Selection of Research Material relating to RiskMetrics Group CDO Manager

Selection of Research Material relating to
RiskMetrics Group CDO Manager
E: europe@riskmetrics.com
W: www.riskmetrics.com
T: 020 7842 0260
F: 020 7842 0269

CONTENTS
1. Introductory Technical Note on the CDO Manager Software.
2. A comparison of stochastic default rate models, Christopher C.
Finger. RiskMetrics Group Working Paper Number 00-02
3. On Default Correlation: A Copula Function Approach, David X.
Li. RiskMetrics Group Working Paper Number 99-07
4. The Valuation of the ith-to-Default Basket Credit Derivatives,
David X. Li. RiskMetrics Group Working Paper.
5. Worst Loss Analysis of BISTRO Reference Portfolio, Toru
Tanaka, Sheikh Pancham, Tamunoye Alazigha, Fuji Bank.
RiskMetrics Group CreditMetrics Monitor April 1999.
6. The Valuation of Basket Credit Derivatives, David X. Li.
RiskMetrics Group CreditMetrics Monitor April 1999.
7. Conditional Approaches for CreditMetrics Portfolio Distributions,
Christopher C. Finger. RiskMetrics Group CreditMetrics Monitor
April 1999.

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The RiskMetrics Group
Working Paper Number 00-02
A comparison of stochastic default rate models
Christopher C. Finger
This draft: August 2000
First draft: July 2000
44Wall St. chris.finger@riskmetrics.com
NewYork, NY 10005 www.riskmetrics.com

A comparison of stochastic default rate models
Christopher C. Finger
August 2000
Abstract
For single horizon models of defaults in a portfolio, the effect of model and distribution choice on the
model results is well understood. Collateralized Debt Obligations in particular have sparked interest in
default models over multiple horizons. For these, however, there has been little research, and there is little
understanding of the impact of various model assumptions. In this article, we investigate four approaches
to multiple horizon modeling of defaults in a portfolio. We calibrate the four models to the same set of
input data (average defaults and a single period correlation parameter), and examine the resulting default
distributions. The differences we observe can be attributed to the model structures, and to some extent,
to the choice of distributions that drive the models. Our results show a significant disparity. In the single
period case, studies have concluded that when calibrated to the same first and second order information,
the various models do not produce vastly different conclusions. Here, the issue of model choice is much
more important, and any analysis of structures over multiple horizons should bear this in mind.
Keywords: Credit risk, default rate, collateralized debt obligations

1 Introduction
In recent years, models of defaults in a portfolio context have been well studied. Three separate
approaches (CreditMetrics, CreditRisk+, and CreditPortfolioView1) were made public in 1997.
Subsequently, researchers2 have examined the mathematical structure of the various models. Each
of these studies has revealed that it is possible to calibrate the models to each other and that the
differences between the models lie in subtle choices of the driving distributions and in the data
sources one would naturally use to feed the models.
Common to all of these models, and to the subsequent examinations thereof, is the fact that the
models describe only a single period. In otherwords, the models describe, for a specific risk horizon,
whether each asset of interest defaults within the horizon. The timing of defaults within the risk
horizon is not considered, nor is the possibility of defaults beyond the horizon. This is not a flaw
of the current models, but rather an indication of their genesis as approaches to risk management
and capital allocation for a fixed portfolio.
Not entirely by chance, the development of portfolio models for credit risk management has coin-cided
with an explosion in issuance of Collateralized Debt Obligations (CDO’s). The performance
of a CDO structure depends on the default behavior of a pool of assets. Significantly, the depen-dence
is not just on whether the assets default over the life of the structure, but also on when the
defaults occur. Thus, while an application of the existing models can give a cursory view of the
structure (by describing, for instance, the distribution of the number of assets that will default over
the structure’s life), a more rigorous analysis requires a model of the timing of defaults.
In this paper, we will survey a number of extensions of the standard single-period models that allow
for a treatment of default timing over longer horizons. We will examine two extensions of the Cred-itMetrics
approach, one that models only defaults over time and a second that effectively accounts
1SeeWilson (1997).
2See Finger (1998), Gordy (2000), and Kolyoglu and Hickman (1998).
1

for rating migrations. In addition, we will examine the copula function approach introduced by Li
(1999 and 2000), as well as a simple version of the stochastic intensity model applied by Duffie
and Garleanu (1998).
We will seek to investigate the differences in the four approaches that arise from model – rather than
data – differences. Thus, we will suppose that we begin with satisfactory estimates of expected
default rates over time, and of the correlation of default events over one period. Higher order
information, such as the correlation of defaults in subsequent periods or the joint behavior of three
or more assets, will be driven by the structure of the models. The analysis of the models will
then illuminate the range of results that can arise given the same initial data. Nagpal and Bahar
(1999) adopt a similar approach in the single horizon context, investigating the range of possible
full distributions that can be calibrated to first and second order default statistics.
In the following section, we present terminology and notation to be used throughout. We proceed to
detail the four models. Finally,we present two comparison exercises: in the first, we use closed form
results to analyze default rate volatilities and conditional default probabilities, while in the second,
we implement Monte Carlo simulations in order to investigate the full distribution of realized default
rates.
2 Notation and terminology
In order to compare the properties of the four models, we will consider a large homogeneous pool
of assets. By homogeneous, we mean that each asset has the same probability of default (first order
statistics) at every time we consider; further, each pair of assets has the same joint probability of
default (second order statistics) at every time.
To describe the first order statistics of the pool, we specify the cumulative default probability qk
– the probability that a given asset defaults in the next k years – for k D 1; 2; : : : T , where T is
the maximum horizon we consider. Equivalently, we may specify the marginal default probability
2

pk – the probability that a given asset defaults in year k. Clearly, cumulative and marginal default
probabilities are related through
qk D qk−1 C pk; for k D 2; : : : ; T: (1)
It is important to distinguish a third equivalent specification, that of conditional default probabilities.
The conditional default probability in year k is defined as the conditional probability that an asset
defaults in year k, given that the asset has survived (that is, has not defaulted) in the first k−1 years.
This probability is given by pk=.1 − qk−1/.
Finally, to describe the second order statistics of the pool, we specify the joint cumulative default
probability qj;k – the probability that for a given pair of assets, the first asset defaults sometime in
the first j years and the second defaults sometime in the first k years – or equivalently, the joint
marginal default probability pj;k – the probability that the first asset defaults in year j and the
second defaults in year k. These two notions are related through
qj;k D qj−1;k−1 C
Xj−1
iD1
pi;k C
Xk−1
iD1
pj;i C pj;k; for j; k D 2; : : : ; T: (2)
In practice, it is possible to obtain first order statistics for relatively long horizons, either by observing
market prices of risky debt and calibrating cumulative default probabilities as in Duffie and Singleton
(1999), or by taking historical cumulative default experience from a study such asKeenan et al (2000)
or Standard Poor’s (2000). Less information is available for second order statistics, however, and
therefore we will assume that we can obtain the joint default probability for the first year (p1;1)3,
but not any of the joint default probabilities for subsequent years. Thus, our exercise will be to
calibrate each of the four models to fixed values of q1; q2; : : : qT and p1;1, and then to compare the
higher order statistics implied by the models.
The model comparison can be a simple task of comparing values of p1;2, p2;2, q2;2, and so on.
However, to make the comparisons a bit more tangible, we will consider the distributions of realized
3This is a reasonable supposition, since all of the single period models mentioned previously essentially require p1;1 as an input.
3

default rates. The term default rate is often used loosely in the literature, without a clear notion
of whether default rate is synonymous with default probability, or rather is itself a random variable.
To be clear, in this article, default rate is a random variable equal to the proportion of assets in
a portfolio that default. For instance, if the random variable X.k/
i is equal to one if the ith asset
defaults in year k, then the year k default rate is equal to
1
n
Xn
iD1
X.k/
i : (3)
For our homogeneous portfolio, the mean year k default rate is simply pk, the marginal default
probability for year k. Furthermore, the standard deviation of the year k default rate (which we will
refer to as the year k default rate volatility) is
q
pk;k − p2
k
C .pk − pk;k/=n: (4)
Of interest to us is the large portfolio limit (that is, n ! 1) of this quantity, normalized by the
default probability. We will refer to this as the normalized year k default volatility, which is given
by
q
pk;k − p2
k
pk
: (5)
Additionally, we will examine the normalized cumulative year k default volatility, which is defined
similarly to the above, with the exception that the default rate is computed over the first k years
rather than year k only. The normalized cumulative default volatility is given by
q
qk;k − q2
k
qk
: (6)
Finally, we will use 8 to denote the standard normal cumulative distribution function. In the
bivariate setting, we will use 82.z1; z2I / to indicate the probability that Z1 z1 and Z2 z2,
where Z1 and Z2 are standard normal random variables with correlation .
In the following four sections, we describe the models to be considered, and discuss in detail the
calibration to our initial data.
4

3 Discrete CreditMetrics extension
In its simplest form, the single period CreditMetrics model, calibrated for our homogeneous port-folio,
can be stated as follows:
(i) Define a default threshold such that 8./ D p1.
(ii) To each asset i, assign a standard normal random variable Z.i/, where the correlation between
distinct Z.i/ and Z.j / is equal to , such that
82.; I / D p1;1: (7)
(iii) Asset i defaults in year 1 if Z.i/ .
The simplest extension of this model to multiple horizons is to simply repeat the one period model.
We then have default thresholds 1; 2; : : : ; T corresponding to each period. For the first period,
we assign standard normal random variables Z.i/
1 to each asset as above, and asset i defaults in the
first period if Z.i/
1 1. For assets that survive the first period, we assign a second set of standard
normal random variables Z.i/
2 , such that the correlation between distinct Z.i/
2 and Z
.j /
2 is but the
variables from one period to the next are independent. Asset i then defaults in the second period
if Z.i/
1 1 (it survives the first period) and Z.i/
2 2. The extension to subsequent periods
should be clear. In the end, the model is specified by the default thresholds 1; 2; : : : ; T and the
correlation parameter .
To calibrate this model to our cumulative default probabilities q1; q2; : : : ; qT and joint default
probability, we begin by setting the first period default threshold:
−1.q1/: (8)
1 D 8
For subsequent periods, we set k such that the probability that Z.k/
i k is equal to the conditional
default probability for period k:
−1
k D 8

qk − qk−1
1 − qk−1

: (9)
5

We complete the calibration by choosing to satisfy (7), with replaced by 1.
The joint default probabilities and default volatilities are easily obtained in this context. For instance,
the marginal year two joint default probability is given by (for distinct i and j ):
p2;2 D P
n
Z.i/
1 1 Z
.j /
1 1 Z.i/
2 2 Z
.j /
2 2
o
D P
n
Z.i/
1 1 Z
.j /
1 1
o
P
n
Z.i/
2 2 Z
.j /
2 2
o
D .1 − 2p1 C p1;1/ 82.2; 2I /: (10)
Similarly, the probability that asset i defaults in the first period, and asset j in the second period is
p1;2 D P
n
Z.i/
1 1 Z
.j /
1 1 Z
.j /
2 2
o
D .p1 − p1;1/ q2 − p1
1 − p1
: (11)
It is then possible to obtain q2;2 using (2) and the default volatilities using (5) and (6).
4 Diffusion-driven CreditMetrics extension
By construction, the discrete CreditMetrics extension above does not allow for any correlation of
default rates through time. For instance, if a high default rate is realized in the first period, this has
no bearing on the default rate in the second period, since the default drivers for the second period
(the Z.i/
2 above) are independent of the default drivers for the first. Intuitively, we would not expect
this behavior from the market. If a high default rate occurs in one period, then it is likely that those
obligors that did not default would have generally decreased in credit quality. The impact would
then be that the default rate for the second period would also have a tendency to be high.
In order to capture this behavior, we introduce a CreditMetrics extension where defaults in con-secutive
periods are not driven by independent random variables, but rather by a single diffusion
process. Our diffusion-driven CreditMetrics extension is described by:
(i) Define default thresholds 1; 2; : : : ; T for each period.
6

(ii) To each obligor, assign a standard Wiener process W.i/, with W.i/
0
D 0, where the instanta-neous
correlation between distinct W.i/ and W.j / is .4
(iii) Obligor i defaults in the first year if W.i/
1 1.
(iv) For k 1, obligor i defaults in year k if it survives the first k − 1 years (that is, W.i/
1
1; : : : ;W.i/
k−1 k−1) and W.i/
k k.
Note that this approach allows for the behavior mentioned above. If the default rate is high in the
first year, this is because many of the Wiener processes have fallen below the threshold 1. The
Wiener processes for non-defaulting obligors will have generally trended downward as well, since
all of the Wiener processes are correlated. This implies a greater likelihood of a high number of
defaults in the second year. In effect, then, this approach introduces a notion of credit migration.
Cases where the Wiener process trends downward but does not cross the default threshold can be
thought of as downgrades, while cases where the process trends upward are essentially upgrades.
To calibrate the first threshold 1, we observe that
P
n
W.i/
1 1
o
D 8.1/; (12)
and thus that 1 is given by (8). For the second threshold, we require that the probability that an
obligor defaults in year two is equal to p2:
P
n
W.i/
1 1 W.i/
2 2
o
D p2: (13)
Since W.i/ is a Wiener process, we know that the standard deviation of W.i/
t is
p
t and that for
s t, the correlation between W.i/
s and W.i/
t is
p
s=t. Thus, given 1, we find the value of 2 that
satisfies
p
2/ − 82.1; 2=
8.2=
p
2I
p
1=2/ D p2: (14)
4Technically, the cross variation process for W.i/ and W.j / is dt .
7

For the kth period, given 1; : : : ; k−1, we calibrate k by solving
P
n
W.i/
1 1 : : : W.i/
k−1 k−1 W.i/
k k
o
D pk; (15)
again utilizing the properties of theWiener processW.i/ to compute the probability on the left hand
side.
We complete the calibration by finding such that the year one joint default probability is p1;1:
P
n
W.i/
1 1 W
.j /
1 1
o
D p1;1: (16)
Since W.i/
.j /
1 each follow a standard normal distribution, and have a correlation of , the
1 and W
solution for here is identical to that of the previous section.
With the calibration complete, it is a simple task to compute the joint default probabilities. For
instance, the joint year two default probability is given by
p2;2 D P
n
W.i/
.j /
1 1 W.i/
1 1 W
.j /
2 2
2 2 W
o
; (17)
where we use the fact that fW.i/
.j /
1 ;W.i/
1 ;W
.j /
2
2 ;W
g follow a multivariate normal distribution with
covariance
CovfW.i/
.j /
1 ;W.i/
1 ;W
.j /
2
2 ;W
g D
0
BB@
1 1
1 1
1 2 2
1 2 2
1
CCA
: (18)
5 Copula functions
A drawback of both the CreditMetrics extensions above is that in a Monte Carlo setting, they require
a stepwise simulation approach. In other words, we must simulate the pool of assets over the first
year, tabulate the ones that default, then simulate the remaining assets over the second year, and so
on. Li (1999 and 2000) introduces an approach wherein it is possible to simulate the default times
directly, thus avoiding the need to simulate each period individually.
The normal copula function approach is as follows:
8

(i) Specify the cumulative default time distribution F, such that F.t/ gives the probability that a
given asset defaults prior to time t .
(ii) Assign a standard normal random variable Z.i/ to each asset, where the correlation between
distinct Z.i/ and Z.j / is .
(iii) Obtain the default time i for asset i through
i D F
−1.8.Z.i///: (19)
Since we are concerned here only with the year in which an asset defaults, and not the precise
timing within the year, we will consider a discrete version of the copula approach:
(i) Specify the cumulative default probabilities q1; q2; : : : ; qT as in Section 2.
(ii) For k D 1; : : : ; T compute the threshold k D 8
−1.qk/. Clearly, 1 2 : : : T .
Define 0 D −1.
(iii) Assign Z.i/ to each asset as above.
(iv) Asset i defaults in year k if k−1 Z.i/ k.
The calibration to the cumulative default probabilities is already given. Further, it is easy to observe5
that the correlation parameter is calibrated exactly as in the previous two sections.
The joint default probabilities are perhaps simplest to obtain for this approach. For example, the
joint cumulative default probability qk;l is given by
qk;l D P
n
Z.i/ k Z.j / l
o
D 82.k; lI /: (20)
5Details are presented in Li (1999) and Li (2000).
9

6 Stochastic default intensity
6.1 Description of the model
The approaches of the three previous sections can all be thought of as extensions of the single
period CreditMetrics framework. Each approach relies on standard normal random variables to
drive defaults, and calibrates thresholds for these variables. Furthermore, it is easy to see that over
the first period, the three approaches are identical; they only differ in their behavior over multiple
periods.
Our fourth model takes a different approach to the construction of correlated defaults over time, and
can be thought of as an extension of the single period CreditRisk+ framework. In the CreditRisk+
model, correlations between default events are constructed through the assets’ dependence on a
common default probability, which itself is a random variable.6 Importantly, given the realization
of the default probability, defaults are conditionally independent. The volatility of the common
default probability is in effect the correlation parameter for this model; a higher default volatility
induces stronger correlations, while a zero volatility produces independent defaults.7
The natural extension of the CreditRisk+ framework to continuous time is the stochastic intensity
approach presented in Duffie and Garleanu (1998) and Duffie and Singleton (1999). Intuitively, the
stochastic intensity model stipulates that in a given small time interval, assets default independently,
with probability proportional to a common default intensity.8 In the next time interval, the intensity
changes, and defaults are once again independent, but with the default probability proportional to
the new intensity level. The evolution of the intensity is described through a stochastic process. In
practice, since the intensity must remain positive, it is common to apply similar stochastic processes
as are utilized in models of interest rates.
6More precisely, assets may depend on different default probabilities, each of which are correlated.
7See Finger (1998), Gordy (2000), and Kolyoglu and Hickman (1998) for further discussion.
8As with our description of the CreditRisk+ model, this is a simplification. The Duffie-Garleanu framework provides for an
intensity process for each asset, with the processes being correlated.
10

For our purposes, we will model a single intensity process h. Conditional on h, the default time
for each asset is then the first arrival of a Poisson process with arrival rate given by h. The Poisson
processes driving the defaults for distinct assets are independent, meaning that given a realization
of the intensity process h, defaults are independent. The Poisson process framework implies that
given h, the probability that a given asset survives until time t is
exp

−
Z
t
0
du hu

: (21)
Further, because defaults are conditionally independent, the conditional probability, given h, that
two assets both survive until time t is
exp

−2
Z
t
0
du hu

: (22)
The unconditional survival probabilities are given by expectations over the process h, so that in
particular, the survival probability for a single asset is given by
1 − qt D Eexp

−
Z
t
0
du hu

: (23)
For the intensity process, we assume that h evolves according to the stochastic differential equation
dht D −.ht − Nh
k/dt C
p
htdWt ; (24)
where W is a Wiener process and Nh
k is the level to which the process trends during year k. (That
is, the mean reversion is toward Nh
1 for t 1, toward Nh
2 for 1 t 2, etc.) Let h0 D Nh
1. Note
Nh
that this is essentially the model for the instantaneous discount rate used in the Cox-Ingersoll-Ross
interest rate model. Note also that in Duffie-Garleanu, there is a jump component to the evolution
of h, while the level of mean reversion is constant.
In order to express the default probabilities implied by the stochastic intensity model in closed
form, we will rely on the following result from Duffie-Garleanu.9 For a process h with h0 D and
9We have changed the notation slightly from the Duffie-Garleanu result, in order to make more explicit the dependence on N h.
11

evolving according to (24) with Nh
k D Nh
for all k, we have
Et exp

−
Z
tCs
t
du hu

exp[x C yhs ] D exp

x C s.y/Nh
C

s.y/ht

; (25)
where Et denotes conditional expectation given information available at time t . The functions s
and

s are given by
s.y/ D
c
s C .a.y/c − d.y//
bcd.y/
log

c C d.y/ebs
c C d

; and (26)

s.y/ D 1 C a.y/ebs
c C d.y/ebs
; (27)
where
c D − C
p
2 C 22
2
; (28)
d.y/ D .1 − cy/
2y − C
p
. 2y − /2 − 2. 2y2 − 2y − 2/
2y2 − 2y − 2
; (29)
a.y/ D .d.y/ C c/y − 1; (30)
b D
−d.y/. C 2c/ C a.y/. 2 − c/
a.y/c − d.y/
: (31)
6.2 Calibration
Our calibration approach for this model will be to fix the mean reversion speed , solve for Nh
1 and
to match p1 and p1;1, and then to solve in turn for Nh
2; : : : ; Nh
T to match p2; : : : ; pT . To begin,
we apply (23) and (25) to obtain
p1 D 1 − exp

1.0/Nh
1 C

1.0/h0

D 1 − exp

[1.0/ C

1.0/]Nh
1

: (32)
To compute the joint probability that two obligors each survive the first year, we must take the
expectation of (22), which is essentially the same computation as above, but with the process h
replaced by 2h. We observe that the process 2h also evolves according to (24) with the same mean
reversion speed , and with Nh
k replaced by 2Nh
k and replaced by
p
2. Thus, we define the
12

s , with replaced by
p
2. We can then compute
the joint one year survival probability:
Eexp

−2
Z
t
0
du hu

D exp
h
2[O
1.0/ C O

1.0/]Nh
1
i
: (33)
Finally, since the joint survival probability is equal to 1 − 2p1 C p1;1, we have
p1;1 D 2p1 − 1 C exp
h
2[O
1.0/ C O

1.0/]Nh
1
i
: (34)
To calibrate and Nh
1 to (32) and (34), we first find the value of such that
1.0/ C O

1.0/
2.O
D log[1 − 2p1 C p1;1]
log[1 − p1]
; (35)
and then set
1 D log[1 − p1]
Nh
1.0/ C

1.0/
: (36)
Note that though the equations are lengthy, the calibration is actually quite straightforward, in that
we only are ever required to fit one parameter at a time.
In order to calibrate Nh
2, we need to obtain an expression for the two year cumulative default
probability q2. To this end, we must compute the two year survival probability
1 − q2 D Eexp

−
Z
2
0
du hu

: (37)
Since the process h does not have a constant level of mean reversion over the first two years, we
cannot apply (25) directly here. However (25) can be applied once we express the two year survival
probability as
1 − q2 D Eexp

−
Z
1
0
du hu

E1 exp

−
Z
2
1
du hu

: (38)
Now given h1, the process h evolves according to (24) from t D 1 to t D 2 with a constant mean
reversion level Nh
2, meaning we can apply (25) to the conditional expectation in (38), yielding
1 − q2 D Eexp

−
Z
1
0
du hu

exp

1.0/Nh
2 C

The same argument allows us to apply (25) again to (39), giving
1 − q2 D exp

1.0/Nh
2 C [1.

1.0//]Nh
1

: (40)
Thus, our calibration for the second year requires setting
2 D 1
Nh
1.0/

log[1 − q2] − [1.

1.0//]Nh
1

: (41)
The remaining mean reversion levels Nh
3; : : : ; Nh
T are calibrated similarly.
6.3 Joint default probabilities
The computation of joint probabilities for longer horizons is similar to (34). The joint probability
that two obligors each survive the first two years is given by
Eexp

−2
Z
2
0
du hu

: (42)
Here, we apply the same arguments as in (38) through (40) to derive
Eexp

−2
Z
2
0
du hu

D exp
h
2O
1.0/Nh
2 C 2[O
1. O

1.0//]Nh
1
i
: (43)
For the joint probability that the first obligor survives the first year and the second survives the first
two years, we must compute
Eexp

−
Z
1
0
du hu

exp

−
Z
2
0
du hu

D Eexp

−2
Z
1
0
du hu

exp

−
Z
2
1
du hu

(44)
The same reasoning yields
Eexp

−
Z
1
0
du hu

exp

−
Z
2
0
du hu

D exp
h
1.0/Nh
2 C 2[O
1. O

1.0/=2/]Nh
1
i
:
(45)
The joint default probabilities p2;2 and p1;2 then follow from (43) and (45).
14

7 Model comparisons – closed form results
Our first set of model comparisons will utilize the closed form results described in the previous
sections. We will restrict the comparisons here to the two period setting, and to second order results
(that is, default volatilities and joint probabilities for two assets); results for multiple periods and
actual distributions of default rates will be analyzed through Monte Carlo in the next section.
For our two period comparisons, we will analyze four sets of parameters: investment and speculative
grade default probabilities10, each with two correlation values. The lowand high correlation settings
will correspond to values of 10% and 40%, respectively, for the asset correlation parameter in
the first three models. For the stochastic intensity model, we will investigate two values for the
mean reversion speed . The slow setting will correspond to D 0:29, such that a random shock
to the intensity process will decay by 25% over the next year; the fast setting will correspond
to D 1:39, such that a random shock to the intensity process will decay by 75% over one year.
Calibration results are presented in Table 1.
We present the normalized year two default volatilities for each model in Figure 1. As defined in (5)
and (6), the marginal and cumulative default volatilities are the standard deviation of the marginal
and cumulative two year default rates of a large, homogeneous portfolio. As we would expect, the
default volatilities are greater in the high correlation cases than in the low correlation cases. Of the
five models tested, the stochastic intensity model with slow mean reversion seems to produce the
highest levels of default volatility, indicating that correlations in the second period tend to be higher
for this model than for the others.
It is interesting to note that of the first three models, all of which are based on the normal distribution
and default thresholds, the copula approach in all four cases has a relatively low marginal default
volatility but a relatively high cumulative default volatility. (The slow stochastic intensity model is
in fact the only other model to show a marginal volatility less than the cumulative volatility.) Note
10Taken from Exhibit 30 of Keenan et al (2000).
15

that the cumulative two year default rate is the sum of the first and second year marginal default
rates, and thus that the two year cumulative default volatility is composed of three terms: the first
and second year marginal default volatilities and the covariance between the first and second years.
Our calibration guarantees that the first year default volatilities are identical across the models.
Thus, the behavior of the copula model suggests a stronger covariance term (that is, a stronger link
between year one and year two defaults) than for either of the two CreditMetrics extensions.
To further investigate the links between default events, we examine conditional probability of a
default in the second year, given the default of another asset. To be precise, for two distinct assets i
and j , we will calculate the conditional probability that asset i defaults in year two, given that asset
j defaults in year one, normalized by the unconditional probability that asset i defaults in year two.
In terms of quantities we have already defined, this normalized conditional probability is equal to
p1;2=.p1p2/. We will also calculate the normalized conditional probability that asset i defaults in
year two, given that asset j defaults in year two, given by p2;2=p2
2. For both of these quantities, a
value of one indicates that the first asset defaulting does not affect the chance that the second asset
defaults; a value of four indicates that the second asset is four times more likely to default if the
first asset defaults than it is if we have no information about the first asset. Thus, the probability
conditional on a year two default can be interpreted as an indicator of contemporaneous correlation
of defaults, and the probability conditional on a year one default as an indicator of lagged default
correlation.
The normalized conditional probabilities under the five models are presented in Figure 2. As we
expect, there is no lagged correlation for the discrete CreditMetrics extension. Interestingly, the
copula and both stochastic intensity models often show a higher lagged than contemporaneous
correlation. While it is difficult to establish much intuition for the copula model, this phenomenon
can be rationalized in the stochastic intensity setting. For this model, any shock to the default
intensity will tend to persist longer than one year. If one asset defaults in the first year, it is most
likely due to a positive shock to the intensity process; this shock then persists into the second year,
where the other asset is more likely to default than normal. Further, shocks are more persistent for the
16

slower mean reversion, explaining why the difference in lagged and contemporaneous correlation
is more pronounced in this case. By contrast, the two CreditMetrics extensions show much higher
contemporaneous than lagged correlation; this lack of persistence in the correlation structure will
manifest itself more strongly over longer horizons.
To this point, we have calibrated the collection of models to have the same means over two periods,
and the same volatilities over one period. We have then investigated the remaining second order
statistics – the second period volatility and the correlation between the first and second periods – that
depend on the particular models. In the next section, we will extend the analysis on two fronts: first,
we will investigate more horizons in order to examine the effects of lagged and contemporaneous
correlations over longer times; second, we will investigate the entire distribution of portfolio defaults
rather than just the second order moments.
8 Model comparisons – simulation results
In this section, we perform Monte Carlo simulations for the five models investigated previously.
In each case, we begin with a homogeneous portfolio of one hundred speculative grade bonds. We
calibrate the model to the cumulative default probabilities in Table 2 and to the two correlation
settings from the previous section. Over 1,000 trials, we simulate the number of bonds that default
within each year, up to a final horizon of six years.11
The simulation procedures are straightforward for the two CreditMetrics extensions and the copula
approach. For the stochastic intensity framework, we simulate the evolution of the intensity process
according to (24). This requires a discretization of (24):
htC1t −.ht − Nh
k/1t C
p
ht
p
1t; (46)
11As we have pointed out before, it is possible to simulate continuous default times under the copula and stochastic intensity
frameworks. In order to compare with the two CreditMetrics extensions, we restrict the analysis to annual buckets.
17

where is a standard normal random variable.12 Given the intensity process path for a particular
scenario, we then compute the conditional survival probability for each annual period as in (21). Fi-nally,
we generate defaults by drawing independent binomial random variables with the appropriate
probability.
The simulation time for the five models is a direct result of the number of timesteps needed. The
copula model simulates the default times directly, and is therefore the fastest. The two CreditMetrics
models require only annual timesteps, and require roughly 50% more runtime than the copula model.
For the stochastic intensity model, the need to simulate over many timesteps produces a runtime
over one hundred times greater than the simpler models.
We first examine default rate volatilities over the six horizons. As in the previous section, we
consider the normalized cumulative default rate volatility. For year k, this is the standard deviation
of the number of defaults that occur in years one through k, divided by the expected number of
defaults in that period. This is essentially the quantity defined in (6), with the exception that
here we consider a finite portfolio. The default volatilities from our simulations are presented
in Figure 3. Our calibration guarantees that the first year default volatilities are essentially the
same. The second year results are similar to those in Figure 1, with slightly higher volatility for
the slow stochastic intensity model, and slightly lower volatility for the discrete CreditMetrics
extension. At longer horizons, these differences are amplified: the slow stochastic intensity and
discrete CreditMetrics models show high and low volatilities, respectively, while the remaining
three models are indistinguishable.
Thought default rate volatilities are illustrative, they do not provide us information about the full dis-tribution
of defaults through time. At the one year horizon, our calibration guarantees that volatility
will be consistent across the five models; the distribution assumptions, however influence the pre-
12Note that while (24) guarantees a non-negative solution for h, the discretized version admits a small probability that htC1t will
be negative. To reduce this possibility, we choose 1t for each timestep such that the probability that htC1t 0 is sufficiently small.
The result is that while we only need 50 timesteps per year in some cases, we require as many as one thousand when the value of
is large, as in the high correlation, fast mean reversion case.
18

cise shape of the portfolio distribution. We see in Table 3 that there is actually very little difference
between even the 1st percentiles of the distributions, particularly in the low correlation case. For
the full six year horizon, Table 4 shows more differences between the percentiles. Consistent with
the default volatility results, the tail percentiles are most extreme for the slow stochastic intensity
model, and least extreme for discrete CreditMetrics. Interestingly, though the CreditMetrics diffu-sion
model shows similar volatility to the copula and fast stochastic intensity models, it produces
less extreme percentiles than these other models. Note also that among distributions with similar
means, the median serves well as an indicator of skewness. The high correlation setting generally,
and the slow stochastic intensity model in particular, show lower medians. For these cases, the
distribution places higher probability on the worst default scenarios as well as the scenarios with
few or no defaults.
The cumulative probability distributions for the six year horizons are presented in Figures 4 through
7. As in the other comparisons, the slow stochastic intensity model is notable for placing large prob-ability
on the very low and high default rate scenarios, while the discrete CreditMetrics extension
stands out as the most benign of the distributions. Most striking, however, is the similarity between
the fast stochastic intensity and copula models, which are difficult to differentiate even at the most
extreme percentile levels.
As a final comparison of the default distributions, we consider the pricing of a simple structure
written on our portfolio. Suppose each of the one hundred bonds in the portfolio has a notional
value of $1 million, and that in the event of a default the recovery rate on each bond is forty percent.
The structure is composed of three elements:
(i) First loss protection. As defaults occur, the protection seller reimburses the structure up to a
total payment of $10 million. Thus, the seller pays $600,000 at the time of the first default,
$600,000 at the time of each of the subsequent fifteen defaults, and $400,000 at the time of
the seventeenth default.
(ii) Second loss protection. The protection seller reimburses the structure for losses in excess of
19

$10 million, up to a total payment of $20 million. This amounts to reimbursing the losses on
the seventeenth through the fiftieth defaults.
(iii) Senior notes. Notes with a notional value of $100 million maturing after six years. The notes
suffer a principal loss if the first and second loss protection are fully utilized – that is, if more
than fifty defaults occur.
For the first and second loss protection, we will estimate the cost of the protection based on a
constant discount rate of 7%. In each scenario, we produce the timing and amounts of the protection
payments, and discount these back to the present time. The price of the protection is then the average
discounted value across the 1,000 scenarios. For the senior notes, we compute the expected principal
loss at maturity, which is used by Moody’s along with Table 5 to determine the notes’ rating.
Additionally, we compute the total amount of protection (capital) required to achieve a rating of A3
(an expected loss of 0.5%) and Aa3 (an expected loss of 0.101%).
We present the first and second loss prices in Table 6, along with the expected loss, current rating,
and required capital for the senior notes. The slow stochastic intensity model yields the lowest
pricing for the first loss protection, the worst rating for the senior notes, and the highest required
capital. The results for the other models are as expected, with the copula and fast mean reversion
models yielding the most similar results.
9 Conclusion
The analysis of Collateralized Debt Obligations, and other structured products written on credit
portfolios, requires a model of correlated defaults over multiple horizons. For single horizon
models, the effect of model and distribution choice on the model results is well understood. For
the multiple horizon models, however, there has been little research.
We have outlined four approaches to multiple horizon modeling of defaults in a portfolio. We
have calibrated the four models to the same set of input data (average defaults and a single period
20

correlation parameter), and have investigated the resulting default distributions. The differences we
observe can be attributed to the model structures, and to some extent, to the choice of distributions
that drive the models. Our results show a significant disparity. The rating on a class of senior
notes under our low correlation assumption varied from Aaa to A3, and under our high correlation
assumption fromA1 to Baa3. Additionally, the capital required to achieve a target investment grade
rating varied by as much as a factor of two.
In the single period case, a number of studies have concluded that when calibrated to the same
first and second order information, the various models do not produce vastly different conclusions.
Here, the issue of model choice is much more important, and any analysis of structures over multiple
horizons should heed this potential model error.
References
Cifuentes, A., Choi, E., andWaite, J. (1998). Stability of rations of CBO/CLO tranches. Moody’s
Investors Service.
Credit Suisse Financial Products. (1997). CreditRisk+: A credit risk management framework.
Duffie, D. and Garleanu, N. (1998). Risk and valuation of Collateralized Debt Obligations. Working
paper. Graduate School of Business, Stanford University.
http://www.stanford.edu/˜duffie/working.htm
Duffie, D. and Singleton, K. (1998). Simulating correlated defaults. Working paper. Graduate
School of Business, Stanford University.
http://www.stanford.edu/˜duffie/working.htm
Duffie, D. and Singleton, K. (1999). Modeling term structures of defaultable bonds. Review of
Financial Studies, 12, 687-720.
21

Finger, C. (1998). Sticks and stones. Working paper. RiskMetrics Group.
http://www.riskmetrics.com/research/working
Gordy, M. (2000). A comparative anatomy of credit risk models. Journal of Banking Finance,
24 (January), 119-149.
Gupton, G., Finger, C., and Bhatia, M. (1997). CreditMetrics – Technical Document. Morgan
Guaranty Trust Co. http://www.riskmetrics.com/research/techdoc
Li, D. (1999). The valuation of basket credit derivatives. CreditMetrics Monitor, April, 34-50.
http://www.riskmetrics.com/research/journals
Li, D. (2000). On default correlation: a copula approach. The Journal of Fixed Income, 9 (March),
43-54.
Keenan, S., Hamilton, D. and Berthault, A. (2000). Historical default rates of corporate bond
issuers, 1920-1999. Moody’s Investors Service.
Kolyoglu, U. and Hickman, A. (1998). Reconcilable differences. Risk, October.
Nagpal, K. and Bahar, R. (1999). An analytical approach for credit risk analysis under correlated de-faults.
CreditMetrics Monitor, April, 51-74. http://www.riskmetrics.com/research/journals
Standard Poor’s. (2000). Ratings performance 1999: Stability Transition.
Wilson, T. (1997). Portfolio Credit Risk I. Risk, September.
Wilson, T. (1997). Portfolio Credit Risk II. Risk, October.
22

Table 1: Calibration results.
Investment grade Speculative grade
Parameter Low correlation High correlation Low correlation High correlation
Inputs
p1 0.16% 0.16% 3.35% 3.35%
p2 0.33% 0.33% 3.41% 3.41%
p1;1 0.0007% 0.0059% 0.1776% 0.5190%
Discrete CreditMetrics extension
1 -2.95 -2.95 -1.83 -1.83
2 -2.72 -2.72 -1.81 -1.81
10% 40% 10% 40%
Diffusion CreditMetrics extension
1 -2.95 -2.95 -1.83 -1.83
2 -3.78 -3.78 -2.34 -2.34
10% 40% 10% 40%
Copula functions
1 -2.95 -2.95 -1.83 -1.83
2 -2.58 -2.58 -1.49 -1.49
10% 40% 10% 40%
Stochastic intensity – slow mean reversion
0.29 0.29 0.29 0.29
0.10 0.37 0.28 0.76
Nh
1 0.16% 0.16% 3.44% 3.67%
Nh
2 1.47% 1.58% 6.06% 12.10%
Stochastic intensity – fast mean reversion
1.39 1.39 1.39 1.39
0.14 0.53 0.40 1.12
Nh
1 0.16% 0.16% 3.44% 3.68%
Nh
2 0.53% 0.55% 4.00% 5.02%
Table 2: Moody’s speculative grade cumulative default probabilities. From Exhibit 30, Keenan et al (2000).
Year 1 2 3 4 5 6
Probability 3.35% 6.76% 9.98% 12.89% 15.57% 17.91%
23

Table 3: One year default statistics. Speculative grade.
CreditMetrics CreditMetrics Stoch. Int. Stoch. Int.
Statistic Discrete Diffusion Copula Slow Fast
Low correlation
Mean 3.37 3.36 3.51 3.20 3.20
St. Dev. 3.15 3.27 3.40 3.03 3.05
Median 3 2 3 3 2
5th percentile 10 9 10 9 10
1st percentile 14 15 15 13 14
High correlation
Mean 3.62 3.24 3.72 3.69 3.56
St. Dev. 7.08 6.32 7.52 6.84 6.73
Median 1 1 1 1 1
Table 4: Six year cumulative default statistics. Speculative grade.
CreditMetrics CreditMetrics Stoch. Int. Stoch. Int.
Statistic Discrete Diffusion Copula Slow Fast
Low correlation
Mean 17.72 16.93 18.04 17.34 18.10
St. Dev. 6.40 8.68 9.66 16.15 9.73
Median 17 16 17 12 16
High correlation
Mean 18.41 17.28 18.61 19.81 20.41
St. Dev. 13.49 17.41 19.27 24.37 19.36
Median 15 12 12 9 13
24

Table 5: Target expected losses for six year maturity. From Chart 3, Cifuentes et al (2000).
Rating Expected loss
Aaa 0.002%
Aa1 0.023%
Aa2 0.048%
Aa3 0.101%
A1 0.181%
A2 0.320%
A3 0.500%
Baa1 0.753%
Baa2 1.083%
Baa3 2.035%
Table 6: Prices (in $M) for first and second loss protection. Expected loss, rating, and required capital ($M)
for senior notes. Speculative grade collateral.
Senior notes
First loss Second loss Exp. loss Rating Capital (Aa3) Capital (A3)
Low correlation
CM Discrete 7.227 1.350 0.000% Aaa 17.3 13.8
CM Diffusion 6.676 1.533 0.017% Aa1 21.6 15.9
Copula 6.788 1.936 0.022% Aa1 24.5 18.0
Stoch. int. – slow 5.533 2.501 0.466% A3 39.8 29.4
Stoch. int. – fast 6.763 1.911 0.038% Aa2 25.7 18.3
High correlation
CM Discrete 6.117 2.698 0.159% A1 32.3 23.6
CM Diffusion 5.144 2.832 0.514% Baa1 41.1 30.2
Copula 5.210 3.200 0.821% Baa2 43.7 34.4
Stoch. int. – slow 4.856 3.307 1.903% Baa3 54.5 46.1
Stoch. int. – fast 5.685 3.500 0.918% Baa2 45.9 35.2
25

Figure 1: Marginal and cumulative year two default volatility.
Marginal
Cumulative
CM CM Copula Stoch int Stoch int
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Investment grade, low correlation
Discrete Diffusion Slow Fast
Marginal
Cumulative
4
3.5
3
2.5
2
1.5
1
0.5
0
Investment grade, high correlation
Marginal
Cumulative
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Speculative grade, low correlation
Marginal
Cumulative
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Speculative grade, high correlation
26

Figure 2: Year two conditional default probability given default of a second asset.
Cond on 1st yr default
Cond on 2nd yr default
2.5
2
1.5
1
0.5
0
Investment grade, low correlation
18
16
14
12
10
8
6
4
2
0
Investment grade, high correlation
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Speculative grade, low correlation
6
5
4
3
2
1
0
Speculative grade, high correlation
27

Figure 3: Normalized cumulative default rate volatilities. Speculative grade.
1 2 3 4 5 6
1 2 3 4 5 6
1.2
1
0.8
0.6
0.4
2.5
2
1.5
1
0.5
Time
Default volatility
High correlation
CM Discrete
CM Diffusion
Copula
St.Int. Slow
St.Int. Fast
0.2
Default volatility
Low correlation
28

Figure 4: Distribution of cumulative six year defaults. Speculative grade, low correlation.
0 10 20 30 40 50 60 70 80 90 100
100%
80%
60%
40%
20%
0
Defaults
Cumulative probability
CM Discrete
CM Diffusion
Copula
St.Int. Slow
St.Int. Fast
29

Figure 5: Distribution of cumulative six year defaults, extreme cases. Speculative grade, low correlation.
100%
96%
92%
88%
84%
80%
20 30 40 50 60 70 80 90 100
Defaults
CM Discrete
CM Diffusion
Copula
St.Int. Slow
St.Int. Fast
30

Figure 6: Distribution of cumulative six year defaults. Speculative grade, high correlation.
0 10 20 30 40 50 60 70 80 90 100
100%
80%
60%
40%
20%
0
Defaults
CM Discrete
CM Diffusion
Copula
St.Int. Slow
St.Int. Fast
31

Figure 7: Distribution of cumulative six year defaults, extreme cases. Speculative grade, high correlation.
100%
96%
92%
88%
84%
80%
20 30 40 50 60 70 80 90 100
Defaults
CM Discrete
CM Diffusion
Copula
St.Int. Slow
St.Int. Fast
32

The RiskMetrics Group
Working Paper Number 99-07
On Default Correlation: A Copula Function Approach
David X. Li
This draft: February 2000
First draft: September 1999
44Wall St.
NewYork, NY 10005
david.li@riskmetrics.com
www.riskmetrics.com

On Default Correlation: A Copula Function Approach
David X. Li
February 2000
Abstract
This paper studies the problem of default correlation. We first introduce a random variable called “time-until-
default” to denote the survival time of each defaultable entity or financial instrument, and define the
default correlation between two credit risks as the correlation coefficient between their survival times.
Then we argue why a copula function approach should be used to specify the joint distribution of survival
times after marginal distributions of survival times are derived from market information, such as risky
bond prices or asset swap spreads. The definition and some basic properties of copula functions are
given. We show that the current CreditMetrics approach to default correlation through asset correlation
is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate
the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and
first-to-default contracts.

1 Introduction
The rapidly growing credit derivative market has created a new set of financial instruments which can be
used to manage the most important dimension of financial risk - credit risk. In addition to the standard
credit derivative products, such as credit default swaps and total return swaps based upon a single underlying
credit risk, many new products are now associated with a portfolio of credit risks. A typical example is the
product with payment contingent upon the time and identity of the first or second-to-default in a given credit
risk portfolio. Variations include instruments with payment contingent upon the cumulative loss before a
given time in the future. The equity tranche of a collateralized bond obligation (CBO) or a collateralized
loan obligation (CLO) is yet another variation, where the holder of the equity tranche incurs the first loss.
Deductible and stop-loss in insurance products could also be incorporated into the basket credit derivatives
structure. As more financial firms try to manage their credit risk at the portfolio level and the CBO/CLO
market continues to expand, the demand for basket credit derivative products will most likely continue to
grow.
Central to the valuation of the credit derivatives written on a credit portfolio is the problem of default
correlation. The problem of default correlation even arises in the valuation of a simple credit default swap
with one underlying reference asset if we do not assume the independence of default between the reference
asset and the default swap seller. Surprising though it may seem, the default correlation has not been well
defined and understood in finance. Existing literature tends to define default correlation based on discrete
events which dichotomize according to survival or nonsurvival at a critical period such as one year. For
example, if we denote
qA = Pr[EA], qB = Pr[EB], qAB = Pr[EAEB]
where EA, EB are defined as the default events of two securities A and B over 1 year. Then the default
correlation ρ between two default events EA and EB, based on the standard definition of correlation of two
random variables, are defined as follows
1

ρ = qAB − qA · √ qB
qA(1 − qA)qB(1 − qB)
. (1)
This discrete event approach has been taken by Lucas [1995]. Hereafter we simply call this definition of
default correlation the discrete default correlation.
However the choice of a specific period like one year is more or less arbitrary. It may correspond with many
empirical studies of default rate over one year period. But the dependence of default correlation on a specific
time interval has its disadvantages. First, default is a time dependent event, and so is default correlation. Let
us take the survival time of a human being as an example. The probability of dying within one year for a
person aged 50 years today is about 0.6%, but the probability of dying for the same person within 50 years is
almost a sure event. Similarly default correlation is a time dependent quantity. Let us now take the survival
times of a couple, both aged 50 years today. The correlation between the two discrete events that each dies
within one year is very small. But the correlation between the two discrete events that each dies within 100
years is 1. Second, concentration on a single period of one year wastes important information. There are
empirical studies which show that the default tendency of corporate bonds is linked to their age since issue.
Also there are strong links between the economic cycle and defaults. Arbitrarily focusing on a one year period
neglects this important information. Third, in the majority of credit derivative valuations, what we need is
not the default correlation of two entities over the next year. We may need to have a joint distribution of
survival times for the next 10 years. Fourth, the calculation of default rates as simple proportions is possible
only when no samples are censored during the one year period1.
This paper introduces a few techniques used in survival analysis. These techniques have been widely applied
to other areas, such as life contingencies in actuarial science and industry life testing in reliability studies,
which are similar to the credit problems we encounter here. We first introduce a random variable called
1A company who is observed, default free, by Moody’s for 5-years and then withdrawn from the Moody’s study must have
a survival time exceeding 5 years. Another company may enter into Moody’s study in the middle of a year, which implies that
Moody’s observes the company for only half of the one year observation period. In the survival analysis of statistics, such incomplete
observation of default time is called censoring. According to Moody’s studies, such incomplete observation does occur in Moody’s
credit default samples.
2

“time-until-default” to denote the survival time of each defaultable entity or financial instrument. Then,
we define the default correlation of two entities as the correlation between their survival times. In credit
derivative valuation we need first to construct a credit curve for each credit risk. A credit curve gives all
marginal conditional default probabilities over a number of years. This curve is usually derived from the
risky bond spread curve or asset swap spreads observed currently from the market. Spread curves and asset
swap spreads contain information on default probabilities, recovery rate and liquidity factors etc. Assuming
an exogenous recovery rate and a default treatment, we can extract a credit curve from the spread curve or
asset swap spread curve. For two credit risks, we would obtain two credit curves from market observable
information. Then, we need to specify a joint distribution for the survival times such that the marginal
distributions are the credit curves. Obviously, this problem has no unique solution. Copula functions used in
multivariate statistics provide a convenient way to specify the joint distribution of survival times with given
marginal distributions. The concept of copula functions, their basic properties, and some commonly used
copula functions are introduced. Finally, we give a few numerical examples of credit derivative valuation to
demonstrate the use of copula functions and the impact of default correlation.
2 Characterization of Default by Time-Until-Default
In the study of default, interest centers on a group of individual companies for each of which there is defined
a point event, often called default, (or survival) occurring after a length of time. We introduce a random
variable called the time-until-default, or simply survival time, for a security, to denote this length of time.
This random variable is the basic building block for the valuation of cash flows subject to default.
To precisely determine time-until-default, we need: an unambiguously defined time origin, a time scale for
measuring the passage of time, and a clear definition of default.
We choose the current time as the time origin to allow use of current market information to build credit
curves. The time scale is defined in terms of years for continuous models, or number of periods for discrete
models. The meaning of default is defined by some rating agencies, such as Moody’s.
3

2.1 Survival Function
Let us consider an existing security A. This security’s time-until-default, TA, is a continuous random variable
which measures the length of time from today to the time when default occurs. For simplicity we just use T
which should be understood as the time-until-default for a specific securityA. Let F(t) denote the distribution
function of T ,
F(t) = Pr(T ≤ t), t ≥0 (2)
and set
S(t) = 1 − F(t) = Pr(T t ), t ≥ 0. (3)
We also assume that F(0) = 0, which implies S(0) = 1. The function S(t) is called the survival function.
It gives the probability that a security will attain age t . The distribution of TA can be defined by specifying
either the distribution function F(t) or the survival function S(t). We can also define a probability density
function as follows
f (t) = F

(t) = −S

(t) = lim
→0+
Pr[t ≤T t + ]

.
To make probability statements about a security which has survived x years, the future life time for this
security is T − x|T x. We introduce two more notations
tqx = Pr[T − x ≤ t |T x], t≥ 0
tpx = 1 − tqx = Pr[T −x t|T x], t≥ 0. (4)
The symbol tqx can be interpreted as the conditional probability that the security A will default within the
next t years conditional on its survival for x years. In the special case of X = 0, we have
tp0 = S(t) x ≥ 0.
4

If t = 1, we use the actuarial convention to omit the prefix 1 in the symbols tqx and tpx , and we have
px = Pr[T −x 1|T x]
qx = Pr[T − x ≤ 1|T x].
The symbol qx is usually called the marginal default probability, which represents the probability of default
in the next year conditional on the survival until the beginning of the year. A credit curve is then simply
defined as the sequence of q0, q1, · · · , qn in discrete models.
2.2 Hazard Rate Function
The distribution function F(t) and the survival function S(t) provide two mathematically equivalent ways
of specifying the distribution of the random variable time-until-default, and there are many other equiva-lent
functions. The one used most frequently by statisticians is the hazard rate function which gives the
instantaneous default probability for a security that has attained age x.
Pr[x T ≤ x + x|T x] = F(x + x) − F(x)
1 − F(x)
≈ f (x)x
1 − F(x)
.
The function
f (x)
1 − F(x)
has a conditional probability density interpretation: it gives the value of the conditional probability density
function of T at exact age x, given survival to that time. Let’s denote it as h(x), which is usually called
the hazard rate function. The relationship of the hazard rate function with the distribution function and
survival function is as follows
5

h(x) = f (x)
1 − F(x)

(x)
S(x)
= −S
. (5)
Then, the survival function can be expressed in terms of the hazard rate function,
S(t) = e
−

t
0 h(s)ds .
Now, we can express tqx and tpx in terms of the hazard rate function as follows
tpx
= e
−

t
0 h(s+x)ds , (6)
tqx = 1 − e
−

t
0 h(s+x)ds .
In addition,
F(t) = 1 − S(t) = 1 − e
−

t
0 h(s)ds ,
and
f (t) = S(t) · h(t). (7)
which is the density function for T .
A typical assumption is that the hazard rate is a constant, h, over certain period, such as [x, x + 1]. In this
case, the density function is
f (t) = he
−ht
6

which shows that the survival time follows an exponential distribution with parameter h. Under this assump-tion,
the survival probability over the time interval [x, x + t ] for 0 t ≤ 1 is
tpx
= 1 − tqx
= e
−

t
0 h(s)ds = e
−ht = (px)t
where px is the probability of survival over one year period. This assumption can be used to scale down the
default probability over one year to a default probability over a time interval less than one year.
Modelling a default process is equivalent to modelling a hazard function. There are a number of reasons why
modelling the hazard rate function may be a good idea. First, it provides us information on the immediate
default risk of each entity known to be alive at exact age t . Second, the comparisons of groups of individuals
are most incisively made via the hazard rate function. Third, the hazard rate function based model can be
easily adapted to more complicated situations, such as where there is censoring or there are several types
of default or where we would like to consider stochastic default fluctuations. Fourth, there are a lot of
similarities between the hazard rate function and the short rate. Many modeling techniques for the short rate
processes can be readily borrowed to model the hazard rate.
Finally, we can define the joint survival function for two entities A and B based on their survival times TA
and TB,
STATB (s, t) = Pr[TA s,TB t].
The joint distributional function is
F(s, t) = Pr[TA ≤ s, TB ≤ t ]
= 1 − STA(s) − STB (t) + STATB (s, t ).
The aforementioned concepts and results can be found in survival analysis books, such as Bowers et al.
[1997], Cox and Oakes [1984].
7

3 Definition of Default Correlations
The default correlation of two entities A and B can then be defined with respect to their survival times TA
and TB as follows
ρAB = √ Cov(TA, TB)
Var(TA)V ar(TB)
= E(√TATB) − E(TA)E(TB)
Var(TA)V ar(TB)
. (8)
Hereafter we simply call this definition of default correlation the survival time correlation. The survival
time correlation is a much more general concept than that of the discrete default correlation based on a one
period. If we have the joint distribution f (s, t) of two survival times TA, TB, we can calculate the discrete
default correlation. For example, if we define
E1 = [TA 1],
E2 = [TB 1],
then the discrete default correlation can be calculated using equation (1) with the following calculation
q12 = Pr[E1E2] =

1
0

1
0
f (s, t)dsdt
q1 =

1
0
fA(s)ds
q2 =

1
0
fB(t)dt .
However, knowing the discrete default correlation over one year period does not allow us to specify the
survival time correlation.
4 The Construction of the Credit Curve
The distribution of survival time or time-until-default can be characterized by the distribution function,
survival function or hazard rate function. It is shown in Section 2 that all default probabilities can be
8

calculated once a characterization is given. The hazard rate function used to characterize the distribution of
survival time can also be called a credit curve due to its similarity to a yield curve. But the basic question is:
how do we obtain the credit curve or the distribution of survival time for a given credit?
There exist three methods to obtain the term structure of default rates:
(i) Obtaining historical default information from rating agencies;
(ii) Taking the Merton option theoretical approach;
(iii) Taking the implied approach using market prices of defaultable bonds or asset swap spreads.
Rating agencies like Moody’s publish historical default rate studies regularly. In addition to the commonly
cited one-year default rates, they also present multi-year default rates. From these rates we can obtain the
hazard rate function. For example, Moody’s (see Carty and Lieberman [1997]) publishes weighted average
cumulative default rates from 1 to 20 years. For the B rating, the first 5 years cumulative default rates in
percentage are 7.27, 13.87, 19.94, 25.03 and 29.45. From these rates we can obtain the marginal conditional
default probabilities. The first marginal conditional default probability in year one is simply the one-year
default probability, 7.27%. The other marginal conditional default probabilities can be obtained using the
following formula:
n+1qx = nqx + npx · qx+n, (9)
which simply states that the probability of default over time interval [0, n + 1] is the sum of the probability
of default over the time interval [0, n], plus the probability of survival to the end of nth year and default in
the following year. Using equation (9) we have the marginal conditional default probability:
qx+n = n+1qx − nqx
1 − nqx
which results in the marginal conditional default probabilities in year 2, 3, 4, 5 as 7.12%, 7.05%, 6.36% and
5.90%. If we assume a piecewise constant hazard rate function over each year, then we can obtain the hazard
rate function using equation (6). The hazard rate function obtained is given in Figure (1).
9

Using diffusion processes to describe changes in the value of the firm, Merton [1974] demonstrated that a
firm’s default could be modeled with the Black and Scholes methodology. He showed that stock could be
considered as a call option on the firm with strike price equal to the face value of a single payment debt.
Using this framework we can obtain the default probability for the firm over one period, from which we
can translate this default probability into a hazard rate function. Geske [1977] and Delianedis and Geske
[1998] extended Merton’s analysis to produce a term structure of default probabilities. Using the relationship
between the hazard rate and the default probabilities we can obtain a credit curve.
Alternatively, we can take the implicit approach by using market observable information, such as asset swap
spreads or risky corporate bond prices. This is the approach used by most credit derivative trading desks. The
extracted default probabilities reflect the market-agreed perception today about the future default tendency of
the underlying credit. Li [1998] presents one approach to building the credit curve from market information
based on the Duffie and Singleton [1996] default treatment. In that paper the author assumes that there exists
a series of bonds with maturity 1, 2, .., n years, which are issued by the same company and have the same
seniority. All of those bonds have observable market prices. From the market price of these bonds we can
calculate their yields to maturity. Using the yield to maturity of corresponding treasury bonds we obtain a
yield spread curve over treasury (or asset swap spreads for a yield spread curve over LIBOR). The credit
curve construction is based on this yield spread curve and an exogenous assumption about the recovery rate
based on the seniority and the rating of the bonds, and the industry of the corporation.
The suggested approach is contrary to the use of historical default experience information provided by rating
agencies such as Moody’s. We intend to use market information rather than historical information for the
following reasons:
• The calculation of profit and loss for a trading desk can only be based on current market information.
This current market information reflects the market agreed perception about the evolution of the market
in the future, on which the actual profit and loss depend. The default rate derived from current market
information may be much different than historical default rates.
• Rating agencies use classification variables in the hope that homogeneous risks will be obtained
10

after classification. This technique has been used elsewhere like in pricing automobile insurance.
Unfortunately, classification techniques omit often some firm specific information. Constructing a
credit curve for each credit allows us to use more firm specific information.
• Rating agencies reacts much slower than the market in anticipation of future credit quality. A typical
example is the rating agencies reaction to the recent Asian crisis.
• Ratings are primarily used to calculate default frequency instead of default severity. However, much
of credit derivative value depends on both default frequency and severity.
• The information available from a rating agency is usually the one year default probability for each rating
group and the rating migration matrix. Neither the transition matrixes, nor the default probabilities
are necessarily stable over long periods of time. In addition, many credit derivative products have
maturities well beyond one year, which requires the use of long term marginal default probability.
It is shown under the Duffie and Singleton approach that a defaultable instrument can be valued as if it is a
default free instrument by discounting the defaultable cash flow at a credit risk adjusted discount factor. The
credit risk adjusted discount factor or the total discount factor is the product of risk-free discount factor and
the pure credit discount factor if the underlying factors affecting default and those affecting the interest rate
are independent. Under this framework and the assumption of a piecewise constant hazard rate function, we
can derive a credit curve or specify the distribution of the survival time.
5 Dependent Models - Copula Functions
Let us study some problems of an n credit portfolio. Using either the historical approach or the market
implicit approach, we can construct the marginal distribution of survival time for each of the credit risks in
the portfolio. If we assume mutual independence among the credit risks, we can study any problem associated
with the portfolio. However, the independence assumption of the credit risks is obviously not realistic; in
reality, the default rate for a group of credits tends to be higher in a recession and lower when the economy
11

is booming. This implies that each credit is subject to the same set of macroeconomic environment, and that
there exists some form of positive dependence among the credits. To introduce a correlation structure into
the portfolio, we must determine how to specify a joint distribution of survival times, with given marginal
distributions.
Obviously, this problem has no unique solution. Generally speaking, knowing the joint distribution of
random variables allows us to derive the marginal distributions and the correlation structure among the
random variables, but not vice versa. There are many different techniques in statistics which allow us to
specify a joint distribution function with given marginal distributions and a correlation structure. Among
them, copula function is a simple and convenient approach. We give a brief introduction to the concept of
copula function in the next section.
5.1 Definition and Basic Properties of Copula Function
A copula function is a function that links or marries univariate marginals to their full multivariate distribution.
For m uniform random variables, U1, U2, · · · ,Um, the joint distribution function C, defined as
C(u1, u2, · · · , um, ρ) = Pr[U1 ≤ u1,U2 ≤ u2, · · · ,Um ≤ um]
can also be called a copula function.
Copula functions can be used to link marginal distributions with a joint distribution. For given univariate
marginal distribution functions F1(x1), F2(x2),· · · , Fm(xm), the function
C(F1(x1), F2(x2), · · · , Fm(xm)) = F(x1, x2, · · · xm),
which is defined using a copula function C, results in a multivariate distribution function with univariate
marginal distributions as specified F1(x1), F2(x2),· · · , Fm(xm).
This property can be easily shown as follows:
12

C(F1(x1), F2(x2), · · · , Fm(xm), ρ) = Pr [U1 ≤ F1(x1),U2 ≤ F2(x2), · · · ,Um ≤ Fm(xm)]
= Pr

F
−1
1 (U1) ≤ x1, F
−1
2 (U2) ≤ x2, · · · , F
−1
m (Um) ≤ xm

= Pr [X1 ≤ x1,X2 ≤ x2, · · · ,Xm ≤ xm]
= F(x1, x2, · · · xm).
The marginal distribution of Xi is
C(F1(+∞), F2(+∞), · · · Fi(xi), · · · , Fm(+∞), ρ)
= Pr [X1 ≤ +∞,X2 ≤ +∞, · · · ,Xi ≤ xi,Xm ≤ +∞]
= Pr[Xi ≤ xi ]
= Fi(xi ).
Sklar [1959] established the converse. He showed that any multivariate distribution function F can be
written in the form of a copula function. He proved the following: If F(x1, x2, · · · xm) is a joint multivariate
distribution function with univariate marginal distribution functions F1(x1), F2(x2),· · · , Fm(xm), then there
exists a copula function C(u1, u2, · · · , um) such that
F(x1, x2, · · · xm) = C(F1(x1), F2(x2), · · · , Fm(xm)).
If each Fi is continuous then C is unique. Thus, copula functions provide a unifying and flexible way to
study multivariate distributions.
For simplicity’s sake, we discuss only the properties of bivariate copula functions C(u, v, ρ) for uniform
random variables U and V , defined over the area {(u, v)|0 u ≤ 1, 0 v ≤ 1}, where ρ is a correlation
parameter. We call ρ simply a correlation parameter since it does not necessarily equal the usual correlation
coefficient defined by Pearson, nor Spearman’s Rho, nor Kendall’s Tau. The bivariate copula function has
the following properties:
(i) Since U and V are positive random variables, C(0, v, ρ) = C(u, 0, ρ) = 0.
13

(ii) Since U and V are bounded above by 1, the marginal distributions can be obtained by C(1, v, ρ) = v,
C(u, 1, ρ) = u.
(iii) For independent random variables U and V , C(u, v, ρ) = uv.
Frechet [1951] showed there exist upper and lower bounds for a copula function
max(0, u + v − 1) ≤ C(u, v) ≤ min(u, v).
The multivariate extension of Frechet bounds is given by Dall’Aglio [1972].
5.2 Some Common Copula Functions
We present a few copula functions commonly used in biostatistics and actuarial science.
Frank Copula The Frank copula function is defined as
C(u, v) = 1
α
ln

1 + (eαu − 1)(eαv − 1)
eα − 1

, −∞ α ∞.
Bivariate Normal
C(u, v) = %2(%
−1(u),%
−1(v), ρ), −1 ≤ ρ ≤ 1, (10)
−1 is the inverse of a
where%2 is the bivariate normal distribution function with correlation coefficient ρ, and%
univariate normal distribution function. As we shall see later, this is the copula function used in CreditMetrics.
Bivariate Mixture Copula Function We can form new copula function using existing copula functions.
If the two uniform random variables u and v are independent, we have a copula function C(u, v) = uv. If
the two random variables are perfect correlated we have the copula function C(u, v) = min(u, v). Mixing
the two copula functions by a mixing coefficient (ρ 0) we obtain a new copula function as follows
14

C(u, v) = (1 − ρ)uv + ρ min(u, v), ifρ 0.
If ρ ≤ 0 we have
C(u, v) = (1 + ρ)uv − ρ(u − 1 + v)(u − 1 + v), if ρ ≤ 0,
where
(x) = 1, if x ≥ 0
= 0, if x 0.
5.3 Copula Function and Correlation Measurement
To compare different copula functions, we need to have a correlation measurement independent of marginal
distributions. The usual Pearson’s correlation coefficient, however, depends on the marginal distributions
(See Lehmann [1966]). Both Spearman’s Rho and Kendall’s Tau can be defined using a copula function only
as follows
ρs = 12

[C(u, v) − uv]dudv,
τ = 4

C(u, v)dC(u, v) − 1.
Comparisons between results using different copula functions should be based on either a common Spear-man’s
Rho or a Kendall’s Tau.
Further examination of copula functions can be found in a survey paper by Frees and Valdez [1988] and a
recent book by Nelsen [1999].
15

5.4 The Calibration of Default Correlation in Copula Function
Having chosen a copula function, we need to compute the pairwise correlation of survival times. Using the
CreditMetrics (Gupton et al. [1997]) asset correlation approach, we can obtain the default correlation of two
discrete events over one year period. As it happens, CreditMetrics uses the normal copula function in its
default correlation formula even though it does not use the concept of copula function explicitly.
First let us summarize how CreditMetrics calculates joint default probability of two credits Aand B. Suppose
the one year default probabilities for A and B are qA and qB. CreditMetrics would use the following steps
• Obtain ZA and ZB such that
qA = Pr[Z ZA]
qB = Pr[Z ZB]
where Z is a standard normal random variable
• If ρ is the asset correlation, the joint default probability for credit A and B is calculated as follows,
Pr[Z ZA,Z ZB] =
ZA
−∞
ZB
−∞
φ2(x, y|ρ)dxdy = %2(ZA,ZB, ρ) (11)
where φ2(x, y|ρ) is the standard bivariate normal density function with a correlation coefficient ρ, and
%2 is the bivariate accumulative normal distribution function.
If we use a bivariate normal copula function with a correlation parameter γ , and denote the survival times
for A and B as TA and TB, the joint default probability can be calculated as follows
Pr[TA 1, TB 1] = %2(%
−1(FA(1)),%
−1(FB(1), γ ) (12)
where FA and FB are the distribution functions for the survival times TA and TB. If we notice that
16

qi = Pr[Ti 1] = Fj (1) and Zi = %
−1(qi) for i = A,B,
then we see that equation (12) and equation (11) give the same joint default probability over one year period
if ρ = γ .
We can conclude that CreditMetrics uses a bivariate normal copula function with the asset correlation as the
correlation parameter in the copula function. Thus, to generate survival times of two credit risks, we use
a bivariate normal copula function with correlation parameter equal to the CreditMetrics asset correlation.
We note that this correlation parameter is not the correlation coefficient between the two survival times. The
correlation coefficient between the survival times is much smaller than the asset correlation. Conveniently,
the marginal distribution of any subset of an n dimensional normal distribution is still a normal distribution.
Using asset correlations, we can construct high dimensional normal copula functions to model the credit
portfolio of any size.
6 Numerical Illustrations
This section gives some numerical examples to illustrate many of the points discussed above. Assume that
we have two credit risks, A and B, which have flat spread curves of 300 bps and 500 bps over LIBOR. These
spreads are usually given in the market as asset swap spreads. Using these spreads and a constant recovery
assumption of 50% we build two credit curves for the two credit risks. For details, see Li [1998]. The two
credit curves are given in Figures (2) and (3). These two curves will be used in the following numerical
illustrations.
6.1 Illustration 1. Default Correlation v.s. Length of Time Period
In this example, we study the relationship between the discrete default correlation (1) and the survival time
correlation (8). The survival time correlation is a much more general concept than the discrete default
17

correlation defined for two discrete default events at an arbitrary period of time, such as one year. Knowing
the former allows us to calculate the latter over any time interval in the future, but not vice versa.
Using two credit curves we can calculate all marginal default probabilities up to anytime t in the future, i.e.
tq0 = Pr[τ t] = 1 − e
−

t
0 h(s)ds ,
where h(s) is the instantaneous default probability given by a credit curve. If we have the marginal default
probabilities tqA
0 and tqB
0 for both A and B, we can also obtain the joint probability of default over the time
interval [0, t] by a copula function C(u, v),
Pr[TA t,TB t] = C(tqA
0 , tqB
0 ).
Of course we need to specify a correlation parameter ρ in the copula function. We emphasize that knowing
ρ would allow us to calculate the survival time correlation between TA and TB.
We can now obtain the discrete default correlation coefficient ρt between the two discrete events that A and
B default over the time interval [0, t] based on the formula (1). Intuitively, the discrete default correlation ρt
should be an increasing function of t since the two underlying credits should have a higher tendency of joint
default over longer periods. Using the bivariate normal copula function (10) and ρ = 0.1 as an example we
obtain Figure (4).
From this graph we see explicitly that the discrete default correlation over time interval [0, t] is a function
of t . For example, this default correlation coefficient goes from 0.021 to 0.038 when t goes from six months
to twelve months. The increase slows down as t becomes large.
6.2 Illustration 2. Default Correlation and Credit Swap Valuation
The second example shows the impact of default correlation on credit swap pricing. Suppose that credit A
is the credit swap seller and credit B is the underlying reference asset. If we buy a default swap of 3 years
18

with a reference asset of credit B from a risk-free counterparty we should pay 500 bps since holding the
underlying asset and having a long position on the credit swap would create a riskless portfolio. But if we
buy the default swap from a risky counterparty how much we should pay depends on the credit quality of the
counterparty and the default correlation between the underlying reference asset and the counterparty.
Knowing only the discrete default correlation over one year we cannot value any credit swaps with a maturity
longer than one year. Figure (5) shows the impact of asset correlation (or implicitly default correlation) on the
credit swap premium. From the graph we see that the annualized premium decreases as the asset correlation
between the counterparty and the underlying reference asset increases. Even at zero default correlation the
credit swap has a value less than 500 bps since the counterparty is risky.
6.3 Illustration 3. Default Correlation and First-to-Default Valuation
The third example shows how to value a first-to-default contract. We assume we have a portfolio of n credits.
Let us assume that for each credit i in the portfolio we have constructed a credit curve or a hazard rate function
for its survival time Ti . The distribution function of Ti is Fi (t). Using a copula function C we also obtain
the joint distribution of the survival times as follows
F(t1, t2, · · · , tn) = C(F1(t1), F2(t2), · · · , Fn(tn)).
If we use normal copula function we have
F(t1, t2, · · · , tn) = %n(%
−1(F1(t1)),%
−1(F2(t2)), · · · ,%
−1(Fn(tn)))
where %n is the n dimensional normal cumulative distribution function with correlation coefficient matrix
-.
To simulate correlated survival times we introduce another series of random variables Y1, Y2, · · · Yn, such
that
Y1 = %
−1(F1(T1)), Y2 = %
−1(F2(T2)), · · · , Yn = %
−1(Fn(Tn)). (13)
19

Then there is a one-to-one mapping between Y and T . Simulating {Ti |i = 1, 2, ..., n} is equivalent to
simulating {Yi |i = 1, 2, ..., n}. As shown in the previous section the correlation between the Y
s is the asset
correlation of the underlying credits. Using CreditManager from RiskMetrics Group we can obtain the asset
correlation matrix -. We have the following simulation scheme
• Simulate Y1, Y2, · · · Yn from an n-dimension normal distribution with correlation coefficient matrix -.
• Obtain T1, T2, · · · Tn using Ti = F
−1
i (N(Yi)), i = 1, 2, · · · , n.
With each simulation run we generate the survival times for all the credits in the portfolio. With this
information we can value any credit derivative structure written on the portfolio. We use a simple structure
for illustration. The contract is a two-year transaction which pays one dollar if the first default occurs during
the first two years.
We assume each credit has a constant hazard rate of h = 0.1 for 0 t +∞. From equation (7) we
know the density function for the survival time is −T he
ht . This shows that the survival time is exponentially
distributed with mean 1/h. We also assume that every pair of credits in the portfolio has a constant asset
correlation σ2.
Suppose we have a constant interest rate r = 0.1. If all the credits in the portfolio are independent, the hazard
rate of the minimum survival time T = min(T1, T2, · · · , Tn) is easily shown to be
hT = h1 + h2 +· · ·+hn = nh.
IfT 2, the present value of the contract is 1 · e
−r·T . The survival time for the first-to-default has a density
function f (t) = hT · e
−hT t , so the value of the contract is given by
2To have a positive definite correlation matrix, the constant correlation coefficient has to satisfy the conditionσ − 1
n−1 .
20

V =

2
0
1 · e
−rtf (t)dt
=

2
0
1 · e
−r thT · e
−hT tdt (14)
= hT
r + hT

1 − e
−2.0·(r+hT )

.
In the general case we use the Monte Carlo simulation approach and the normal copula function to obtain
the distribution of T . For each simulation run we have one scenario of default times t1, t2, · · · tn, from which
we have the first-to-default time simply as t = min(t1, t2, · · · tn).
Let us examine the impact of the asset correlation on the value of the first-to-default contract of 5-assets. If
σ = 0, the expected payoff function, based on equation (14), should give a value of 0.5823. Our simulation of
50,000 runs gives a value of 0.5830. If all 5 assets are perfectly correlated, then the first-to-default of 5-assets
should be the same as the first-to-default of 1-asset since any one default induces all others to default. In this
case the contract should worth 0.1648. Our simulation of 50,000 runs produces a result of 0.1638. Figure
(6) shows the relationship between the value of the contract and the constant asset correlation coefficient.
We see that the value of the contract decreases as the correlation increases. We also examine the impact of
correlation on the value of the first-to-default of 20 assets in Figure (6). As expected, the first-to-default of
5 assets has the same value of the first-to-default of 20 assets when the asset correlation approaches to 1.
7 Conclusion
This paper introduces a few standard technique used in survival analysis to study the problem of default
correlation. We first introduce a random variable called “the time-until-default” to characterize the default.
Then the default correlation between two credit risks is defined as the correlation coefficient between their
survival times. In practice we usually use market spread information to derive the distribution of survival
times. When it comes to credit portfolio studies we need to specify a joint distribution with given marginal
distributions. The problem cannot be solved uniquely. The copula function approach provides one way of
21

specifying a joint distribution with known marginals. The concept of copula functions, their basic properties
and some commonly used copula functions are introduced. The calibration of the correlation parameter used
in copula functions against some popular credit models is also studied. We have shown that CreditMetrics
essentially uses the normal copula function in its default correlation formula even though CreditMetrics does
not use the concept of copula functions explicitly. Finally we show some numerical examples to illustrate the
use of copula functions in the valuation of credit derivatives, such as credit default swaps and first-to-default
contracts.
References
[1] Bowers, N. L., JR., Gerber, H. U., Hickman, J. C., Jones, D. A., and Nesbitt, C. J., Actuarial Mathe-matics,
2nd Edition, Schaumberg, Illinois, Society of Actuaries, (1997).
[2] Carty, L. and Lieberman, D. Historical Default Rates of Corporate Bond Issuers, 1920-1996, Moodys
Investors Service, January (1997).
[3] Cox, D. R. and Oakes, D. Analysis of Survival Data, Chapman and Hall, (1984).
[4] Dall’Aglio, G., Frechet Classes and Compatibility of Distribution Functions, Symp. Math., 9, (1972),
pp. 131-150.
[5] Delianedis, G. and R. Geske, Credit Risk and Risk Neutral Default Probabilities: Information about
Rating Migrations and Defaults,Working paper, The Anderson School at UCLA, (1998).
[6] Duffie, D. and Singleton, K. Modeling Term Structure of Defaultable Bonds,Working paper, Graduate
School of Business, Stanford University, (1997).
[7] Frechet, M. Sur les Tableaux de Correlation dont les Marges sont Donnees, Ann. Univ. Lyon, Sect. A 9,
(1951), pp. 53-77.
[8] Frees, E.W. and Valdez, E., 1998, Understanding Relationships Using Copulas, North American Actu-arial
Journal, (1998), Vol. 2, No. 1, pp. 1-25.
22

[9] Gupton, G. M., Finger, C. C., and Bhatia, M. CreditMetrics – Technical Document, NewYork: Morgan
Guaranty Trust Co., (1997).
[10] Lehmann, E. L. Some Concepts of Dependence, Annals of Mathematical Statistics, 37, (1966), pp.
1137-1153.
[11] Li, D. X., 1998, Constructing a credit curve, Credit Risk, A RISK Special report, (November 1998), pp.
40-44.
[12] Litterman, R. and Iben, T. Corporate BondValuation and theTerm Structure of Credit Spreads, Financial
Analyst Journal, (1991), pp. 52-64.
[13] Lucas, D. Default Correlation and Credit Analysis, Journal of Fixed Income, Vol. 11, (March 1995),
pp. 76-87.
[14] Merton, R. C. On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of
Finance, 29, pp. 449-470.
[15] Nelsen, R. An Introduction to Copulas, Springer-Verlag NewYork, Inc., 1999.
[16] Sklar, A., Random Variables, Joint Distribution Functions and Copulas, Kybernetika 9, (1973), pp.
449-460.
23

Figure 1: Hazard Rate Function of B Grade Based on Moody’s Study (1997)
Hazard Rate Function
1 2 3 4 5 6
0.075
0.070
0.065
0.060
hazard rate
Years
24

Figure 2: Credit Curve A
0.045 0.050 0.055 0.060 0.065 0.070 0.075
Credit Curve A: Instantaneous Default Probability
09/10/1998 09/10/2000 09/10/2002 09/10/2004 09/10/2006 09/10/2008 09/10/2010
Date
Hazard Rate
(Spread = 300 bps, Recovery Rate = 50%)
25

Figure 3: Credit Curve B
0.08 0.09 0.10 0.11 0.12
Credit Curve B: Instantaneous Default Probability
09/10/1998 09/10/2000 09/10/2002 09/10/2004 09/10/2006 09/10/2008 09/10/2010
Date
Hazard Rate
(Spread = 500 bps, Recovery Rate = 50%)
26

Figure 4: The Discrete Default Correlation v.s. the Length of Time Interval
1 3 5 7 9
Length of Period (Years)
0.30
0.25
0.20
0.15
Discrete Default Correlation
Discrete Default Correlation v. s. Length of Period
27

Figure 5: Impact of Asset Correlation on the Value of Credit Swap
-1.0 -0.5 0.0 0.5 1.0
Asset Correlation
500
480
460
440
420
400
Default Swap Value
Asset Correlation v. s. Default Swap Premium
28

Figure 6: The Value of First-to-Default v. s. Asset Correlation
0.1 0.3 A0s.5set Correl0a.7tion
1.0
First-to-Default Premium
0.8
0.6
0.4
0.2
The Value of First-to-Default v. s. the Asset Correlation
5-Asset
20-Asset
29

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