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# transprobbytotals

Estimate transition probabilities using totals structure input

## Syntax

[transMat,sampleTotals] = transprobbytotals(totals)
[transMat,sampleTotals] = transprobbytotals(totals,
Name,Value)

## Description

[transMat,sampleTotals] = transprobbytotals(totals) estimates transition probabilities using a totals structure input.

[transMat,sampleTotals] = transprobbytotals(totals,
Name,Value)
estimates transition probabilities using a totals structure input with additional options specified by one or more Name,Value pair arguments.

transprobbytotals is useful for removing outlier information, obtaining bootstrapped confidence intervals, or computing transition probability estimates for different periodicity parameters (1-year transitions, 2-year transitions, etc.) in an efficient manner.

## Input Arguments

 totals This can be:totalsVec — A sparse vector of size 1-by-nRatings1.totalsMat — A sparse matrix of size nRatings1-by-nRatings2 with nRatings1 ≤ nRatings2.algorithm — A string with values 'duration' or 'cohort'. For the 'duration' algorithm, totalsMat(i,j) contains the total transitions observed out of rating i into rating j (all the diagonal elements are 0). The total time spent on rating i is stored in totalsVec(i). For example, you have three rating categories, Investment Grade (IG), Speculative Grade (SG) and Default (D), and the following information:```Total time spent IG SG D in rating: 4859.09 1503.36 1162.05 Transitions IG SG D out of (row) IG 0 89 7 into (column): SG 202 0 32 D 0 0 0```Then:```totals.totalsVec = [4859.09 1503.36 1162.05] totals.totalsMat = [ 0 89 7 202 0 32 0 0 0] totals.algorithm = 'duration'``` For the 'cohort' algorithm, totalsMat(i,j) contains the total transitions observed from rating i to rating j, and totalsVec(i) is the initial count in rating i. For example, given the following information:```Initial count IG SG D in rating: 4808 1572 1145 Transitions IG SG D from (row) IG 4721 80 7 to (column): SG 193 1347 32 D 0 0 1145``` Then: ```totals.totalsVec = [4808 1572 1145] totals.totalsMat = [4721 80 7 193 1347 32 0 0 1145 totals.algorithm = 'cohort'``` Common totals structures are the optional output arguments from transprob:sampleTotals — A single structure summarizing the totals information for the whole dataset.idTotals — A struct array with the totals information at the ID level.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

 'snapsPerYear' Integer indicating the number of credit-rating snapshots per year to be considered for the estimation. Values are 1, 2, 3, 4, 6, or 12. This argument is only used with the cohort algorithm. Default: 1 — One snapshot per year 'transInterval' Length of the transition interval, in years. Default: 1 — One-year transition probabilities

## Output Arguments

 transMat Matrix of transition probabilities in percent. The size of the transition matrix is nRatings1-by-nRatings2. sampleTotals Structure with fields:totalsVec — A vector of size 1-by-nRatings1.totalsMat — A matrix of size nRatings1-by-nRatings2 with nRatings1 ≤ nRatings2.algorithm — A string with values 'duration' or 'cohort'. If totals is a struct array, sampleTotals contains the aggregated information. That is, sampleTotals.totalsVec is the sum of totals(k).totalsVec over all k, and similarly for totalsMat. When totals is itself a single structure, sampleTotals and totals are the same.

## Examples

expand all

### Estimate Transition Probabilities Using a totals Structure Input

Use historical credit rating input data from Data_TransProb.mat and transprob to generate input for transprobbytotals:

```load Data_TransProb

% Call TRANSPROB with three output arguments
[transMat, sampleTotals, idTotals] = transprob(data);
transMat
```
```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

```

Suppose companies 4 and 27 are outliers and you want to remove them from the pre-processed idTotals struct array and estimate the new transition probabilities.

```idTotals([4 27]) = [];
[transMat1, sampleTotals1] = transprobbytotals(idTotals);
transMat1
```
```transMat1 =

Columns 1 through 7

93.1172    5.8427    0.8231    0.1763    0.0377    0.0012    0.0001
1.6213   93.1501    4.3584    0.6614    0.1631    0.0055    0.0004
0.1239    2.9027   92.2297    4.0628    0.5367    0.0661    0.0028
0.0236    0.2313    5.0070   90.1825    3.7986    0.4734    0.0642
0.0216    0.1134    0.6357    5.7959   88.9866    3.4497    0.2920
0.0010    0.0062    0.1081    0.8697    7.3367   86.7217    2.5171
0.0002    0.0011    0.0120    0.2591    1.4340    4.3034   81.3027
0         0         0         0         0         0         0

Column 8

0.0017
0.0397
0.0753
0.2193
0.7050
2.4395
12.6875
100.0000

```

Obtain the 1-year, 2-year, 3-year, 4-year, and 5-year default probabilities, without the outlier information (i.e., using sampleTotals1).

```DefProb = zeros(7,5);
for t = 1:5
transMatTemp = transprobbytotals(sampleTotals1,'transInterval',t);
DefProb(:,t) = transMatTemp(1:7,8);
end
DefProb
```
```DefProb =

0.0017    0.0070    0.0159    0.0285    0.0450
0.0397    0.0828    0.1299    0.1813    0.2377
0.0753    0.1606    0.2567    0.3640    0.4831
0.2193    0.4675    0.7430    1.0445    1.3700
0.7050    1.4668    2.2759    3.1232    4.0000
2.4395    4.9282    7.4071    9.8351   12.1847
12.6875   23.1184   31.7177   38.8282   44.7266

```

## More About

expand all

### Cohort Estimation

The cohort algorithm estimates the transition probabilities based on a sequence of snapshots of credit ratings at regularly spaced points in time. If the credit rating of a company changes twice between two snapshot dates, the intermediate rating is overlooked and only the initial and final ratings influence the estimates. For more information, see Algorithms.

### Duration Estimation

Unlike the cohort algorithm, the duration algorithm estimates the transition probabilities based on the full credit ratings history, looking at the exact dates on which the credit rating migrations occur. There is no concept of snapshots in this method, and all credit rating migrations influence the estimates, even when a company's rating changes twice within a short time. For more information, see Algorithms.

## References

Hanson, S., T. Schuermann, "Confidence Intervals for Probabilities of Default," Journal of Banking & Finance, Elsevier, vol. 30(8), pages 2281–2301, August 2006.

Löffler, G., P. N. Posch, Credit Risk Modeling Using Excel and VBA, West Sussex, England: Wiley Finance, 2007.

Schuermann, T., "Credit Migration Matrices," in E. Melnick, B. Everitt (eds.), Encyclopedia of Quantitative Risk Analysis and Assessment, Wiley, 2008.

## See Also

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