In this section, the way from raw data to estimated rating transition probabilities is described. First, migration events of the same kind are counted. The resulting migration counts are transformed into migration rates, which are used as estimates for the unknown transition probabilities. These estimates are complemented with estimated standard errors for two cases, for independence and for a special correlation structure.
We assume that credits or credit obligors are rated in
categories ranging from 1, the best rating category, to the
category
containing defaulted credits. The raw data consist
of a collection of migration events. The
observed migration events
form a
matrix with rows
We assume that each observation is a realization of a
random variable
with conditional probability
distribution
If the migration events are
independent, i. e., the
variables
are stochastically
independent,
is the observed value of the binomially
distributed random variable
The case of dependent rating migrations raises new problems. In this
context,
is distributed as sum of
correlated
Bernoulli variables, see (4.1), indicating for each
credit with initial rating
a migration to
by 1. If these
Bernoulli variables are pairwise correlated with correlation
, then the variance
of the unbiased estimator
for
is (Huschens and Locarek-Junge; 2000, p. 44)
The law of large numbers fails also if the correlations are different with either a common positive lower bound, or non vanishing positive average correlation or constant correlation blocks with positive correlations in each block (Finger; 1998, p. 5). This failing of the law of large numbers may not surprise a time series statistician, who is familiar with mixing conditions to ensure mean ergodicity of stochastic processes (Davidson; 1994, chapter 14). In statistical words, in the case of non-zero correlation the relative frequency is not a consistent estimator of the Bernoulli parameter.
The parameters may be modeled in consistent way in
the framework of a threshold normal model with a single parameter
(Gupton et al.; 1997; Kim; 1999; Basel Committee on Banking
Supervision; 2001). This
model specifies a special dependence structure based on a
standard multinormal distribution for a vector
with equicorrelation matrix
(Mardia et al.; 1979, p. 461), where
(
) is the
standardized asset return and
is the number of obligors. The
parameter
may be interpreted as a mean asset return
correlation. In this model each pair of variables
with
and
is
bivariate normally distributed with density function
Given
we can estimate the migration correlation
by the restricted Maximum-Likelihood estimator
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The quantlet
VaRRatMigCount
can be used to compute migration
counts from migration events, where d is the number of
categories including default and e is the
data
matrix containing
migration events. The result is assigned to
the variable counts, which is the
matrix of migration counts.
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The quantlet
VaRRatMigRate
computes migration rates and
related estimated standard errors for
periods from an input
matrix of migration counts and a given correlation parameter.
Here, c is a
array of
-period
migration counts and rho is a non-negative correlation
parameter as used in (4.6). For rho
the
independent case is computed.
The calculation uses stochastic integration in order to determine
the probability from (4.6). The accuracy of the
applied Monte Carlo procedure is controlled by the input
parameter s. For
the sample size is at least
. This guarantees that the user-specified value
s is an upper bound for the standard deviation of the
Monte Carlo estimator for
. Note that with increasing
accuracy (i. e. decreasing s) the computational effort
increases proportional to
.
The result is assigned to the variable b, which is a list containing:
The matrices b.nstart and b.nend have components given
by (4.2) and (4.3). The matrices b.etp, b.emc, and b.esd contain the
,
, and
from (4.4),
(4.7), and (4.9) for
and
. The estimates
are given
only for
. The matrix b.etv contains the
from (4.8) for
.
Note that
and
.