# transprob

Estimate transition probabilities from credit ratings data

## Syntax

## Description

`[`

constructs a transition matrix from historical data of credit ratings.`transMat`

,`sampleTotals`

,`idTotals`

] = transprob(`data`

)

`[`

adds optional name-value pair arguments. `transMat`

,`sampleTotals`

,`idTotals`

] = transprob(___,`Name,Value`

)

## Examples

### Construct a Transition Matrix From a Table of Historical Data of Credit Ratings

Using the historical credit rating table as input data from `Data_TransProb.mat`

display the first ten rows and compute the transition matrix:

```
load Data_TransProb
data(1:10,:)
```

`ans=`*10×3 table*
ID Date Rating
____________ _______________ _______
{'00010283'} {'10-Nov-1984'} {'CCC'}
{'00010283'} {'12-May-1986'} {'B' }
{'00010283'} {'29-Jun-1988'} {'CCC'}
{'00010283'} {'12-Dec-1991'} {'D' }
{'00013326'} {'09-Feb-1985'} {'A' }
{'00013326'} {'24-Feb-1994'} {'AA' }
{'00013326'} {'10-Nov-2000'} {'BBB'}
{'00014413'} {'23-Dec-1982'} {'B' }
{'00014413'} {'20-Apr-1988'} {'BB' }
{'00014413'} {'16-Jan-1998'} {'B' }

```
% 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

Using the historical credit rating table input data from `Data_TransProb.mat`

, compute the transition matrix using the `cohort`

algorithm:

%Estimate transition probabilities with 'cohort' algorithm transMatCoh = transprob(data,'algorithm','cohort')

`transMatCoh = `*8×8*
93.1345 5.9335 0.7456 0.1553 0.0311 0 0 0
1.7359 92.9198 4.5446 0.6046 0.1560 0 0 0.0390
0.1268 2.9716 91.9913 4.3124 0.4711 0.0544 0 0.0725
0.0210 0.3785 5.0683 89.7792 4.0379 0.4627 0.0421 0.2103
0.0221 0.1105 0.6851 6.2320 88.3757 3.6464 0.2873 0.6409
0 0 0.0761 0.7230 7.9909 86.1872 2.7397 2.2831
0 0 0 0.3094 1.8561 4.5630 80.8971 12.3743
0 0 0 0 0 0 0 100.0000

Using the historical credit rating data with ratings investment grade (`'IG'`

), speculative grade (`'SG'`

), and default (`'D'`

), from `Data_TransProb.mat`

display the first ten rows and compute the transition matrix:

dataIGSG(1:10,:)

`ans=`*10×3 table*
ID Date Rating
____________ _______________ ______
{'00011253'} {'04-Apr-1983'} {'IG'}
{'00012751'} {'17-Feb-1985'} {'SG'}
{'00012751'} {'19-May-1986'} {'D' }
{'00014690'} {'17-Jan-1983'} {'IG'}
{'00012144'} {'21-Nov-1984'} {'IG'}
{'00012144'} {'25-Mar-1992'} {'SG'}
{'00012144'} {'07-May-1994'} {'IG'}
{'00012144'} {'23-Jan-2000'} {'SG'}
{'00012144'} {'20-Aug-2001'} {'IG'}
{'00012937'} {'07-Feb-1984'} {'IG'}

transMatIGSG = transprob(dataIGSG,'labels',{'IG','SG','D'})

`transMatIGSG = `*3×3*
98.6719 1.2020 0.1261
3.5781 93.3318 3.0901
0 0 100.0000

Using the historical credit rating data with numeric ratings for investment grade (`1`

), speculative grade (`2`

), and default (`3`

), from `Data_TransProb.mat`

display the first ten rows and compute the transition matrix:

dataIGSGnum(1:10,:)

`ans=`*10×3 table*
ID Date Rating
____________ _______________ ______
{'00011253'} {'04-Apr-1983'} 1
{'00012751'} {'17-Feb-1985'} 2
{'00012751'} {'19-May-1986'} 3
{'00014690'} {'17-Jan-1983'} 1
{'00012144'} {'21-Nov-1984'} 1
{'00012144'} {'25-Mar-1992'} 2
{'00012144'} {'07-May-1994'} 1
{'00012144'} {'23-Jan-2000'} 2
{'00012144'} {'20-Aug-2001'} 1
{'00012937'} {'07-Feb-1984'} 1

`transMatIGSGnum = transprob(dataIGSGnum,'labels',{1,2,3})`

`transMatIGSGnum = `*3×3*
98.6719 1.2020 0.1261
3.5781 93.3318 3.0901
0 0 100.0000

### Create a Transition Matrix Using a Cell Array for Historical Data of Credit Ratings

Using a MATLAB® table containing the historical credit rating cell array input data (`dataCellFormat`

) from `Data_TransProb.mat`

, estimate the transition probabilities with default settings.

```
load Data_TransProb
transMat = transprob(dataCellFormat)
```

`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

Using the historical credit rating cell array input data (`dataCellFormat`

), compute the transition matrix using the `cohort`

algorithm:

%Estimate transition probabilities with 'cohort' algorithm transMatCoh = transprob(dataCellFormat,'algorithm','cohort')

`transMatCoh = `*8×8*
93.1345 5.9335 0.7456 0.1553 0.0311 0 0 0
1.7359 92.9198 4.5446 0.6046 0.1560 0 0 0.0390
0.1268 2.9716 91.9913 4.3124 0.4711 0.0544 0 0.0725
0.0210 0.3785 5.0683 89.7792 4.0379 0.4627 0.0421 0.2103
0.0221 0.1105 0.6851 6.2320 88.3757 3.6464 0.2873 0.6409
0 0 0.0761 0.7230 7.9909 86.1872 2.7397 2.2831
0 0 0 0.3094 1.8561 4.5630 80.8971 12.3743
0 0 0 0 0 0 0 100.0000

### Visualize Transitions Data for `transprob`

This example shows how to visualize credit rating transitions that are used as an input to the `transprob`

function. The example also describes how the `transprob`

function treats rating transitions when the company data starts after the start date of the analysis, or when the end date of the analysis is after the last transition observed.

**Sample Data**

Set up fictitious sample data for illustration purposes.

data = {'ABC','17-Feb-2015','AA'; 'ABC','6-Jul-2017','A'; 'LMN','12-Aug-2014','B'; 'LMN','9-Nov-2015','CCC'; 'LMN','7-Sep-2016','D'; 'XYZ','14-May-2013','BB'; 'XYZ','21-Jun-2016','BBB'}; data = cell2table(data,'VariableNames',{'ID','Date','Rating'}); disp(data)

ID Date Rating _______ _______________ _______ {'ABC'} {'17-Feb-2015'} {'AA' } {'ABC'} {'6-Jul-2017' } {'A' } {'LMN'} {'12-Aug-2014'} {'B' } {'LMN'} {'9-Nov-2015' } {'CCC'} {'LMN'} {'7-Sep-2016' } {'D' } {'XYZ'} {'14-May-2013'} {'BB' } {'XYZ'} {'21-Jun-2016'} {'BBB'}

The `transprob`

function understands that this panel-data format indicates the dates when a new rating is assigned to a given company. `transprob`

assumes that such ratings remain unchanged, unless a subsequent row explicitly indicates a rating change. For example, for company `'ABC'`

, `transprob`

understands that the `'A'`

rating is unchanged for any date after `'6-Jul-2017'`

(indefinitely).

**Compute Transition Matrix and Transition Counts**

The `transprob`

function returns a transition probability matrix as the primary output. There are also optional outputs that contain additional information for how many transitions occurred. For more information, see `transprob`

for information on the optional outputs for both the `'cohort'`

and the `'duration'`

methods.

For illustration purposes, this example allows you to pick the `StartYear`

(limited to `2014`

or `2015`

for this example) and the `EndYear`

(`2016`

or `2017`

). This example also uses the `hDisplayTransitions`

helper function (see the Local Functions section) to format the transitions information for ease of reading.

StartYear = 2014; EndYear = 2017; startDate = datetime(StartYear,12,31,'Locale','en_US'); endDate = datetime(EndYear,12,31,'Locale','en_US'); RatingLabels = ["AAA","AA","A","BBB","BB","B","CCC","D"]; [tm,st,it] = transprob(data,'startDate',startDate,'endDate',endDate,'algorithm','cohort','labels',RatingLabels);

The transition probabilities of the `TransMat`

output indicate the probability of migrating between ratings. The probabilities are expressed in %, that is, they are multiplied by 100.

`hDisplayTransitions(tm,RatingLabels,"Transition Matrix")`

Transition Matrix AAA AA A BBB BB B CCC D ___ __ ___ ___ __ _ ___ ___ AAA 100 0 0 0 0 0 0 0 AA 0 50 50 0 0 0 0 0 A 0 0 100 0 0 0 0 0 BBB 0 0 0 100 0 0 0 0 BB 0 0 0 50 50 0 0 0 B 0 0 0 0 0 0 100 0 CCC 0 0 0 0 0 0 0 100 D 0 0 0 0 0 0 0 100

The transition counts are stored in the `sampleTotals`

optional output and indicate how many transitions occurred between ratings for the entire sample (that is, all companies).

`hDisplayTransitions(st.totalsMat,RatingLabels,"Transition counts, all companies")`

Transition counts, all companies AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 1 1 0 0 0 0 0 A 0 0 0 0 0 0 0 0 BBB 0 0 0 1 0 0 0 0 BB 0 0 0 1 1 0 0 0 B 0 0 0 0 0 0 1 0 CCC 0 0 0 0 0 0 0 1 D 0 0 0 0 0 0 0 1

The third output of `transprob`

is `idTotals`

that contains information about transitions at an ID level, company by company (in the same order that the companies appear in the input data).

Select a company to display the transition counts and a corresponding visualization of the transitions. The `hPlotTransitions`

helper function (see the Local Functions section) shows the transitions history for a company.

CompanyID = "ABC"; UniqueIDs = unique(data.ID,'stable'); [~,CompanyIndex] = ismember(CompanyID,UniqueIDs); hDisplayTransitions(it(CompanyIndex).totalsMat,RatingLabels,strcat("Transition counts, company ID: ",CompanyID))

Transition counts, company ID: ABC AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 1 1 0 0 0 0 0 A 0 0 0 0 0 0 0 0 BBB 0 0 0 0 0 0 0 0 BB 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0

hPlotTransitions(CompanyID,startDate,endDate,data,RatingLabels)

To understand how `transprob`

handles data when the first observed date is after the start date of the analysis, or whose last observed date occurs before the end date of the analysis, consider the following example. For company `'ABC'`

suppose that the analysis has a start date of `31-Dec-2014`

and end date of `31-Dec-2017`

. There are only two transitions reported for this company for that analysis time window. The first observation for `'ABC'`

happened on `17-Feb-2015`

. So the `31-Dec-2015`

snapshot is the first time the company is observed. By `31-Dec-2016`

, the company remained in the original `'AA'`

rating. By `31-Dec-2017`

, a downgrade to `'A'`

is recorded. Consistent with this, the transition counts show one transition from `'AA'`

to `'AA'`

(from the end of 2015 to the end of 2016), and one transition from `'AA'`

to `'A'`

(from the end of 2016 to the end of 2017). The plot shows the last rating as a dotted red line to emphasize that the last rating in the data is extrapolated indefinitely into the future. There is no extrapolation into the past; the company's history is ignored until a company rating is known for an entire transition period (`31-Dec-2015`

through `31-Dec-2016`

in the case of `'ABC'`

).

**Compute Transition Matrix Containing NR (Not Rated) Rating**

Consider a different sample data containing only a single company `'DEF'`

. The data contains transitions of company `'DEF'`

from `'A'`

to `'NR'`

rating and a subsequent transition from `'NR'`

to `'BBB'`

.

dataNR = {'DEF','17-Mar-2011','A'; 'DEF','24-Mar-2014','NR'; 'DEF','26-Sep-2016','BBB'}; dataNR = cell2table(dataNR,'VariableNames',{'ID','Date','Rating'}); disp(dataNR)

ID Date Rating _______ _______________ _______ {'DEF'} {'17-Mar-2011'} {'A' } {'DEF'} {'24-Mar-2014'} {'NR' } {'DEF'} {'26-Sep-2016'} {'BBB'}

`transprob`

treats `'NR'`

as another rating. The transition matrix below shows the estimated probability of transitioning into and out of `'NR'`

.

StartYearNR = 2010; EndYearNR = 2018; startDateNR = datetime(StartYearNR,12,31,'Locale','en_US'); endDateNR = datetime(EndYearNR,12,31,'Locale','en_US'); CompanyID_NR = "DEF"; RatingLabelsNR = ["AAA","AA","A","BBB","BB","B","CCC","D","NR"]; [tmNR,~,itNR] = transprob(dataNR,'startDate',startDateNR,'endDate',endDateNR,'algorithm','cohort','labels',RatingLabelsNR); hDisplayTransitions(tmNR,RatingLabelsNR,"Transition Matrix")

Transition Matrix AAA AA A BBB BB B CCC D NR ___ ___ ______ ___ ___ ___ ___ ___ ______ AAA 100 0 0 0 0 0 0 0 0 AA 0 100 0 0 0 0 0 0 0 A 0 0 66.667 0 0 0 0 0 33.333 BBB 0 0 0 100 0 0 0 0 0 BB 0 0 0 0 100 0 0 0 0 B 0 0 0 0 0 100 0 0 0 CCC 0 0 0 0 0 0 100 0 0 D 0 0 0 0 0 0 0 100 0 NR 0 0 0 50 0 0 0 0 50

Display the transition counts and corresponding visualization of the transitions.

`hDisplayTransitions(itNR.totalsMat,RatingLabelsNR,strcat("Transition counts, company ID: ",CompanyID_NR))`

Transition counts, company ID: DEF AAA AA A BBB BB B CCC D NR ___ __ _ ___ __ _ ___ _ __ AAA 0 0 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 0 0 A 0 0 2 0 0 0 0 0 1 BBB 0 0 0 2 0 0 0 0 0 BB 0 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 NR 0 0 0 1 0 0 0 0 1

hPlotTransitions(CompanyID_NR,startDateNR,endDateNR,dataNR,RatingLabelsNR)

To remove the `'NR'`

from the transition matrix, use the `'excludeLabels'`

name-value input argument in `transprob`

. The list of labels to exclude may or may not be specified in the name-value pair argument `labels`

. For example, both `RatingLabels`

and `RatingLabelsNR`

generate the same output from `transprob`

.

[tmNR,stNR,itNR] = transprob(dataNR,'startDate',startDateNR,'endDate',endDateNR,'algorithm','cohort','labels',RatingLabelsNR,'excludeLabels','NR'); hDisplayTransitions(tmNR,RatingLabels,"Transition Matrix")

Transition Matrix AAA AA A BBB BB B CCC D ___ ___ ___ ___ ___ ___ ___ ___ AAA 100 0 0 0 0 0 0 0 AA 0 100 0 0 0 0 0 0 A 0 0 100 0 0 0 0 0 BBB 0 0 0 100 0 0 0 0 BB 0 0 0 0 100 0 0 0 B 0 0 0 0 0 100 0 0 CCC 0 0 0 0 0 0 100 0 D 0 0 0 0 0 0 0 100

Display the transition counts and corresponding visualization of the transitions.

`hDisplayTransitions(itNR.totalsMat,RatingLabels,strcat("Transition counts, company ID: ",CompanyID_NR))`

Transition counts, company ID: DEF AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 0 A 0 0 2 0 0 0 0 0 BBB 0 0 0 2 0 0 0 0 BB 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0

hPlotTransitions(CompanyID_NR,startDateNR,endDateNR,dataNR,RatingLabels)

Consistent with the previous plot, the transition counts still show two transitions from `'A'`

to `'A'`

(from the end of 2012 to the end of 2014), and two transitions from `'BBB'`

to `'BBB'`

(from the end of 2017 to the end of 2019).

However, different from the previous plot, specifying `'NR'`

using the `'excludeLabels'`

name-value input argument of `transprob`

removes any transitions into and out of the `'NR'`

rating.

**Local Functions**

function hDisplayTransitions(TransitionsData,RatingLabels,Title) % Helper function to format transition information outputs TransitionsAsTable = array2table(TransitionsData,... 'VariableNames',RatingLabels,'RowNames',RatingLabels); fprintf('\n%s\n\n',Title) disp(TransitionsAsTable) end function hPlotTransitions(CompanyID,startDate,endDate,data,RatingLabels) % Helper function to visualize transitions between ratings Ind = string(data.ID)==CompanyID; DatesOriginal = datetime(data.Date(Ind),'Locale','en_US'); RatingsOriginal = categorical(data.Rating(Ind),flipud(RatingLabels(:)),flipud(RatingLabels(:))); stairs(DatesOriginal,RatingsOriginal,'LineWidth',2) hold on; % Indicate rating extrapolated into the future (arbitrarily select 91 % days after endDate as the last date on the plot) endDateExtrap = endDate+91; if endDateExtrap>DatesOriginal(end) DatesExtrap = [DatesOriginal(end); endDateExtrap]; RatingsExtrap = [RatingsOriginal(end); RatingsOriginal(end)]; stairs(DatesExtrap,RatingsExtrap,'LineWidth',2,'LineStyle',':') end hold off; % Add lines to indicate the snapshot dates % transprob uses 1 as the default for 'snapsPerYear', hardcoded here for simplicity % The call to cfdates generates the exact same snapshot dates that transprob uses snapsPerYear = 1; snapDates = cfdates(startDate-1,endDate,snapsPerYear)'; yLimits = ylim; for ii=1:length(snapDates) line([snapDates(ii) snapDates(ii)],yLimits,'Color','m') end title(strcat("Company ID: ",CompanyID)) end

### Visualize Transitions Data for `transprob`

This example shows how to visualize credit rating transitions that are used as an input to the `transprob`

function. The example also describes how the `transprob`

function treats rating transitions when the company data starts after the start date of the analysis, or when the end date of the analysis is after the last transition observed.

**Sample Data**

Set up fictitious sample data for illustration purposes.

data = {'ABC','17-Feb-2015','AA'; 'ABC','6-Jul-2017','A'; 'LMN','12-Aug-2014','B'; 'LMN','9-Nov-2015','CCC'; 'LMN','7-Sep-2016','D'; 'XYZ','14-May-2013','BB'; 'XYZ','21-Jun-2016','BBB'}; data = cell2table(data,'VariableNames',{'ID','Date','Rating'}); disp(data)

ID Date Rating _______ _______________ _______ {'ABC'} {'17-Feb-2015'} {'AA' } {'ABC'} {'6-Jul-2017' } {'A' } {'LMN'} {'12-Aug-2014'} {'B' } {'LMN'} {'9-Nov-2015' } {'CCC'} {'LMN'} {'7-Sep-2016' } {'D' } {'XYZ'} {'14-May-2013'} {'BB' } {'XYZ'} {'21-Jun-2016'} {'BBB'}

The `transprob`

function understands that this panel-data format indicates the dates when a new rating is assigned to a given company. `transprob`

assumes that such ratings remain unchanged, unless a subsequent row explicitly indicates a rating change. For example, for company `'ABC'`

, `transprob`

understands that the `'A'`

rating is unchanged for any date after `'6-Jul-2017'`

(indefinitely).

**Compute Transition Matrix and Transition Counts**

The `transprob`

function returns a transition probability matrix as the primary output. There are also optional outputs that contain additional information for how many transitions occurred. For more information, see `transprob`

for information on the optional outputs for both the `'cohort'`

and the `'duration'`

methods.

For illustration purposes, this example allows you to pick the `StartYear`

(limited to `2014`

or `2015`

for this example) and the `EndYear`

(`2016`

or `2017`

). This example also uses the `hDisplayTransitions`

helper function (see the Local Functions section) to format the transitions information for ease of reading.

StartYear = 2014; EndYear = 2017; startDate = datetime(StartYear,12,31,'Locale','en_US'); endDate = datetime(EndYear,12,31,'Locale','en_US'); RatingLabels = ["AAA","AA","A","BBB","BB","B","CCC","D"]; [tm,st,it] = transprob(data,'startDate',startDate,'endDate',endDate,'algorithm','cohort','labels',RatingLabels);

The transition probabilities of the `TransMat`

output indicate the probability of migrating between ratings. The probabilities are expressed in %, that is, they are multiplied by 100.

`hDisplayTransitions(tm,RatingLabels,"Transition Matrix")`

Transition Matrix AAA AA A BBB BB B CCC D ___ __ ___ ___ __ _ ___ ___ AAA 100 0 0 0 0 0 0 0 AA 0 50 50 0 0 0 0 0 A 0 0 100 0 0 0 0 0 BBB 0 0 0 100 0 0 0 0 BB 0 0 0 50 50 0 0 0 B 0 0 0 0 0 0 100 0 CCC 0 0 0 0 0 0 0 100 D 0 0 0 0 0 0 0 100

The transition counts are stored in the `sampleTotals`

optional output and indicate how many transitions occurred between ratings for the entire sample (that is, all companies).

`hDisplayTransitions(st.totalsMat,RatingLabels,"Transition counts, all companies")`

Transition counts, all companies AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 1 1 0 0 0 0 0 A 0 0 0 0 0 0 0 0 BBB 0 0 0 1 0 0 0 0 BB 0 0 0 1 1 0 0 0 B 0 0 0 0 0 0 1 0 CCC 0 0 0 0 0 0 0 1 D 0 0 0 0 0 0 0 1

The third output of `transprob`

is `idTotals`

that contains information about transitions at an ID level, company by company (in the same order that the companies appear in the input data).

Select a company to display the transition counts and a corresponding visualization of the transitions. The `hPlotTransitions`

helper function (see the Local Functions section) shows the transitions history for a company.

CompanyID = "ABC"; UniqueIDs = unique(data.ID,'stable'); [~,CompanyIndex] = ismember(CompanyID,UniqueIDs); hDisplayTransitions(it(CompanyIndex).totalsMat,RatingLabels,strcat("Transition counts, company ID: ",CompanyID))

Transition counts, company ID: ABC AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 1 1 0 0 0 0 0 A 0 0 0 0 0 0 0 0 BBB 0 0 0 0 0 0 0 0 BB 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0

hPlotTransitions(CompanyID,startDate,endDate,data,RatingLabels)

To understand how `transprob`

handles data when the first observed date is after the start date of the analysis, or whose last observed date occurs before the end date of the analysis, consider the following example. For company `'ABC'`

suppose that the analysis has a start date of `31-Dec-2014`

and end date of `31-Dec-2017`

. There are only two transitions reported for this company for that analysis time window. The first observation for `'ABC'`

happened on `17-Feb-2015`

. So the `31-Dec-2015`

snapshot is the first time the company is observed. By `31-Dec-2016`

, the company remained in the original `'AA'`

rating. By `31-Dec-2017`

, a downgrade to `'A'`

is recorded. Consistent with this, the transition counts show one transition from `'AA'`

to `'AA'`

(from the end of 2015 to the end of 2016), and one transition from `'AA'`

to `'A'`

(from the end of 2016 to the end of 2017). The plot shows the last rating as a dotted red line to emphasize that the last rating in the data is extrapolated indefinitely into the future. There is no extrapolation into the past; the company's history is ignored until a company rating is known for an entire transition period (`31-Dec-2015`

through `31-Dec-2016`

in the case of `'ABC'`

).

**Compute Transition Matrix Containing NR (Not Rated) Rating**

Consider a different sample data containing only a single company `'DEF'`

. The data contains transitions of company `'DEF'`

from `'A'`

to `'NR'`

rating and a subsequent transition from `'NR'`

to `'BBB'`

.

dataNR = {'DEF','17-Mar-2011','A'; 'DEF','24-Mar-2014','NR'; 'DEF','26-Sep-2016','BBB'}; dataNR = cell2table(dataNR,'VariableNames',{'ID','Date','Rating'}); disp(dataNR)

ID Date Rating _______ _______________ _______ {'DEF'} {'17-Mar-2011'} {'A' } {'DEF'} {'24-Mar-2014'} {'NR' } {'DEF'} {'26-Sep-2016'} {'BBB'}

`transprob`

treats `'NR'`

as another rating. The transition matrix below shows the estimated probability of transitioning into and out of `'NR'`

.

StartYearNR = 2010; EndYearNR = 2018; startDateNR = datetime(StartYearNR,12,31,'Locale','en_US'); endDateNR = datetime(EndYearNR,12,31,'Locale','en_US'); CompanyID_NR = "DEF"; RatingLabelsNR = ["AAA","AA","A","BBB","BB","B","CCC","D","NR"]; [tmNR,~,itNR] = transprob(dataNR,'startDate',startDateNR,'endDate',endDateNR,'algorithm','cohort','labels',RatingLabelsNR); hDisplayTransitions(tmNR,RatingLabelsNR,"Transition Matrix")

Transition Matrix AAA AA A BBB BB B CCC D NR ___ ___ ______ ___ ___ ___ ___ ___ ______ AAA 100 0 0 0 0 0 0 0 0 AA 0 100 0 0 0 0 0 0 0 A 0 0 66.667 0 0 0 0 0 33.333 BBB 0 0 0 100 0 0 0 0 0 BB 0 0 0 0 100 0 0 0 0 B 0 0 0 0 0 100 0 0 0 CCC 0 0 0 0 0 0 100 0 0 D 0 0 0 0 0 0 0 100 0 NR 0 0 0 50 0 0 0 0 50

Display the transition counts and corresponding visualization of the transitions.

`hDisplayTransitions(itNR.totalsMat,RatingLabelsNR,strcat("Transition counts, company ID: ",CompanyID_NR))`

Transition counts, company ID: DEF AAA AA A BBB BB B CCC D NR ___ __ _ ___ __ _ ___ _ __ AAA 0 0 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 0 0 A 0 0 2 0 0 0 0 0 1 BBB 0 0 0 2 0 0 0 0 0 BB 0 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 NR 0 0 0 1 0 0 0 0 1

hPlotTransitions(CompanyID_NR,startDateNR,endDateNR,dataNR,RatingLabelsNR)

To remove the `'NR'`

from the transition matrix, use the `'excludeLabels'`

name-value input argument in `transprob`

. The list of labels to exclude may or may not be specified in the name-value pair argument `labels`

. For example, both `RatingLabels`

and `RatingLabelsNR`

generate the same output from `transprob`

.

[tmNR,stNR,itNR] = transprob(dataNR,'startDate',startDateNR,'endDate',endDateNR,'algorithm','cohort','labels',RatingLabelsNR,'excludeLabels','NR'); hDisplayTransitions(tmNR,RatingLabels,"Transition Matrix")

Transition Matrix AAA AA A BBB BB B CCC D ___ ___ ___ ___ ___ ___ ___ ___ AAA 100 0 0 0 0 0 0 0 AA 0 100 0 0 0 0 0 0 A 0 0 100 0 0 0 0 0 BBB 0 0 0 100 0 0 0 0 BB 0 0 0 0 100 0 0 0 B 0 0 0 0 0 100 0 0 CCC 0 0 0 0 0 0 100 0 D 0 0 0 0 0 0 0 100

Display the transition counts and corresponding visualization of the transitions.

`hDisplayTransitions(itNR.totalsMat,RatingLabels,strcat("Transition counts, company ID: ",CompanyID_NR))`

Transition counts, company ID: DEF AAA AA A BBB BB B CCC D ___ __ _ ___ __ _ ___ _ AAA 0 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 0 A 0 0 2 0 0 0 0 0 BBB 0 0 0 2 0 0 0 0 BB 0 0 0 0 0 0 0 0 B 0 0 0 0 0 0 0 0 CCC 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0

hPlotTransitions(CompanyID_NR,startDateNR,endDateNR,dataNR,RatingLabels)

Consistent with the previous plot, the transition counts still show two transitions from `'A'`

to `'A'`

(from the end of 2012 to the end of 2014), and two transitions from `'BBB'`

to `'BBB'`

(from the end of 2017 to the end of 2019).

However, different from the previous plot, specifying `'NR'`

using the `'excludeLabels'`

name-value input argument of `transprob`

removes any transitions into and out of the `'NR'`

rating.

**Local Functions**

function hDisplayTransitions(TransitionsData,RatingLabels,Title) % Helper function to format transition information outputs TransitionsAsTable = array2table(TransitionsData,... 'VariableNames',RatingLabels,'RowNames',RatingLabels); fprintf('\n%s\n\n',Title) disp(TransitionsAsTable) end function hPlotTransitions(CompanyID,startDate,endDate,data,RatingLabels) % Helper function to visualize transitions between ratings Ind = string(data.ID)==CompanyID; DatesOriginal = datetime(data.Date(Ind),'Locale','en_US'); RatingsOriginal = categorical(data.Rating(Ind),flipud(RatingLabels(:)),flipud(RatingLabels(:))); stairs(DatesOriginal,RatingsOriginal,'LineWidth',2) hold on; % Indicate rating extrapolated into the future (arbitrarily select 91 % days after endDate as the last date on the plot) endDateExtrap = endDate+91; if endDateExtrap>DatesOriginal(end) DatesExtrap = [DatesOriginal(end); endDateExtrap]; RatingsExtrap = [RatingsOriginal(end); RatingsOriginal(end)]; stairs(DatesExtrap,RatingsExtrap,'LineWidth',2,'LineStyle',':') end hold off; % Add lines to indicate the snapshot dates % transprob uses 1 as the default for 'snapsPerYear', hardcoded here for simplicity % The call to cfdates generates the exact same snapshot dates that transprob uses snapsPerYear = 1; snapDates = cfdates(startDate-1,endDate,snapsPerYear)'; yLimits = ylim; for ii=1:length(snapDates) line([snapDates(ii) snapDates(ii)],yLimits,'Color','m') end title(strcat("Company ID: ",CompanyID)) end

### Create Exposure-Based Transition Matrix From Historical Data of Credit Ratings with Exposures

This example shows how to load a historical credit rating table and then use `transprob`

to compute the exposure-based transition matrix. The sample totals and ID totals are weighted by the exposure. For more information on the computation of transition probabilities with general weights, which specializes to exposure-based probabilities, see Algorithms.

```
load Data_TransProb
dataExposures(1:10,:)
```

`ans=`*10×4 table*
ID Date Rating Exposure
_____ ___________ _______ ________
10283 10-Nov-1984 {'CCC'} 8500
10283 12-May-1986 {'B' } 8500
10283 29-Jun-1988 {'CCC'} 8500
10283 12-Dec-1991 {'D' } 8500
13326 09-Feb-1985 {'A' } 7500
13326 24-Feb-1994 {'AA' } 7500
13326 10-Nov-2000 {'BBB'} 8500
14413 23-Dec-1982 {'B' } 8500
14413 20-Apr-1988 {'BB' } 8500
14413 16-Jan-1998 {'B' } 8500

The `"Weight"`

column is the fourth column, and in this example, it is the loan's exposure on an observation date. Note that the `transprob`

function also supports more general weights that are only required to be nonnegative and real.

**Use transprob With duration Algorithm**

Use `transprob`

to estimate the exposure based transition probabilities with default settings.

[transMatExposures,sampleTotalsExposures,idTotalsExposures] = transprob(dataExposures);

Display the exposure-based transition matrix using the default settings.

transMatExposures

`transMatExposures = `*8×8*
92.9124 6.1143 0.7937 0.1300 0.0470 0.0013 0.0001 0.0011
1.6083 93.2741 4.2951 0.6416 0.1552 0.0056 0.0005 0.0197
0.1205 3.1292 92.0483 4.0680 0.4639 0.0845 0.0034 0.0822
0.0190 0.2259 5.0466 90.1037 3.8386 0.4605 0.0819 0.2239
0.0219 0.1085 0.5943 5.9012 89.2276 3.2159 0.2776 0.6530
0.0010 0.0057 0.0654 1.0355 7.7249 85.9825 2.6259 2.5591
0.0002 0.0012 0.0131 0.3450 1.4889 4.1707 81.6593 12.3218
0 0 0 0 0 0 0 100.0000

Display the exposure-based sample totals with default settings that use the `duration`

algorithm.

sampleTotalsExposures.totalsVec

ans =1×810^{7}× 1.6488 2.8735 2.8788 2.5613 2.4690 1.3158 0.6614 2.1678

sampleTotalsExposures.totalsMat

`ans = `*8×8*
0 1081500 116000 17000 7000 0 0 0
496000 0 1326500 170000 42000 0 0 5000
29000 971000 0 1280500 118500 22000 0 22500
4000 38500 1417500 0 1090000 113000 21000 53500
5500 25500 116500 1620500 0 902500 65500 153000
0 0 3000 116000 1156500 0 411000 332500
0 0 0 21500 100500 327500 0 896000
0 0 0 0 0 0 0 0

sampleTotalsExposures.algorithm

ans = 'duration'

Display the exposure-based ID totals for the second obligor (ID `13326`

) with default settings that use the `duration`

algorithm.

idTotalsExposures(2).totalsVec

`ans = `*1x8 sparse double row vector (3 nonzeros)*
1.0e+04 *
(1,2) 5.0328
(1,3) 6.7808
(1,4) 3.6445

idTotalsExposures(2).totalsMat

`ans = `*8x8 sparse double matrix (2 nonzeros)*
(3,2) 7500
(2,4) 7500

idTotalsExposures(2).algorithm

ans = 'duration'

**Use transprob With Cohort Algorithm**

Use `transprob`

to estimate the exposure-based transition probabilities with the `cohort`

algorithm.

`[transMatCohExposures,sampleTotalsCohExposures,idTotalsCohExposures] = transprob(dataExposures,algorithm="cohort");`

Display the exposure-based transition matrix when using the `cohort`

algorithm.

transMatCohExposures

`transMatCohExposures = `*8×8*
92.9468 6.1934 0.7124 0.1044 0.0430 0 0 0
1.7148 93.0587 4.4778 0.5811 0.1497 0 0 0.0178
0.1393 3.1653 91.8358 4.2990 0.4017 0.0786 0 0.0803
0.0160 0.4148 5.1063 89.7052 4.0382 0.4529 0.0521 0.2144
0.0227 0.1054 0.5992 6.3851 88.6102 3.4198 0.2707 0.5868
0 0 0.0231 0.8706 8.3320 85.5894 2.7311 2.4538
0 0 0 0.4250 1.9731 4.3181 81.1793 12.1044
0 0 0 0 0 0 0 100.0000

Display the exposure-based sample totals when using the `cohort`

algorithm.

sampleTotalsCohExposures.totalsVec

`ans = `*1×8*
16283500 28049500 28006500 24949500 24197000 12980000 6588500 20952000

sampleTotalsCohExposures.totalsMat

`ans = `*8×8*
15135000 1008500 116000 17000 7000 0 0 0
481000 26102500 1256000 163000 42000 0 0 5000
39000 886500 25720000 1204000 112500 22000 0 22500
4000 103500 1274000 22381000 1007500 113000 13000 53500
5500 25500 145000 1545000 21441000 827500 65500 142000
0 0 3000 113000 1081500 11109500 354500 318500
0 0 0 28000 130000 284500 5348500 797500
0 0 0 0 0 0 0 20952000

sampleTotalsCohExposures.algorithm

ans = 'cohort'

Display the exposure-based ID totals for the second obligor (ID `13326`

) when using the `cohort`

algorithm

idTotalsCohExposures(2).totalsVec

`ans = `*1x8 sparse double row vector (3 nonzeros)*
(1,2) 45000
(1,3) 75000
(1,4) 34000

idTotalsCohExposures(2).totalsMat

`ans = `*8x8 sparse double matrix (5 nonzeros)*
(2,2) 37500
(3,2) 7500
(3,3) 67500
(2,4) 7500
(4,4) 34000

idTotalsCohExposures(2).algorithm

ans = 'cohort'

The `duration`

algorithm and the `cohort`

algorithm produce similar transition matrices. In each case, the totals are weighted by the exposures. For additional details, see Algorithms.

## Input Arguments

`data`

— Credit migration data

table | cell array of character vectors | preprocessed data structure

Using `transprob`

to estimate transition
probabilities given credit ratings historical data (that is, credit
migration data), the `data`

input can be one of the following:

An

`nRecords`

-by-`3`

MATLAB^{®}table containing the historical credit ratings data of the form:Or anID Date Rating __________ _____________ ______ '00010283' '10-Nov-1984' 'CCC' '00010283' '12-May-1986' 'B' '00010283' '29-Jun-1988' 'CCC' '00010283' '12-Dec-1991' 'D' '00013326' '09-Feb-1985' 'A' '00013326' '24-Feb-1994' 'AA' '00013326' '10-Nov-2000' 'BBB' '00014413' '23-Dec-1982' 'B'

`nRecords`

-by-`4`

MATLAB table containing weights and the historical credit ratings data of the form:where each row contains an ID (column 1), a date (column 2), a credit rating (column 3), and an optional weight (column 4). Column 3 is the rating assigned to the corresponding ID on the corresponding date. All information corresponding to the same ID must be stored in contiguous rows. Sorting this information by date is not required, but recommended for efficiency. When using a MATLAB table input, the names of the columns are irrelevant, but the ID, date, rating information, and weights are assumed to be in the first, second, third, and fourth columns, respectively. Also, when using a table input, the first and third columns can be categorical arrays, and the second can be a datetime array. The following summarizes the supported data types for table input:ID Date Rating Weight __________ _____________ ______ _____ '00010283' '10-Nov-1984' 'CCC' 1 '00010283' '12-May-1986' 'B' 1.4 '00010283' '29-Jun-1988' 'CCC' 1.8 '00010283' '12-Dec-1991' 'D' 0.2 '00013326' '09-Feb-1985' 'A' 0 '00013326' '24-Feb-1994' 'AA' 2 '00013326' '10-Nov-2000' 'BBB' 1.7 '00014413' '23-Dec-1982' 'B' 1.1

Data Input Type ID (1st Column) Date (2nd Column) Rating (3rd Column) Weight (Optional 4th Column) Table Numeric array

Cell array of character vectors

String array

Categorical array

Numeric array

Cell array of character vectors

String array

Datetime array

Numeric array

Cell array of character vectors

String array

Categorical array

Numeric array with nonnegative values

For an example of using the

`data`

input argument with an optional fourth column for`Weight`

, see Create Exposure-Based Transition Matrix From Historical Data of Credit Ratings with Exposures.**Note**If no weights are provided in a fourth column of the

`data`

, the default is to set all weights equal to`1`

. In this case, the weighted transition matrix output agrees with the ordinary, count-based transition matrix.An

`nRecords`

-by-`3`

cell array of character vectors with the historical credit ratings data of the form:Or an'00010283' '10-Nov-1984' 'CCC' '00010283' '12-May-1986' 'B' '00010283' '29-Jun-1988' 'CCC' '00010283' '12-Dec-1991' 'D' '00013326' '09-Feb-1985' 'A' '00013326' '24-Feb-1994' 'AA' '00013326' '10-Nov-2000' 'BBB' '00014413' '23-Dec-1982' 'B'

`nRecords`

-by-`4`

cell array of character vectors if weights are included with the historical credit ratings data of the form:where each row contains an ID (column 1), a date (column 2), a credit rating (column 3), and an optional weight (Column 4). Column 3 is the rating assigned to the corresponding ID on the corresponding date. All information corresponding to the same ID must be stored in contiguous rows. Sorting this information by date is not required, but recommended for efficiency. IDs, dates, and ratings are stored in character vector format, but they can also be entered in numeric format. The following summarizes the supported data types for cell array input:'00010283' '10-Nov-1984' 'CCC' '1.2' '00010283' '12-May-1986' 'B' '1' '00010283' '29-Jun-1988' 'CCC' '1.2' '00010283' '12-Dec-1991' 'D' '0.2' '00013326' '09-Feb-1985' 'A' '1.7' '00013326' '24-Feb-1994' 'AA' '1.3' '00013326' '10-Nov-2000' 'BBB' '1' '00014413' '23-Dec-1982' 'B' '1.8'

Data Input Type ID (1st Column) Date (2nd Column) Rating (3rd Column) Weight (Optional 4th Column) Cell Numeric elements

Character vector elements

Numeric elements

Character vector elements

Numeric elements

Character vector elements

Numeric elements with nonnegative values

**Note**If no weights are provided in a fourth column of the

`data`

, the default is to set all weights equal to`1`

. In this case, the weighted transition matrix output agrees with the ordinary, count-based transition matrix.A preprocessed data structure obtained using

`transprobprep`

. This data structure contains the fields`'idStart'`

,`'numericDates'`

,`'numericRatings'`

,`'Weights'`

(optional) , and`'ratingsLabels'`

.

**Data Types: **`table`

| `cell`

| `struct`

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **```
transMat =
transprob(data,'algorithm','cohort')
```

`algorithm`

— Estimation algorithm

`'duration'`

(default) | character vector with values are `'duration'`

or
`'cohort'`

Estimation algorithm, specified as the comma-separated pair
consisting of `'algorithm'`

and a character vector with
a value of `'duration'`

or `'cohort'`

.

**Data Types: **`char`

`endDate`

— End date of the estimation time window

latest date in `data`

(default) | datetime array | string array | date character vector | serial date number

End date of the estimation time window, specified as the
comma-separated pair consisting of `'endDate'`

and a
scalar datetime, string, date character vector, or serial date number.
The `endDate`

cannot be a date before the
`startDate`

.

**Data Types: **`char`

| `double`

| `string`

| `datetime`

`labels`

— Credit-rating scale

`{'AAA','AA','A','BBB','BB','B','CCC','D'}`

(default) | cell array of character vectors

Credit-rating scale, specified as the comma-separated pair
consisting of `'labels'`

and a
`nRatings`

-by-`1`

, or
`1`

-by-`nRatings`

cell array of
character vectors.

`labels`

must be consistent with the ratings
labels used in the third column of `data`

. Use a
cell array of numbers for numeric ratings, and a cell array for
character vectors for categorical ratings.

**Note**

When the input argument `data`

is a
preprocessed data structure obtained from a previous call to
`transprobprep`

, this optional input for
`'labels`

is unused because the labels
in the `'ratingsLabels'`

field of `transprobprep`

take priority.

**Data Types: **`cell`

`snapsPerYear`

— Number of credit-rating snapshots per year

`1`

(default) | numeric values are `1`

, `2`

,
`3`

, `4`

,
`6`

, or `12`

Number of credit-rating snapshots per year to be considered for
the estimation, specified as the comma-separated pair consisting of
`'snapsPerYear'`

and a numeric value of
`1`

, `2`

, `3`

,
`4`

, `6`

, or
`12`

.

**Note**

This parameter is only used with the
`'cohort'`

`algorithm`

.

**Data Types: **`double`

`startDate`

— Start date of the estimation time window

earliest date in `data`

(default) | datetime array | string array | date character vector | serial date number

Start date of the estimation time window, specified as the
comma-separated pair consisting of `'startDate'`

and a
scalar datetime, string, date character vector, or serial date
number.

**Data Types: **`char`

| `double`

| `string`

| `datetime`

`transInterval`

— Length of the transition interval in years

`1`

(one year transition
probability) (default) | numeric

Length of the transition interval, in years, specified as the
comma-separated pair consisting of `'transInterval'`

and a numeric value.

**Data Types: **`double`

`excludeLabels`

— Label that is excluded from the transition probability computation

`''`

(do not exclude any label) (default) | numeric | character vector | string

Label that is excluded from the transition probability computation,
specified as the comma-separated pair consisting of
`'excludeLabels'`

and a character vector, string,
or numerical rating.

If multiple labels are to be excluded,
`'excludeLabels'`

must be a cell array containing
all of the labels for exclusion. The type of the labels given in
`'excludeLabels'`

must be consistent with the data
type specified in the `labels`

input.

The list of labels to exclude may or may not be specified in
`labels`

.

**Data Types: **`double`

| `char`

| `string`

## Output Arguments

`transMat`

— Matrix of transition probabilities in percent

matrix

Matrix of transition probabilities in percent, returned as a
`nRatings`

-by-`nRatings`

transition matrix.

`sampleTotals`

— Structure with sample totals

structure

Structure with sample totals, returned with fields:

`totalsVec`

— A vector of size`1`

-by-`nRatings`

.`totalsMat`

— A matrix of size`nRatings`

-by-`nRatings`

.`algorithm`

— A character vector with values`'duration'`

or`'cohort'`

.

For the `'duration'`

algorithm,
`totalsMat`

(*i*,*j*)
contains the total weight which transitioned out of rating
*i* into rating *j* (all the diagonal
elements are zero). The total weighted time spent on rating
*i* is stored in
`totalsVec`

(*i*). If the default
weights are used,
`totalsMat`

(*i*,*j*)
contains the total transitions out of rating *i* into
rating *j* and
`totalsVec`

(*i*) stores the total time
spent on rating *i*.

For example, if there are 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

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 weight which is transitioned out of rating
*i* to rating *j*, and
`totalsVec`

(*i*) is the initial weight
in rating *i*. If the default weights are used, then
`totalsMat`

(*i*,*j*)
contains the total transitions out of rating *i* into
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

totals.totalsVec = [4808 1572 1145] totals.totalsMat = [4721 80 7 193 1347 32 0 0 1145 totals.algorithm = 'cohort'

`idTotals`

— IDs totals

struct array

IDs totals, returned as a struct array of size
`nIDs`

-by-`1`

, where
*n*IDs is the number of distinct IDs in column 1 of
`data`

when this is a table or cell array or,
equivalently, equal to the length of the `idStart`

field
minus 1 when `data`

is a preprocessed data structure from
`transprobprep`

. For each ID
in the sample, `idTotals`

contains one structure with the
following fields:

`totalsVec`

— A sparse vector of size`1`

-by-`nRatings`

.`totalsMat`

— A sparse matrix of size`nRatings`

-by-`nRatings`

.`algorithm`

— A character vector with values`'duration'`

or`'cohort'`

.

These fields contain the same information described for the output
`sampleTotals`

, but at an ID level. For example, for
`'duration'`

,
`idTotals`

(*k*).`totalsVec`

contains the total time that the *k*-th company spent on
each rating.

## More About

### 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.

### Duration Estimation

Unlike the cohort method, 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.

## Algorithms

### Cohort Estimation

The algorithm first determines a sequence
*t _{0},...,t_{K}*
of snapshot dates. The elapsed time, in years, between two consecutive snapshot
dates

*t*and

_{k-1}*t*is equal to

_{k}`1`

/ *ns*, where

*ns*is the number of snapshots per year. These

*K*+

`1`

dates
determine *K*transition periods.

The algorithm computes $${N}_{i}^{n}$$, the number of transition periods in which obligor
*n* starts at rating *i*. These are added up
over all obligors to get *N _{i}*, the number of
obligors in the sample that start a period at rating

*i*. The number periods in which obligor

*n*starts at rating

*i*and ends at rating

*j*, or migrates from

*i*to

*j*, denoted by$${N}_{ij}^{n}$$, is also computed. These are also added up to get $${N}_{ij}^{}$$, the total number of migrations from

*i*to

*j*in the sample.

The estimate of the transition probability from *i* to
*j* in one period, denoted by$${P}_{ij}^{}$$, is

$${P}_{ij}^{}=\frac{Nij}{Ni}$$

These probabilities are arranged in a one-period transition matrix
*P _{0}*, where the

*i,j*entry in

*P*is

_{0}*P*.

_{i}_{j}If the number of snapshots per year *ns* is 4 (quarterly
snapshots), the probabilities in *P _{0}* are
3-month (or 0.25-year) transition probabilities. You may, however, be interested in
1-year or 2-year transition probabilities. The latter time interval is called the
transition interval, Δ

*t*, and it is used to convert

*P*into the final transition matrix,

_{0}*P*, according to the formula:

$$P={P}_{0}^{ns\ast \u25b3t}$$

For example, if *ns* = `4`

and
Δ*t* = `2`

, *P* contains the
two-year transition probabilities estimated from quarterly snapshots.

When weights are provided, the calculation is similar. In this case, the number $${N}_{i}^{n}$$ is equal to the sum of the starting weights of transition periods
in which obligor *n* starts at rating *i*.
These are added up over all obligors to get
*N _{i}*. The quantity $${N}_{ij}^{n}$$ is computed as the sum of the starting weights of transition
periods in which obligor

*n*starts at rating

*i*and ends at rating

*j*. These are added up to get $${N}_{ij}^{}$$. The remainder of the computation proceeds as above.

**Note**

For the cohort algorithm, optional output arguments
`idTotals`

and `sampleTotals`

from
`transprob`

contain the following information:

`idTotals(n).totalsVec`

= $$({N}_{i}^{n})\forall i$$`idTotals(n).totalsMat`

= $$({N}_{i,j}^{n})\forall ij$$`idTotals(n).algorithm`

=`'cohort'`

`sampleTotals.totalsVec`

= $$({N}_{i}^{})\forall i$$`sampleTotals.totalsMat`

= $$({N}_{i,j}^{})\forall ij$$`sampleTotals.algorithm`

=`'cohort'`

For efficiency, the vectors and matrices in
`idTotals`

are stored as sparse arrays.

When ratings must be excluded (see the `excludeLabels`

name-value input argument), all transitions involving the excluded ratings are
removed from the sample. For example, if the `'NR'`

rating must be
excluded, any transitions into `'NR'`

and out of
`'NR'`

are excluded from the sample. The total counts for all
other ratings are adjusted accordingly. For more information, see Visualize Transitions Data for transprob.

### Duration Estimation

The algorithm computes $${T}_{i}^{n}$$, the total time that obligor *n* spends in rating
*i* within the estimation time window. These quantities are
added up over all obligors to get $${T}_{i}^{}$$, the total time spent in rating *i*,
collectively, by all obligors in the sample. The algorithm also computes $${T}_{ij}^{n}$$, the number times that obligor *n* migrates from
rating *i* to rating *j*, with
*i* not equal to *j*, within the estimation
time window. And it also adds them up to get $${T}_{ij}^{}$$, the total number of migrations, by all obligors in the sample,
from the rating *i* to *j*, with
*i* not equal to *j*.

To estimate the transition probabilities, the duration algorithm first computes a
generator matrix $$\Lambda $$. Each off-diagonal entry of this matrix is an estimate of the
transition rate out of rating *i* into rating *j*,
and is

$${\lambda}_{ij}^{}=\frac{{T}_{ij}^{}}{{T}_{i}^{}},i\ne j$$

The diagonal entries are computed as:

$${\lambda}_{ii}^{}=-{\displaystyle \sum _{j\ne i}^{}}{\lambda}_{ij}^{}$$

With the generator matrix and the transition interval Δ*t* (e.g.,
Δ*t* = `2`

corresponds to two-year transition
probabilities), the transition matrix is obtained as $$P=\mathrm{exp}(\Delta t\Lambda )$$, where *exp* denotes matrix exponentiation
(`expm`

in MATLAB).

When weights are provided, the calculation is similar. In this case, the number $${T}_{i}^{n}$$ is the total weighted time that obligor *n*
spends in rating *i* within the estimation window. In general, $${T}_{i}^{n}$$ will be a sum of terms, each of which is the length of a period
the obligor spent in rating *i* times the obligor's starting
weight during that period. The quantities $${T}_{i}^{n}$$ are added up over all obligors to get $${T}_{i}^{}$$, the total weighted time spent in rating *i*,
collectively, by all obligors in the sample. The quantity $${T}_{ij}^{n}$$ is computed as the sum of the weights of periods in which obligor
*n* starts at rating *i* and ends at
rating *j*, with *i* not equal
to *j*. The remainder of the computation proceeds as
above.

**Note**

For the duration algorithm, optional output arguments
`idTotals`

and `sampleTotals`

from
`transprob`

contain the following information:

`idTotals(n).totalsVec`

= $$({T}_{i}^{n})\forall i$$`idTotals(n).totalsMat`

= $$({T}_{i,j}^{n})\forall ij$$`idTotals(n).algorithm`

=`'duration'`

`sampleTotals.totalsVec`

= $$({T}_{i}^{})\forall i$$`sampleTotals.totalsMat`

= $$({T}_{i,j}^{})\forall ij$$`sampleTotals.algorithm`

=`'duration'`

For efficiency, the vectors and matrices in
`idTotals`

are stored as sparse arrays.

When ratings must be excluded (see the `excludeLabels`

name-value input argument), all transitions involving the exclude ratings are
removed from the sample. For example, if the `'NR'`

rating must be
excluded, any transitions into `'NR'`

and out of
`‘NR’`

are excluded from the sample. The total time spent in
`'NR'`

(or any other excluded rating) is also removed.

## References

[1] Hanson, S., T. Schuermann. "Confidence Intervals for Probabilities of
Default." *Journal of Banking & Finance.* Vol. 30(8),
Elsevier, August 2006, pp. 2281–2301.

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

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

## Version History

**Introduced in R2010b**

### R2024a: Support for weights as optional fourth column in `data`

`transprob`

supports an optional fourth column for weights in the
`data`

input.

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