Credit quality thresholds, specified as a
M
-by-N
matrix of credit
quality thresholds.
In each row, the first element must be Inf
and the
entries must satisfy the following monotonicity
condition:
thresh(i,j) >= thresh(i,j+1), for 1<=j<N
The M
-by-N
input
thresh
and the
M
-by-N
output
trans
are related as follows. The thresholds
thresh
(i,j)
are critical values of a standard normal distribution
z, such
that:
trans(i,N) = P[z < thresh(i,N)],
trans(i,j) = P[z < thresh(i,j)] - P[z < thresh(i,j+1)], for 1<=j<N
Any given row in the output matrix trans
determines a probability distribution over a discrete set of
N
ratings 'R1'
,
...
, 'RN'
, so that for any row
i
trans
(i,j) is
the probability of migrating into
'Rj'
.
trans
can be a standard transition matrix, with
M
≤ N
, in which case row
i contains the transition probabilities for
issuers with rating 'Ri'
. But
trans
does not have to be a standard transition
matrix. trans
can contain individual transition
probabilities for a set of M
-specific issuers, with
M
> N
.
For example, suppose that there are only N
=3
ratings, 'High'
, 'Low'
, and
'Default'
, with these credit quality
thresholds:
High Low Default
High Inf -2.0814 -3.1214
Low Inf 2.4044 -1.7530
The
matrix of transition probabilities is
then:
High Low Default
High 98.13 1.78 0.09
Low 0.81 95.21 3.98
This means the probability of default for 'High'
is
equivalent to drawing a standard normal random number smaller than
−3.1214, or 0.09%. The probability that a 'High'
ends
up the period with a rating of 'Low'
or lower is
equivalent to drawing a standard normal random number smaller than
−2.0814, or 1.87%. From here, the probability of ending with a
'Low'
rating
is:
P[z<-2.0814] - P[z<-3.1214] = 1.87% - 0.09% = 1.78%
And
the probability of ending with a
'High'
rating
is:
where 100% is the
same as:
Data Types: double