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
`'R`*j*'

.
`trans`

can be a standard transition matrix, with
`M`

≤ `N`

, in which case row
*i* contains the transition probabilities for
issuers with rating `'R`*i*'

. 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`