Estimate transition probabilities from credit ratings data
[
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
)
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
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
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
data
— Credit migration dataUsing 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'
Data Input Type | ID (1st Column) | Date (2nd Column) | Rating (3rd Column) |
---|---|---|---|
Table |
|
|
|
An nRecords
-by-3
cell array of character vectors containing 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'
Data Input Type | ID (1st Column) | Date (2nd Column) | Rating (3rd Column) |
---|---|---|---|
Cell |
|
|
|
A preprocessed data structure obtained using transprobprep
. This data structure contains
the fields'idStart'
,
'numericDates'
,
'numericRatings'
, and
'ratingsLabels'
.
Data Types: table
| cell
| struct
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
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 windowdata
(default) | character vector | serial date number | datetimeEnd date of the estimation time window, specified as the
comma-separated pair consisting of 'endDate'
and a
date character vector, serial date number, or datetime object. The
endDate
cannot be a date before the
startDate
.
Data Types: char
| double
| datetime
'labels'
— Credit-rating scale{'AAA','AA','A','BBB','BB','B','CCC','D'}
(default) | cell array of character vectorsCredit-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 year1
(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 windowdata
(default) | character vector | serial date number | datetimeStart date of the estimation time window, specified as the
comma-separated pair consisting of 'startDate'
and a
date character vector, serial date number, or datetime object.
Data Types: char
| double
| datetime
'transInterval'
— Length of the transition interval in years1
(one year transition
probability) (default) | numericLength 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 | stringLabel 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
transMat
— Matrix of transition probabilities in percentMatrix of transition probabilities in percent, returned as a
nRatings
-by-nRatings
transition matrix.
sampleTotals
— Structure with sample totalsStructure 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 transitions observed out of rating
i into ratingj (all the
diagonal elements are zero). The total time spent on rating
i is stored in
totalsVec
(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 transitions observed from rating i
to rating j, and
totalsVec
(i) is the initial
count in rating i. For example, given the following
information:
Initial count IG SG D in rating: 4808 1572 1145 Transitions IG SG D from (row) IG 4721 80 7 to (column): SG 193 1347 32 D 0 0 1145
totals.totalsVec = [4808 1572 1145] totals.totalsMat = [4721 80 7 193 1347 32 0 0 1145 totals.algorithm = 'cohort'
idTotals
— IDs totalsIDs 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.
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.
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.
The algorithm first determines a sequence
t0,...,tK
of snapshot dates. The elapsed time, in years, between two consecutive snapshot
dates tk-1 and
tk 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 , the number of transition periods in which obligor n starts at rating i. These are added up over all obligors to get Ni, 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, is also computed. These are also added up to get , 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, is
These probabilities are arranged in a one-period transition matrix P0, where the i,j entry in P0 is Pij.
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:
For example, if ns = 4
and
Δt = 2
, P contains the
two-year transition probabilities estimated from quarterly snapshots.
Note
For the cohort algorithm, optional output arguments
idTotals
and sampleTotals
from
transprob
contain the following information:
idTotals(n).totalsVec
=
idTotals(n).totalsMat
=
idTotals(n).algorithm
=
'cohort'
sampleTotals.totalsVec
=
sampleTotals.totalsMat
=
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.
The algorithm computes , 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 , the total time spent in rating i, collectively, by all obligors in the sample. The algorithm also computes , 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 , 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 . Each off-diagonal entry of this matrix is an estimate of the transition rate out of rating i into rating j, and is
The diagonal entries are computed as:
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 , where exp denotes matrix exponentiation
(expm
in MATLAB).
Note
For the duration algorithm, optional output arguments
idTotals
and sampleTotals
from
transprob
contain the following information:
idTotals(n).totalsVec
=
idTotals(n).totalsMat
=
idTotals(n).algorithm
=
'duration'
sampleTotals.totalsVec
=
sampleTotals.totalsMat
=
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.
[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.
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