# 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

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### 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 ```

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### 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.