Documentation

transprobtothresholds

Convert from transition probabilities to credit quality thresholds

Syntax

thresh = transprobtothresholds(trans)

Description

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

Input Arguments

trans

M-by-N matrix with transition probabilities, in percent. 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 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' 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<Inf]

Output Arguments

thresh

M-by-N matrix of credit quality thresholds.

Examples

expand all

Transform Transition Probabilities Into Credit Quality Thresholds

Use historical credit rating input data from Data_TransProb.mat. Load input data from file Data_TransProb.mat.

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

References

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

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