# transprobfromthresholds

Convert from credit quality thresholds to transition probabilities

## Description

example

trans = transprobfromthresholds(thresh) transforms credit quality thresholds into transition probabilities

## Examples

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Use historical credit rating input data from Data_TransProb.mat, estimate transition probabilities with default settings.

% Estimate transition probabilities with default settings
transMat = transprob(data)
transMat = 8×8

93.1170    5.8428    0.8232    0.1763    0.0376    0.0012    0.0001    0.0017
1.6166   93.1518    4.3632    0.6602    0.1626    0.0055    0.0004    0.0396
0.1237    2.9003   92.2197    4.0756    0.5365    0.0661    0.0028    0.0753
0.0236    0.2312    5.0059   90.1846    3.7979    0.4733    0.0642    0.2193
0.0216    0.1134    0.6357    5.7960   88.9866    3.4497    0.2919    0.7050
0.0010    0.0062    0.1081    0.8697    7.3366   86.7215    2.5169    2.4399
0.0002    0.0011    0.0120    0.2582    1.4294    4.2898   81.2927   12.7167
0         0         0         0         0         0         0  100.0000

Obtain the credit quality thresholds.

thresh = transprobtothresholds(transMat)
thresh = 8×8

Inf   -1.4846   -2.3115   -2.8523   -3.3480   -4.0083   -4.1276   -4.1413
Inf    2.1403   -1.6228   -2.3788   -2.8655   -3.3166   -3.3523   -3.3554
Inf    3.0264    1.8773   -1.6690   -2.4673   -2.9800   -3.1631   -3.1736
Inf    3.4963    2.8009    1.6201   -1.6897   -2.4291   -2.7663   -2.8490
Inf    3.5195    2.9999    2.4225    1.5089   -1.7010   -2.3275   -2.4547
Inf    4.2696    3.8015    3.0477    2.3320    1.3838   -1.6491   -1.9703
Inf    4.6241    4.2097    3.6472    2.7803    2.1199    1.5556   -1.1399
Inf       Inf       Inf       Inf       Inf       Inf       Inf       Inf

Recover the transition probabilities.

trans = transprobfromthresholds(thresh)
trans = 8×8

93.1170    5.8428    0.8232    0.1763    0.0376    0.0012    0.0001    0.0017
1.6166   93.1518    4.3632    0.6602    0.1626    0.0055    0.0004    0.0396
0.1237    2.9003   92.2197    4.0756    0.5365    0.0661    0.0028    0.0753
0.0236    0.2312    5.0059   90.1846    3.7979    0.4733    0.0642    0.2193
0.0216    0.1134    0.6357    5.7960   88.9866    3.4497    0.2919    0.7050
0.0010    0.0062    0.1081    0.8697    7.3366   86.7215    2.5169    2.4399
0.0002    0.0011    0.0120    0.2582    1.4294    4.2898   81.2927   12.7167
0         0         0         0         0         0         0  100.0000

## Input Arguments

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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 MN, 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:
100%-1.87% = 98.13%
where 100% is the same as:
P[z<Inf]

Data Types: double

## Output Arguments

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Matrix of transition probabilities in percent, returned as a M-by-N matrix.

## References

[1] Gupton, G. M., C. C. Finger, and M. Bhatia. “CreditMetrics.” Technical Document, RiskMetrics Group, Inc., 2007.

## Version History

Introduced in R2011b