transprob

Estimate transition probabilities from credit ratings data

Syntax

[transMat,sampleTotals,idTotals] = transprob(data)

[transMat,sampleTotals,idTotals] = transprob(___,Name,Value)

Description

[transMat,sampleTotals,idTotals] = transprob(data) constructs a transition matrix from historical data of credit ratings.

[transMat,sampleTotals,idTotals] = transprob(___,Name,Value) adds optional name-value pair arguments.

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 = 1×8
107 ×

    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 = 1×8 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 = 8×8 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 = 1×8 sparse double row vector (3 nonzeros)
   (1,2)          45000
   (1,3)          75000
   (1,4)          34000

idTotalsCohExposures(2).totalsMat

ans = 8×8 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:

   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'    

  Or an nRecords-by-4 MATLAB table containing weights and the historical credit ratings data of the form:

   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    

  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:

  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:

  '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'  

  Or an nRecords-by-4 cell array of character vectors if weights are included with the historical credit ratings data of the form:

  '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'

  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:

  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

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 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

Then

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 nIDs 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

Transition Probabilities

Transition probabilities quantify the probability of transitioning from one state (or condition) to another within a defined time frame.

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 t₀,...,t_K of snapshot dates. The elapsed time, in years, between two consecutive snapshot dates t_k-1 and t_k is equal to 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 P₀, where the i,j entry in P₀ is P_ij.

If the number of snapshots per year ns is 4 (quarterly snapshots), the probabilities in P₀ 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, Δt, and it is used to convert P₀ into the final transition matrix, P, according to the formula:

$$P=P_0^{ns\ast △t}$$

For example, if ns = 4 and Δt = 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 Δt (e.g., Δt = 2 corresponds to two-year transition probabilities), the transition matrix is obtained as $$P=\exp (\Delta t\Lambda )$$, 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.