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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 -1.7530```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`]`

## 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.

```load Data_TransProb % 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.