first-to-default CDS provides a payoff
only when the first default occurs. A second-to-default CDS provides a
payoff only when the second default occurs. More generally, an nth-to-
default CDS provides a payoff only when the nth default occurs. Payoffs
are calculated in the same way as for a regular CDS. After the relevant
default has occurred, there is a settlement. The swap then terminates and
there are no further payments by either party.
13.7 COLLATERALIZED DEBT OBLIGATIONS
A collateralized debt obligation (CDO) is a way of creating securities with
widely different risk characteristics from a portfolio of debt instruments.
8
Repos are structured to minimize credit risk in a similar way. A company requiring
short-term funds sells securities to the lender and agrees to buy them back at a later time
at a slightly higher price. The difference between the prices is the interest the lender earns.
If the borrower defaults, the lender keeps the securities. If the lender defaults, the
borrower keeps the funds.
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Figure 13.3 Collateralized debt obligation.
An example is shown in Figure 13.3, where four types of securities (or
tranches) are created from a portfolio of bonds. The first tranche has 5% of
the total bond principal and absorbs all credit losses from the portfolio
during the life of the CDO until they have reached 5% of the total bond
principal. The second tranche has 10% of the principal and absorbs all
losses during the life of the CDO in excess of 5% of the principal up to a
maximum of 15% of the principal. The third tranche has 10% of the
principal and absorbs all losses in excess of 15% of the principal up to a
maximum of 25% of the principal. The fourth tranche has 75% of the
principal and absorbs all losses in excess of 25% of the principal. The
yields in Figure 13.3 are the rates of interest paid to tranche holders. These
rates are paid on the balance of the principal remaining in the tranche after
losses have been paid. Consider the first tranche. Initially the return of
35% is paid on the whole amount invested by the tranche holders. But after
losses equal to 1 % of the total bond principal have been experienced, the
tranche holders have lost 20% of their investment and the return is paid on
only 80% of the original amount invested.
9
Tranche 1 is referred to as the
9
When a bond with principal Q defaults and a recovery of QR is made, the usual
arrangement is that a loss of (1 - Q)R is sustained by the most junior tranche. An
amount QR is paid to the most senior tranche and this tranche's principal is reduce
by QR.
Credit Derivatives 313
equity tranche. A default loss of 2.5% on the bond portfolio translates into
a loss of 50% of the tranche's principal. Tranche 4 is usually given an Aaa
rating. Defaults on the bond portfolio must exceed 25% before the holders
of this tranche are responsible for any credit losses.
The creator of the CDO normally retains the equity tranche and sells
the remaining tranches in the market. A CDO provides a way of creating
high-quality debt from average-quality (or even low-quality) debt.
Synthetic CDOs
The CDO in Figure 13.3 is referred to as a cash CDO. An alternative
structure which has become popular is a synthetic CDO, where the creator
of the CDO sells a portfolio of credit default swaps to third parties. It
then passes the default risk on to the synthetic CDO's tranche holders.
Analogously to Figure 13.3, the first tranche might be responsible for the
payoffs on the credit default swaps until they have reached 5% of the total
notional principal; the second tranche might be responsible for the pay-
offs between 5% and 15% of the total notional principal; and so on. The
income from the credit default swaps is distributed to the tranches in a
way that reflects the risk they are bearing. For example, the first tranche
might get 3,000 basis points; the second tranche 1,000 basis points, and
so on. As with a cash CDO, losses on defaults would be netted against
the principal to determine the amount on which interest is paid.
Single-Tranche Trading
In Section 13.2 we discussed the portfolios of 125 companies that are used
to generate CDX and iTraxx indices. The market uses these portfolios to
define standard CDO tranches. The trading of these standard tranches is
known as single-tranche trading. A single-tranche trade is an agreement
where one side agrees to sell protection against losses on a tranche and the
other side agrees to buy the protection. The tranche is not part of a
synthetic CDO, but cash flows are calculated in the same way as if it were
part of a synthetic CDO. The tranche is referred to as "unfunded" because
it has not been created by selling credit default swaps or buying bonds.
In the case of the CDX NA IG index, the equity tranche covers losses
between 0% and 3% of the principal. The second tranche, which is
referred to as the mezzanine tranche, covers losses between 3% and 7%.
The remaining tranches cover losses from 7% to 10%, 10% to 15%, and
15% to 30%. In the case of the iTraxx Europe index, the equity tranche
covers losses between 0% and 3%. The mezzanine tranche covers losses
314 Chapter 13
Table 13.6 Five-year CDX IG NA and iTraxx Europe tranches on
August 30, 2005. Quotes are in basis points except for 0-3% tranche.
Source: Reuters
CDX IG NA
Tranche 0-3% 3-7% 7-10% 10-15% 15-30%
Quote 40% 127 35.5 20.5 9.5
iTraxx Europe
Tranche 0-3% 3-6% 6-9% 9-12% 12-22%
Quote 24% 81 26.5 15 9
between 3% and 6%. The remaining tranches cover losses from 6% to 9%,
9% to 12%, and 12% to 22%.
Table 13.6 shows the mid-market quotes for the five-year CDX and
iTraxx tranches on August 30, 2005. On that date the CDX index level was
50 basis points and the iTraxx index was 36.375 basis points. For example,
the mid-market price of mezzanine protection for the CDX IG NA was
127 basis points per year, while that for iTraxx Europe was 81 basis points
per year. Note that the equity tranche is quoted differently from the other
tranches. The market quote of 40% for CDX means that the protection
seller receives an initial payment of 40% of the principal plus a spread of
500 basis points per year. Similarly, the market quote of 24% for iTraxx
means that the protection seller receives an initial payment of 24% of the
principal plus a spread of 500 basis points per year.
13.8 VALUATION OF A BASKET CDS AND CDO
The spread for an nth-to-default CDS or the tranche of a CDO is critically
dependent on default correlation. Suppose that a basket of 100 reference
entities is used to define a five-year nth-to-default CDS and that each
reference entity has a risk-neutral probability of defaulting during the five
years equal to 2%. When the default correlation between the reference
entities is zero, the binomial distribution shows that the probability of one
or more defaults during the five years is 86.74% and the probability often
or more defaults is 0.0034%. A first-to-default CDS is therefore quite
valuable, whereas a tenth-to-default CDS is worth almost nothing.
As the default correlation increases the probability of one or more
defaults declines and the probability of ten or more defaults increases.
In the limit where the default correlation between the reference entities
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315
is perfect, the probability of one or more defaults equals the probability
of ten or more defaults and is 2%. This is because in this extreme
situation the reference entities are essentially the same. Either they all
default (with probability 2%) or none of them default (with prob-
ability 98%).
The valuation of a tranche of a CDO is similarly dependent on default
correlation. If the correlation is low, the junior equity tranche is very risky
and the senior tranches are very safe. As the default correlation increases,
the junior tranches become less risky and the senior tranches become more
risky. In the limit where the default correlation is perfect the tranches are
equally risky.
Using the Gaussian Copula Model of Time to Default
The one-factor Gaussian copula model of time to default presented in
Section 6.5 has become the standard market model for valuing an nth-to-
default CDS or a tranche of a CDO.
Consider a portfolio of N companies, each having a probability Q(T)
of defaulting by time T. From equation (6.11), the probability of
default, conditional on the level of the factor F, is
The trick to valuing an nth-to-default CDS or a CDO is to calculate
expected cash flows conditional on F and then integrate over F. The
advantage of this is that, conditional on F, defaults are independent. The
Business Snapshot 13.3 Correlation Smiles
Credit derivatives dealers imply default correlations from the spreads on
tranches. The compound correlation is the correlation that prices a particular
tranche correctly. The base correlation is the correlation that prices all tranches
up to a certain level of seniority correctly. If all implied correlations were the
same, we could deduce that market prices are consistent with the one-factor
Gaussian copula model for time to default. In practice, we find that compound
correlations exhibit a "smile" with the correlations for the most junior (equity)
and senior tranches higher than those for intermediate tranches. The base
correlations exhibit a "skew" where the correlation increases with the level of
seniority considered.
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probability of exactly k defaults by time T, conditional on F, is
Derivatives dealers calculate the implied copula correlation in
equation (13.1) from the spreads quoted in the market for tranches of
CDOs and tend to quote these rather than the spreads themselves (see
Business Snapshot 13.3). This is similar to the practice in options markets
of quoting Black-Scholes implied volatilities rather than dollar prices.
SUMMARY
Financial institutions use credit derivatives to actively manage their
credit risks. They use them to transfer credit risk from one company
to another and to diversify credit risk by swapping one type of exposure
for another.
The most common credit derivative is a credit default swap. This is a
contract where one company buys insurance against another company
defaulting on its obligations. The payoff is usually the difference between
the face value of a bond issued by the second company and its value
immediately after a default. Credit default swaps can be analyzed by
calculating the present value of the expected payments and the present
value of the expected payoff.
A forward credit default swap is an obligation to enter into a particular
credit default swap on a particular date. A credit default swap option is the
right to enter into a particular credit default swap on a particular date.
Both cease to exist if the reference entity defaults before the date.
A total return swap is an instrument where the total return on a
portfolio of credit-sensitive assets is exchanged for LIBOR plus a spread.
Total return swaps are often used as financing vehicles. A company
wanting to purchase a portfolio of bonds approaches a financial institu-
tion, who buys the bonds on its behalf. The financial institution then
enters into a total return swap where it pays the return on the bonds to
the company and receives LIBOR plus a spread. The advantage of this
type of arrangement is that the financial institution reduces its exposure
to defaults by the company.
An nth-to-default CDS is defined as a CDS that pays off when the nth
default occurs in a portfolio of companies. In a collateralized debt
obligation, a number of different securities are created from a portfolio
Credit Derivatives 317
of corporate bonds or commercial loans. There are rules for determining
how credit losses are allocated to the securities. The result of the rules is
that securities with both very high and very low credit ratings are created
from the portfolio. A synthetic collateralized debt obligation creates a
similar set of securities from credit default swaps. The standard market
model for pricing both an nth-to-default CDS and tranches of a CDO is
the one-factor Gaussian copula model for time to default.
FURTHER READING
Andersen, L., J. Sidenius, and S. Basu, "All Your Hedges in One Basket," Risk,
November 2003.
Andersen, L., and J. Sidenius, "Extensions to the Gaussian Copula: Random
Recovery and Random Factor Loadings," Journal of Credit Risk, 1, No. 1
(Winter 2004): 29-70.
Das, S., Credit Derivatives: Trading & Management of Credit & Default Risk.
Singapore: Wiley, 1998.
Hull, J. C, and A. White, "Valuation of a CDO and nth to Default Swap
without Monte Carlo Simulation," Journal of Derivatives, 12, No. 2 (Winter
2004): 8-23.
Hull, J. C, and A. White, "The Perfect Copula," Working Paper, University of
Toronto.
Laurent, J.-P., and J: Gregory, "Basket Default Swaps, CDOs and Factor
Copulas," Working Paper, ISFA Actuarial School, University of Lyon, 2003.
Li, D. X., "On Default Correlation: A Copula Approach," Journal of Fixed
Income, March 2000: 43-54.
Tavakoli, J. M., Credit Derivatives: A Guide to Instruments and Applications.
New York: Wiley, 1998.
Schonbucher, P. J., Credit Derivatives Pricing Models. Wiley, 2003.
QUESTIONS AND PROBLEMS (Answers at End of Book)
13.1. Explain the difference between a regular credit default swap and a binary
credit default swap.
13.2. A credit default swap requires a semiannual payment at the rate of 60 basis
points per year. The principal is $300 million and the credit default swap is
settled in cash. A default occurs after 4 years and 2 months, and the
calculation agent estimates that the price of the cheapest deliverable bond
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is 40% of its face value shortly after the default. List the cash flows and
their timing for the seller of the credit default swap.
13.3. Explain the two ways a credit default swap can be settled.
13.4. Explain how a cash CDO and a synthetic CDO are created.
13.5. Explain what a first-to-default credit default swap is. Does its value
increase or decrease as the default correlation between the companies
in the basket increases? Explain your answer.
13.6. Explain the difference between risk-neutral and real-world default
probabilities.
13.7. Explain why a total return swap can be useful as a financing tool.
13.8. Suppose that the risk-free zero curve is flat at 7% per annum with
continuous compounding and that defaults can occur halfway through
each year in a new 5-year credit default swap. Suppose that the recovery
rate is 30% and the default probabilities each year conditional on no
earlier default is 3%. Estimate the credit default swap spread. Assume
payments are made annually.
13.9. What is the value of the swap in Problem 13.8 (per dollar of notional
principal) to the protection buyer if the credit default swap spread is
150 basis points?
13.10. What is the credit default swap spread in Problem 13.8 if it is a binary
CDS?
13.11. How does a 5-year nth-to-default credit default swap work? Consider a
basket of 100 reference entities where each reference entity has a prob-
ability of defaulting in each year of 1%. As the default correlation
between the reference entities increases, what would you expect to
happen to the value of the swap when (a) n = 1 and (b) n = 25. Explain
your answer.
13.12. How is the recovery rate of a bond usually defined?
13.13. Show that the spread for a new plain vanilla CDS should be (1 - R)
times the spread for a similar new binary CDS, where R is the recovery
rate.
13.14. A company enters into a total return swap where it receives the return on
a corporate bond paying a coupon of 5% and pays LIBOR. Explain the
difference between this and a regular swap where 5% is exchanged for
LIBOR.
13.15. Explain how forward contracts and options on credit default swaps are
structured.
13.16. "The position of a buyer of a credit default swap is similar to the position
of someone who is long a risk-free bond and short a corporate bond-
Explain this statement.
Credit Derivatives 319
13.17. Why is there a potential asymmetric information problem in credit
default swaps?
13.18. Does valuing a CDS using real-world default probabilities rather than
risk-neutral default probabilities overstate or understate the value of the
protection? Explain your answer.
13.19. What is the difference between a total return swap and an asset swap?
13.20. Suppose that in a one-factor Gaussian copula model the 5-year prob-
ability of default for each of 125 names is 3% and the pairwise copula
correlation is 0.2. Calculate, for factor values of —2, — 1, 0, 1, and 2,
(a) the default probability conditional on the factor value and (b) the
probability of more than 10 defaults conditional on the factor value.
13.21. What is a CDO squared? How about a CDO cubed?
ASSIGNMENT QUESTIONS
13.22. Suppose that the risk-free zero curve is flat at 6% per annum with
continuous compounding and that defaults can occur at times 0.25 years,
0.75 years, 1.25 years, and 1.75 years in a 2-year plain vanilla credit
default swap with semiannual payments. Suppose that the recovery rate is
20% and the unconditional probabilities of default (as seen at time zero)
are 1% at times 0.25 years and 0.75 years, and 1.5% at times 1.25 years
and 1.75 years. What is the credit default swap spread? What would the
credit default spread be if the instrument were a binary credit default
swap?
13.23. Assume that the default probability for a company in a year, conditional
on no earlier defaults is and the recovery rate is R. The risk-free interest
rate is 5% per annum. Default always occurs halfway through a year.
The spread for a 5-year plain vanilla CDS where payments are made
annually is 120 basis points and the spread for a 5-year binary CDS
where payments are made annually is 160 basis points. Estimate R and
13.24. Explain how you would expect the yields offered on the various tranches
in a CDO to change when the correlation between the bonds in the
portfolio increases.
13.25. Suppose that (a) the yield on a 5-year risk-free bond is 7%, (b) the yield on
a 5-year corporate bond issued by company X is 9.5%, and (c) a 5-year
credit default swap providing insurance against company X defaulting
costs 150 basis points per year. What arbitrage opportunity is there in this
situation? What arbitrage opportunity would there be if the credit default
spread were 300 basis points instead of 150 basis points? Give two reasons
why arbitrage opportunities such as those you have identified are less than
perfect.
Operational Risk
In 1999, bank supervisors announced plans to assign capital for opera-
tional risk in the new Basel II regulations. This met with some opposition
from banks. The chairman and CEO of one major international bank
described it as "the dopiest thing I have ever seen". However, bank super-
visors persisted. They argued that operational risk was a major issue for
banks. They pointed out that during a ten-year period more than 100
operational risk losses, each exceeding $100 million, had occurred. Some of
these losses, listed by the categories used by the Bank for International
Settlements, are:
Internal fraud: Allied Irish Bank, Barings, and Daiwa lost $700 million,
$1 billion, and $1.4 billion, respectively, from fraudulent trading.
External fraud: Republic New York Corp. lost $611 million because of
fraud committed by a custodial client.
Employment practices and workplace safety: Merrill Lynch lost $250 mil-
lion in a legal settlement regarding gender discrimination.
Clients, products, & business practices: Household International lost
$484 million from improper lending practices; Providian Financial Cor-
poration lost $405 million from improper sales and billing practices.
Damage to physical assets: Bank of New York lost $140 million because of
damage to its facilities related to the September 11, 2001, terrorist attack.
Business disruption and system failures: Solomon Brothers lost $303 mil-
lion from a change in computing technology.