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We mentioned in Section 3.2 that bank regulatory considerations spelled
out in the 1988 Basel Bank Capital Accord”also known as the Basel I
Accord”played an important role in the early days of the credit derivatives
markets. In particular, banks™ use of credit derivatives was, at least initially,
signi¬cantly motivated by the desire to better align the notion of regulatory
capital”the share of risk-adjusted assets that bank regulators require to be
set aside to support risk-taking activities”with that of economic capital”
a prudent bank™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 contract”maturity of the CDS vs. that of
the loan, de¬nition of credit events, etc.”e¬ectively granted it appropriate
protection against default-related losses on the loans.
The interplay between bank regulatory issues and banks™ 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, banks™ other (non-regulatory-
driven) uses of credit derivatives”such as applications to risk management
and portfolio diversi¬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 e¬ort culminated with the
New Basel Capital Accord”also known as the Basel II Accord. The Accord
was ¬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 ¬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 accord™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 re¬‚ective of these orga-
nizations™ somewhat unique risk pro¬les. We say “unique” 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 di¬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 banks™ 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 Poor™s and
Moody™s”the so-called standardized approach”or on two methods derived
from banks™ own internal ratings systems”the so-called internal ratings-
based approaches”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
speci¬ed in the Basel I Accord”see Table 3.1 in Chapter 3 for a comparison.
In particular, whereas the original accord made no distinction among cor-
porate borrowers, and di¬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 banks™ 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 Speci¬ed in the Basle II Accord for Selected Obligorsa
(percent)


Obligor Credit Rating
AAA A+ BBB+ BB+ Below B’ Unrated
to AA’ to A’ to BBB’ to B’

Sovereign 0 20 50 100 150 100
Bankb 20 50 50 100 150 50
Corporate 20 50 100 100 150 100
a As speci¬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 e¬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, banks™ 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 banks™ 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 obligor”the BBB-rated entity”
with another”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 o¬setting, impli-
cations for future developments in the credit derivatives market. On one
hand, banks™ 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 banks™ 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 o¬sets a bank™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 Settlements™ 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 accord™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:
• Australia: Australian Prudential Regulation Authority
(www.apra.gov.au);
• Canada: O¬ce of the Superintendent of Financial Institutions
(www.os¬-bsif.gc.ca);
• France: Commission Bancaire (www.commission-bancaire.org);
• Germany: Budesanstalt fur Finanzdienstleistungsaufsicht
(www.ba¬n.de);
• Ireland: Central Bank of Ireland and Financial Services Authority
(www.centralbank.ie);
• Italy: Banca D™Italia (www.bancaditalia.it);

• Japan: Bank of Japan (www.boj.or.jp);
• Sweden: Swedish Financial Supervisory Authority (www.¬.se);
• United Kingdom: Financial Services Authority (www.fsa.gov.uk);

• United States: Board of Governors of the Federal Reserve Sys-
tem (www.federalreserve.gov), O¬ce of the Comptroller of the
Currency (www.occ.treas.gov), Federal Deposit Insurance Cor-
poration (www.fdic.gov), and the O¬ce of Thrift Supervision
(www.ots.treas.gov).
Appendix A
Basic Concepts from Bond Math




We brie¬‚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 ’ .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
de¬ned as the constant annual rate at which the bond™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 ’ 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 ’ 1]
Y1

and, as j tends to in¬nity, we obtain the following expression for the yield
to maturity:
(∞)
= ’ 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
’1
(j)
Yn =j (A.2)
price

and the continuously compounded yield is:

price
Yn = ’ log
(∞)
/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 = e’nYn
(∞)
[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 today™s dollars.
For instance, the present value of $1 to be received in one-year™s time is,
by de¬nition, simply today™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 ¬‚ows of a corporate security using
prices of zero-coupon bonds derived from the security issuer™s yield curve.


A.4 Coupon-paying Bonds
Consider a n-year bond that pays a ¬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
+ ’(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 issuer™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 “par” 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 (today™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-year™s time, is simply the price of the ¬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 ’ 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
1’ 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:
n’1
1’ 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 today™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

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

• 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 ¬‚ow at time t. Subsequent cash ¬‚ows are as follows:
• Time T1 : Pay $1 to the buyer of the T1 -bond sold at t;

• 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 ) ’ 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 de¬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 ’ T1 = .5 year, which corresponds to the following expression
for F (t, T1 , T2 ):

1 Z(t, T1 )
’1
F (t, T1 , T2 ) = (A.9)
T2 ’ 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 ’T1 ) =
Z(t, T2 )

Thus:
log(Z(t, T1 )) ’ log(Z(t, T2 ))
f (t, T1 , T2 ) = (A.10)
T2 ’ T1


A.7 Forward Interest Rates and Bond Prices
The rate f (t, T ) such that

log(Z(t, T )) ’ log(Z(t, T + ∆T )) ‚ log(Z(t, T ))
f (t, T ) ≡ lim =’
∆T ’0 ∆T ‚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 in¬nitesimal time increment.
Integrating both sides of equation (A.11), we obtain:

Z(t, T ) = e’
T
f (t,v)dv
(A.12)
t




Given the de¬nition of the continuously compounded yield to maturity
as the rate R(t, T ) such that Z(t, T ) = e’R(t,T )(T ’t) :

T
f (t, v)dv
t
R(t, T ) = (A.13)
T ’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
de¬ned as

F (x) = Prob[X ¤ x] (B.1)

F (x) is also commonly called the distribution function of X. We highlight
two of its basic properties below:
• Because F (x) is a probability: 0 ¤ F (x) ¤ 1;
• F (x) is a non-decreasing function of x, i.e., if x1 < x2 then if X ¤ x1
we also have X ¤ x2 , and thus:

F (x1 ) = Prob[X ¤ x1 ] ¤ Prob[X ¤ 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 ¤x


where the sum is computed over all xi such that xi ¤ 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 ) ≥ 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)
’∞

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) ≥ 0 for all real x

f (x)dx = 1
’∞

In the light of (B.4), it can be shown that:
b
Prob[a ¤ X ¤ b] = F (b) ’ 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 ¤ b] = Prob[a ¤ X < b] = Prob[a ¤ X ¤ b]

We also highlight the following useful relationship:

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

where the approximation error is negligible for su¬ciently small values of
dx. Indeed, given (B.4), from classic calculus we know that, for values of x
for which ‚F (x) exists:
‚x

F (x + ds) ’ F (x) ‚F (x)
lim = = f (x) (B.8)
dx ‚x
dx’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 de¬ned as
n
p(xi )(xi ’ E[X])2
V [X] = (B.10)
i=1

If X is a continuous random variable we have

E[X] = sf (s)ds (B.11)
’∞

and

(s ’ E[X])2 f (s)ds
V [X] = (B.12)
’∞
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 outcomes”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 ω, the probability function of X
can be written as

p(x) = ω x (1 ’ ω)1’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
xω x (1 ’ ω)1’x = (1)ω + (0)(1 ’ ω) = ω
E[X] =
x=0

1
(x ’ ω)2 ω x (1 ’ ω)1’x = ω(1 ’ ω)
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 ω.
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!
ω y (1 ’ ω)n’y
b(y; n, ω) = (B.14)
y!(n ’ y)!

where y denotes the possible values of Y ”y = 0, 1, 2, . . . , n”and Y is said
to be binomially distributed.
The c.d.f. of the binomial distribution is
y
B(y; n, ω) ≡ Prob[Y ¤ y] = b(s; n, ω) 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
independence”for instance, a default by one company changes the default
probabilities of the others”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 “arrival” 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 »
per year, with » > 0.
From a credit risk modeling perspective, in most cases all that we care
about is the ¬rst occurrence of default. Nonetheless, in some applications”
such as when modeling defaults and corporate reorganizations”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 “the
beginning of time” 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 satis¬ed”
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 »s;

3. the probability of more than one default during a su¬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) ≡ Prob[x defaults in a one-year interval]

»x e’»
= 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 »:

»x e’»
E[X] = x =» (B.17)
x!
x=0

The variance of X can also be shown to be ».
More generally, let us change the reference time interval to be s years”
1
where s > 0”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 », then the mean arrival of defaults in a s-year interval
is »s. As a result, we can write:

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

(»s)x e’»s
= for x = 0, 1, 2, . . . (B.18)
x!
provided, of course, conditions 1 through 3 are satis¬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) ≡ Prob[0 defaults in a s-year interval]

= e’»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) = »s e’»s

which, consistent with condition 2, is approximately »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 ¬rst default by a given
reference entity. Let „ denote the time of ¬rst default, which can be thought
¯
of as a continuous random variable with c.d.f. G(s). (Alternatively, „ can
also be thought of as the “waiting time” until the ¬rst default, as seen from
¯
time 0.) Clearly, for s < 0, G(s) = 0, given that a waiting time cannot be
negative. For s ≥ 0 we can write

G(s) ≡ Prob[„ ¤ s] = 1 ’ Prob[„ > s]
¯

but note that Prob[„ > 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 ’ e’»s
¯ (B.20)

which can be thought of as the unconditional probability of a ¬rst default
by time s.
¯
The unconditional p.d.f. of „ , de¬ned as ‚ G(s) , can be written as:
‚s


g (s) = »e’»s
¯ (B.21)

Readers with some familiarity with statistics will recognize g (s) as the p.d.f.
¯
1
of an exponentially distributed random variable with mean » and variance
1
»2 . Thus, the time of ¬rst default is exponentially distributed when defaults
occur according to a Poisson process.
The conditional probability of ¬rst default by time s, given no default
through time t, for s ≥ t, can be written as

Gt (s) ≡ Prob[„ ¤ s|„ > t]

= 1 ’ Prob[„ > s|„ > t]

Prob[„ > s]
=1’
Prob[„ > t]

= 1 ’ e’»(s’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 „ , given no default through time t, is the
function gt (s) such that

gt (s) = e’»(s’t) » (B.23)

which is simply ‚Gt (s) .
‚s
Armed with (B.23), we can, for instance, compute the expected time of
the ¬rst default, conditional on no default through time t:

E[„ |„ > t] = s gt (s)ds (B.24)
t

1
which can be shown to be equal to t + » .



B.8 The Normal Distribution
If X is a continuous random variable that is normally distributed with
mean µ and variance σ 2 , its p.d.f. is

1
e’[(x’µ)/σ] /2
2
f (x; µ, σ) = √ (B.25)
2πσ

and the normal c.d.f. is accordingly given by
x
1
e’[(s’µ)/σ] /2 ds
2

F (x) = (B.26)
2πσ
’∞


If we de¬ne Y ≡ X’µ , then it can be shown that Y is normally dis-
σ
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) = √ e’y /2
2
(B.27)


The standard normal c.d.f. is
y
1
√ e’s /2 ds
2
N (y) = (B.28)

’∞

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

x’µ
F (x; µ, σ) = N = N (y) (B.29)
σ

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(’y) (B.30)

which can be shown mathematically, or veri¬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 (’y) = 1 ’ 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 µ and variance σ 2 , then Y ≡ eX
is lognormally distributed. The p.d.f. of Y is

1
e’(log(y)’µ) /(2σ )
2 2
f (y) = √ for y > 0 (B.32)
2πσy

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 µ and variance σ 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) ≡ Prob[Y ¤ y] = Prob[log(Y ) ¤ 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 ¤ log(y)], and thus:

log(y) ’ µ
F (y) = N (B.33)
σ

The mean and variance of Y are:
2
E[Y ] = eµ+σ /2
(B.34)
2 2
V [Y ] = e2(µ+σ ) ’ e2µ+σ (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 ¬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 de¬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 ¤ x1 , X2 ¤ x2 , . . . , Xn ¤ xn ] (B.37)

i.e., F (x1 , . . . , xn ) is the probability that X1 ¤ x1 , and X2 ¤ x2 , . . . , and
Xn ¤ 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
’∞ ’∞ ’∞
(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
de¬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 di¬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 µ1
and µ2 and variances σ1 and σ2 , respectively, and ρ is the correlation coef-
¬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 , ρ) =
1 ’ ρ2
2πσ1 σ2
2 2
x1 ’µ1 x1 ’µ1 x2 ’µ2 x2 ’µ2
’ 2(1’ρ2 ) ’2ρ
1
+
—e
σ1 σ1 σ2 σ2
(B.41)

and the bivariate normal c.d.f. has the usual de¬nition
x2 x1
N2 (x1 , x2 , ρ) = n2 (s1 , s2 , ρ)ds1 ds2 (B.42)
’∞ ’∞

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.
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Index




asset correlation market participants 23“5
basket default swaps 247“8 OECD 31“5
FTD 247“8 regulatory capital
PDS 255“6 management 31“5
asset swaps 53“65 Basel II Capital Accord,
bond piece 59 regulatory issues 300“2
credit risk 57“8 Basel II Risk Weights and
diagram 55 Credit Derivatives,
embedded interest rate regulatory issues 302“3
swap 59“62 basket default swaps 99“106
cf. FRNs 62“5 alternative approaches 248
hedge funds 57 asset correlation 247“8
market size 20
basic features 239“40
mechanisms 54“6
extensions 248
cf. par ¬‚oaters 62“5
FTD 99“106, 240“8
uses 56“8
mechanisms 99“101
valuation considerations

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