transprobbytotals

Estimate transition probabilities using totals structure input

Syntax

[transMat,sampleTotals] = transprobbytotals(totals)

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

Description

[transMat,sampleTotals] = transprobbytotals(totals) estimates transition probabilities using a totals structure input. transprobbytotals is useful for removing outlier information, obtaining bootstrapped confidence intervals, or computing transition probability estimates for different periodicity parameters (1-year transitions, 2-year transitions, and so on) efficiently.

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

Examples

Estimate Transition Probabilities Using a totals Structure Input

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

load Data_TransProb

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

transMat = 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

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

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

transMat1 = 8×8

   93.1172    5.8427    0.8231    0.1763    0.0377    0.0012    0.0001    0.0017
    1.6213   93.1501    4.3584    0.6614    0.1631    0.0055    0.0004    0.0397
    0.1239    2.9027   92.2297    4.0628    0.5367    0.0661    0.0028    0.0753
    0.0236    0.2313    5.0070   90.1825    3.7986    0.4734    0.0642    0.2193
    0.0216    0.1134    0.6357    5.7959   88.9866    3.4497    0.2920    0.7050
    0.0010    0.0062    0.1081    0.8697    7.3367   86.7217    2.5171    2.4395
    0.0002    0.0011    0.0120    0.2591    1.4340    4.3034   81.3027   12.6875
         0         0         0         0         0         0         0  100.0000

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

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

DefProb = 7×5

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

Input Arguments

totals — Total transitions observed

structure | struct array

Total transitions observed, specified as a structure, or a struct array of length nTotals, with fields:

• totalsVec — A sparse vector of size 1-by-nRatings1.

• totalsMat — A sparse matrix of size nRatings1-by-nRatings2 with nRatings1 ≤ nRatings2.

• 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 rating j (all the diagonal elements are 0). The total time spent on rating i is stored in totalsVec(i). For example, you have three rating categories, Investment Grade (IG), Speculative Grade (SG), and Default (D), and the following information:

Total time spent    IG       SG       D
in rating:       4859.09  1503.36  1162.05
 
Transitions             IG   SG    D
out of (row)       IG    0   89    7
into (column):     SG  202    0   32
                    D    0    0    0

Then:

totals.totalsVec = [4859.09  1503.36  1162.05]
totals.totalsMat = [  0   89    7
                    202    0   32
                      0    0    0]
totals.algorithm = 'duration'

For the 'cohort' algorithm, totalsMat(i,j) contains the total transitions observed from rating i to rating j, and totalsVec(i) is the initial count in rating i. For example, given the following information:

Initial count       IG     SG     D
in rating:        4808   1572   1145
 
Transitions         IG     SG     D
from (row)    IG  4721     80      7
to (column):  SG   193   1347     32
               D     0      0   1145

Then:

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

Common totals structures are the optional output arguments from transprob:

• sampleTotals — A single structure summarizing the totals information for the whole dataset.

• idTotals — A struct array with the totals information at the ID level.

Data Types: struct | structure

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 = transprobbytotals(Totals1,'transInterval',5)

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

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

Output Arguments

transMat — Matrix of transition probabilities in percent

matrix

Matrix of transition probabilities in percent, returned as a nRatings1-by-nRatings2 transition matrix.

sampleTotals — Structure with sample totals

structure

Structure with sample totals, returned with fields:

• totalsVec — A vector of size 1-by-nRatings1.

• totalsMat — A matrix of size nRatings1-by-nRatings2 with nRatings1 ≤ nRatings2.

• algorithm — A character vector with values 'duration' or 'cohort'.

If totals is a struct array, sampleTotals contains the aggregated information. That is, sampleTotals.totalsVec is the sum of totals(k).totalsVec over all k, and similarly for totalsMat. When totals is itself a single structure, sampleTotals and totals are the same.

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.

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