Measures of Joint Default Dependence Risk based on Copulas
Aihua Huang
1,a
and Wende Yi
2,b
1
Finance Department, Chongqing University of Arts and Sciences, Chongqing, 402160, China
2
School of Mathematics and Big Data, Chongqing University of Arts and Sciences, Chongqing, 402160, China
Keywords: Default, Copula, Conditional Dependence Probability, Financing Indexes.
Abstract: This paper studies the problem of forecasting joint default. The default is the result that the credit rating of an
obligor, determined by obligor’s operating situation and financing state, decreases to some certain degree.
The dependence relationship of financing indexes is investigated to judge the credit rating of an obligor and
the conditional dependence probability and probability density functions are proposed. A member of
conditional dependence risk relationships is completely characterized by the marginal distribution and the
copulas of random variables. These results can be applied to investigate the conditional dependence structure
and the conditional dependence measure of obligor’s assets and of the defaults among obligors.
1 INTRODUCTION
In the economic and financial market environment, a
default would have a chain effect on. The default is
the result that the credit rating of an obligor,
determined by obligor’s operating situation and
financing state, decreases to some certain degree. A
number of studies have investigated credit risk about
financial market and default correlation of obligors.
The KMV (Kealhofer, 1998) and Credit Metrics
(Gupton, et al., 1997) Models are the most important
and widely used industry models. A core assumption
of the KMV and Credit Metrics Models is the
multivariate normality of the latent variables, where
the latent variables often interpreted as the value of
the obligor’s assets. In these models default of an
obligor occurs if the latent variables fall below some
threshold which often interpreted as the value of the
obligor’s liabilities. Defaults are predictable since the
values of assets are continuous process. Indeed, at
any time investors know the nearness of the assets to
the default threshold, so that they are warned in
advance when a default is imminent. However, for
bond prices and credit spreads, prices converge
continuously to their default-contingent values can
not appear at all. This means that they fail to be
consistent in particular with the observed contagion
phenomena, although the existing structural
approaches provide important insights into the
relation between firms’ fundamentals and correlated
default events as well as practically most valuable
tools (Kay, 2004). A benchmark study was provided
on the basis of time to default in credit scoring using
survival analysis and identifying hidden patterns in
credit risk survival data using Generalised Additive
Models (Dirick, 2017,
Claeskens, 2017, Baesens, 2017,
Djeundje, 2019, Crook, 2019).
The default of an obligor is an asymptotical
accumulating process of firm’s assets decreasing. The
default will occur when the operating situation and
financing are distressed to some certain degree. The
indexes characterizing the credit rating of an obligor
are dependence on each other. It is useful of judging
the probability of default of an obligor and running
the credit risk to investigate the dependence structure
of indexes.
In this paper we provide a dependence model of
multivariate indexes based on copula functions for
forecasting the obligor’s default and the conditional
dependence relationship of some indexes. Based on
the properties of probability, we present the
conditional dependence probability and density
functions.
2 COPULAS AND THE LATENT
VARIABLE MODEL
Copulas are simply the joint distribution function of
random vectors with standard uniform marginal
distributions. The most important result in the copula
654
Huang, A. and Yi, W.
Measures of Joint Default Dependence Risk based on Copulas.
DOI: 10.5220/0011203700003440
In Proceedings of the International Conference on Big Data Economy and Digital Management (BDEDM 2022), pages 654-657
ISBN: 978-989-758-593-7
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

framework is due to (Sklar, 1959). That is, the copula
connects a multivariate distribution to its margins in
such a way that it captures the entire dependence
structure in the multivariate distribution. Their value
in statistics is that they provide a way of
understanding how marginal distributions of single
risks are coupled together to form joint distributions
of groups of risks.
Let
F
be a joint distribution function with
continuous margins
1
,,
m
FF
. Then there exists a
unique copula
[
]
[
]
:0,1 0,1
m
C →
such that
()()()
()
111
,, ,,
mmm
Fx x CFx F x=
(1)
holds. Conversely, if
C
is a copula and
1
,,
m
FF
are distribution functions, then the function
F
by (1)
is a joint distribution function with margins
1
,,
m
FF
.
For example, in the credit application, if the latent
variables
()
1
,,
m
XX X= 
have a multivariate Gaussian
distribution with correlation matrix
ρ
, then the
copula of
X
may be represented by
() ()()
()
Ga 1 1
11
,, ,,
mm
Cu u u u
ρρ
=Φ Φ Φ
－－
,
where
ρ
Φ
denotes the joint distribution function of a
standard
m −
dimensional normal random vector
with correlation matrix
ρ
, and
Φ
is the distribution
function of univariate standard normal.
Ga
C
ρ
is called
as the Gaussian copula which characterizes the
dependence structure of the latent variables.
(David 2000) studied the default correlation via
the copula function approach. In his model the
random vector
()
1
,,
T
m
XX X= 
are interpreted as time-
until-default which implicate the survival time of
each defaultable entity or financial instrument, and
the thresholds
1
,,
m
D
D
are all set to take the value
T
, the time horizon. (Frey 2001, McNeil 2001)
proposed a latent variable model combine to copula
functions to investigate the credit. For random vector
X
with continuous marginal distributions,
deterministic cut-off levels vector
1
,,
m
DD
and the
binary random vector
()
1
,,
T
m
YY , such that the
following relationship holds:
1
iii
YXD=⇔ ≤
.
We define the
1
i
Y =
as default of obligor
i
at time
t
and
0
i
Y =
as non-default.
3 DEFAULT FORECASTING
MODELS OF OBLIGOR
In this section, we take into account the defaultable
probability of an obligor. Suppose that the credit
quality of the obligor at time
t
is completely
determined by its financing index
()
1
,,
T
tmt
XX X= 
which is a m− dimensional observable random
vector with continuous marginal distributions
i
F
and density functions
i
f
,
1, ,im= 
, respectively.
The financing indexes correlations are calibrated by
assuming that they follow a factor model, where the
underlying factors are interpreted as a set of
macroeconomic variables. Let
1
,,
m
D
D 
be a vector of
deterministic cut-off levels of financing indexes,
respectively, for determining the thresholds
i
D
,
1, ,im= 
, an option pricing technique based on
historical firm value data is used. Under normal
operation, commonly
it i
X
D>
,
1, ,im= 
, when the
credit rating of the obligor decreases, some financing
indexes become
kk
it i
X
D≤
,
{
}
{
}
1
,, 1,,
k
ii m⊂
.
Moreover, the procedure of the credit rating
decreasing is an asymptotic process. The lower the
credit rating is, the higher the probability of obligor’s
default is. Let
()
1
,,
T
tmt
XX X= 
have joint
distribution
H
such that
()
{
}
12 1 1 2 2
,,, , ,,
mtt mtm
H
xx x PX xX x X x=≤≤ ≤
,
By Sklar’s theorem, there exists a copula function
such that
()()()()
()
12 11 2 2
,,, , ,,
mmm
H
xx x CFx Fx F x=
.
So that copula
C
represents the probability of
event
1122
,,,
tt mtm
X
xX x X x≤≤ ≤
and simultaneously
describes the dependence structure of financing
indexes. The joint probability of all financing indexes
being less than the cut-off levels can be calculated by
{
}
1122
,,,
tt mtm
P
XDXD X D≤≤ ≤
() ( ) ( )
()
11 2 2
,,,
mm
CF D F D F D= 
.
Now we consider that
j
financing indexes
1
,,
j
it it
XX
among
1
,,
tmt
X
X
are more than the cut-off
levels and others
mj− financing indexes are less
than the cut-off levels. For simplify representing, we
Measures of Joint Default Dependence Risk based on Copulas
655

take into account the probability
{
}
11 1 1
,, ,
tjtjjtjmtm
PX D X DX D X D
++
>>≤ ≤
,
{
}
11 1 1
,, ,
tjtjjtjmtm
PX D X DX D X D
++
>>≤ ≤
()
()
()
()
()
()
()
11 11
1,,1,,, sgncc,,,
jj mm jj mm
CFDFD CFDFD
++ ++
=−

 
(2)
Where
() ( )
()
()
11 2 2
c,,,
jj
FD FD FD= 
,
()
1
ii
FD=
or
()
ii
FD
,
1,,ij= 
,
()
()
()
11 ',
sgn c
-1, 1 ' .
ii
ii
i f F D f or an odd number of i s
if F D for aneven number of i s
≠


=

≠


，
ji ,,2,1 =
.
By the properties of copula functions, we know
the
()
()
()
11
1, , 1, , ,
jj mm
CFDFD
++

is the marginal copula
of the copula
C
of
X
t
(and then is the copula of
variables
()
1
,,
j
tmt
XX
+

). Others are similar.
One is often interested in estimating or
forecasting certain conditional probability, such as
under the condition of some financing indexes lower
than the cut-off level values, calculating the
probability of which the remains are more than the
cut-off levels.
{
}
11 1 1
,,
tjtjjtjmtm
PX D X DX D X D
++
>>≤ ≤ 
{
}
{}
11 1 1
11
,, ,
,
tjtjjtjmtm
jt j mt m
PX D X DX D X D
PX D X D
++
++
>>≤ ≤
=
≤≤


()
()
()
()
()
()
()
()
()
()
11 11
11
1,,1,,, sgncc,,,
1,,1,,,
jj mm jj mm
jj mm
CFDFD CFDFD
CFDFD
++ ++
++
−
=

 

(3)
In similarly, we can obtain the conditional
probability
{
}
11 1 1
,, ,,
tjtjjtjmtm
PX D X DX D X D
++
≤≤≤ ≤
.
Furthermore, the conditional probability of
()
11
,,
j
tj mtm
XD XD
++
≤≤
under the condition
()
11
,,
tjtj
XD XD>>
.
{
}
11 11
,,
jt j mt m t jt j
PX D X D X D X D
++
≤≤>>
{
}
{}
11 1 1
11
,, ,
,
tjtjjtjmtm
tjtj
PX D X DX D X D
PX D X D
++
>>≤ ≤
=
>>


()
()
()
()
()
()
()
() ( )
11 11
1,,1,,, sgncc,,,
1sgncc,1,,1
jj mm jj mm
CFDFD CFDFD
C
++ ++
−
=
−


 

(4)
According to the definition and properties of
probability, we can obtain the conditional probability
density function
()
11
1
,,
jm j
jm
XXXX
f
xx
+
+


{
}
1
111 1 11
0
1
0
,,
lim
j
m
j
jt j j m mtm mt jt j
x
jm
x
Px X x x x X x x X D X D
xx
+
+
+
+++ +
Δ→
+
Δ→
< ≤ +Δ < ≤ +Δ > >
=
ΔΔ



()
()
()
()
()
()
()
() ( )
1,, 1 1 1,, 1 1
1,,1,,, sgncc,,,
1sgnc c,1,,1
jm j j mm jm j j mm
CFxFx CFxFx
C
+++ +++
−
=
−



 

()
()
11
j
jmm
f
xfx
++
⋅⋅⋅
. (5)
where
()
()
1
1, , 1
1
,,,
,,,
mj
jm
jm j m
jm
Cu u
Cuu
uu
−
+
++
+
∂
=
∂∂




.
In this section, we achieve some dependence
conditional probability functions and its density
functions under the conditions of some financing
indexes lower or more than the cut-off levels. These
results are very useful in credit risk management,
since the risk analysts need to analyse the conditional
dependence structure and conditional dependence
measure of financing indexes according to given
some conditions. Especially, the conditional
dependence risk probability and density functions
have particular meaning when the threshold values
equal some certain values such as
0
i
D=
and
()
ii
D
VaR X
α
=
.
In realistic application, it is very difficult that the
default of an obligor is exactly forecasted by the
obligor’s credit rating or its operating situation. The
default is a result which is affected by many factors.
Obligors have the same credit rating or similar
operating situation, but they have different credit
results. The default is capable of contagion among
obligors. Whenever an obligor suddenly defaults,
investors learn about the default threshold of closely
associated business partner obligors. This updating
leads to “contagious” jumps in credit spreads of
business partners.
BDEDM 2022 - The International Conference on Big Data Economy and Digital Management
656

4 CONCLUSIONS
In this paper, we have studied the dependence
structure of financing indexes of obligor, and the
conditional dependence risk probability and the
conditional dependence risk density functions. We
mainly focus on the scenarios under the conditions
such as
()
11
,,
jt j mt m
X
DXD
++
≤≤
and
()
11
,,
tjtj
XD XD>>
.
A member of conditional dependence risk
relationships is completely characterized by the
marginal distribution and the copulas of random
variables. These results can be applied to investigate
the conditional dependence structure and the
conditional dependence measure of obligor’s assets
and of the defaults among obligors.
FUNDING
This work was sponsored by Project Supported by the
Program for Innovation Team Building at Institutions
of Higher Education in Chongqing (Grant
No.KJTD201321) and the National Social Science
Fund of China (Grant No. 15XJY023).
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