5.3 Dependence modelling
To formalize the ratings-based approach, we
characterize each exposure
by a four-dimensional stochastic
vector
where for obligor
- (1)
is the driving stochastic process
for defaults and rating migrations,
- (2)
represent the initial and end-of-period rating
category,
- (3)
represents the credit loss (end-of-period exposure
value).
In this context
(which is, with reference to the Merton model,
often interpreted as
a proxy of the obligor's underlying
equity) is used to obtain the end-of-period state of
the obligor.
If we assume
rating classes, we obtain cut-off points
using the matrix of
transition probabilities
together with a distributional assumption on
.
Then, obligor
changes from rating
to rating
if the
variable
falls in the range
.
Our default-mode
framework implies two rating classes,
default resp. no-default, labeled as
resp. 0
(and thus only a single cut-off point
obtained from the probability of default).
Furthermore, interpreting
as the individual loss function,
(no default) and according to
our zero recovery assumption
.
To illustrate the methodology we plot in Figure 5.1 two
simulated drivers
and
together with the corresponding
cut-off points
and
.
Figure:
Two simulated driver
and the corresponding cut-off points
for default.
XFGSCP01.xpl
|
5.3.1 Factor modelling
In a typical credit
portfolio model dependencies of individual obligors are modelled
via dependencies of the underlying latent variables
. In the typical portfolio analysis the vector
is embedded in a factor model, which allows for easy
analysis of correlation, the typical measure of dependence. One
assumes that the underlying variables
are driven by a vector
of common factors. Typically, this vector is assumed to be
normally distributed (see e.g. JP Morgan (1997)). Thus,
with
a
-dimensional
normal vector and
independent normally distributed random variables, independent
also from
, we define
 |
(5.1) |
Here
describes the exposure of obligor
to factor
,
i.e. the so-called factor loading, and
is the
volatility of the idiosyncratic risk contribution.
In such a framework one can easily interfere default correlation
from the correlation of the underlying drivers
. To do so,
we define default indicators
where
is the cut-off point for default of obligor
.
The individual default probabilities are
and the joint default probability is
If we denote by
the correlation of the underlying
latent variables and by
the default
correlation of obligors
and
, then we obtain for the default correlation the simple
formula
 |
(5.2) |
Under the assumption that
are bivariate normal, we obtain for the
joint default probability
where
is bivariate normal density with
correlation coefficient
. Thus, asset (factor) correlation influences default correlation by entering
in joint default probability. Within the Gaussian framework we can
easily evaluate the above quantities, see (5.1).
Table 5.1:
Effect of asset correlation on default correlation
Asset correlation |
Default correlation |
|
|
*[-2pt]
0.1 |
0.0094 |
0.2 |
0.0241 |
0.3 |
0.0461 |
|
We see, that under our modelling assumption default correlation is of an order of magnitude smaller than
asset correlation (which is also supported by empirical evidence).
5.3.2 Copula modelling
As an alternative approach to the factor assumption, we can model each of the
underlying variables independently and subsequently use a copula to generate
the dependence structure. (For basic facts on copulae we refer the reader to
Chapter 2 and the references given there.)
So, suppose we have specified the individual distributions
of the variables
and a copula
for the dependence
structure. Then, for any subgroup of obligors
,
we have for the joint default probability
where we denote by
the
-dimensional margin
of
. In particular, the joint default probability of two
obligors is now
To study the effect of different copulae on default correlation,
we use the following examples of copulae (further details on
these copulae can be found in Embrechts et al. (2001)).
- 1.
- Gaussian copula:
Here
denotes the joint distribution function of the
-variate normal with linear correlation matrix
,
and
the inverse of the distribution function of the
univariate standard normal.
- 2.
-copula:
where
denotes the distribution function of an
-variate
-distributed random vector with parameter
and
linear correlation matrix
. Furthermore,
is the
univariate
-distribution function with parameter
.
- 3.
- Gumbel copula:
where
. This class of copulae is a sub-class of the class of
Archimedean copulae. Furthermore, Gumbel copulae have applications in multivariate
extreme-value theory.
In Table 5.2
joint default probabilities of two obligors are reported using three types of
obligors with individual default probabilities roughly corresponding to
rating classes A,B,C. We assume that underlying variables
are univariate normally
distributed and model the joint dependence structure using the above
copulae.
Table 5.2:
Copulae and default probabilities
Copula |
Default probability |
|
class A (
) |
class B (
) |
class C (
) |
|
|
|
|
*[-2pt]
Gaussian |
6.89 |
3.38 |
52.45 |
 |
46.55 |
7.88 |
71.03 |
 |
134.80 |
15.35 |
97.96 |
Gumbel,  |
57.20 |
14.84 |
144.56 |
Gumbel,  |
270.60 |
41.84 |
283.67 |
|
The computation shows that
and Gumbel copulae have higher joint default probabilities
than the Gaussian copula (with obvious implication for default correlation,
see equation (5.2)). To explain the reason for this we need the concept of
tail dependence:
DEFINITION 5.1
Let

and

be continuous random variables with distribution
functions

and

. The coefficient of upper tail dependence of

and

is
![$\displaystyle \lim_{u \rightarrow 1} \textrm{P}[Y > G^{-1}(u) \vert X >F^{-1}(u)]= \lambda_U$](xfghtmlimg1003.gif) |
(5.3) |
provided that the limit
![$ \lambda_U\in[0,1]$](xfghtmlimg1004.gif)
exists. If
![$ \lambda_U
\in (0,1]$](xfghtmlimg1005.gif)
,

and

are said to be asymptotically dependent in
the upper tail; if

,

and

are said to be asymptotically independent in
the upper tail.
For continuous distributions
and
one can replace (5.3)
by a version involving the bivariate copula directly:
 |
(5.4) |
Lower tail dependence, which is more relevant to our current purpose,
is defined in a similar way. Indeed, if
 |
(5.5) |
exists, then
exhibits lower tail dependence if
, and lower tail independence if
.
It can be shown that random variables linked by Gaussian copulae have no tail-dependence, while
the use of
and the Gumbel copulae results in tail-dependence. In fact, in case of the
copula, we have increasing tail dependence with decreasing
parameter
, while for the Gumbel family tail dependence increases
with increasing parameter
.