M byN matrix with transition
probabilities, in percent. Entries cannot be negative and cannot exceed
100, and all rows must add up to 100.
Any given row in the M byN 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:
