Transition probabilities in percent, specified as a
M-by-N matrix. Entries cannot
be negative and cannot exceed 100, and all rows must add up to
100.
Any given row in the M-by-N
input matrix trans determines a probability
distribution over a discrete set of N ratings. If the
ratings are
'R1',...,'RN',
then for any row i
trans(i,j) is
the probability of migrating into 'Rj'. If
trans is a standard transition matrix, then
M ≦ N and 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.
The credit quality 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
This implies that thresh(i,1)
= Inf, for all i. For example,
suppose that there are only N=3 ratings,
'High', 'Low', and
'Default', with the following transition
probabilities:
High Low Default
High 98.13 1.78 0.09
Low 0.81 95.21 3.98
The matrix of credit quality thresholds
is:
High Low Default
High Inf -2.0814 -3.1214
Low Inf 2.4044 -1.7530
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:
