ńņš. 5 |

tract into an insurance contract. Transformers owe their existence to the

fact that many jurisdictions place regulatory constraints on the ability of

insurance companies to enter into derivatives contracts, which eļ¬ectively

prevent insurers from selling protection in simple CDSs. Thus, a trans-

former is set up in a jurisdiction where such regulatory barriers are not in

place, such as in Bermuda. For instance, the transformer can, on one hand,

sell protection in a CDS contract with a bank/dealer and, on the other

hand, enter into a largely oļ¬setting credit insurance contract with an insur-

ance company that would otherwise be barred from the credit derivatives

market.

14

Synthetic CDOs

Synthetic collateralized debt obligations are structured ļ¬nancial products

that closely mimic the risk and cash ļ¬‚ow characteristics of traditional (cash-

funded) collateralized debt obligations. This āmimickingā is done through

the use of credit derivatives, such as credit default swaps and portfolio

default swaps, and that is why synthetic CDOs are part of any broad

discussion of credit derivatives.

To explain synthetic CDOs, we ļ¬rst go over the general nature of the

instruments they are designed to mimic, and thus we start this chapter

with a brief overview of traditional CDOs. As we shall see, these are instru-

ments that allow one to redistribute the credit risk in a given portfolio into

tranches with diļ¬erent risk characteristics and, in the process, meet the

risk appetites of diļ¬erent investors.

14.1 Traditional CDOs

The basic idea behind traditional CDOs is quite simple and can be illus-

trated with a nonļ¬nancial example. Imagine a small number of water

reservoirs located on the slope of a mountain. The reservoirs are emptied

and then reļ¬lled at the beginning of each month. When each reservoir is

ļ¬lled to its capacity, the water in it cascades into the one just below, and in

this way a single water source can in principle reļ¬ll all reservoirs simply by

feeding the pool at the highest elevation. We say in principle because this

assumes that the source has enough water to ļ¬ll all reservoirs to capacity,

134 14. Synthetic CDOs

otherwise one or more reservoirs down the mountain may be left empty or

only partially ļ¬lled.

What do cascading water reservoirs have to do with collateralized debt

obligations? Replace the monthly water replenishments with the monthly

or quarterly cash ļ¬‚ows generated by a portfolio of, say, business loans and

think of the individual reservoirs as investors with diļ¬erent claims on these

payments, and you arrive at the main idea behind a CDO. The reservoirs at

higher elevations, which are the ones to be replenished ļ¬rst, correspond to

investors with most senior claims on the cash ļ¬‚ows of the portfolioā”they

are ļ¬rst in line to collect their share of the āwater.ā The reservoirs in lower

elevations, which receive their water allotments only after the reservoirs up

the mountain have received their full due, are analogous to investors with

junior (subordinated) claims to the cash ļ¬‚ows of the portfolio. The reservoir

at the foot of the mountain is akin to a ļ¬rst-loss investor: Should the water

source come short of its promised amount, the lowest reservoir will be the

ļ¬rst to dry up.

To sum up, collateralized debt obligations are essentially securities with

diļ¬erent levels of seniority and with interest and principal payments that

are backed by the cash ļ¬‚ows of an underlying portfolio of debt instruments.

When the debt instruments are loans, the CDO is often called a CLOā”

a collateralized loan obligationā”if they are bonds, the CDO becomes a

CBOā”a collateralized bond obligation.

14.1.1 How Does It Work?

Let us look at a very simple and stylized CDO structure, represented

schematically in Figure 14.1.1 Consider a CDO issuer with a portfolio of

loans with a total face value of, say, $100 million. (As noted above, we

could call this a CLO, but we will stick to the more general terminology

to emphasize that this example would work just as well with a CBO.)

To fund the purchase of the loan portfolio, the issuer sells debt obliga-

tions (notes) to investors. The stream of payments promised by these notes

is, in turn, backed by the cash ļ¬‚ows generated by the loan portfolio. The

ļ¬gure depicts the relatively common case where the CDO issuer is a special

purpose vehicle (Chapter 13).

Suppose both the loans that make up the collateral and the resulting

notes make monthly payments. Each month, the issuer (the SPV) receives

1

The CDO structure examined here is used to highlight only the basic features of

CDOs. More complex structures are not uncommon. For instance, many CDOs allow

for the addition, removal, and substitution of assets in the collateral poolā”so-called

ramp-ups, removals, and replenishmentsā”during the life of the structure. In addition,

the basic waterfall structure described above is often complicated by ācoverage tests,ā

a topic that is brieļ¬‚y discussed in Chapter 21. Goodman and Fabozzi (2002)[34] discuss

CDO structures in some detail.

14.1 Traditional CDOs 135

FIGURE 14.1. Diagram of a Simple CDO

the payments due on the loans and passes them through to the investors

who bought the notes (net of any administrative charges). Again, a key

aspect of a CDO is that the notes have diļ¬erent coupons to reļ¬‚ect various

levels of seniority and risk. In particular, in each month, any income paid

by the underlying loans is used ļ¬rst to meet payments owed to the holders

of the most senior notes. Once, the holders of those notes are paid, investors

in the second most senior notes are paid up, and this process continues until

the holders of the most junior notes receive their share of the portfolioā™s

cash ļ¬‚ow. (This is the cascading pool structure that we described above.)

The payments owed to the various notes are such that, in the absence of

default in the underlying loans, the net monthly cash ļ¬‚ow generated by the

loans is just enough to pay all investors, from the most senior to the most

junior. In the event of default in one or more of the loans in the portfolio,

however, the holders of the most junior notes will receive less than their

total payment due, which implies that their ācoupon + principalā cash

ļ¬‚ow shown in Figure 14.1 is essentially a residual amount after more senior

investors and administrative fees are paid. Naturally, because the most

junior investors are long the ļ¬rst-loss piece of the CDO, their expected

return is higher than the expected returns that correspond to the more

senior notes in the structure.

In the parlance of the CDO world, each level of seniority of the notes

issued under the CDO is called a tranche. CDO tranches are typically

136 14. Synthetic CDOs

rated by major credit rating agenciesā”such as Moodyā™s, Standard and

Poorā™s, and FitchRatings. In the case of a CDO with, say, four levels of

seniority, one typically refers to the ļ¬rst-loss notes as the equity tranche,

to the second- and third-loss notes as the subordinated mezzanine and

senior mezzanine tranches, respectively, and to the most senior notes as,

simply, the senior tranche. In the example depicted in Figure 14.1, the

CDO involved three tranches: an equity tranche with a total par value of

$5 million, a mezzanine tranche initially valued at $15 million, and a senior

tranche with notes with a total face value of $80 million.

In many instances, the institution that accumulated the loans that make

up the collateral held by the SPV buys back at least part of the equity

tranche. This is done for a number of reasons. For instance, the high degree

of credit risk associated with certain ļ¬rst-loss pieces may make them harder

to sell in the open market. In addition, by staying on as a ļ¬rst-loss investor,

the institution may hope to address potential CDO investorsā™ concerns

about moral hazard and adverse selection problems.

To understand the moral hazard problem associated with CDOs, consider

the example of a commercial bank that intends to securitize the business

loans on its books. One manifestation of the moral hazard problem is the

concern that the bank may make riskier loans than otherwise if it knows

that it can then transfer all of the associated credit risk to CDO investors.

The adverse selection problem is related to the notion that, relative to

investors, the bank may have an informational advantage when it comes to

evaluating the reference entities represented in its own portfolio. As a result,

investors may become concerned about the so-called ālemonā phenomenon,

or the possibility that they may end up buying debt securities that reference

entities with particular problems and risks that are known only by the bank.

By becoming a ļ¬rst-loss investor in the CDO, the bank in this example

hopes to dispel or at least mitigate such worries.

14.1.2 Common Uses: Balance-sheet and Arbitrage CDOs

CDOs have a wide range of applications in the ļ¬nancial markets and tend to

be classiļ¬ed according to the ultimate goals of their sponsors. For instance,

from the perspective of a commercial bank, CDOs make it possible to

transfer a large portfolio of loans oļ¬ the bankā™s balance sheet in a single

transaction with a repackaging vehicle. Such CDOs are commonly called

balance-sheet CDOs, and, indeed, historically, banksā™ desire to free up

regulatory capital through balance-sheet CDOs was an important driver

of CDO market activity in the 1990s. The banks would sell the assets to

sponsored SPVs, which would then securitize them and place them with

institutional investors, as broadly outlined in Figure 14.1.

In recent years, a substantial share of CDO issuance has been driven not

so much by banksā™ balance-sheet management needs, but by investor

14.2 Synthetic Securitization 137

demands for leveraged credit risk exposures.2 These CDOs are often ref-

erred to as arbitrage CDOs in that the institutions behind the issuance are,

for instance, attempting to enhance their return on the underlying assets by

becoming ļ¬rst-loss (equity) investors in the newly created structures. Most

of these arbitrage CDOs are actively managed and thus the CDO investors

are exposed both to credit risk and to the particular trading strategy fol-

lowed by the CDO manager. Insurance companies, asset managers, and

some banks are among the main equity investors in arbitrage CDOs.

Over and above their general application to balance-sheet management

and return enhancement, CDOs can be used to create some liquidity in

what would otherwise be essentially illiquid assets. For instance, many

bank loans are inherently illiquid, in part because each sale may require

approval by the borrower. Once those loans are securitized, however, their

underlying risk characteristics become more tradable as the SPV-issued

notes are bought and sold in the marketplace. In addition, as noted, the

CDO structure allows, through the tranching process, the creation of new

assets with speciļ¬c proļ¬les that may better match the individual needs and

risk tolerances of institutional investors.

14.1.3 Valuation Considerations

Three main factors enter into the pricing of the various tranches of a CDO:

the degree of default correlation among the debt instruments in the col-

lateral pool, the credit quality of the individual debt instruments, and the

tranching structure of the CDO. These are essentially the same factors

that we discussed in Chapters 9 and 10, where we examined key multi-

name credit derivatives. In addition, and quite naturally, these are some of

the main variables taken into account by the major credit-rating agencies

when assessing the risk embedded in individual CDO structures. We will

see more about CDO valuation in Part IV of this book.

14.2 Synthetic Securitization

Having reviewed the basics of traditional CDOs, understanding the

mechanics of synthetic CDOs becomes relatively straightforward, espe-

cially if one is already familiar with portfolio default swaps (Chapter 10).

Figure 14.2 illustrates a simple synthetic balance-sheet CDO structure.

The ļ¬gure shows a commercial bank (labeled sponsoring bank) with a loan

2

Similar to ļ¬rst-loss investors in baskets and portfolio default swaps (Chapters 9 and

10), equity investors in CDOs are exposed to the credit risk in the entire collateral pool

even though their maximum loss is substantially smaller than the total par value of

the CDO.

138 14. Synthetic CDOs

FIGURE 14.2. Diagram of a Simple Synthetic CDO

portfolio of $100 million (the reference assets). The bank wants to shed

the credit risk associated with the portfolio, but, rather than selling the

loans to a repackaging vehicle (labeled SPV in the ļ¬gure), the bank opts

for selling only the credit risk associated with the portfolio and for keeping

the loans on its balance sheet.

The risk transfer is done via a portfolio default swap where the SPV is

the counterparty and where the sponsoring entity buys protection against,

say, any losses in excess of 2 percent of the portfolio. Alternatively, the risk

transfer could be done via a series of single-name credit default swaps. (As

in the case of traditional CDOs, the sponsor tends to keep a small ļ¬rst-loss

piece, 2 percent in this case, partly in order to address investorsā™ potential

concerns about moral hazard and adverse selection problems.) The bank in

Figure 14.2 makes periodic premium payments to the SPV, and the SPV

promises to stand ready to step in to cover any default-related losses that

exceed 2 percent of the portfolio, just as in any typical portfolio default

swap agreement.

What does the SPV do next? As in the traditional CDO structure, the

SPV issues notes to various classes of investors (three in the example in

Figure 14.2), where each class corresponds to claims with a given level

of seniority toward the SPVā™s cash ļ¬‚ows. Because the portfolio default

swap is an unfunded structure, however, the cash ļ¬‚ows it generates (the

protection premiums) cannot possibly fully compensate the investors both

14.2 Synthetic Securitization 139

for their funding costs (the SPV-issued notes are fully funded investments)

and for the credit risk embedded in the reference portfolio. To make up for

this shortfall, the SPV invests the proceeds of the note sales in high-grade

assets, typically AAA-rated instruments. The SPV then uses these assets

both as collateral for its obligations toward the sponsoring bank and the

investors and, through the income that they generate, as a funding source

to supplement the coupon payments promised by the notes. The collateral

is shown in the lower part of the ļ¬gure.

Provided there are no defaults in the reference portfolio, the sponsoring

bank keeps on making full premium payments to the SPV, which in turn

uses that income, along with the cash ļ¬‚ow generated by the AAA-rated

collateral, to meet the all coupon payments owed to the note investors.

At the maturity date of the CDO notes, the portfolio swap is terminated

and the SPV liquidates the collateral to repay the investorsā™ principal in full.

The CDO investors absorb all default-related losses in excess of the

ļ¬rst-loss piece retained by the sponsoring bank, starting with the equity

investors, as in a traditional CDO. Suppose, for instance that, after the

bankā™s ļ¬rst-loss piece is exhausted, an additional default takes place.

A common approach is for the SPV to liquidate part of its collateral in order

to cover the sponsoring bankā™s losses and for the par value of the notes held

by the equity investors to be reduced accordingly. Again, this is analogous

to the portfolio default swap arrangement discussed in Chapter 10.

While Figure 14.2 shows the mechanics of a synthetic balance-sheet CDO,

the structure of a synthetic arbitrage CDO would be similar. Salient dif-

ferences would include the facts that the sponsoring entity could be, for

instance, an asset manager, rather than the commercial bank featured in

Figure 14.2, and that the SPV could potentially be selling protection to

a number of buyers in the credit derivatives market, instead of just to its

sponsor. Similar to the traditional CDO market, arbitrage-motivated deals

have come to dominate new issuance ļ¬‚ows in the synthetic CDO market in

recent years.

14.2.1 Common Uses: Why Go Synthetic?

A powerful rationale for using synthetic, as opposed to traditional, CDOs

relates to the fact that the latter does not require the sponsoring bank in

a balance-sheet CDO to sell any of the loans in the reference portfolio or,

especially in the case of arbitrage CDOs, the SPV to source loans and secu-

rities in various markets. To take on the sponsoring bankā™s perspective, as

we have argued before in this book, selling loans can be both potentially

problematic for maintaining bank relationships and costly in terms of the

legal steps involved in the borrower approval and notiļ¬cation process. The

situation here is entirely analogous to an example, discussed in Section 3.1,

involving single-name credit default swaps and loan sales, except that now

140 14. Synthetic CDOs

we are dealing with a potentially large number of reference entities in one

single transaction: Synthetic CDOs allow a bank to sell anonymously the

credit risk associated with the loans held on its books. Through the syn-

thetic CDO, the bank essentially securitizes the credit risk in the portfolio,

whereas through a traditional CDO both the credit risk and the loans are

securitized.

Another rationale for using synthetic CDOs stems from banksā™ hedging

needs regarding potential exposures through undrawn credit facilities, such

as back-up lines of credit oļ¬ered to investment-grade entities that issue

commercial paper (CP). For instance, XYZ Corp. may have an option to

borrow $100 million from AZZ Bank in case it is unable to roll over its

CP obligations. Should XYZ run into unexpected diļ¬culties that lead it to

draw down on the credit facility, it may well be too late (or too expensive)

for AZZ Bank to hedge its XYZ exposure. Thus, the bank has a $100 million

exposure to XYZ that does not quite ļ¬t the most common form of the

traditional CDO model. Instead, the bank could decide to take preemptive

measures and, for instance, refashion a synthetic CDO to include a $100

million notional exposure to XYZ Corp.3

14.2.2 Valuation Considerations for Synthetic CDOs

Valuation considerations are similar to those involving traditional CDOs.

Default correlation, the credit quality of the individual entities represented

in the reference portfolio, and the details of the tranching structure are

important factorsā”see Chapter 21. In addition, the legal structure of the

SPV, as well as the credit quality of the SPVā™s collateral and of the spon-

soring bank may also play a role. Regarding the latter two factors, their

importance stems from the fact that the SPV depends on the incomes gen-

erated by the collateral and on the premiums paid by the sponsoring bank

to fund the cash ļ¬‚ows owed to the note investors. Similar to traditional

CDOs, the tranches of synthetic CDOs can be, and typically are, rated by

the major credit-rating agencies.

14.2.3 Variations on the Basic Structure

Synthetic CDOs are commonly structured to require even less securitization

than the example shown in Figure 14.2. One such structure, motivated

by an example provided by Oā™Kane (2001)[63], is shown in Figure 14.3.

In this example, the sponsoring bank enters into two separate portfolio

3

Obviously, a synthetic CDO is not the only alternative available to the bank.

Protection bought through a single-name CDS written on XYZ Corp. would be another

possibility.

14.2 Synthetic Securitization 141

FIGURE 14.3. Diagram of a Variation on the Simple Synthetic CDO Structure

default swaps: a second-loss contract with the SPV, covering losses between

2 percent and 10 percent of the portfolio, and a third-loss contract with an

OECD bank covering any losses in excess of 10 percent. Only the contract

with the SPV is securitized and sold oļ¬ into diļ¬erent tranches to investors

following the same pattern described in the discussion of Figure 14.2. In this

particular example, the bank obtained substantial credit protection while

securitizing only 8 percent of the total portfolio.

An alternative synthetic CDO structure to the one illustrated in

Figures 14.2 and 14.3 is the unfunded synthetic CDO, where investors put

up no cash at the inception of the transaction and, similar to a single-name

credit default swap, receive only the premiums passed through by the SPV.

In this case, there is no SPV collateral involved and an investor would only

be called upon to make a payment under the terms of the contract if and

when default-related losses in the underlying portfolio fall within the range

covered by his or her tranche. Such a structure is akin to a collection of

portfolio default swaps written on the reference portfolio, with each swap

corresponding to a diļ¬erent tranche of the synthetic CDO.

Before we end this chapter, we should once again note that, while the

examples discussed in this chapter centered on CDOs that referenced pools

of loans, CDOs are also commonly set up to reference bond portfolios.

Part III

Introduction to Credit

Modeling I: Single-Name

Defaults

143

15

Valuing Defaultable Bonds

Before we start exploring speciļ¬c models for pricing credit derivatives, we

will pause to introduce some notation and, in the process, review a few

important concepts. Those familiar with risk-neutral probabilities and the

risk-neutral valuation approach, two of the most important topics discussed

in this chapter, may still want to at least glance through the next few pages

to familiarize themselves with the notation that will be used throughout

this part of the book.

In this chapter we shall assume that riskless interest rates are determin-

istic. This is done for expositional purposes only so we can avoid certain

technical details related to the discussion of the risk-neutral valuation in

Section 15.2.1 Stochastic interest rates are discussed in Chapter 17.

15.1 Zero-coupon Bonds

Let Z(t, T ) denote todayā™s (time-t) price of a riskless zero-coupon bond that

pays out $1 at a future time T . If R(t, T ) is the continuously compounded

1

In this non-technical overview of the valuation of defaultable bonds, we make no

formal distinction between so-called risk-neutral and forward-risk-neutral probabilities.

Indeed, when riskless interest rates are deterministic, these two probability measures

coincide. Readers interested in pursuing these technical details further could consult

Neftci (2002)[62] and Baxter and Rennie (2001)[6] for very accessible discussions of the

main issues involved. More mathematically oriented readers may also be interested in

the discussion in Bjork (1998)[7].

146 15. Valuing Defaultable Bonds

FIGURE 15.1. Payout Scenarios for a Zero-Recovery Zero-Coupon Bond

yield to maturity on this bond, we have2

Z(t, T ) = eā’R(t,T )(T ā’t) (15.1)

We can think of Z(t, T ) as reļ¬‚ecting the time-value of money, or todayā™s

(time-t) value of $1 that will be received for sure at time T . Note that,

for positive interest rates, even though the terminal payout of this bond

is never in question, its value today is only a fraction of that payout, i.e.,

Z(t, T ) < 1.

Now consider another bond with the same maturity and face value as

the above riskless security, but this bond is subject to default risk. In par-

ticular assume that there is a nonzero probability that the bond issuer

will default, in which case the bond will have a recovery value of $0.

As shown in Figure 15.1, this bond has two possible cash ļ¬‚ows associ-

ated with it: At time T , it will either pay $1 (no default) or nothing

(default). We shall assume that the actual probability of no default by this

issuer at time T , conditional on information available at time t, is P (t, T ),

which is also called the survival probability of the issuer. (Needless to say,

this probability is also conditional on the issuer not having defaulted by

time t.)

2

Up until now, we have been working on discretely compounded yieldsā”see, e.g.,

Chapter 4ā”a concept with which most people tend to be familiar. Many of the credit

risk models considered in this part of the book, however, are cast in terms of continuously

compounded yields, and hence our switch to continuous compounding. The relationship

between discretely and continuously compounded yields is reviewed in Appendix A,

where we also discuss basic concepts related to bond yields and bond prices.

15.2 Risk-neutral Valuation and Probability 147

From the perspective of time t, it is useful to think of this risky bond as

a lottery that, at time T , will pay either $1, with probability P (t, T ), or $0,

with probability 1 ā’ P (t, T ). Let Z0 (T, T ) be the risky bondā™s payout at

d

time T .3 While this amount is unknown at time t, its expected value can

be computed based on knowledge of P (t, T ). Suppose that the expected

payout of the bond/lottery is Y , i.e.,

Y ā” Et Z0 (T, T ) = P (t, T ) Ć— 1 + [1 ā’ P (t, T )] Ć— 0 = P (t, T )

d

(15.2)

where Et [.] denotes an expectation formed on the basis of information

available at time t, given the survival probability P (t, T ).

15.2 Risk-neutral Valuation and Probability

How do we compute the present value of this bond/lottery? First, as with

the riskless bond, there is the time-value of money. The payout of the lot-

tery, if any, will only be made at the future date T , and a dollar tomorrow

is less valuable than a dollar today. Second, and unlike the riskless bond,

there is a chance that the lottery may not pay out at all (the bond issuer

may default). As a result, one may want to discount the promised pay-

ment further when assessing the current value of the bond. There are two

equivalent ways of thinking about this discounting. One can apply a higher

discount rate to the promised payment of $1,

Z0 (t, T ) = eā’[R(t,T )+S(t,T )](T ā’t) = Z(t, T )eā’S(t,T )(T ā’t)

d

(15.3)

where, for S(t, T ) > 0, the promised payout of the risky bond is now

discounted based on the higher rate R(t, T ) + S(t, T ).

Alternatively, one can think of the artiļ¬cial āprobabilityā Q(t, T )ā”also

conditional on information available at time tā”which is such that the risk-

free rate can be relied upon to discount the defaultable bondā™s expected

future payment:

Z0 (t, T ) = eā’R(t,T )(T ā’t) [Q(t, T ) Ć— 1 + (1 ā’ Q(t, T )) Ć— 0]

d

= Z(t, T )Q(t, T ) (15.4)

where 1 ā’ Q(t, T ) is the āprobabilityā attached to a default by the bond

issuer, as shown in Figure 15.2.

3 d

Note that Z0 (t, T ), the time-t price of the zero-recovery, zero-coupon defaultable

bond that matures at time T , corresponds to the variable D(t, T ), which we introduced

in Chapter 4. We use this new notation in this part of the book because it is more

commonly used in the modeling literature.

148 15. Valuing Defaultable Bonds

FIGURE 15.2. Payout Scenarios for a Zero-Recovery Zero-Coupon Bond

Note that default probability 1ā’Q(t, T ) in equation (15.4)ā”or the prob-

ability that the lottery will not pay outā”will generally not coincide with

the actual default probability 1ā’P (t, T ) featured in equation (15.2). Why?

Because investors may be risk averse. Here is how we will characterize a risk

averse investor. Suppose the investor is given two alternative investment

opportunities at time t: (i) a promise to receive Y dollars for sure at time

T and (ii) a chance to enter into the above lottery, which, based on the

actual default probability 1 ā’ P (t, T ), has an expected payout of Y . For

the purposes of this book, we will say that, if the investor is risk averse,

he or she will choose the āsure thing,ā rather than take the risk associated

with the lottery.4

Now note that the time-t value of a promise to pay Y dollars for sure at

time T is simply Z(t, T )Y . The time-t value of the lottery can be thought of

as the time-t price of a zero-recovery zero-coupon bond with a face value of

$1 and an actual probability of default 1 ā’ P (t, T ). This is simply Z0 (t, T ).

d

Thus, if the marketplace is composed mainly of risk averse investors, the

former (the āsure thingā) will be more valued than the latter (the lottery):

d

Z(t, T )Y > Z0 (t, T ) (15.5)

Using (15.2) and (15.4), we can rewrite the above as

Z(t, T )P (t, T ) > Z(t, T )Q(t, T ) (15.6)

4

Formal deļ¬nitions of risk aversion can be found in most ļ¬nancial economics

textbooks, such as LeRoy and Werner (2001)[54] and Huang and Litzenberger (1988)[40].

15.2 Risk-neutral Valuation and Probability 149

which implies that P (t, T ) > Q(t, T ). This is a key result that bears

repeating,

d

Risk aversion => Z(t, T )Y > Z0 (t, T ) => P (t, T ) > Q(t, T ) (15.7)

In words, when investors are risk averse, the actual survival probability

used in the valuation of a risky bondā”equation (15.4)ā”is higher than the

artiļ¬cial survival probability associated with the bond.

What if investors are risk neutral? In the context of this book, we shall

say that an investor is risk neutral if he or she is indiļ¬erent between the

riskfree promise of a payout of Y at time T and a risky bond/lottery with

an expected payout of Y , also to be made at time T . Thus, in a market

composed of risk-neutral investors, the prices of the riskfree promise and

the lottery should be the same

d

Risk neutrality => Z(t, T )Y = Z0 (t, T ) => P (t, T ) = Q(t, T ) (15.8)

and again using equations (15.2) and (15.4) we arrive at the result that,

when investors are risk neutral, the probabilities that are used in the valua-

tion of the risky bond in equation (15.4) are indeed the actual probabilities

associated with default events related to the bond issuer.5

15.2.1 Risk-neutral Probabilities

Note that the artiļ¬cial probabilities used in equation (15.4) were such that

investors would be willing to discount the uncertain payout of the risky

bond at the riskless rate. As we saw in the previous section, this is essen-

tially what a risk-neutral investor would do. Consider the case of risk-averse

investors. Intuitively, what we did was ask ourselves the following question:

By how much do we have to inļ¬‚ate the default probability 1ā’P (t, T ) so that

a risk-averse investor would be willing to behave as if she were risk neutral?

The resulting artiļ¬cial default probability 1ā’Q(t, T ) is commonly called the

risk-neutral probability of default, which will only coincide with the actual

probability of default if the world were populated by risk-neutral investors.

Ė d

Let Et Z0 (T, T ) denote the expected payout of the risky bond, com-

puted on the basis of the risk-neutral probabilities Q(t, T ) and 1 ā’ Q(t, T )

and of information available as of time t, i.e.,

Et Z0 (T, T ) = Q(t, T ) Ć— 1 + [1 ā’ Q(t, T )] Ć— 0 = Q(t, T )

Ė d

We will deļ¬ne Ļ„ as the default time, unknown at time t, of the bond

issuer. If Ļ„ is independent of the riskfree rate embedded in Z(t, T ), which

5

We have left out the case of risk-loving investors. These are the ones who would value

the lottery more highly than the sure thing, which would imply that P (t, T ) < Q(t, T ).

150 15. Valuing Defaultable Bonds

is clearly the case thus far given that we are assuming that the riskless rate

is deterministic, the risk-neutral valuation formula for this bond, essentially

given by equation (15.4), can then be written as6

Ė

d d

Z0 (t, T ) = Z(t, T )Et Z0 (T, T ) (15.9)

The above equation is a very important result, and we will come back to it

in many instances throughout the remainder of this book. We can simplify

this basic result further in the context of zero-recovery bonds. In particular,

when R(t, T ) and Ļ„ are independent, and going back to equation (15.4),

we can write

Price of zero-recovery zero-coupon risky bond

equals

price of comparable riskless bond

times

risk-neutral survival probability of risky bond issuer

where a comparable riskless bond is one with the same maturity date and

face value as the risky bond, and the survival probability, Q(t, T ), refers to

the risky-neutral probability of no default by the risky bond issuer by the

maturity date of the bond, T .7

15.3 Coupon-paying Bonds

Thus far we have limited ourselves to zero-coupon bonds. Extending

the above results to coupon-bearing bonds is relatively straightforward.

6

For models that allow riskless rates to be stochastic, the simplifying assumption

of independence between riskless interest rates and the default process may be a more

reasonable approximation for highly rated entities than for lower-rated ones, at least as

far as results based on empirical (as opposed to risk-neutral) probabilities are concerned.

For instance, Duļ¬ee (1998)[21] and Collin-Dufresne, Goldstein, and Martin (2001)[14]

found only a relatively weak relationship between changes in US Treasury yields and

changes in credit spreads, but the sensitivity of spreads to interest rates was reported

to increase with decreases in credit quality.

7

An alternative way to write the valuation formula for a risky zero-coupon bond is

to make use of the indicator function 1{Ļ„ >T } , which is one if the time of default, Ļ„ , falls

beyond the end of the horizon, T , and zero otherwise. Using this notation:

Ė

d

Z0 (t, T ) = Z(t, T )Et 1{Ļ„ >T }

15.3 Coupon-paying Bonds 151

In particular, one can think of a coupon-paying bond as a portfolio of zero-

coupon bonds. To see this, consider the case of a newly issued two-year

defaultable bond that makes coupon payments annually and that promises

the full repayment of its face value at maturity. Assume a ļ¬xed coupon

rate of C, a face value of $1, and a zero recovery rate. If the bond issuer

does not default over the next 2 years, it will pay C in one year and 1 + C

two years from now.

Note that this coupon-paying bond is equivalent to a portfolio with two

zero-coupon bonds, one that matures in one year and has a face value of C

and the other maturing in two years with a face value of 1 + C. Thus, as we

discussed in Chapter 1, if we know the price of this replicating portfolio of

zero-coupon bonds, we know the price of the coupon-paying bond. Using

notation consistent with that introduced in the previous section we can

write the time-t value, V B (t, t + 2), of this two-year bond as

V B (t, t + 2) = Z0 (t, t + 1)C + Z0 (t, t + 2)(1 + C)

d d

(15.10)

where t denotes the current year, and t + 1 and t + 2 are dates one and two

years from today, respectively.

Here we can invoke the risk-neutral valuation formula discussed in

Section 15.2 and write

V B (t, t + 2) = Z(t, t + 1)Q(t, t + 1)C + Z(t, t + 2)Q(t, t + 2)(1 + C)

(15.11)

where Q(t, t + i) is the risk-neutral probability, as seen at time t, that the

bond issuer will not default by time t + i, or the bond issuerā™s survival

probability through time t + i.

We can generalize the above results to a coupon-bearing defaultable bond

that matures at some future date TN , has a face value of F , and makes

coupon payments CF on dates T1 , T2 , . . . , TN . The value of such a bond at

time t, t < T1 , is

N

B

V (t, TN ) = Z(t, Ti )Q(t, Ti )C + Z(t, TN )Q(t, TN ) F (15.12)

i=1

and we can now see that the discount factors D(t, Ti ) used, for instance,

in Chapters 4 and 5, were essentially given by the product of the corre-

sponding āriskless discount factor,ā Z(t, Ti ), and the survival probability

associated with bond issuer, Q(t, Ti ).

152 15. Valuing Defaultable Bonds

15.4 Nonzero Recovery

We can bring the framework outlined in the chapter a bit closer to reality

by considering the case of coupon-bearing bonds that are subject to default

risk but that have a nonzero recovery value upon default. Let X, 0 ā¤ X < 1,

denote the recovery value of the bond. To avoid unnecessary complica-

tions at this point, assume that X is nonrandom and that the bond holder

receives X on the coupon payment date that immediately follows a default.8

Let Probt [default between Tiā’1 and Ti ] denote the (risk-neutral) proba-

bility of a default occurring between coupon payment dates Tiā’1 and Ti ,

based on all information available at time t. Intuitively, it is not hard to

see that this probability should be equal to the probability of surviving

through time Tiā’1 minus the probability of surviving through Ti :

Probt [default between Tiā’1 and Ti ] = Q(t, Tiā’1 ) ā’ Q(t, Ti ) (15.13)

For instance, if the probability of surviving through Tiā’1 is 50 percent and

the probability of surviving through Ti is 47 percent, then there is a 3

percent probability that the bond issuer will default between Tiā’1 and Ti ,

where, for the reminder of this chapter, all references to probabilities are

made in the risk-neutral sense.

Should the bond issuer default between Tiā’1 and Ti , the bond holder

will receive X at time Ti . What is the time-t value of the recovery pay-

ment received on that date? We can think of this payment as the present

discounted value of a zero-coupon bond that pays X at time Ti with prob-

ability [Q(t, Tiā’1 ) ā’ Q(t, Ti )] and zero otherwise. Using the risk-neutral

valuation framework, the value of such a hypothetical bond would be

Z(t, Ti )[Q(t, Tiā’1 ) ā’ Q(t, Ti )]X

which is the present discounted value of the recovery payment associated

with a default between times Tiā’1 and Ti .

For a nonzero recovery bond with N payment dates, there are N possible

dates for the recovery payment X to take place, each corresponding to a

diļ¬erent default scenario. For instance, if the bond issuer defaults between

T2 and T3 , the bond holder receives X at time T3 , which, in the absence

of uncertainty, has a present value of Z(t, T3 )X at time t. If, instead,

a default were to occur at a later period, say between T7 and T8 , the

time-t value of the bondā™s recovery value would be Z(t, T8 )X in a world

without uncertainty. As a result, once uncertainty is factored back into the

computation, one can think of todayā™s (time t) value of the bondā™s recovery

8

See Arvanitis and Gregory (2001)[2] for the more general case where the recovery

payment occurs immediately upon default.

15.5 Risky Bond Spreads 153

payment as a weighted sum of all the recovery payments associated with all

possible default scenarios, where the weights are given by the risk-neutral

probabilities of each scenario actually taking place:

N

Z(t, Ti )[Q(t, Tiā’1 ) ā’ Q(t, Ti )]X

REC

V (t) = (15.14)

i=1

Given (15.14), writing down an expression for the total value of the bond

is relatively simple. This is just the sum of the present discounted values

of the bondā™s coupon and principal payments, equation (15.12), and the

bondā™s recovery payment:

N

B

V (t, TN ) = Z(t, Ti )Q(t, Ti )C + Z(t, TN )Q(t, TN ) F

i=1

N

Z(t, Ti )[Q(t, Tiā’1 ) ā’ Q(t, Ti )]X

+ (15.15)

i=1

Upon closer inspection, one can see in equation (15.15) that the market

value of a coupon-bearing defaultable bond is simply the probability-

weighted sum of the present values of all possible cash ļ¬‚ows associated

with the bond.

15.5 Risky Bond Spreads

The price of a risky bond is often communicated in the marketplace in terms

of the spread between the bondā™s yield and some benchmark yield, such as a

swap rate or the yield on a government security with comparable maturity.

For instance, a newly issued ten-year corporate bond in the United States

might be said to be trading at, say, 156 basis points over the yield on the

most-recently issued ten-year US Treasury note.

Our intuition should tell us that the yield spread on a given risky bond

should be closely related to how the probability of default associated with

the issuer of that bond compares to the default probability, if any, associ-

ated with the benchmark bond. Indeed, as noted in Section 15.2, one can

think of the pricing of a risky bond as the process of ļ¬nding the spread

S(t, T ) over the riskless rate so that the bondā™s promised future cash ļ¬‚ow

is appropriately discounted to reļ¬‚ect both the time-value of money and

the credit risk embedded in the bond. In particular, bringing together

equations (15.3) and (15.4), we obtain

Z0 (t, T ) = eā’R(t,T )(T ā’t) Q(t, T ) = eā’[R(t,T )+S(t,T )](T ā’t)

d

(15.16)

154 15. Valuing Defaultable Bonds

Equation (15.16) leads to a simple relation between the yield spread and

the (risk-neutral) probability of no default associated with a zero-coupon,

zero-recovery risky bond:

Q(t, T ) = eā’S(t,T )(T ā’t) (15.17)

and the reader can verify that, for values of S(t, T ) that are not too high,

1 ā’ Q(t, T ) and S(t, T ) will be approximately equal when the period under

consideration is one year (T ā’t = 1). More generally, the annualized default

probability 1ā’Q(t,T ) is approximately equal to the risky bond spread S(t, T )

T ā’t

when S(t, T ) is not too large.9

It is relatively straightforward to examine the relationship between yield

spreads and default probabilities in coupon-bearing bonds and in bonds

with nonzero recovery values. For instance, in the case of a one-year zero-

coupon bond, T ā’ t = 1, with a recovery value X, equation (15.4) becomes

Z d (t, t + 1) = Z(t, t + 1) [Q(t, t + 1) + (1 ā’ Q(t, t + 1))X] (15.18)

d

where we dropped the 0 subscript on Z0 (.) to indicate that this is a nonzero

recovery bond.

Combining (15.18) with (15.3) leads to

1 ā’ eā’S(t,t+1)

1 ā’ Q(t, t + 1) = (15.19)

1ā’X

And, for S(t, t + 1) not too large, it can be seen that

S(t, t + 1)

1 ā’ Q(t, t + 1) ā (15.20)

1ā’X

15.6 Recovery Rates

Thus far we have taken the recovery value of the bond, denoted above as

X, to be a nonrandom parameter in the basic bond valuation expression

9

If we had cast our default probabilities in a continuous-time framework, we would

have found that the risk spread and the annualized risk-neutral default probability are

one and the same in the case of no recovery value. To see this, consider the above

example where one is interested in the probability of default between todayā™s date, t,

and some future date T . If Ī»dt denotes the default probability over a short time period

[t, t + dt], assuming that no default has occurred before t, it can be shown that the

probability of no default occurring between t and T tends to eā’Ī»(T ā’t) as dt becomes

inļ¬nitesimally small, assuming that Ī» is constant for this issuer between t and T . Thus,

the default probability over the period becomes 1 ā’ eā’Ī»(T ā’t) , leading to the result that

S(t, T ) = Ī». We will examine this continuous-time case in greater detail in Chapter 17.

15.6 Recovery Rates 155

TABLE 15.1

Historical Recovery Value Statistics (1970ā“1999)a

Seniority/ 1st 3rd Std

security Min quartile Median Mean quartile Max dev

Sr. sec. loans 15.00 60.00 75.00 69.91 88.00 98.00 23.47

Eq. trust bds 8.00 26.25 70.63 59.96 85.00 103.00 31.08

Sr. sec. bds 7.50 31.00 53.00 52.31 65.25 125.00 25.15

Sr. unsec. bds 0.50 30.75 48.00 48.84 67.00 122.60 25.01

Sr. sub. bds 0.50 21.34 35.50 39.46 53.47 123.00 24.59

Sub. bds 1.00 19.62 30.00 33.17 42.94 99.13 20.78

Jr. sub. bds 3.63 11.38 16.25 19.69 24.00 50.00 13.85

Pref. stocks 0.05 5.03 9.13 11.06 12.91 49.50 9.09

Source: Moodyā™s Investors Service

a Prices of defaulted instruments approximately one month after default, expressed as a

percent of the instrumentā™s par value. Abbreviations: Eq. = equipment, Sr. = senior,

sec. = secured, sub. = subordinated, Jr. = junior, pref. = preferred, bds = bonds.

in equation (15.15). Yet, the recovery value of a defaultable bond is an

important source of uncertainty in the valuation process. Predicting recov-

ery ratesā”the recovery value expressed as a percentage of the par value of

the bondā”is particularly diļ¬cult in light of the relative sparseness of the

underlying default data: Despite the large number of corporate defaults

in the early 2000s, defaults are still relatively rare events, which makes it

harder to conduct statistical analysis and develop models of recovery values.

Thus, practical applications of credit risk models often involve experimen-

tation with a range of recovery values and reliance on the credit analystā™s

judgment regarding the recovery rates corresponding to particular debtors,

as opposed to heavy reliance on pure statistical models of recovery.10

As shown in Table 15.1, extracted from Keenan, Hamilton, and

Berthault (2000)[49], the data that do exist on recovery rates suggest that

they can vary substantially both across and within levels of seniority and

security. For instance, the table shows that, while the mean recovery rate

for senior unsecured bonds over the 1970ā“1999 period was 48.84 percent,

the corresponding standard deviation was 25.01 percent, and actual recov-

ery rates ranged from 0.5 percent to 122.6 percent just within this level of

seniority. (Recall that senior unsecured debt instruments are often speciļ¬ed

10

In Chapter 17 we brieļ¬‚y review the recovery assumptions embedded in the main

types of credit risk models.

156 15. Valuing Defaultable Bonds

as deliverable instruments in physically settled credit default swapsā”see

Chapter 6.)

The table also shows the outer limits of the central 50 percent of obser-

vations on recovery rates for each of the security/seniority classesā”the

columns labeled ļ¬rst and third quartiles. Again focusing on the case of

senior unsecured bonds, we see that these central observations involve

recovery rates ranging from 30.75 percent to 67 percent. In part, variations

in recovery rates reļ¬‚ect diļ¬erences in the capital structures of default-

ing ļ¬rmsā”e.g. public vs. private debt composition (Hamilton and Carty

(1999)[38])ā”and the fact that recovery rates seem to have a noticeable

cyclical componentā”they tend to be higher during good economic times

and low during lean times. Nonetheless, these factors explain only a portion

of the observed variation in recovery rates. The one pattern that does show

through strongly in the data is that recovery rates tend to be monotonically

increasing with the level of seniority and security of the debt instrument.

16

The Credit Curve

We saw in Chapter 15 that risk-neutral survival probabilities are key ele-

ments in the pricing of ļ¬nancial market instruments that involve credit risk.

For instance, we showed that the fair market price of a defaultable bond

that makes payments at the future dates Ti and Ti+1 depends importantly

on the risk-neutral probabilities Q(t, Ti ) and Q(t, Ti+1 ) that the bond issuer

will not default by Ti and Ti+1 , respectively, where these probabilities are

conditional on all information available at time t and, naturally, on the

issuer having survived through time t.1 More generally, for any given issuer

or reference entity, one can imagine an entire term structure of survival

probabilities, which is one way of thinking of the credit curve. In simpler

words, the credit curve is the relationship that tells us the risk-neutral

survival probabilities of a given reference entity over various time horizons.

One can derive risk-neutral survival probabilities from the prices of liquid

credit market instruments and then use such probabilities to price other,

less liquid or more complex, instruments. In this chapter we describe a rel-

atively straightforward framework for inferring survival probabilities from

quoted credit default swap (CDS) premiums. We focus on three progres-

sively simpler methods: one that can handle any shape of the term structure

of CDS premiums, one built on the assumption of a ļ¬‚at term structure

of CDS premiums, and one based on a simple rule of thumb for quick,

back-of-the-envelope calculations for highly rated reference entities.

1

Unless otherwise stated, all probabilities conditional on information available at

time t will henceforth also be conditional on the issuer having survived through time t.

158 16. The Credit Curve

Armed with the CDS-implied credit curve for two hypothetical reference

entities, Sections 16.2 and 16.3 use a simple illustrative framework to go

over two practical applications. In Section 16.2 we illustrate how to mark to

market an existing credit default swap position. In Section 16.3 we revisit

the problem of valuing a principal-protected note (PPN), which was the

subject of Chapter 11. In particular, we describe how to price a PPN based

on premiums quoted in the CDS market. Section 16.4 highlights some of

the limitations of the simpliļ¬ed methodology described in this chapter.

16.1 CDS-implied Credit Curves

Consider a CDS with a notional amount of $1, written at time t on a

given reference entity, and with premium payment dates at [T1 , T2 , . . . , Tn ].

Let Sn be the corresponding annualized CDS premium. For simplicity,

assume that, in the event of default, the protection seller will pay 1 ā’ X

at the premium payment date immediately following the default, where X,

0 ā¤ X < 1, is the recovery rate. To keep things even simpler, we will ignore

the question of accrued premiums, or the fact that the protection buyer

would be paying any premium accrued between the last payment date and

the date of default.2

As seen in Chapter 6, we can think of a CDS as having two ālegsā: The

premium leg is made up of the periodic payments made by the protection

buyer; the protection leg is the default-contingent payment made by the

protection seller. Given the discussion in Chapter 15, assuming that, based

on risk-neutral probabilities, the occurrence of defaults is independent of

the riskfree interest rate embedded in the prices of riskless bonds, the

present discounted value of the premium leg can be written as:

n

PV(premiums)t = Z(t, Tj )Q(t, Tj )Ī“j Sn (16.1)

j=1

2

This simplifying assumption of no accrued premiums is relatively innocuous for

highly rated reference entities, but can have more signiļ¬cant eļ¬ects for riskier entities.

Typically, one addresses the issue of accrued premiums by adding half an accrual period

to the premium leg of the swap, which amounts to assuming that, should a default occur,

it will on average take place midway through the period. In this case, equation (16.1)

would become

n n

Ī“j S n

Probt [Tjā’1 < Ļ„ ā¤ Tj ]

PV(premiums)t = Z(t, Tj )Q(t, Tj )Ī“j Sn +

2

j=1 j=1

where Ļ„ is the time of default, and Probt [.] denotes a risk-neutral probability conditional

on all information available at time t.

16.1 CDS-implied Credit Curves 159

where Z(t, Tj ) is the time-t price of a riskless zero-coupon bond that

matures at Tj with a face value of $1; Q(t, Tj ) is the reference entityā™s

survival probability through time Tj , or the risk-neutral probability that

the reference entity will not have defaulted by Tj ; and Ī“j is the accrual fac-

tor for the jth premium payment (the number of days between the (j ā’1)th

and jth premium payment dates divided by the number of days in the year,

based on the appropriate day-count convention).

Equation (16.1) shows that there are two elements to discounting future

premiums. The logic here is similar to that in Chapter 15. When comput-

ing the present value of a future payment, ļ¬rst, there is the time-value of

money, captured by Z(t, Tj ), and, second, one must take account of the

fact that a future premium due, say, at Tj will only be received if the

reference entity has not defaulted by then, and, conditional on all informa-

tion available at time t, the risk-neutral probability of that happening is

Q(t, Tj ).3

The present value of the protection leg can be written in a similar way:

n

Z(t, Tj )Probt [Tjā’1 < Ļ„ ā¤ Tj ](1 ā’ X)

PV(protection)t = (16.2)

j=1

where Ļ„ is the time of default, and Probt [Tjā’1 < Ļ„ ā¤ Tj ] denotes the prob-

ability, conditional on information available at time t, that the reference

entity will default between Tjā’1 and Tj . The intuition behind (16.2) is

clear: One does not know whether and when a default will occur, but there

is some probability Probt [Tjā’1 < Ļ„ ā¤ Tj ] that the reference entity will

default during the interval [Tjā’1 , Tj ], in which case the protection seller

would have to pay 1 ā’ X at Tj , which is worth Z(t, Tj )(1 ā’ X) in todayā™s

dollars. As a result, the present value of the protection leg of the CDS is

the probability-weighted sum of all possible default scenarios.

From Chapter 15, equation (15.13), we know that we can rewrite (16.2) as

n

Z(t, Tj )[Q(t, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X)

PV(protection)t = (16.3)

j=1

16.1.1 Implied Survival Probabilities

It typically costs nothing to enter into a standard CDS so it must be that

Sn is such that the expected present discounted value of the premiums

paid by the protection buyer equals the expected present discounted value

3

In this part of the book we sidestep the issue of counterparty credit risk, or the fact

that the protection seller may default on its obligations under the CDS agreement. We

will revisit this issue in Part IV.

160 16. The Credit Curve

of the protection payment made by protection seller. We can then equate

the expressions in (16.1) and (16.3) and solve for the probabilities Q(t, Tj ).

In particular, after some manipulation, we can write:

nā’1

Z(t, Tj )[LQ(t, Tjā’1 ) ā’ (L + Ī“j Sn )Q(t, Tj )]

j=1

Q(t, Tn ) =

Z(t, Tn )(L + Ī“n Sn )

Q(t, Tnā’1 )L

+ (16.4)

(L + Ī“n Sn )

where L ā” (1 ā’ X).

Suppose now that you observe CDS premium quotes for a reference entity

covering all dates involved in the above expression, i.e., the markets tell

you the vector [S1 , S2 , . . . , Sn ].4 From (16.4), and for a given recovery rate,

X, it can be shown that

L

Q(t, T1 ) =

L + Ī“ 1 S1

and the other survival probabilities can be computed recursively. For

instance, given the value for Q(t, T1 ) implied by S1 , as well as the term

structure of riskless discount rates, Z(t, Ti ),

Z(t, T1 )[L ā’ (L + Ī“1 S2 )Q(t, T1 )] Q(t, T1 )L

Q(t, T2 ) = +

Z(t, T2 )(L + Ī“2 S2 ) (L + Ī“2 S2 )

Of course, given the survival probabilities Q(t, Tn ), n = 1, 2, . . . , we can

compute probabilities of default within speciļ¬c horizons. For instance, the

probability of default before period Tn is trivially given by 1 ā’ Q(t, Tn ).

At this point we should make one additional remark about the above

method for computing survival probabilities from quoted credit default

swap premiums. Note that the expression for the survival probabilities

involves the riskfree discount factors Z(t, Ti ), for i = 1, 2, . . . , n, which are

meant to represent only the time value of money. In practice, however, these

discount factors are often derived from the term structure of LIBOR and

swap rates, which corresponds to the funding costs of the main participants

4

In a more realistic situation, the observed CDS premiums may not exactly corre-

spond to the ones in expression (16.4). For instance, premiums may be quoted only for

contracts maturing at dates T1 , T3 , T5 , etc. In practice, what one can do in these cases

is to use interpolation methods to obtain premiums for the desired maturities from the

ones actually seen in the marketplace. Interpolation methods are widely used in the

context of yield curve modeling. See, e.g., the book by James and Webber (2000)[44]

and the several references therein.

16.1 CDS-implied Credit Curves 161

of the credit default swap market.5 This is consistent with the practice

of interpreting CDS premiums and asset swap spreads as being roughly

analogous to the yield spread of a risky ļ¬‚oater issued by the reference

entity over a ļ¬‚oater that pays out LIBOR ļ¬‚at (see Chapter 6).

16.1.2 Examples

We shall illustrate the derivation of CDS-implied default probabilities using

two numerical examples. Figure 16.1 shows the term structure of CDS

Maturity 1.0 2.0 3.0 4.0 5.0

AZZ 29 39 46 52 57

XYZ 9,100 7,800 7,400 6,900 6,500

Memo:

Z(t, T ) 0.9803 0.9514 0.9159 0.8756 0.8328

FIGURE 16.1. Two Hypothetical CDS Curves

5

Hull (2003)[41] describes a simple approach for obtaining zero-coupon bond prices,

Z(t, Ti ), from yields on coupon-paying riskless bonds. See also Appendix A in this book.

162 16. The Credit Curve

TABLE 16.1

Implied Survival Probabilities over Various Time Horizons

(percent)

Horizon AZZ Bank XYZ Corp.

One year 99.42 49.72

Two years 98.45 30.60

Three years 97.26 18.87

Four years 95.88 14.10

Five years 94.37 11.52

Note. Assumed recovery rates: AZZ Bank, 50 percent; XYZ

Corp., 10 percent.

premiums for two hypothetical reference entities at opposite ends of the

credit spectrum: AZZ Bank, which is assumed to be a highly rated insti-

tution, and XYZ Corp., a corporation that is seen as very likely to default

in the near term.

For simplicity, we assume that the premiums in both agreements are

paid once a year (Ī“j = 1 for all j). As can be seen in the ļ¬gure, while

the ļ¬ve-year CDS premium for AZZ Bank is under 60 basis points, that

for XYZ Corp. is assumed to be 6,500 basis points. Also noteworthy is

the pronounced negative slope of the XYZ Corp. curve, which is typical

of companies perceived to have a high likelihood of default. The quotes

plotted are shown in the table at the bottom of the ļ¬gure so that the

reader can verify the calculations that follow.

Table 16.1 shows implied survival probabilities for AZZ Bank and XYZ

Corp. over the next ļ¬ve years, computed as in equation (16.4) and based on

zero-coupon bond prices shown in the memo line in the table at the bottom

of Figure 16.1. Judged from the perspective of CDS market participants,

the risk-neutral odds that XYZ Corp. will still be around in one yearā™s

time are about even, whereas the corresponding risk-neutral probability for

AZZ Bank is nearly 100 percent. Over the ļ¬ve-year horizon, the survival

probabilities fall to almost 95 percent for AZZ Bank and 11.5 percent for

XYZ Corp.

As indicated in the footnote in the table, these calculations assume recov-

ery rates of 50 percent and 10 percent for AZZ Bank and XYZ Corp.,

respectively. We will examine how the results would diļ¬er under diļ¬erent

recovery rates in what follows.

16.1.3 Flat CDS Curve Assumption

The calculation of implied survival probabilities simpliļ¬es substantially

if one is willing to assume a ļ¬‚at CDS curve. In particular, the survival

16.1 CDS-implied Credit Curves 163

TABLE 16.2

Implied Survival Probabilities over Various Time Horizons

under Flat CDS Curve Assumption

(percent)

Horizon AZZ Bank XYZ Corp.

One year 98.87 58.06

Two years 97.76 33.71

Three years 96.66 19.58

Four years 95.57 11.37

Five years 94.49 6.60

Note. Assumed recovery rates: AZZ Bank, 50 percent; XYZ

Corp., 10 percent.

probability Q(t, Tn ) reduces to Q(t, T1 )n . (The reader can verify this by

going back to the derivation of Q(t, T2 ) in Section 16.1.1.)

As can be seen in Table 16.2, despite the fact that the ātrueā curves for

AZZ Bank and XYZ Corp. are non-ļ¬‚at, the ļ¬‚at curve assumption generates

survival probabilities that are not too far oļ¬ the ones shown in Table 16.1.

The calculations in Table 16.2 assume that the credit curves of AZZ Bank

and XYZ Corp. are ļ¬‚at at their respective ļ¬ve-year CDS premium levels

and thus the survival probabilities for XYZ Corp., which has the steepest

CDS curve, are aļ¬ected the most by the ļ¬‚at curve assumption.

16.1.4 A Simple Rule of Thumb

From the calculations done above, and assuming ļ¬‚at CDS curves with

premiums equal to the ļ¬ve-year quotes shown in Figure 16.1, we have that

the probability of default by T1 is

1ā’X S5

1 ā’ Q(t, T1 ) = 1 ā’ ā (16.5)

1 ā’ X + S5 1ā’X

and thus we can approximately think of the CDS premium as a measure

of the probability of default over the next year under the assumption of

no recovery. Close inspection of equation (16.5), however, shows that the

goodness of this approximation hinges on S5 being small enough, a topic

that we also discussed in the context of equation (15.20). Indeed, while this

rule of thumb would place the probability of default by AZZ Bank within

the next year at 1.14 percent, which is close to the number based on the ļ¬‚at

credit curve assumption, the corresponding number for XYZ Corp. would

be about 72 percent, compared to 42 percent in the ļ¬‚at curve scenario.

164 16. The Credit Curve

TABLE 16.3

Implied Survival Probabilities for AZZ Bank over Various Time

Horizons and under Alternative Recovery Rate Assumptions

(percent)

Horizon X = 0.20 X = 0.50 X = 0.65

One year 99.64 99.42 99.18

Two years 99.03 98.45 97.80

Three years 98.28 97.26 96.12

Four years 97.40 95.88 94.17

Five years 96.44 94.37 92.06

16.1.5 Sensitivity to Recovery Rate Assumptions

Table 16.3 shows how the survival probabilities for AZZ Bank are aļ¬ected

by alternative recovery rate assumptions. Because there is great uncertainty

surrounding actual recovery rates in the event of default, implied survival

probabilities are sometimes best reported in terms of ranges corresponding

to alternative values of the recovery rate. The diļ¬erences in the results

reported in the table would be more dramatic for lower-rated reference

entities.

16.2 Marking to Market a CDS Position

Although a credit default swap agreement typically has zero market value

at its inception, that generally does not remain true throughout the life

of the agreement, especially as the credit quality of the reference entity

may change. Marking a CDS position to market is the act of determining

todayā™s value of a CDS agreement that was entered into at some time in

the past.

One can use equations (16.1) and (16.3) to write an explicit expression

for the value of a CDS contract to a protection buyer:

n

Z(t, Tj )[Q(t, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X)

CDS

V (t, Tn ) =

j=1

n

ā’ Z(t, Tj )Q(t, Tj )Ī“j Sn

j=1

n

Z(t, Tj ){[Q(t, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X) ā’ Q(t, Tj )Ī“j Sn }

=

j=1

(16.6)

16.2 Marking to Market a CDS Position 165

where the ļ¬rst term to the right of the ļ¬rst equal sign is simply the expected

present value of the protection payment that the seller has committed to

make in the event of a default by the reference entity, and the second term

is the expected present value of the stream of future premium payments

owed under the contract. The value of the contract to the protection seller

is simply the negative of this expression.

Consider now a credit default swap that was written on XYZ Corp.ā”one

of the hypothetical reference entities examined in the previous section. The

contract was entered into exactly one year ago with an original maturity

of ļ¬ve years. Todayā™s CDS quotes for XYZ Corp. are shown in the table at

the bottom of Figure 16.1. Let us imagine that XYZ Corp. was perceived

to have a substantially more favorable proļ¬t outlook one year ago (and

hence higher survival probabilities) than today. As a result, we assume

that the CDS premium that was written into the year-old contract is 500

basis points, as opposed to the 6,500 basis points demanded by protection

sellers today.

Given the signiļ¬cant deterioration in the prospects for XYZ Corp., the

protection seller in the year-old contract is collecting a premium that is

well below the going market rate. This contract then has negative market

value to the seller of protection. The protection buyer, on the other hand, is

holding a contract with positive market value as she is paying only 500 basis

points per year per dollar of notional amount while a brand new contract

with the same remaining maturity would command an annual premium of

6,900 basis points per dollar of notional amount.

How can we value the year-old contract? One approach is simply to derive

the survival probabilities Q(.) implied by the current CDS premiums for

XYZ Corp. (as we did in Section 16.1) and to put them into equation (16.6),

along with the riskfree discount factors, the assumed recovery rate, and

the premium written into the contract. A simpler approach is to think

of the problem of valuing the year-old contract as that of computing its

āreplacement cost.ā In particular, how much would it cost to replace the

year-old contract, which now has a remaining maturity of four years, with

a brand new four-year contract? To put it simply,

replacement cost = value of new contract ā’ value of old contract (16.7)

where, given that the value of the new contract is zero by construction, the

replacement cost of the old contract is simply the negative of its market

value. The values of the old and new contractsā”denoted below as Vtold and

Vtnew ā”can be written as

4

Z(t, Tj ){[Q(t, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X) ā’ Q(t, Tj )Ī“j So }

Vtold =

j=1

(16.8)

166 16. The Credit Curve

4

Z(t, Tj ){[Q(t, Tjā’1 ) ā’ Q(t, Tj )](1 ā’ X) ā’ Q(t, Tj )Ī“j Sn }

Vtnew =

j=1

(16.9)

where So and Sn are the premiums written into the old and new contracts,

respectively.

Substituting equations (16.8) and (16.9) into (16.7) we arrive at the result

that we can write the market value of the year-old contract as a function

of the diļ¬erence between the premium written in that contract and the

four-year premium currently quoted in the marketplace:

4

Z(t, Tj )Q(t, Tj )Ī“j [Sn ā’ So ]

Vtold = (16.10)

j=1

which is a simpler expression than (16.6).

To sum up, to mark to market an existing CDS position one can rely

on current CDS premiums to obtain implied survival probabilities, as

described in Section 16.1, and use either (16.6) or (16.10) to determine

the market value of the position. Either way, the year-old CDS contract

written on XYZ Corp. with an original maturity of ļ¬ve years would now

be worth 68.8 cents per dollar of notional amount to the protection buyer

or about $6.9 million for a contract with a notional amount of $10 million!6

16.3 Valuing a Principal-protected Note

Credit default swaps have become so liquid for certain reference entities

that prices quoted in the CDS market are often used as the basis for valuing

other credit-based instruments that reference those entities. In this section

we illustrate how this can be done when valuing a principal-protected note

(PPN), a credit market instrument that we discussed in Chapter 11. We

will continue to work with the two examples introduced in Section 16.1.

As we saw in Chapter 11, a PPN is a coupon-paying note written on

a particular reference entity and sold to an investor by a highly rated

third party. In its simplest form, the note guarantees the return of its face

value at its maturity date, even if the reference entity has defaulted in its

obligations by then. The coupon payments themselves are stopped in the

6

Note that, in the example considered here, So is the premium written into the year-

old contract and Sn is the going premium for a four-year CDS contract written on XYZ

Corp. (6,900 basis points, according to the table in Figure 16.1).

16.3 Valuing a Principal-protected Note 167

event of default by the reference entity. In Chapter 11 we examined a ļ¬xed-

coupon PPN and showed that one can think of it as a portfolio consisting

of a riskless zero-coupon bond and a risky annuity that pays a ļ¬xed coupon

PPN = Riskless zero-coupon bond + Risky annuity (16.11)

where the payments made by the annuity are contingent on the reference

entity remaining solvent during the life of the PPN.

In practice, the price of the embedded riskless zero-coupon bond may be

derived directly from the swap curve, which incorporates the credit quality

of the AA-rated entities that are major sellers of PPNs. Valuing the above

described PPN then essentially means determining the coupon that will be

paid by the risky annuity. To see this, we can go back to equation (11.3)

and recast it in terms of the notation introduced in Chapter 15.

ļ£® ļ£¹

n

V P P N (t, Tn ) = ļ£°Z(t, Tn ) + Z(t, Tj )Q(t, Tj )Ī“j RP P N ļ£» F (16.12)

j=1

where t and Tn are, respectively, todayā™s date and the maturity date of the

PPN, and the vector [T1 , T2 , . . . , Tn ] contains the coupon payment dates

of the PPN in the case of no default by the reference entity. F is the

face value of the PPN, and RP P N is its coupon rate. Two key points

are worth emphasizing here about equation (16.12): Z(t, Tn ) corresponds

to the credit quality of the PPN issuer (or a riskfree rate if the issuer

poses no counterparty credit risk) and Q(t, Tj ) corresponds to the survival

probabilities of the reference entity (not the PPN issuer).

PPNs are typically issued at their par value so pricing a brand new PPN

amounts to ļ¬nding the value of RP P N that makes V P P N (t, Tn ) in (16.12)

equal to F . This is simply

1 ā’ Z(t, Tn )

RP P N = (16.13)

n

Z(t, Tj )Q(t, Tj )Ī“j

j=1

16.3.1 Examples

Suppose we want to price a family of PPNs written on the hypothetical ref-

erence entities introduced at the beginning of this chapter. (Each āfamilyā

references only a single reference entity.) For simplicity, we assume that

the notes pay coupons annually and that no accrued interest is paid in the

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