The purpose here is to generate portfolios with given marginals
(normal) and the above copulae. We focus on the Gaussian and -copula
case.
For the generation of an -variate Normal with linear
correlation matrix
,
,
we apply the quantlet
gennorm
.
To obtain realizations from a Gaussian copula we simply have to transform the
marginals:
The implementation of these algorithms in
XploRe
is very
straightforward. Indeed, using the quantlet
normal
we can
generate normally distributed random variables.
Naturally all the distribution functions needed are also
implemented,
cdfn
,
cdft
etc.
We simulate standard portfolios of size with all obligors
belonging to one rating class. We use three rating classes, named
A,B,C with default probabilities
roughly
corresponding to default probabilities from standard rating
classes, Ong (1999), p. 77.
For our first simulation exercise we assume that the underlying
variables are normally distributed within a single factor
framework, i.e.
in (5.1).
The factor loadings
in (5.1) are constant and chosen
so that the correlation for the underlying latent variables
is
, which
is a standard baseline value for credit portfolio simulations,
Kiesel et al. (1999). To generate different degrees
of tail correlation, we link the individual assets together using
a Gaussian, a
and a
-copula as implemented in
VaRcredN
and
VaRcredTcop
.
|
The default driver are normal for all obligors
in both quantlets.
p denotes the default probability
of an individual obligor and
rho is the asset correlation
. opt is an optional
list
parameter consisting of opt.alpha, the
significance level for VaR estimation and opt.nsimu, the number of
simulations.
Both quantlets return a
list
containing the mean, the
variance and the opt.alpha-quantile of the portfolio default distribution.
|
To assess the effects of increased correlation within parts of
the portfolio, we change the factor loading within parts of our portfolio.
We assume a second factor, i.e. in (5.1), for a sub-portfolio of
obligors increasing the
correlation of the latent variables
within the sub-portfolio to
.
In the simulation below, the quantlets
VaRcredN2
and
VaRcredTcop2
are used.
|
The number of obligors in the first (second) subportfolio is
d1 (d2). rho1 (rho2)
is the asset correlation generated by the first (second) factor.
The other parameters correspond to the parameters in
VaRcredN
and
VaRcredTcop
.
Such a correlation cluster might be
generated by a sector or regional exposure for a real portfolio. Again, degrees
of tail correlation are generated by using a Gaussian, a and a
-copula.
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Our simulation results indicate that the degree of tail dependence of the underlying copula plays a major role as a credit risk characteristicum. Thus, while analysis of the driving factors for the underlying variables (obligor equity, macroeconomic variables, ..) remains an important aspect in modelling credit risky portfolio, the copula linking the underlying variables together is of crucial importance especially for portfolios of highly rated obligors.