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:

**Data Types: **`double`