In the multi-period case, transitions in credit ratings are also
characterized by rating transition matrices. The -period
transition matrix is labeled
. Its generic
element
gives the rating transition probability
from rating
to
over the
periods. For the sake
of simplicity the one-period transition matrix
is shortly denoted by
in the following. This
transition matrix is considered to be of type
. The
last row contains
expressing the absorbing
default state. Multi-period transition matrices can be
constructed from one-period transition matrices under the
assumption of the Markov property.
Let
be a discrete-time stochastic process
with countable state space. It is called a first-order
Markov chain if
Transferred to rating transitions, homogeneity and the Markov
property imply constant one-period transition matrices
independent of the time
, i. e.
obeys time-stability. Then the one-period
transition
matrix
contains the non-negative rating transition
probabilities
The two-period transition matrix is then calculated by ordinary
matrix multiplication,
.
Qualitatively, the composition of the portfolio after one period
undergoes the same transitions again. Extended for
periods
this reads as
The one-period transition matrix
is unknown and must
be estimated. The estimator
is associated
with estimation errors which consequently influence the estimated
multi-period transition matrices. The traditional approach to quantify this influence turns out to be
tedious since it is difficult to obtain the distribution of
, which could characterize the
estimation errors. Furthermore, the distribution of
, with
Assuming a homogeneous first-order Markov chain
, the rating transitions are generated from
the unknown transition matrix
. In the spirit of the
bootstrap method, the unknown transition matrix
is substituted by the estimated transition matrix
, containing transition rates. This then allows
to draw a bootstrap sample from the multinomial distribution
assuming independent rating migrations,
Then the bootstrap sample
is used to estimate a bootstrap transition matrix
with generic elements
according
We can now access the distribution of
by Monte Carlo sampling, e. g.
samples are drawn and
labeled
for
. Then
the distribution of
estimates the
distribution of
. This is justified since
the consistency of this bootstrap estimator has been proven by
Basawa et al. (1990). In order to characterize the
distribution of
, the standard deviation
which is the bootstrap estimator of
, is estimated by
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For time homogeneity, the -period rating transition matrices
are obtained by the quantlet
XFGRatMig5.xpl
(q = XFGRatMig5(p, m)).
It computes all
multi-period transition matrices given the one-period
matrix p. Note that the output q is a
array, which can be directly visualized by
XFGRatMig6.xpl
(XFGRatMig6(q))
returning a graphical output. To visualize
-period
transition matrices each with
elements for
, we plot
aggregated values
A typical example is shown in Figure 4.2 for the
one-year transition matrix given in
Nickell et al. (2000, p. 208), which uses Moody's unsecured
bond ratings between 31/12/1970 and 31/12/1997. According
(4.21), aggregated values are plotted for
. Thereby, the transition matrix is condensed for simplicity
to
with only 4 basic rating categories, see the
example in Section 4.2.4. Again, the last
category stands for defaulted credits. Estimation errors are
neglected in Figure 4.2.
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Bootstrapping is performed by the quantlet
VaRRatMigRateM
.
It takes as input counts, the
matrix of
migration counts, from which the bootstrap sample is generated.
Further, m denotes the number of periods and B
the number of generated bootstrap samples. The result is assigned
to the variable out, which is a list of the following
output:
In the following the bootstrapping is illustrated in an example.
As estimator
we use the
rating
transition matrix of small and medium-sized German bank
borrowers from Machauer and Weber (1998, p. 1375), shown in
Table 4.1. The data cover the period from
January 1992 to December 1996.
With the quantlet
VaRRatMigRateM
the
-period transition
probabilities are estimated by
and the
bootstrap estimators of their standard deviations are calculated.
This calculations are done for 1, 5 and 10 periods and
Monte Carlo steps. A part of the resulting
output is summarized in Table 4.2, only default
probabilities are considered. Note that the probabilities in
Table 4.1 are rounded and the following
computations are based on integer migration counts
.
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Based on the techniques presented in the last sections we can now
tackle the problem of portfolio migration, i. e. we can assess
the distribution of credits over the
rating categories
and its evolution over periods
. Here, a
stationary transition matrix
is assumed. The
randomly changing number of credits in category
at time
is labeled by
and allows to define non-negative
portfolio weights
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For periods the multi-period transition matrix
has to be used, see Section 4.3.1. Hence, (4.22) and
(4.23) are modified to
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