ńņš. 10 |

We mentioned in Section 3.2 that bank regulatory considerations spelled

out in the 1988 Basel Bank Capital Accordā”also known as the Basel I

Accordā”played an important role in the early days of the credit derivatives

markets. In particular, banksā™ use of credit derivatives was, at least initially,

signiļ¬cantly motivated by the desire to better align the notion of regulatory

capitalā”the share of risk-adjusted assets that bank regulators require to be

set aside to support risk-taking activitiesā”with that of economic capitalā”

a prudent bankā™s own estimate of the needed capital reserve. For instance,

in that chapter we examined the case of a hypothetical bank that made a

loan to a corporation and subsequently bought protection from an OECD

bank in the CDS market in a contract that referenced that corporation. The

protection-buying bank would essentially see its capital charge drop from

8 percent of the total exposure associated with the corporate loan to only

1.6 percent of the exposure, provided the bank could demonstrate to its

regulators that the terms of the contractā”maturity of the CDS vs. that of

the loan, deļ¬nition of credit events, etc.ā”eļ¬ectively granted it appropriate

protection against default-related losses on the loans.

The interplay between bank regulatory issues and banksā™ usage of credit

default swaps and related instruments has changed considerably since the

formative days of the credit derivatives market, however. Not only have

banks increased their use of credit derivatives for reasons other than just

regulatory capital management, but the regulatory environment itself has

also evolved. We discussed, in Chapter 3, banksā™ other (non-regulatory-

driven) uses of credit derivativesā”such as applications to risk management

and portfolio diversiļ¬cation. We will now take a quick look at the new

300 25. A Primer on Bank Regulatory Issues

regulatory environment facing banks and examine its possible implications

for the future development of the credit derivatives market.

Similar to the treatment of documentation considerations in Chapter 24,

the coverage of regulatory issues provided in this chapter is of a very broad

nature and thus does not constitute advice about what is permissible under

current international accords or even under the banking laws of any single

country.

25.1 The Basel II Capital Accord

In 1999, member countries of the Basel Committee on Banking Supervision

formally started working on a new set of guidelines and standards for their

national bank regulators and supervisors.1 That eļ¬ort culminated with the

New Basel Capital Accordā”also known as the Basel II Accord. The Accord

was ļ¬nalized at the Basel Committee level in 2004, and the process now

is largely in the hands of signatory countries, which have agreed to start

implementing the terms of the new accord by the end of 2006. The pro-

cess towards implementation will likely include further deliberation at the

national level, and, possibly, additional ļ¬ne-tuning in the language of the

accord at the international level. As it was the case with Basel I, the Basel II

Accord only provides a framework for the regulation of bank capital. Each

national authority then has the discretion to adapt the accordā™s stipulations

to its own needs and reality.

A main drive behind the genesis of the Basel II Accord has been the need

to strengthen the regulatory framework for large banking organizations,

especially those that are active in the international markets, and, in the

process, devise capital requirements that are more reļ¬‚ective of these orga-

nizationsā™ somewhat unique risk proļ¬les. We say āuniqueā because, more

so than their smaller cousins, large banking organizations have tended to

rely on newer instruments and approaches, including credit derivatives, in

measuring and managing risk. Many of these instruments either did not

exist or were not widely used when Basel I was conceived and thus were

not directly mentioned by the framers of that accord. In addition, the new

Basel Accord seeks to address some of the limitations of its predecessor,

1

The Basel Committee on Banking Supervision was established in 1974. Its members

include representatives of central banks or other supervisory authorities from Belgium,

Canada, France, Germany, Italy, Japan, Luxembourg, the Netherlands, Spain, Sweden,

Switzerland, the United Kingdom, and the United States. The committee has no supra-

national supervisory authority and its conclusions have no legal force. Its goals are to

formulate and propose standards and guidelines to encourage and facilitate the adop-

tion of best practices regarding bank regulation among member countries (see Federal

Reserve Board (2003))[27].

25.1 The Basel II Capital Accord 301

such as, as noted in Chapter 3, the lack of granularity in the way the

original accord assigned risk weights to diļ¬erent categories of borrowers.

As we saw in Chapter 2, despite the phenomenal growth of the credit

derivatives market, as well as the fact that banks are major users and deal-

ers of credit derivatives, these instruments represent only a small portion

of the total notional amount of derivatives at banks. Thus, it is both log-

ical and natural that the scope and motivation for revamping the Basel I

Accord goes well beyond issues directly related to the emergence of the

credit derivatives market. Given the subject matter of this book, however,

we shall highlight those aspects of the new accord that are most relevant for

credit derivatives, especially credit default swaps. Towards the end of the

chapter we provide a list of basic sources for readers interested in delving

deeper into bank regulatory issues involving credit derivatives.

Similar to the 1988 Accord, the provisions of the Basel II Accord call for

regulatory capital to be determined according to the credit risk associated

with the range of debt instruments held by a bank. But the new accord

is much more discriminating than its predecessor with regard to the credit

quality of a banksā™ debtors when it comes to determining the appropriate

risk weights that go into the calculation of regulatory capital.2 The com-

putation of regulatory capital is analogous to that devised under Basel I,

in that it involves multiplying the product of the regulatory risk weight

and the basic capital requirement of 8 percent by the notional exposure, as

discussed in Chapter 3.

The new accord allows the risk weights to be based on either ratings

provided by outside credit-rating agencies, such as Standard and Poorā™s and

Moodyā™sā”the so-called standardized approachā”or on two methods derived

from banksā™ own internal ratings systemsā”the so-called internal ratings-

based approachesā”provided, in the case of the latter, those systems have

been explicitly approved by the bank regulator. National bank regulators

are free to specify which risk weighting approach(es) are applicable to their

jurisdiction and under which conditions.

Based on the standardized approach, Table 25.1 shows Basel II risk

weights that correspond to sovereign, bank, and corporate (non-bank) bor-

rowers. These weights stand in sharp contrast to the weighting scheme

speciļ¬ed in the Basel I Accordā”see Table 3.1 in Chapter 3 for a comparison.

In particular, whereas the original accord made no distinction among cor-

porate borrowers, and diļ¬erentiated banks and sovereign borrowers only

according to their OECD status, the new accord allows for a closer cor-

respondence between regulatory risk weights and the credit quality of

2

Similar to the discussion of Basel I, in Chapter 3, we limit ourselves here to the regu-

latory treatment of obligations on banksā™ banking books. The new accord also addresses

requirements associated with obligations held on the trading book (see Basel Committee

on Bank Regulation (2003)[1]).

302 25. A Primer on Bank Regulatory Issues

TABLE 25.1

Risk Weights Speciļ¬ed in the Basle II Accord for Selected Obligorsa

(percent)

Obligor Credit Rating

AAA A+ BBB+ BB+ Below Bā’ Unrated

to AAā’ to Aā’ to BBBā’ to Bā’

Sovereign 0 20 50 100 150 100

Bankb 20 50 50 100 150 50

Corporate 20 50 100 100 150 100

a As speciļ¬ed in the April 2003 version of the Consultative Document on the New

Basel Capital Accord (Basel Committee on Banking Supervision, 2003[1]). Weights

shown are for the so-called standardized approach to weighting banking book expo-

sures.

b The Accord also provides for an alternative risk weighting of claims on banks. The

alternative weights are based on the credit assessment of the country in which the

bank is incorporated (see Basel Committee on Banking Supervision, 2003[1]).

obligors, a provision that helped narrow the gap that existed in the orig-

inal accord between regulatory and economic capital. For instance, under

Basel I, loans to corporations rated either AAA or B would be subject to

the same regulatory risk weight of 100 percent, despite the fact that these

corporations are nearly at opposite ends of the credit quality spectrum.

In contrast, the Basel II Accord assigns risk weights of 20 and 100 percent

to these obligors, respectively.

25.2 Basel II Risk Weights and Credit

Derivatives

With regard to credit default swaps, the new accord generally allows

protection-buying banks to continue to substitute the risk weight of the

protection seller for that of the reference entity in instances where the

bank is long an obligation of that entity and where the bank can demon-

strate that eļ¬ective protection has been bought. Thus, banks continue to

have an incentive to seek regulatory capital relief in the credit derivatives

market, but that incentive is arguably not as strong as under the Basel I

provisions, given the closer alignment between Basel II risk weights and

true economic risk.

One might be tempted to conclude that the closer alignment between

regulatory and economic capital might lead banks to become less important

users of credit derivatives. Yet, as we saw in Chapter 3, even before the

25.3 Suggestions for Further Reading 303

adoption of Basel II, banksā™ use of such derivatives had been increasingly

driven by nonregulatory considerations, such as the need to manage credit

risk exposures in response to greater scrutiny by investors. Indeed, given

its closer correspondence between regulatory and economic capital, Basel

II may well end up reinforcing banksā™ incentives to use credit derivatives

as a risk management tool.

Another implication of the greater granularity of the risk weighting

scheme of the Basel II Accord regards the attractiveness of non-banks as

potential sellers of protection to banks. Under Basel I, banks would be

granted no capital relief, say, for buying protection from an AA-rated

corporation in a credit default swap intended to hedge exposure to a

BBB-rated corporate reference entity. They would simply be replacing

exposure to one 100-percent risk-weighted obligorā”the BBB-rated entityā”

with anotherā”the AA-rated protection seller. Under Basel II, however, this

same contract would result, assuming other regulatory conditions are met,

in a decrease in the corresponding regulatory risk weight from 100 percent

to 20 percent, a development that bodes well for the greater participation

of highly-rated nonbank entities as protection sellers in credit derivatives.

(A similar argument applies to highly rated non-OECD banks.)

On the whole, Basel II has multiple, and sometimes oļ¬setting, impli-

cations for future developments in the credit derivatives market. On one

hand, banksā™ incentives to use of credit derivatives purely as a regulatory

capital management tool may wane further. On the other hand, the new

accord is clearly intended to reinforce banksā™ incentives to monitor closely

and manage their credit risk exposures, which could be accomplished with

greater use of instruments such as credit derivatives. Outside the banking

sector, the risk weighting provisions of the new accord may well promote

greater participation of other market players, such as highly-rated corpo-

rations, as sellers of protection, which may help facilitate the transfer of

credit risk from banks to other institutions.

25.3 Suggestions for Further Reading

A complete discussion of the full regulatory treatment of credit derivatives

under the Basel II Accord would go well beyond the scope of this book.

Indeed, we have omitted the discussion of several topics, such as the treat-

ment of basket swaps and synthetic securitization. In addition, the new

accord covers situations where the default protection bought in a credit

derivative contract only partially oļ¬sets a bankā™s exposure to the reference

entity, such as is the case when the maturity of a loan to the reference

entity is longer than that of the corresponding credit default swap. Read-

ers interested in these and other related topics are encouraged to consult

304 25. A Primer on Bank Regulatory Issues

the text of the accord itself, which can be downloaded from the Bank for

International Settlementsā™ website (www.bis.org).

While the focus of this chapter has been on international bank regulatory

standards as they relate to credit derivatives, we should once again remind

the reader that the Basel II Accord, like its predecessor, only provides a

general framework for national bank regulators in signatory countries. The

implementation of the accordā™s provisions varies from country to country

according to local concerns and market characteristics. Moreover, credit

derivatives have regulatory implications that go beyond the setting of capi-

tal requirements, and these too vary across national borders. The following

is a partial list of national bank regulator websites, some of which provide

further on-line information on issues related to credit derivatives in their

respective jurisdictions:

ā¢ Australia: Australian Prudential Regulation Authority

(www.apra.gov.au);

ā¢ Canada: Oļ¬ce of the Superintendent of Financial Institutions

(www.osļ¬-bsif.gc.ca);

ā¢ France: Commission Bancaire (www.commission-bancaire.org);

ā¢ Germany: Budesanstalt fur Finanzdienstleistungsaufsicht

(www.baļ¬n.de);

ā¢ Ireland: Central Bank of Ireland and Financial Services Authority

(www.centralbank.ie);

ā¢ Italy: Banca Dā™Italia (www.bancaditalia.it);

ā¢ Japan: Bank of Japan (www.boj.or.jp);

ā¢ Sweden: Swedish Financial Supervisory Authority (www.ļ¬.se);

ā¢ United Kingdom: Financial Services Authority (www.fsa.gov.uk);

ā¢ United States: Board of Governors of the Federal Reserve Sys-

tem (www.federalreserve.gov), Oļ¬ce of the Comptroller of the

Currency (www.occ.treas.gov), Federal Deposit Insurance Cor-

poration (www.fdic.gov), and the Oļ¬ce of Thrift Supervision

(www.ots.treas.gov).

Appendix A

Basic Concepts from Bond Math

We brieļ¬‚y discuss in this appendix some of the basic bond and term-

structure concepts featured in the book. Hull (2003)[41] provides further

details on these concepts at the introductory level.

A.1 Zero-coupon Bonds

A zero-coupon bond, as its name suggests, is a bond that makes no coupon

payments. Its yield comes from the fact that it is sold at a discount from

its face value. For instance, a one-year zero-coupon bond initially sold for

95 cents on the dollar can be said to have an annual yield of 5.26 percent,

or (1 ā’ .95)/.95. Indeed, it is straightforward to see that, for this one-year

bond:

1 = .95(1 + 0.0526)

i.e., the annual yield is the annual rate at which the price of the bond will

have to grow so that it will converge to its face value at the maturity date

of the bond. This is also called the yield-to-maturity of this bond.

The yield-to-maturity, Yn , of a n-year zero-coupon bond is likewise

deļ¬ned as the constant annual rate at which the bondā™s price will have

to grow so that the bond will be valued at par at maturity:

face value = (1 + Yn )n price (A.1)

306 Appendix A. Basic Concepts from Bond Math

A.2 Compounding

The zero-coupon yields discussed above assume annual compounding. If we

were to compound twice per annum, the yield-to-maturity of the one-year

zero-coupon bond considered above would be:

(2)

= 2[(1/.95)1/2 ā’ 1] = 0.0520

Y1

(2)

or 5.20 percent, where the above equation comes from the fact that Y1 is

such that

2

(2)

Y

1 = .95 1 + 1

2

For compounding j times per annum, the one-year yield is:

(j)

= j[(1/.95)1/j ā’ 1]

Y1

and, as j tends to inļ¬nity, we obtain the following expression for the yield

to maturity:

(ā)

= ā’ log(.95) = 0.05129

Y1

which is the continuously compounded yield-to-maturity for the one-year

bond.

Generalizing for n-year bonds, the yield-to-maturity with compounding

j times per year is

1/(jn)

face value

ā’1

(j)

Yn =j (A.2)

price

and the continuously compounded yield is:

price

Yn = ā’ log

(ā)

/n (A.3)

face value

The last two equations trivially allow us to write the price of a zero-

coupon bond in terms of its yield-to-maturity. With compounding j times

per annum:

face value

price of n-year bond = (A.4)

jn

(j)

Yn

1+

j

A.4 Coupon-paying Bonds 307

and, with continuous compounding:

price of n-year bond = eā’nYn

(ā)

[face value] (A.5)

A.3 Zero-coupon Bond Prices as Discount

Factors

Zero-coupon bond prices can be thought of as discount factors that can

be applied to future payments in order to express them in todayā™s dollars.

For instance, the present value of $1 to be received in one-yearā™s time is,

by deļ¬nition, simply todayā™s price of a zero-coupon bond with a face value

of $1.

If there is no chance that a given future payment will be missed (no

default risk) then the receiver of that payment should discount it on the

basis of the price of a zero-coupon bond that involves no credit risk.

By the same token, future payments that are subject to default risk should

be discounted with prices of zero-coupon bonds that are subject to com-

parable default risk. This is essentially what we do in equation (4.2) in the

text, where we discounted the future cash ļ¬‚ows of a corporate security using

prices of zero-coupon bonds derived from the security issuerā™s yield curve.

A.4 Coupon-paying Bonds

Consider a n-year bond that pays a ļ¬xed annual coupon C at dates

T1 , T2 , . . . , Tn . In addition, the bond pays its face value F at its matu-

rity date Tn . The bond has no default risk. Given the discount-factor

interpretation of zero-coupon bond prices, we can write:

n

V (t, Tn ) = Z(t, Th )C + Z(t, Tn )F (A.6)

h=1

where V (t, Tn ) is the price of the coupon-paying bond, and Z(t, Th ) is the

price of a riskless zero-coupon bond that pays $1 at date Th .

If the coupon-paying bond has some default risk, the discount factors

should correspond to the prices of zero-coupon bonds issued by the same

entity issuing the coupon-paying bond. Assuming that the bond has no

recovery value upon default, we can write:

n

+ ā’(t, Tn ) =

V0d d d

Z0 (t, Th )C + Z0 (t, Tn )F (A.7)

h=1

308 Appendix A. Basic Concepts from Bond Math

TABLE A.1

Par Yield Curve for a Hypothetical Issuera

Maturity Yield to Maturity

(years) (percent)

1 2.0

2 2.2

3 2.5

4 2.7

a Yieldsshown are compounded annually and

coupons are paid once a year.

where V d (t, Tn ) is the price of the defaultable coupon-paying bond, and

d

Z0 (t, Th ) is the price of a defaultable zero-coupon bond that promises to

pay $1 at date Th . Thus, given the zero-coupon curve for a given issuer, we

can price other debt instruments of that same issuer, as we do, for instance,

in Chapter 11 when valuing principal-protected notes.

A.5 Inferring Zero-coupon Yields from the

Coupon Curve

Zero-coupon bonds with long maturities are rarely issued in practice.

Instead, actual longer-dated bonds traded in the marketplace are of the

coupon-paying variety. Using (A.7), however, we can back out the zero-

coupon yields and prices embedded in an issuerā™s coupon curve. To see

how this can be done, we consider a simple numerical example for a hypo-

thetical issuer. Table A.1 summarizes the main inputs. The table shows

the āparā yield curve for the issuer, where by par we mean that the prices

of the bonds shown are all equal to their face values, or, equivalently, that

the yield-to-maturity of each bond is equal to its respective coupon rate.

We show four bonds, with maturities ranging from one to four years, each

with a face value of $1 and a zero recovery rate.1

The time-t (todayā™s) price of the one-year zero-coupon bond associated

with this issuer, which would be used to discount payments to be made in

1

In a more realistic setting, the issuer may not have outstanding bonds along the

entire maturity spectrum, and one may need to resort to interpolation and smooth-

ing methods. See, for instance, James and Webber (2000)[44] for a discussion of these

methods.

A.6 Forward Rates 309

one-yearā™s time, is simply the price of the ļ¬rst bond shown. Thus, given

d

1 = Z0 (t, t + 1)(1.02)

we have

d

Z0 (t, t + 1) = 1/1.02 = 0.9804

To derive the price of the two-year zero-coupon bond, recall that the

price of the two-year coupon-paying bond can be written as

d d

1 = Z0 (t, t + 1)0.022 + Z0 (t, t + 2)(1.022)

Thus

1 ā’ Z0 (t, t + 1)0.022

d

d

Z0 (t, t + 2) =

1.022

d

where the value of Z0 (t, t + 1) was derived in the prior step.

For the three-year bond:

d d d

1 = Z0 (t, t + 1)0.025 + Z0 (t, t + 2)0.025 + Z0 (t, t + 3)(1.025)

and thus

2

1ā’ d

Z0 (t, t + h)0.025

d h=1

Z0 (t, t + 3) =

1.025

Generalizing, the price of a n-year zero-coupon bond can be written as

a function of the prices of shorter-dated zeros and the coupon of a n-year

coupon-paying bond sold by the same issuer:

nā’1

1ā’ d

Z0 (t, t + h)Cn

d h=1

Z0 (t, t + n) = (A.8)

1 + Cn

where Cn is the coupon payment of the n-year bond.

A.6 Forward Rates

A forward rate F (t, T1 , T2 ) is the annual interest rate agreed upon in the

marketplace at todayā™s date (time t) for lending and borrowing during a

future period [T1 , T2 ], but involving no net cash outlay at time t. One

can obtain the fair value of F (t, T1 , T2 ) entirely in terms of time-t prices

of zero-coupon bonds. Consider the following simultaneous transactions at

time t:

310 Appendix A. Basic Concepts from Bond Math

ā¢ sell one zero-coupon bond that matures at time T1 ;

ā¢ buy Z(t,T1 )

zero-coupon bonds that mature at time T2 .

Z(t,T2 )

Assume that the face value of the bonds is $1. Note that this transaction

results in no net cash ļ¬‚ow at time t. Subsequent cash ļ¬‚ows are as follows:

ā¢ Time T1 : Pay $1 to the buyer of the T1 -bond sold at t;

ā¢ Time T2 : Receive $ Z(t,T1 )

from the issuer of the T2 -bonds bought

Z(t,T2 )

at time t.

Thus we can see that the time-t transaction involves no initial cost and

is akin to contracting to lend $1 at time T1 in exchange for receiving

$ Z(t,T1 ) at time T2 . The implicit interest payment in this forward loan

Z(t,T2 )

is Z(t,T1 ) ā’ 1. As already noted, the annualized interest rate implicit in

Z(t,T2 )

this forward loan, which we denote below as F (t, T1 , T2 ), is deļ¬ned as the

time-t forward rate for the future time period [T1 , T2 ]. For instance, in

Chapter 4, we examine an example where the length of the future period is

6 months, T2 ā’ T1 = .5 year, which corresponds to the following expression

for F (t, T1 , T2 ):

1 Z(t, T1 )

ā’1

F (t, T1 , T2 ) = (A.9)

T2 ā’ T1 Z(t, T2 )

Using equation (A.5), the time-t continuously compounded forward rate

for the future period [T1 , T2 ] is the rate f (t, T1 , T2 ) that solves the following

equation:

Z(t, T1 )

ef (t,T1 ,T2 )(T2 ā’T1 ) =

Z(t, T2 )

Thus:

log(Z(t, T1 )) ā’ log(Z(t, T2 ))

f (t, T1 , T2 ) = (A.10)

T2 ā’ T1

A.7 Forward Interest Rates and Bond Prices

The rate f (t, T ) such that

log(Z(t, T )) ā’ log(Z(t, T + āT )) ā‚ log(Z(t, T ))

f (t, T ) ā” lim =ā’

āT ā’0 āT ā‚T

(A.11)

A.7 Forward Interest Rates and Bond Prices 311

is called the instantaneous continuously compounded forward rate. It can

be thought of as the time-t interest rate that applies to the future period

[T, T + dt], where dt is an inļ¬nitesimal time increment.

Integrating both sides of equation (A.11), we obtain:

Z(t, T ) = eā’

T

f (t,v)dv

(A.12)

t

Given the deļ¬nition of the continuously compounded yield to maturity

as the rate R(t, T ) such that Z(t, T ) = eā’R(t,T )(T ā’t) :

T

f (t, v)dv

t

R(t, T ) = (A.13)

T ā’t

which says that the time-t yield-to-maturity on a zero-coupon bond can

be thought of as the average of all time-t instantaneous forward rates that

span the remaining life of the bond.

Appendix B

Basic Concepts from Statistics

This appendix provides a brief review of some key statistical concepts used

in the text. We have skipped over some technical details and mathematical

proofs involving these concepts. Bain and Engelhardt (1987)[5], Hogg and

Tanis (1983)[39], and Grimmett and Stirzaker (1998)[36] provide a more

complete coverage of the topics discussed herein.

B.1 Cumulative Distribution Function

The cumulative distribution function (c.d.f.) of a random variable X is

deļ¬ned as

F (x) = Prob[X ā¤ x] (B.1)

F (x) is also commonly called the distribution function of X. We highlight

two of its basic properties below:

ā¢ Because F (x) is a probability: 0 ā¤ F (x) ā¤ 1;

ā¢ F (x) is a non-decreasing function of x, i.e., if x1 < x2 then if X ā¤ x1

we also have X ā¤ x2 , and thus:

F (x1 ) = Prob[X ā¤ x1 ] ā¤ Prob[X ā¤ x2 ] = F (x2 )

314 Appendix B. Basic Concepts from Statistics

B.2 Probability Function

A random variable X is said to be discrete if it can only take on one of the

discrete values x1 , x2 , x3 , . . .. If X is a discrete random variable:

F (x) = p(xi ) (B.2)

xi ā¤x

where the sum is computed over all xi such that xi ā¤ x, and where p(x) is

called the probability function of X.

The function p(x) assigns a probability to each one of the possible values

of X:

p(x) = Prob[X = x] x = x1 , x2 , . . . (B.3)

Note that, for p(x) to be a probability function, it must satisfy the following

two properties:

p(xi ) ā„ 0 for all xi

p(xi ) = 1

all xi

B.3 Probability Density Function

Let X be a continuous random variable, in that it can take any value, for

instance, in the interval [x1 , xn ]. The c.d.f. of X can be written as:

x

F (x) = f (s)ds (B.4)

ā’ā

where f (x) is called the probability density function (p.d.f.) of X.

Similar to the probability function in the discrete case, a p.d.f. must

satisfy the following conditions

f (x) ā„ 0 for all real x

ā

f (x)dx = 1

ā’ā

In the light of (B.4), it can be shown that:

b

Prob[a ā¤ X ā¤ b] = F (b) ā’ F (a) = f (x)dx (B.5)

a

B.4 Expected Value and Variance 315

Because F (x) is a continuous function:

Prob[X = a] = 0 (B.6)

which is consistent with the basic result from classic calculus that

a

f (x)dx = 0. Thus:

a

Prob[a < X < b] = Prob[a < X ā¤ b] = Prob[a ā¤ X < b] = Prob[a ā¤ X ā¤ b]

We also highlight the following useful relationship:

Prob[x < X ā¤ x + dx] = F (x + dx) ā’ F (x) ā f (x)dx (B.7)

where the approximation error is negligible for suļ¬ciently small values of

dx. Indeed, given (B.4), from classic calculus we know that, for values of x

for which ā‚F (x) exists:

ā‚x

F (x + ds) ā’ F (x) ā‚F (x)

lim = = f (x) (B.8)

dx ā‚x

dxā’0

B.4 Expected Value and Variance

The expected value of a discrete random variable is simply the probability-

weighted average of all of its possible values. If the possible values of X are

x1 , x2 , . . . , xn , its expected value, E[X], is given by

n

E[X] = p(xi )xi (B.9)

i=1

where p(.) is the probability function of X.

The variance of X is likewise deļ¬ned as

n

p(xi )(xi ā’ E[X])2

V [X] = (B.10)

i=1

If X is a continuous random variable we have

ā

E[X] = sf (s)ds (B.11)

ā’ā

and

ā

(s ā’ E[X])2 f (s)ds

V [X] = (B.12)

ā’ā

316 Appendix B. Basic Concepts from Statistics

Note. The square root of the variance of X is called the standard devia-

tion of X.

B.5 Bernoulli Trials and the Bernoulli

Distribution

Throughout this book, we often deal with situations where a company can

be in only one of two possible states at some given future date, default

or survival, and where there are probabilities associated with each state.

Statistically, one way to characterize such situations is through the concept

of Bernoulli trials.

A Bernoulli trial is a random experiment that can result in only one of

two possible outcomesā”e.g., a given company will either default or survive.

A sequence of independent Bernoulli trials is one where the probabilities

associated with the two outcomes are the same from trial to trial.

Let X be a random variable associated with a Bernoulli trial. For

instance:

X = 1 if the company defaults and X = 0 if it survives

If the probability of default is denoted as Ļ, the probability function of X

can be written as

p(x) = Ļ x (1 ā’ Ļ)1ā’x x = 0, 1 (B.13)

and one can say that X has a Bernoulli distribution.

The expected value and variance of X are:

1

xĻ x (1 ā’ Ļ)1ā’x = (1)Ļ + (0)(1 ā’ Ļ) = Ļ

E[X] =

x=0

1

(x ā’ Ļ)2 Ļ x (1 ā’ Ļ)1ā’x = Ļ(1 ā’ Ļ)

V [X] =

x=0

B.6 The Binomial Distribution

Consider a sequence of n independent Bernoulli trials. For instance, given n

corporate borrowers, each trial may involve either the default or survival of

an individual borrower, where defaults among the n borrowers are mutually

independent. Let the default probability for each borrower be denoted as Ļ.

B.7 The Poisson and Exponential Distributions 317

Let Y be the random variable that represents the number of defaults among

the n borrowers over a given period of time. The probability function of Y is

n!

Ļ y (1 ā’ Ļ)nā’y

b(y; n, Ļ) = (B.14)

y!(n ā’ y)!

where y denotes the possible values of Y ā”y = 0, 1, 2, . . . , nā”and Y is said

to be binomially distributed.

The c.d.f. of the binomial distribution is

y

B(y; n, Ļ) ā” Prob[Y ā¤ y] = b(s; n, Ļ) y = 0, 1, . . . , n (B.15)

s=0

which, continuing with our example, is the probability that at most y

companies will default over a given time horizon.

In Part IV we use the results just derived to examine expected default-

related losses in an equally weighted homogeneous portfolio where defaults

among the issuers represented in the portfolio are mutually independent.

With default independence the question of how many issuers are likely

to default or survive reduces to a sequence of independent Bernoulli tri-

als, in which case the binomial distribution applies. Without default

independenceā”for instance, a default by one company changes the default

probabilities of the othersā”we cannot directly appeal to results based on

the binomial distribution.

B.7 The Poisson and Exponential Distributions

Most reduced-form models make use of the Poisson distribution to charac-

terize the āarrivalā process of defaults over time for a given borrower.1 For

this borrower, let X be the discrete random variable that corresponds to

the number of defaults arriving (occurring) over a given continuous time

interval. We assume that defaults occur randomly at the mean rate of Ī»

per year, with Ī» > 0.

From a credit risk modeling perspective, in most cases all that we care

about is the ļ¬rst occurrence of default. Nonetheless, in some applicationsā”

such as when modeling defaults and corporate reorganizationsā”one may

also be interested in the notion of second default, third default, etc.

1

We will measure time in terms of non-negative real numbers, with 0 denoting āthe

beginning of timeā and time t representing the present time. The unit of measurement

will be years so that, for instance, t = 1.25 means that we are one and a quarter years

away from the beginning of time.

318 Appendix B. Basic Concepts from Statistics

If X is Poisson distributed, the following conditions must be satisļ¬edā”

see, for instance, Hogg and Tanis (1983)[39]:

1. the numbers of defaults occurring in nonoverlapping time intervals

are independent;

2. the probability of exactly one default occurring during a short time

interval of length s is approximately Ī»s;

3. the probability of more than one default during a suļ¬ciently short

time period is essentially zero.

Assuming the above conditions are met, the probability function of X

can be shown to be:

p(x; 1) ā” Prob[x defaults in a one-year interval]

Ī»x eā’Ī»

= for x = 0, 1, 2, . . . (B.16)

x!

which, indeed, has the property, stated above, that the expected number

of defaults during a time interval of one year is Ī»:

ā

Ī»x eā’Ī»

E[X] = x =Ī» (B.17)

x!

x=0

The variance of X can also be shown to be Ī».

More generally, let us change the reference time interval to be s yearsā”

1

where s > 0ā”so that, for instance, s = 12 indicates a time interval of

about a month. If the mean arrival rate of defaults for this borrower in a

one-year interval is Ī», then the mean arrival of defaults in a s-year interval

is Ī»s. As a result, we can write:

p(x; s) ā” Prob[x defaults in a s-year interval]

(Ī»s)x eā’Ī»s

= for x = 0, 1, 2, . . . (B.18)

x!

provided, of course, conditions 1 through 3 are satisļ¬ed.

Thus, using time 0 as our vantage point, the unconditional probability

that no default will take place during a time interval of length s years is:

p(0; s) ā” Prob[0 defaults in a s-year interval]

= eā’Ī»s (B.19)

B.7 The Poisson and Exponential Distributions 319

Similarly, the unconditional probability of exactly one default occurring

during a time interval of length s years is

p(1; s) = Ī»s eā’Ī»s

which, consistent with condition 2, is approximately Ī»s for small s.

As discussed in Chapter 17, one concept that is of great interest in the

pricing of credit derivatives is the expected time of ļ¬rst default by a given

reference entity. Let Ļ„ denote the time of ļ¬rst default, which can be thought

ĀÆ

of as a continuous random variable with c.d.f. G(s). (Alternatively, Ļ„ can

also be thought of as the āwaiting timeā until the ļ¬rst default, as seen from

ĀÆ

time 0.) Clearly, for s < 0, G(s) = 0, given that a waiting time cannot be

negative. For s ā„ 0 we can write

G(s) ā” Prob[Ļ„ ā¤ s] = 1 ā’ Prob[Ļ„ > s]

ĀÆ

but note that Prob[Ļ„ > s] is simply the probability, as seen at time 0, that

there will be no default by time s. This probability is given by equation

(B.19). Thus, we can write:

G(s) = 1 ā’ eā’Ī»s

ĀÆ (B.20)

which can be thought of as the unconditional probability of a ļ¬rst default

by time s.

ĀÆ

The unconditional p.d.f. of Ļ„ , deļ¬ned as ā‚ G(s) , can be written as:

ā‚s

g (s) = Ī»eā’Ī»s

ĀÆ (B.21)

Readers with some familiarity with statistics will recognize g (s) as the p.d.f.

ĀÆ

1

of an exponentially distributed random variable with mean Ī» and variance

1

Ī»2 . Thus, the time of ļ¬rst default is exponentially distributed when defaults

occur according to a Poisson process.

The conditional probability of ļ¬rst default by time s, given no default

through time t, for s ā„ t, can be written as

Gt (s) ā” Prob[Ļ„ ā¤ s|Ļ„ > t]

= 1 ā’ Prob[Ļ„ > s|Ļ„ > t]

Prob[Ļ„ > s]

=1ā’

Prob[Ļ„ > t]

= 1 ā’ eā’Ī»(sā’t) (B.22)

where, to arrive at (B.22), we used the Bayes rule and equation (B.19).

320 Appendix B. Basic Concepts from Statistics

The conditional p.d.f. of Ļ„ , given no default through time t, is the

function gt (s) such that

gt (s) = eā’Ī»(sā’t) Ī» (B.23)

which is simply ā‚Gt (s) .

ā‚s

Armed with (B.23), we can, for instance, compute the expected time of

the ļ¬rst default, conditional on no default through time t:

ā

E[Ļ„ |Ļ„ > t] = s gt (s)ds (B.24)

t

1

which can be shown to be equal to t + Ī» .

B.8 The Normal Distribution

If X is a continuous random variable that is normally distributed with

mean Āµ and variance Ļ 2 , its p.d.f. is

1

eā’[(xā’Āµ)/Ļ] /2

2

f (x; Āµ, Ļ) = ā (B.25)

2ĻĻ

and the normal c.d.f. is accordingly given by

x

1

eā’[(sā’Āµ)/Ļ] /2 ds

2

ā

F (x) = (B.26)

2ĻĻ

ā’ā

If we deļ¬ne Y ā” Xā’Āµ , then it can be shown that Y is normally dis-

Ļ

tributed with a mean of zero and a variance of one. Y is commonly called

the standardized value of X, and its density is referred to as the standard

normal p.d.f.

1

n(y) = ā eā’y /2

2

(B.27)

2Ļ

The standard normal c.d.f. is

y

1

ā eā’s /2 ds

2

N (y) = (B.28)

2Ļ

ā’ā

which is used extensively in this book, especially in discussion of structural

credit risk models in Part III and in the treatment of portfolio credit risk

issues in Part IV.

B.9 The Lognormal Distribution 321

A key result regarding the normal distribution is that

xā’Āµ

F (x; Āµ, Ļ) = N = N (y) (B.29)

Ļ

so one can rely exclusively on N (.) when handling normally distributed

variables. This is especially convenient because there is no analytical for-

mula for the normal c.d.f., and one often has to rely on tabulated values.

In addition, most statistical and mathematical software have embedded

functions to generate values of N (.) for any given y. Alternatively, Hull

(2003)[41] provides a simple approximation that is very accurate.

A useful property of the standard normal distribution, and one that we

rely upon in Part IV, is that of symmetry. By symmetry we mean that the

standard normal p.d.f. is such that for any real number y:

n(y) = n(ā’y) (B.30)

which can be shown mathematically, or veriļ¬ed visually if one recalls that

n(.) has a bell-shape and is perfectly symmetric around its mean of zero.

It can also be shown that, again for any real number y:

N (ā’y) = 1 ā’ N (y) (B.31)

a result that we use in the derivation of the loss distribution of a large

homogeneous portfolio in Chapter 19.

B.9 The Lognormal Distribution

If X is normally distributed, with mean Āµ and variance Ļ 2 , then Y ā” eX

is lognormally distributed. The p.d.f. of Y is

1

eā’(log(y)ā’Āµ) /(2Ļ )

2 2

f (y) = ā for y > 0 (B.32)

2ĻĻy

The distribution of Y is called lognormal because, given Y = eX and the

fact that X is normally distributed, log(Y ) is itself normally distributed

with mean Āµ and variance Ļ 2 .

The lognormal c.d.f. can be expressed in terms of the normal c.d.f. In

particular, given that log(Y ) is an increasing function of Y , it can be shown

that

F (y) ā” Prob[Y ā¤ y] = Prob[log(Y ) ā¤ log(y)]

322 Appendix B. Basic Concepts from Statistics

but the term on the right-hand side of the above equation can be rewritten

as Prob[X ā¤ log(y)], and thus:

log(y) ā’ Āµ

F (y) = N (B.33)

Ļ

The mean and variance of Y are:

2

E[Y ] = eĀµ+Ļ /2

(B.34)

2 2

V [Y ] = e2(Āµ+Ļ ) ā’ e2Āµ+Ļ (B.35)

We use the lognormal distribution in Chapters 17 and 18 when dis-

cussing structural models of credit risk and in the valuation of credit

options, respectively. Lognormality of the value of individual ļ¬rms is also

an important assumption in the basic portfolio credit risk model discussed

in Part IV.

B.10 Joint Probability Distributions

The joint probability function of the discrete random variables

X1 , X2 , . . . , Xn is deļ¬ned as the function f (.) such that

f (x1 , x2 , . . . , xn ) = Prob[X1 = x1 , X2 = x2 , . . . , X3 = x3 ] (B.36)

for all possible values x1 , x2 , . . . , xn of X1 , X2 , . . . , Xn . In words,

f (x1 , x2 , . . . , xn ) is the probability that X1 = x1 and X2 = x2 , . . . , and

Xn = xn .

The joint c.d.f. of these Xi s is the function F (.) such that

F (x1 , x2 , . . . , xn ) = Prob[X1 ā¤ x1 , X2 ā¤ x2 , . . . , Xn ā¤ xn ] (B.37)

i.e., F (x1 , . . . , xn ) is the probability that X1 ā¤ x1 , and X2 ā¤ x2 , . . . , and

Xn ā¤ xn .

The joint p.d.f. of the continuous random variables Y1 , Y2 , . . . , Yn is the

function g(.) such that the joint c.d.f. G(.) can be written as

yn y2 y1

G(y1 , y2 , . . . , yn ) = ... g(s1 , s2 , . . . , sn )ds1 ds2 . . . dsn

ā’ā ā’ā ā’ā

(B.38)

for all possible values of y1 , y2 , . . . , yn .

B.12 The Bivariate Normal Distribution 323

B.11 Independence

The random variables Y1 , Y2 , . . . , Yn are independent if and only if one of

the following holds:

G(y1 , y2 , . . . , yn ) = G1 (y1 )G2 (y2 ), . . . , Gn (yn ) (B.39)

g(y1 , y2 , . . . , yn ) = g1 (y1 )g2 (y2 ), . . . , gn (yn ) (B.40)

where Gi (yi ) and gi (yi ) are the c.d.f. and p.d.f., respectively, of Yi . (This

deļ¬nition applies to both discrete and continuous random variables.)

A closely related concept is that of serial independence. For instance,

if R is a random variable that changes its value over time, R is serially

independent if Rt , the value of R at time t, is independent of Rs , its value

at time s, for s diļ¬erent from t.

Note: For normally distributed variables, events that are uncorrelated are

also independent and vice versa. For such variables, the terms uncorrelation

and independence can generally be used interchangeably.

B.12 The Bivariate Normal Distribution

If X1 and X2 are two normally distributed random variables with means Āµ1

and Āµ2 and variances Ļ1 and Ļ2 , respectively, and Ļ is the correlation coef-

ļ¬cient of X1 and X2 , then the joint distribution of X1 and X2 is bivariate

normal.

The bivariate normal p.d.f. is

1

n2 (x1 , x2 , Ļ) =

1 ā’ Ļ2

2ĻĻ1 Ļ2

2 2

x1 ā’Āµ1 x1 ā’Āµ1 x2 ā’Āµ2 x2 ā’Āµ2

ā’ 2(1ā’Ļ2 ) ā’2Ļ

1

+

Ć—e

Ļ1 Ļ1 Ļ2 Ļ2

(B.41)

and the bivariate normal c.d.f. has the usual deļ¬nition

x2 x1

N2 (x1 , x2 , Ļ) = n2 (s1 , s2 , Ļ)ds1 ds2 (B.42)

ā’ā ā’ā

Similar to the univariate normal, there is no analytical formula for the

bivariate c.d.f. Hull (2003)[41] describes a numerical approximation that is

reasonably accurate.

We use the bivariate normal distribution in Chapter 19, in the treatment

of default correlation, in Chapter 21, when discussing premiums on port-

folio default swaps, and in Chapter 23, in the valuation of credit default

swaps that are subject to counterparty credit risk.

Bibliography

[1] Basel Committee on Banking Supervision. The New Basel

Capital Accord, Consultative Document. April 2003.

[2] A. Arvanitis and J. Gregory. Credit: The Complete Guide to

Pricing, Hedging and Risk Management. Risk Books, 2001.

[3] A. Arvanitis, J. Gregory, and J. Laurent. Building Models for

Credit Spreads. Journal of Derivatives, 1:27ā“43, 1999.

[4] British Bankers Association. Credit Derivatives Report

2000/2002. British Bankers Association, 2002.

[5] L. Bain and M. Engelhardt. Introduction to Probability and

Mathematical Statistics. PSW, 1987.

[6] M. Baxter and A. Rennie. Financial Calculus: An Introduction to

Derivative Pricing. Cambridge University Press, 2001.

[7] T. Bjork. Arbitrage Theory in Continuous Time. Oxford University

Press, 1998.

[8] F. Black. The Pricing of Commodity Futures Contracts. Journal of

Financial Economics, 3:167ā“79, 1976.

[9] F. Black and M. Scholes. The Pricing of Options and Corporate

Liabilities. Journal of Political Economy, 81:637ā“54, 1973.

326 Bibliography

[10] R. Black and J. Cox. Valuing Corporate Securities: Some Eļ¬ects

of Bond Indenture Provisions. Journal of Finance, 31:351ā“67, 1976.

[11] A. Bomfim. Credit Derivatives and their Potential to Synthesize

Riskless Assets. Journal of Fixed Income, December: 6ā“16, 2002.

[12] A. Cifuentes and G. Oā™Connor. The binomial expansion method

applied to cbo/clo analysis. Technical report, Moodyā™s Investors

Service, December 1996.

[13] A. Cifuentes and C. Wilcox. The double binomial method and

its applications to a special case of cbo structures. Technical report,

Moodyā™s Investors Service, March 1998.

[14] P. Collin-Dufresne, R. Goldstein, and J. Martin. The Deter-

minants of Credit Spread Changes. Journal of Finance, 56:2177ā“207,

2001.

[15] J. Cox, J. Ingersoll, and S. Ross. A Theory of the Term Structure

of Interest Rates. Econometrica, 53:385ā“407, 1985.

[16] P. Crosbie. Modeling Default Risk. KMV Corporation, 2002.

[17] M. Crouhy, D. Galai, and R. Mark. A Comparative Analysis

of Current Credit Risk Models. Journal of Banking and Finance,

24:59ā“117, 2000.

[18] S. Das. Credit Derivatives and Credit-Linked Notes. John Wiley &

Sons, second edition, 2000.

[19] S. Das and P. Tufano. Pricing Credit Sensitive Debt when Interest

Rates, Credit Ratings and Credit Spreads are Stochastic. Journal of

Financial Engineering, 5:161ā“98, 1996.

[20] G. Delianedis and R. Geske. Credit Risk and Risk Neutral Default

Probabilities: Information about Rating Migrations and Defaults.

Working Paper 19-98, Anderson Graduate School of Business, Uni-

versity of California, Los Angeles, 1998.

[21] G. Duffee. The Relationship between Treasury Yields and Corpo-

rate Bond Yield Spreads. Journal of Finance, 53:2225ā“41, 1998.

[22] D. Duffie and N. Garleanu. Risk and Valuation of Collateralized

Debt Obligations. Financial Analysts Journal, 57:41ā“62, 2001.

[23] D. Duffie and D. Lando. Term Structures of Credit Spreads with

Incomplete Accounting Information. Econometrica, 69:633ā“64, 2001.

Bibliography 327

[24] D. Duffie and K. Singleton. Modeling Term Structures of

Defaultable Bonds. Review of Financial Studies, 12:687ā“720, 1999.

[25] D. Duffie and K. Singleton. Credit Risk. Princeton University

Press, 2003.

[26] P. Embrechts, A. McNeil, and D. Strautman. Correlation and

Dependency in Risk Management: Properties and Pitfalls. Working

Paper, ETH Zurich, 1999.

[27] Federal Reserve Board. Capital Standards for Banks: The

Evoluing Basel Accord. Federal Reserve Bulletin, September 2003;

395ā“405.

[28] FitchRatings. Credit derivatives: Risk management or risk? Tech-

nical report, FitchRatings, March 2003.

[29] J. Fons. Using Default Rates to Model the Term Structure of Credit

Risk. Financial Analysts Journal, Sept/Oct: 25ā“33, 1994.

[30] R. Geske. The Valuation of Corporate Liabilities as Compound

Options. Journal of Financial and Quantitative Analysis, 12:541ā“52,

1977.

[31] R. Geske and R. Johnson. The Valuation of Corporate Liabili-

ties as Compound Options: A Correction. Journal of Financial and

Quantitative Analysis, 19:231ā“2, 1977.

[32] K. Giesecke. Default and Information. Working paper, Cornell

University, 2001.

[33] K. Giesecke and L. Goldberg. Forecasting Default in the Face of

Uncertainty. Working paper, Cornell University, 2004.

[34] L. Goodman and F. Fabozzi. Collateralized Debt Obligations:

Structures and Analysis. Wiley, 2002.

[35] M. Gordy. A Comparative Anatomy of Credit Risk Models. Journal

of Banking and Finance, 24:119ā“49, 2000.

[36] G. Grimmett and D. Stirzaker. Probability and Random Pro-

cesses. Oxford University Press, 1998.

[37] G. Gupton, C. Finger, and M. Bhatia. CreditMetricsā“technical

document. Technical report, Morgan Guaranty Trust Co., 1997.

[38] D. Hamilton and L. Carty. Debt recoveries for corporate bank-

ruptcies. Technical report, Moodyā™s Investors Service, June 1999.

328 Bibliography

[39] R. Hogg and E. Tanis. Probability and Statistical Inference.

Macmillan, 1983.

[40] C. Huang and R. Litzenberger. Foundations for Financial

Economics. Prentice Hall, 1988.

[41] J. Hull. Options, Futures, and Other Derivatives. Prentice Hall, ļ¬fth

edition, 2003.

[42] J. Hull and A. White. Valuing Credit Default Swaps II: Modeling

Default Correlations. Journal of Derivatives, 8:12ā“22, 2001.

[43] P. J. Hunt and J. E. Kennedy. Financial Derivatives in Theory

and Practice. John Wiley & Sons, 2000.

[44] J. James and N. Webber. Interest Rate Modelling. Wiley, 2000.

[45] R. Jarrow, D. Lando, and S. Turnbull. A Markov Model for the

Term Structure of Credit Risk Spreads. Review of Financial Studies,

10:481ā“523, 1997.

[46] R. Jarrow and S. Turnbull. Pricing Options on Financial Securi-

ties Subject to Default Risk. Journal of Finance, 50:53ā“86, 1995.

[47] R. Jarrow and F. Yu. Counterparty Risk and the Pricing of

Defaultable Securities. Journal of Finance, 56:1765ā“99, 2001.

[48] E. Jones, S. Mason, and E. Rosenfeld. Contingent Claims Anal-

ysis of Corporate Capital Structures: An Empirical Investigation.

Journal of Finance, 39:611ā“25, 1984.

[49] S. Keenan, D. Hamilton, and A. Berthault. Historical default

rates of corporate bond issuers, 1920ā“1999. Technical report,

Moodyā™s Investors Service, January 2000.

[50] M. Kijima and K. Komoribayashi. A Markov Chain Model for

Valuing Credit Risk Derivatives. Journal of Derivatives, 6:97ā“108,

1998.

[51] D. Lando. Cox Processes and Credit-Risky Securities. Review of

Derivatives Research, 2:99ā“120, 1998.

[52] D. Lando and T. Skodeberg. Analyzing Rating Transitions and

Rating Drift with Continuous Observations. Journal of Banking and

Finance, 26:423ā“44, 2002.

[53] H. Leland. Corporate Debt Value, Bond Covenants, and Optimal

Capital Structure. Journal of Finance, 49:371ā“87, 1994.

Bibliography 329

[54] S. LeRoy and J. Werner. Principles of Financial Economics.

Cambridge University Press, 2001.

[55] D. Li. On Default Correlation: A Copula Function Approach. Journal

of Fixed Income, March: 43ā“54, 2000.

[56] F. Longstaff and E. Schwartz. A Simple Approach to Valuing

Risky Fixed and Floating Rate Debt. Journal of Finance, 50:789ā“819,

1995.

[57] D. Lucas. Default Correlation and Credit Analysis. Journal of Fixed

Income, March: 76ā“87, 1995.

[58] D. Madan and H. Unal. A Two-factor Hazard Rate Model for Pric-

ing Risky Debt and the Term Structure of Credit Spreads. Journal

of Financial and Quantitative Analysis, 35:43ā“65, 2000.

[59] R. Merton. On the Pricing of Corporate Debt: The Risk Structure

of Interest Rates. Journal of Finance, 29:449ā“70, 1974.

[60] T. Mikosch. Elementary Stochastic Calculus with Finance in View.

World Scientiļ¬c, 1999.

[61] M. Musiela and M. Rutkowski. Martingale Methods in Financial

Modeling. Springer Verlag, 1998.

[62] S. Neftci. An Introduction to the Mathematics of Financial Deriva-

tives. Academic Press, 2002.

[63] D. Oā™Kane. Credit derivatives explained: Market, products, and

regulations. Technical report, Lehman Brothers, March 2001.

[64] D. Oā™Kane and L. Schlogl. Modeling credit: Theory and practice.

Technical report, Lehman Brothers, February 2001.

[65] G. Pan. Equity to Credit Pricing. Risk, November: 107ā“10, 2001.

[66] N. Patel. Credit Derivatives Survey: Flow Business Booms. Risk

Magazine, February: 20ā“3, 2003.

[67] Credit Suisse Financial Products. Creditrisk+, a credit risk

management framework. Technical report, Credit Suisse Financial

Products, 1997.

[68] D. Rule. The Credit Derivatives Market: Its Development and Pos-

sible Implications for Financial Stability. Financial Stability Review

(Bank of England), June: 117ā“40, 2001.

330 Bibliography

[69] O. Sarig and A. Warga. Some Empirical Estimates of the Risk

Structure of Interest Rates. Journal of Finance, XLIV:1351ā“60, 1989.

[70] P. Schonbucher. Term Structure Modeling of Defaultable Bonds.

Review of Derivatives Research, 2:161ā“92, 1998.

[71] P. Schonbucher. A Tree Implementation of a Credit Spread Model

for Credit Derivatives. Working paper, University of Bonn, 1999.

[72] D. Shimko, N. Tejima, and D. Van Deventer. The Pricing of

Risky Debt when Interest Rates are Stochastic. Journal of Fixed

Income, 3:58ā“65, 1993.

[73] O. Vasicek. Probability of Loss on Loan Portfolio. KMV Corpora-

tion, 1987.

[74] P. Wilmott, S. Howison, and J. Dewynne. The Mathematics of

Financial Derivatives: A Student Introduction. Cambridge University

Press, 1999.

[75] Y. Yoshizawa. Moodyā™s approach to rating synthetic CDOs. Tech-

nical report, Moodyā™s Investors Service, July 2003.

[76] C. Zhou. A Jump-Diļ¬usion Approach to Modeling Credit Risk and

Valuing Defaultable Securities. Finance and Economics Discussion

Series 1997-15, Board of Governors of the Federal Reserve System,

1997.

Index

asset correlation market participants 23ā“5

basket default swaps 247ā“8 OECD 31ā“5

FTD 247ā“8 regulatory capital

PDS 255ā“6 management 31ā“5

asset swaps 53ā“65 Basel II Capital Accord,

bond piece 59 regulatory issues 300ā“2

credit risk 57ā“8 Basel II Risk Weights and

diagram 55 Credit Derivatives,

embedded interest rate regulatory issues 302ā“3

swap 59ā“62 basket default swaps 99ā“106

cf. FRNs 62ā“5 alternative approaches 248

hedge funds 57 asset correlation 247ā“8

market size 20

basic features 239ā“40

mechanisms 54ā“6

extensions 248

cf. par ļ¬‚oaters 62ā“5

FTD 99ā“106, 240ā“8

uses 56ā“8

mechanisms 99ā“101

valuation considerations

ńņš. 10 |