# Documentation

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# transprobfromthresholds

Convert from credit quality thresholds to transition probabilities

## Syntax

trans = transprobfromthresholds(thresh)

## Description

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

## Input Arguments

 thresh 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. 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 -1.7530The 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

## Output Arguments

 trans M-by-N matrix with transition probabilities, in percent.

## Examples

collapse all

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 =

Columns 1 through 7

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

Column 8

0.0017
0.0396
0.0753
0.2193
0.7050
2.4399
12.7167
100.0000

Obtain the credit quality thresholds.

thresh = transprobtothresholds(transMat)
thresh =

Columns 1 through 7

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

Column 8

-4.1413
-3.3554
-3.1736
-2.8490
-2.4547
-1.9703
-1.1399
Inf

Recover the transition probabilities.

trans = transprobfromthresholds(thresh)
trans =

Columns 1 through 7

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

Column 8

0.0017
0.0396
0.0753
0.2193
0.7050
2.4399
12.7167
100.0000

## References

Gupton, G. M., C. C. Finger, and M. Bhatia. "CreditMetrics." Technical Document, RiskMetrics Group, Inc., 2007.