# transprobtothresholds

Convert from transition probabilities to credit quality thresholds

## Description

example

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

## Examples

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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 = 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

## Input Arguments

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Transition probabilities in percent, specified as a M-by-N matrix. Entries cannot be negative and cannot exceed 100, and all rows must add up to 100.

Any given row in the M-by-N input matrix trans determines a probability distribution over a discrete set of N ratings. If the ratings are 'R1',...,'RN', then for any row i trans(i,j) is the probability of migrating into 'Rj'. If trans is a standard transition matrix, then MN and 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.

The credit quality 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

This implies that thresh(i,1) = Inf, for all i. For example, suppose that there are only N=3 ratings, 'High', 'Low', and 'Default', with the following transition probabilities:

High   Low   Default
High  98.13   1.78   0.09
Low    0.81  95.21   3.98
The matrix of credit quality thresholds is:
High    Low    Default
High    Inf   -2.0814   -3.1214
Low     Inf    2.4044   -1.7530

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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Credit quality thresholds, 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