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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 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' will end 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

Use historical credit rating input data from Data_TransProb.mat:

```% Load input data from file Data_TransProb.mat.

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

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
```
```% Get credit quality thresholds
thresh = transprobtothresholds(transMat)```
```thresh =

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 transition probabilities
trans = transprobfromthresholds(thresh)```
```trans =

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```