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DOI: https://doi.Org/10.17323/j.jcfr.2073-0438.15.4.2021.66-87 JEL classification: B40, G21, F65
(cc)
Approaches to Building Default Probability Models for Financial Instruments of Project Financing at Long Time Horizons
Project financing is one of the priority tools for stimulating the country's economic growth around the world, which allows the implementation of large-scale and capital-intensive projects, providing favorable credit conditions with insufficient creditworthiness of the project beneficiaries.
As a rule, project financing instruments are long-term (10-30 years, depending on the type of transaction), so this asset class is interesting for the implementation of the task of building long-term models for assessing credit risk associated with the introduction in 2018 of the new international financial reporting standard IFRS 9 "Financial Instruments". The new standard requires financial institutions to calculate their expected credit loss (ECL) at the time of granting loans and other banking products exposed to credit risk, taking into account different time horizons, which significantly changes the traditional approaches to assessing credit risk by commercial banks.
As part of this work, a model was built to assess the long-term probability of default for the portfolio of assets of a Russian commercial bank belonging to the project finance segment in accordance with the requirements of the International Financial Reporting standard IFRS 9 "Financial Instruments". At present, the topic of this work is extremely relevant and may be of interest both for commercial banks that are faced with the problem of improving credit risk assessment models
Keywords: IFRS 9, expected credit losses, credit risk assessment stages, project finance, pdlife-time For citation: Vasilieva, A. Approaches to Building Default Probability Models for Financial Instruments of Project Financing at Long Time Horizons. Korporativnye finansy = Journal of Corporate Finance Research.2021;15(4): 66-87. doi: https://doi.org/10.17323/j.jcfr.2073-0438.15.4.2021.66-87
The journal is an open access journal which means that everybody can read, download, copy, distribute, print, search, or link to the full texts of these articles in accordance with CC Licence type: Attribution 4.0 International (CC BY 4.0 http://creativecommons.org/licenses/by/4.0/).
Alfia Vasilieva
Associate Director of the Project and Structured Finance Ratings Group, Analytical Credit Rating Agency (ACRA), Moscow, Russia, alfiavaf@mail.ru, ORCID
Abstract
Introduction
Project financing (PF) is a method of long-term borrowing for large projects by means of "financial engineering" based on loan on cash flow created only by the project without recourse to the borrower. The fundamental feature of project financing is that for implementation of a certain project a special project enterprise is established (SPV, SPE) which attracts resources (not only funds) for project implementation, implements the project and squares accounts with creditors and project investors using the funds (cash flows) generated by the project itself [1]. For decades project financing was a preferable way of financing of large-scale infrastructure projects all over the world. A series of studies emphasized its importance, especially for the countries with emerging market economies and accentuated the correlation between investment in infrastructure and economic growth.
Long-termness is a distinctive feature of project financing transactions - implementation of some projects takes 30 years. Large-scale capital-intensive projects usually require significant initial investments and generate the revenue sufficient only to cover expenses in the long-term. Thus, some authors point out in their papers that on average
Figure 1. Global Volume of the Project Financing Market
loans aimed at financing projects have a longer maturity than other syndicated credits [5-7]. This distinctive feature makes this assets class an interesting research object as a part of the task of constructing long-term models of assessing credit risk according to requirements of IFRS 9.
In the recent 15-20 years investors' interest to transactions of project financing has been growing across the globe. First of all, it is due to financing of infrastructure facilities' construction under public-private partnership (PPP): from 1999 through 2019 the volume and number of performed transactions increased more than twice (Figure 1). Projects are increasingly financed by issue of bonds backed by cash-flows from implementation of infrastructure projects based on PPP. According to international practices default of this tool occurs less frequently even in economic turbulence periods [8].
As on September 2020 over 3,000 PPP projects were implemented in Russia. Their total value exceeds 4.5 trillion roubles and the share of private investments amounts to 3.1 trillion roubles (69%). The total value of PPP projects amounts to 44% of expenses for infrastructure which have been planned for implementation of national projects in 2019 [8].
450
400
350
300
260
200
150
15 100
T3
ZD so
c
-Q 0
Ill
i jeoa
1.400 1,200 1.000
1999 3000 3001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 201! 2013 2014 2015 2016 2017 201S
loans bonds number of transactions
Source: Moody's, ACRA.
Review of Methods and Models of Assessment of Credit Risk of Assets at a Long Time Horizon On the basis of PD estimates MPD values are calculated for each life period of an agreement. Then they are used to calculate ECL for the whole lifetime of the agreement.
The model is calibrated on the basis of the PIT (point in time) principle:
(1)
PDpIT =ßt .PD™,
where PDt calibration;
PIT
default probability on the basis of the PIT
PDt -
bration;
default probability on the basis of the TTC cali-
pt - scaling coefficient defined on the basis of data about the current default rate of the portfolio
The following methods are applied to evaluate PD:
a) Use of external data about defaults;
b) Methods based on the migration matrix;
c) Methods based on approximation of historic default levels;
d) Approach based on exponential curve extrapolation (a simplified approach).
Use of external data on defaults. This method implies a PD assessment based on migration of ratings information on which is provided by external rating agencies (S&P, Moody's, Fitch Ratings, ACRA).
In case the Bank has no statistics to build a migration matrix using internal data the migration matrix built on external data is used. Depending on the purpose of modelling statistics of one or several rating agencies may be applied.
In case of inversion in the data of external matrices the matrix is adjusted (by experts or applying mathematical methods of reducing the function to a monotone function) PD assessment on the basis of migration matrices. The migration matrix is a square matrix which elements contain probabilities of change (transition probabilities) of the rating category of a corresponding Borrower.
M =
Pii
Pn-1,1 0
0
Pin
pn —1,n i
(1)
where p^ - probability of transition to the j rating category in a certain period of time provided it belongs to the i rating category.
The Bank uses the rating scale of internal credit ratings to build the migration matrix (Appendix 1).
The Bank does not set upper and lower bounds of the values of the probability of default. According to IFRS 9 assessment of the probability of default is unbiassed. Subsequently, the conservatism concept enshrined in the model of probability of default assessment as per IRB Basel II [10] cannot be used to calculate PD in accordance with IFRS 9 and as part of upgrade of IRB of PD models in order to meet requirements of IFRS 9 such material adjustments are excluded (adjustment "PG not less than 0.03%" established in accordance with 483-P is also excluded) [4].
This is with the exception of the RF rating adjustment (the borrower's rating is not higher than the RF rating): this adjustment is preserved.
Depending on availability of data when constructing a migration matrix enlarged or initial rating categories may be used (for example, combining of ratings 7-, 7, 7+ into one category).
Assessment of probabilities of transition is determined by cohort analysis:
~ N (1)
Pi, =-T^T , (2)
J Ni (t -1)
where NiJ (t) - number of migrations from state I into state j in the t period;
Ni (t -1) - number of transactions in state I in the t - 1 period.
The probability of default at a 1-year horizon.
A one-year migration matrix M0 is constructed on the basis of statistics of observations for 12 calendar months. Shorter periods may be used in order to take into consideration the most relevant information. An average one-year migration matrix is calculated by computing the arithmetic mean of one-year migration matrices obtained every quarter (month).
A one-year probability of default (PDt) for each rating category is defined as the likelihood of transition into the
state of "10-default". In the migration matrix (PDt) is indicated in the last column of the one-year transition matrix. If statistical frequency of defaults does not correspond to the probability of default in each rating grade in the Bank's master scale scaling is performed.
The adjustments performed are recorded in the Report on the Model Development.
When assessing the PD indicator on the basis of migration matrices the following main assumptions are contemplated:
• further transitions to rating grades depend only on the current rating but not on previous ratings (property of Markov process);
• probabilities of migration do not depend on a certain time point, i.e. the transition rates are unchanged in time (homogeneity) [9].
A formular to calculate the probability of default for the lifetime of a financial instrument:
MT = Mj T , (3)
where T - lifetime of a financial instruments.
The column in the multiyear matrix which shows a probability of transition into default is the cumulative probability of default of a certain period (cPD). Use of the migration matrix allows to take into consideration complete information on migration of ratings when calculating the probability of default for the lifetime.
Profiles of cumulative PDs are made by evaluating parameters of cumulative DR distribution.
On the basis of the Weilbull distribution:
Parameters of the Weilbull distribution k and X are assessed using a linear regression of double logarithm of the survival function. The survivorship function is defined by the following formula:
S(t).1- F(t;k,X) , (4)
where F(t;K,X) is a two-parameter Weilbull distribution function.
F( t,K,X) = cDR ( t,K,X) =
1 — e w ,t > 0
0 ,t < 0.
(5)
where k > 0 defines the shape of the distribution function. k < 1 is indicative of a decrease of default rate in time,
k = 1 points at stability of default rate in time, k > 1 is indicative of increase of default rate in time; X > 0 is a scale parameter which regulates survivorship time [11].
On the basis of the modified Weilbull distribution:
• Modeling of cumulative PD is made by selecting such distribution parameters which describe most accurately the behaviour of cumulative default rates.
A two-parameter modified Weibull distribution function is as follows:
F( t,a,ß) = cDR ( t,a,ß) =
1 - e
-(a tß)
t > 0
(1 - e-1)
0 , t < 0
(6)
where a and P < 0 are parameters of the modified Weibull distribution;
cDR (t,a,p) is the cumulative default rate per year [11].
Construction of the model for the project financing segment
The target segment of this model is customers pertaining to the area of applying the Project Financing models in accordance with the bank's methodology. When developing of the present model the default definition stated in the section Terms and Definitions was used. The fact of assigning to a borrower of the "10-default" rating was taken into consideration as default characteristic.
At the date of developing the model Lt PD of the Project Financing segment 1,171 observations were available (key "TIN + reporting date", 325 default observations) for 116 borrowers since April 2009. In the segment of Project Financing (Developers) 1,334 observations were available (key "TIN + reporting date", 565 default observations) for 141 borrowers since April 2009. Moreover, the sample contains observations concerning 15 borrowers pertaining to both segments.
In view of the above the approach to modeling of Lt PD on the basis of the migration matrices constructed of consolidated internal data on changing ratings for both segments was considered for the segments of Project Financing and Project Financing (Developers).
When combining rating groups the number of observations in a rating group amounted from 292 to 890 observations (the largest number of observations was in the "default" rating group and the smallest number of observations was in "good" rating groups). As long as it is important for the segments with the borrowers characterized by credit risk above average to have a number of observations in "bad" ratings sufficient for rating migration analysis we made the conclusion on applicability of the approach to modeling of Lt PD by means of constructing the rating migration matrix on the basis of internal data.
The approach to getting multiyear PD using rating migration matrices. Computation of an average one-year migration matrix.
The migration matrix is indicative of the likelihood that a borrower with a certain rating as at the beginning of the year will have the following:
• the same rating (shown on the principal diagonal);
• rating with a higher probability of default (in the migration matrix such values are indicated above the principal diagonal);
• rating with a lower probability of default (in the migration matrix such values are indicated under the principal diagonal);
• rating of default state (in the migration matrix such values are indicated in column D, default).
When computing a one-year rating migration matrix using the data for a period exceeding 1 year probabilities of migration of the final one-year migration matrix are obtained by averaging probabilities of migration of several matrices. Averaging is performed by calculating the arithmetic mean.
A one-year migration matrix with averaged probabilities of migration is applied as the basic matrix to compute multi-year matrices.
When empiric default rates deviate from model ones (PD of the master scale) PD of the basic migration matrix (the last column - an average one-year default rate, DR) is adjusted to harmonize with PD of the Bank's master scale.
PD adjustment of the basic migration matrix is also necessary in case of inversions (PD of "bad" ratings is lower than PD of "good" ratings).
Adjustment may be performed both by means of calculating the coefficient by which the actually obtained DR is to be multiplied / divided and by means of permutation of PD of a corresponding master scale into the last column of the basic migration matrix. If the basic migration matrix is constructed on the basis of rating groups the weighted average of the number of observations in the rating group of PD of the master scale is calculated for the last column.
After adjustment of the values of the last column the transition probabilities of the basic matrix are adjusted in such a way that the sum of probabilities of transition of each line was 100% (by proportional change of probabilities of rating transitions of each line).
Adjustment example
Table 1. Values before adjustment
- e
2
3
4
5
2 15.75% 78.91% 4.94% 0.40% 0.00% 0.00%
Table 2. Values after adjustment
15.55% 77.
0.39% 0.00%
1.29%
2
3
4
5
2
Sum of probabilities of transition (except for default) before adjustment = 100%, sum of probabilities of transition after adjustment (except for default) = 98.71%.
Adjusted probability of transition from rating 2 to rating 1
is calculated by the following formula:
15.55% = 15.75% • (98.71% / 100%).
Other probabilities in the line are adjusted in the same way.
Calculation of cumulative PD estimates
Estimates of cumulative PD are obtained using migration matrices by means of raising a one-year migration matrix to the corresponding power. For example, in order to get a cumulative PD for N years it is necessary to raise the matrix to the N power in accordance with the Formula (1):
M = MN, (7)
where M is a migration matrix for N years; M0 - a one-year migration matrix.
The column in the multiyear matrix indicating a probability of transition to the default state is the cumulative probability of default of a corresponding period (cPD ). Advantages of use of a complete migration matrix as opposed to use of just the column indicating the probability of default when calculating the cumulative probability of default consist in recording of complete information on rating migration when the probability of default is calculated for several years [12-15].
Adjustment of Probabilities of Transition of Ratings of the Basic Migration Matrix
A probability of rating transition should decrease mono-tonically when moving from the principal diagonal to extreme columns of the migration matrix. It means that the probability of transition of ratings to neighboring rating groups is higher in comparison to the probability of rating transition "skipping" 2 or 3 ratings.
Probabilities of transition are adjusted using mathematical methods (for example, approximation of nonmonotonic data series by a monotone function). In practice parameters of the function (exponential, logarithmic) used for adjustment may be selected applying Excel ("search for solution" package, trend adding).
After obtaining the cumulative probability of default for consolidated rating groups it is converted into the conditional probability of default to calculate conditional PD
for each rating inside rating groups applying logarithmic interpolation.
On the basis of the conditional PD obtained at the previous stage final marginal PD is calculated (exclusive of forecasting information).
The above transitions of the probability of default profiles are performed by the following formulae.
The cumulative PD is determined as:
^ ícPDt , +(1 -cPDt ,>PDt,i > 0 /x cPDt =<! t-1 v tu (8)
t [ 0,/ = 0.
The marginal PD is determined as:
MPDt = PDt .(1 - cPDt-1 ) = cPDt - cPDt-1. (9)
Due to a non-linear character of change of PD ratings when moving along a rating scale it is not recommended to apply the linear approach to interpolation. See below the approach to interpolation which takes into consideration the non-linear character of PD. Interpolation consists of several main stages. 1st stage.
Each rating is assigned a numerical value (Table 3). Table 3. Numerical values of ratings
Rating Numerical value
1+ 1
1 2
1- 3
2+ 4
2 5
2- 6
3+ 7
3 8
3- 9
4+ 10
4 11
4- 12
5+ 13
14
Rating Numerical value ^H Rating Numerical value
5- 15 8 23 6+ 16 8- 24 6 17 9 25
6- 18 10 26 7+ 19
1........................................................................20.................................... 2nd stage.
......................................................................................................................... Average-weighted ratings and PD corresponding to them
7- 21 expressed in terms of numerical values are calculated.
8+ 22 Weights are the number of observations in each corresponding rating (Table 4).
Table 4. Example of calculated actual ratings and cPD corresponding to them
Actual average numerical value of the rating group PD-1 PD-2 PD-3 PD-4 PD-5 PD-6
19.7 11.74% 11.10% 10.88% 10.53% 10.11% 10.11%
23.9 31.97% 29.02% 25.34% 21.64% 18.22% 18.22%
Table 5. cPD of the "8+" rating obtained by interpolation
Numerical value of the rating PD-1 PD-2 PD-3 PD-4 PD-5 PD-6
22 19.37% 18.69% 17.21% 15.56% 13.91% 13.91%
3rd stage. PD is calculated for each rating on the basis of corresponding PD of rating groups.
For the j rating PDjt for the period of t (te[1, N], teN) is calculated as per the following formula:
PDj,t = PDi,t -
PD
i+1,t
PD
(10)
where b j is a numerical value of the j rating which is between the numerical values of the rating groups i and i +1;
ai ,a;+j are numerical values of rating groups i and i +1 respectively;
PDlt, PDj+1t are conditional probabilities of default calculated applying migration matrices for the t period for rating groups i and i +1 respectively (Table 5).
At the next stage adjustment is performed: PD of the first year is set equal to PD of the master scale. Forecasting macroeconomic information is taken into consideration by adjusting conditional PD of the 1st and 2nd year with consideration to the forecasted one-year default rate for 2 years (2 values of DR) from the date of reports.
TTC PD is the probability of default average for the whole economic cycle which assessment is based on all information available about the borrower. TTC PD is stable in time
and is not correlated to the economic cycle. The calculated transition probability and default probability obtained after multiplication of the one-year migration matrix according to art. 7.1.3 are mean values computed on the basis of rating results for approximately 8 years (01.04.200901.07.2017) which covers various stages of the economic cycle, i.e. they are TTC PD estimates.
In order to take into consideration forecasting macroeco-nomic information it is necessary to adjust estimates of TTC PD ratings obtained for the model with consideration to the forecasted default rate.
PIT calibration is performed on the basis of Bayes' formula where the rating PD is scaled according to the forecasted default rate and CDT.
In order to convert one-year PD values Bayes' formula is applied.
PDN
(1 - CDT )*DRNeW*PDj
CDT*(1 - DRNew) *(1 - PDi )+(1 - CDT) *DRNew *PDj , (U)
where PDiNew - PIT PD of the i rating which corresponds to a new forecasting default rate DRNew; PDi - conditional PD of rating grade i; DRNew - forecasting default rate;
CDT - average default rate calculated by the economic cycle.
The data source for developing the model is the corporate data warehouse (area of corporate data warehouse which stores the information of the data warehouse of CRM) with a set of presentations which contain data from various points of view (loan portfolio, agreement information etc.).
Data from the segments of Project Financing and Project Financing (Developers) was used for analysis. Analysis comprises 114 customers from the model of Project Financing (994 observations, 315 of them are default ones) and 128 customers from the model of Project Financing (Developers) (1,082 observations, 506 of them are default ones).
Data from the segments of Project Financing and Project Financing (Developers) was used for analysis (hereafter - the PF portfolio). Due to a significant reduction in the number of borrowers in the PF portfolio in 2015-2017 in comparison to 2009-2013 we decided to analyze the PF portfolio as one segment without division into PF and PF (Developers).
See detailed information on the model in the Table 6. The first 4 lines are the models developed after October 2011, the last 2 lines are the models developed before October 2011 (in 2009).
Table 6. Number of unique records with a breakdown into models (PF portfolio) Grouping of observations
No. Model Number of records
Project Financing (Developers) v.2.0 (archive) 625
Project Financing v.2.0 (archive) 527
Project Financing (Developers) v.2.1 69
Project Financing v.2.1 55
Project Financing (Developers) v.1 (archive) 640
Project Financing v.1 (archive) 563
Due to insufficient number of observations for individual ratings we made rating groups of 1 to 9 ratings in order to construct migration matrices.
When making rating groups we took into consideration the following:
Table 7. Number of observations related to grades/groups
Rating group Rating grade
1+ 1
• rating groups comprise ratings close in terms of risk level;
• the number of observations in a rating group should be sufficient to model probabilities of ratings transition (Table 7).
Number of observations in Number of observations in a rating grade a rating group
1- -2+ 2
345 2- - 330
3+ -31
3- -
4+ 9
4 12
Rating group Rating grade Number of observations in a rating grade Number of observations in a rating group
4- 38
345 5+ 91 330
5 97
5- 82
6+ 139
6 6 165 401
6- 97
7+ 115
7 7 110 292
7- 67
8+ 57
89 8 266 593
8- 73
9 197
Analysis of data in terms of assigning to a stage of the project As a part of model development we studied existence of dependence between the default rate and a project stage. When assessing borrowers' projects assigned to the segments of Project Financing and Project Financing (Developers) in the Bank the basic rating defining module comprises the factor of the Project Stage. Since the wording of factors changed in re-development of models we decided
Table 8. Description of values of the factor of the Project Stage
to combine the observations data in three groups: 1) A (initial financing); 2) B (work performance); 3) C (completion) (Table 8).
Analysis showed that over 65% observations pertain to the C stage (completion) and 18% - to the B stage (work performance). Apart from that, 84.3% of projects were at the completion (C) stage out of 312 observations as at 20152017 (Table 9).
model depending on the model version
Project version
Values of the factor of the Project Stage Designation of the factor of model the Project Stage model
Project Financing v.1
A: Initial financing - <idea> level
B: Preparation and beginning of work <planning> level
A
C: Work performance
D: Ready business - <completion> level
A: Initial financing - <idea> level
B: Preparation and beginning of work <planning> level
B C
A
Project Financing v. 2
C: Work performance B
D: Ready business - <completion> level C
P . t . Values of the factor of the Project Stage Designation of the factor of
model the Project Stage model
A: Construction "from greenfield" A
B: Ditch / foundation
..............................................................................................................................B
C: House case constructed
Project Financing (Developers) v.1 D: Finishing
E: Finishing constructing / restructuring of an existing building C
F: Repairs of an existing building for further use
Development project's stage (in percent)
Project Financing (Developers) v.2 < 20% A
> 20% < 70% B
> 70% c
Table 9. Number of observations by the type of the project stage
Project stage Number of observations Share of observations in the total amount, %
A 134 5.6
B 459 18.3
C 1,632 65.2
(Empty) 280 11.2
Total 2,505 100
On the basis of the results of studies we took the decision not to divide the initial sample into project stages and not to determine individual models for various stages for the following reasons:
1) The major part of observations pertain to the stage of project completion (65%).
2) The rating calculated on the basis of the one-year default probability model takes into consideration the fact of project affiliation to a certain stage.
3) A single sample will allow to develop a stable PD Lifetime model.
Results of modeling of TTC LT PD before taking into consideration forecasting macroeconomic information
Basic Prerequisites
When constructing one-year migration matrix we adopted the following prerequisites:
• default is an absorbing state, i.e. getting out of the default state is not considered;
• in case of several ratings calculated on the basis of the same reports we used for calculation the rating with the last date of calculation;
• within the period (one year) we eliminated migrations into the state of "no rating (no re-rating)", i.e. if as at the beginning of the considered period
a customer was assigned a rating and at the end of the year there was no information on the calculated rating such rating was considered in the calculation as remaining in the same rating. The prerequisite was introduced to meet the modeling purposes - the event of "no re-rating" was not simulated, change of the rating while the borrower is in the Bank's portfolio is simulated.
• assigning of "10" rating to the borrower was considered as an event of default (provided it did not equal 10 as at the previous date).
Average (Basic) One-Year Migration Matrix
Calculation of the basic migration matrix
One-year matrices were calculated as follows:
• a one-year probability that a borrower with a certain rating as at the beginning of the year will in one year have the same or a different rating was calculated per quarters for a one-year interval (one-year matrices were calculated per quarters);
• data was analyzed from 01.04.2009 to 01.07.2017 (30 matrices in total);
• the matrix obtained by averaging of 30 matrices was taken as the basic one-year matrix;
Table 10. Average one-year migration matrix
• for the PF portfolio the data was combined in rating groups 345 (3, 4+, 4, 4-, 5+, 5, 5-), 6 (6+, 6, 6-), 7 (7+, 7, 7-), 89 (8+; 8; 8- and 9). It was necessary to consolidate ratings into groups due to an insufficient number of observations in individual rating grades (Table 10).
Adjustment of the Last Column of the Basic Migration Matrix
Due to an insufficient number of default observations we adjusted probabilities of default on the basis of weighted PD of corresponding ratings in the master scale of the Bank. PD are weighted by the number of observations in each rating (Table 11 and 12).
Risk category 345 6 7 89 10
345 77.6% 13.5% 4.4% 1.3% 3.2%
6 18.3% 41.5% 23.9% 9.8% 6.5%
7 2.8% 13.7% 45.6% 30.5% 7.3%
89 3.7% 2.0% 6.6% 72.5% 15.2%
10 0.0% 0.0% 0.0% 0.0% 100.0%
Table 11. Weighted PD for rating groups
Rating group Rating grade Number of observations in the rating grade Rating PD PD* number of questionaries PD of the rating group
1+ - 0.01% -
1 - 0.02% -
1- - 0.04% -
2+ - 0.08% -
2 - 0.16% -
2- - 0.32% -
3+ - 0.45% -
345 3 1 0.58% 0.006 2.42%
3- - 0.75% -
4+ 9 0.96% 0.086
4 12 1.23% 0.148
4- 38 1.58% 0.600
5+ 91 2.03% 1.847
5 97 2.61% 2.532
5- 82 3.36% 2.755
Rating group Rating grade Number of observations in the rating grade Rating PD PD* number of questionaries PD of the rating group
6+ 139 4.31% 5.991
66 165 5.54% 9.141 5.50%
6- 97 7.12% 6.906
7+ 67 9.14% 10.511
77 110 11.74% 12.914 11.48%
7- 115 15.08% 10.104
8+ 73 19.37% 11.041
8 266 24.89% 66.207 30.60%
8- 57 31.97% 23.338
9 197 41.06% 80.888
Table 12. Basic one-year migration matrix (after reducing the master scale to PD)
Risk category 345 6 7 89 10
345 78.3% 13.6% 4.4% 1.3% 2.4%
6 18.5% 42.0% 24.2% 9.9% 5.5%
7 2.7% 13.1% 43.6% 29.2% 11.5%
89 3.1% 1.6% 5.4% 59.3% 30.6%
10 0.0% 0.0% 0.0% 0.0% 100.0%
Reducing the Basic Migration Matrix to the Monotone Type
The basic matrix is reduced to the monotone type against the principal diagonal by using smoothing functions. Values of transition probability are adjusted line-by-line except for the values in the last column and the principal diagonal. In order to eliminate zero probabilities of transition and non-monotonic values above the principal diagonal we used the decreasing function y = a-exp(-b-t). Its parameters were selected applying the Search for Solution package in Excel (Table 13).
Table 13. The basic one-year migration matrix
(after reducing to PD of the master scale and reducing to the monotone type)
Risk category 345 6 7 89 10
345 78.3% 13.6% 4.3% 1.4% 2.4%
6 18.5% 42.0% 24.2% 9.9% 5.5%
7 0.0% 15.7% 43.6% 29.2% 11.5%
89 2.0% 3.1% 5.0% 59.3% 30.6%
10 0.0% 0.0% 0.0% 0.0% 100.0%
Results of calculation of cumulative PD
The cumulative PDs were calculated by raising to power of the adjusted basic one-year migration matrix (Table 14 and Figure 2). Multiyear matrices were calculated for the period not exceeding 5 years.
Table 14. The cumulative probability of default
Risk category 1st year 2nd year 3rd year 4th year 5th year
345 2.4% 6.0% 10.8% 16.7% 23.2%
6 5.5% 14.0% 23.7% 33.1% 41.6%
7 11.5% 26.3% 39.6% 50.5% 59.2%
89 30.6% 49.6% 61.9% 70.2% 76.2%
10 100.0% 100.0% 100.0% 100.0% 100.0%
Figure 2. Cumulative probabilities of default calculated on the basis of the basic matrix after adjustments
Reducing to the Master Scale, Results' Interpolation
Cumulative probabilities of default were transformed into conditional probabilities of default in accordance with dependence [12] for further reducing of TTC PD for the first year to the Bank's master scale and calculation of the conditional PD for each rating inside rating groups by means of logarithmic interpolation.
PD ( t ) =
cPD ( t )- cPD ( t -1) 1 - cPD (t -1) '
(12)
The conditional PDs for rating categories 345, 6, 7, 89 were calculated by means of logarithmic interpolation. The conditional PDs for ratings 1+, 1, 1-, 2+, 2, 2-, 3+, 3, 3- were fixed at the Bank's master scale level.
Table 15. The conditional probabilities of default obtained after reducing to the master scale and interpolation
Risk category 1st year 2nd year 3rd year 4th year 5th year
1+ 0.01% 0.01% 0.01% 0.01% 0.01%
1 0.02% 0.02% 0.02% 0.02% 0.02%
1- 0.04% 0.04% 0.04% 0.04% 0.04%
2+ 0.08% 0.08% 0.08% 0.08% 0.08%
2 0.16% 0.16% 0.16% 0.16% 0.16%
2- 0.32% 0.32% 0.32% 0.32% 0.32%
3+ 0.45% 0.45% 0.45% 0.45% 0.45%
3 0.58% 0.58% 0.58% 0.58% 0.58%
3- 0.75% 0.75% 0.75% 0.75% 0.75%
4+ 0.96% 1.06% 1.15% 1.21% 1.26%
Risk category 1st year 2nd year 3rd year 4th year 5th year
4 1.23% 1.51% 1.76% 1.96% 2.11%
4- 1.58% 2.14% 2.70% 3.17% 3.53%
5+ 2.03% 3.65% 5.17% 6.58% 7.76%
5 2.61% 4.16% 5.77% 7.20% 8.32%
5- 3.36% 5.44% 7.28% 8.67% 9.63%
6+ 4.31% 7.11% 9.17% 10.44% 11.15%
6 5.54% 9.05% 11.28% 12.33% 12.72%
6- 7.12% 11.40% 13.47% 14.24% 14.35%
7+ 9.14% 14.04% 15.81% 16.23% 16.02%
7 11.74% 16.72% 18.07% 18.09% 17.56%
7- 15.08% 19.39% 19.79% 19.18% 18.23%
8+ 19.37% 22.02% 21.40% 20.17% 18.83%
8 24.89% 27.31% 24.42% 21.96% 19.89%
8- 31.97% 28.40% 25.02% 22.30% 20.09%
9 41.06% 32.26% 27.05% 23.45% 20.76%
Marginal PDs were calculated on the basis of the conditional PDs (Table 15).
For ratings 1+, 1, 1-, 2+, 2, 2-, 3+, 3, 3- mPD were fixed at the Bank's master scale level.
For ratings 7, 7-, 8+, 8, 8- and 9 values of marginal PDs were adjusted in order to eliminate intersections (mPD should not decrease when the rating moves from 1+ to 9). The final values of mPD (without regard to forecasting information) are presented in Table 16 and Figure 3.
Table 16. Marginal TTC profiles of multiyear default probabilities for the PF portfolio (adjusted)
Risk category 1st year 2nd year 3rd year 4th year 5th year
1+ 0.01% 0.01% 0.01% 0.01% 0.01%
1 0.02% 0.02% 0.02% 0.02% 0.02%
1- 0.04% 0.04% 0.04% 0.04% 0.04%
2+ 0.08% 0.08% 0.08% 0.08% 0.08%
2 0.16% 0.16% 0.16% 0.16% 0.16%
2- 0.32% 0.32% 0.32% 0.32% 0.32%
3+ 0.45% 0.45% 0.45% 0.44% 0.44%
3 0.58% 0.58% 0.57% 0.57% 0.57%
3- 0.75% 0.74% 0.74% 0.73% 0.73%
4+ 0.96% 1.05% 1.13% 1.17% 1.20%
4 1.23% 1.49% 1.71% 1.87% 1.97%
Risk category 1st year 2nd year 3rd year 4th year 5th year
4- 1.58% 2.11% 2.60% 2.97% 3.21%
5+ 2.03% 3.58% 4.88% 5.89% 6.49%
5 2.61% 4.05% 5.39% 6.33% 6.79%
5- 3.36% 5.25% 6.65% 7.34% 7.46%
6+ 4.31% 6.81% 8.15% 8.43% 8.06%
6 5.54% 8.55% 9.69% 9.40% 8.50%
6- 7.12% 10.58% 11.08% 10.14% 8.77%
7+ 9.14% 12.76% 12.35% 10.67% 8.82%
7 11.74% 14.75% 13.28% 10.90% 8.82%
7- 15.08% 16.46% 13.55% 10.90% 8.82%
8+ 19.37% 17.75% 13.55% 10.90% 8.82%
8 24.89% 20.51% 13.55% 10.90% 8.82%
8- 31.97% 20.51% 13.55% 10.90% 8.82%
9 41.06% 20.51% 13.55% 10.90% 8.82%
Figure 3. The marginal default probabilities (adjusted) for the PF portfolio
Comparative analysis of the results of development of the basic model for the Construction and Rental Business segments with the results of development of the basic model for project financing
Comparison of obtained mPD estimates for project financing (exclusive of forecasting information) to estimates for the Construction and Rental Business segments revealed
that for "good" rating grades (3+, 3, 3-, 4+, 4, 4-, 4+, 4, 4-, 5+, 5, 5-) the obtained estimates are better (the probability of default is lower) than for the Construction and Rental Business segments. This may be due to the fact that the PF portfolio contains a third less observations in "good" rating grades (Table 17).
For "average" and "bad" rating grades the obtained estimates are a little worse (the probability of default is higher) than for the Construction and Rental Business segments. It should be noted that the PF portfolio contains 2.5 times
as much observations in "bad" rating grades (8+, 8, 8-, 9) as in the Construction and Rental Business segments. It is important to take this feature into consideration when assessing the final mPD value in order to calculate ECL.
Table 17. Number of observations for rating grades
Rating grades Number of observations for the PF Number of observations for the Construction
portfolio and Rental Business segments
2-, 3+, 3, 3-, 4+, 4, 4-, 4+, 4, 4-, 5+, 5, 5- 274 979
6+, 6, 6- 278 457
7+, 7, 7- 228 258
8+, 8, 8-, 9 475 178
10 821 414
Total
2,076
2,286
The effect of adding the obtained mPD estimates of the PF portfolio (instead of the estimates for the Construction and Rental Business segments) to calculation of ECL for the corporate portfolio as of 01.01.2018 amounted to 0.9 million roubles (or + 0.1 %).
Thus, the computed estimates do not lead to overvaluation of the ECL amount and show the specific character of the PF portfolio in the best way. Therefore, they should be used in ECL calculation in the Bank.
Table 18. The final conditional PDs which indicate the probability of default taking into consideration influence of mac-roeconomic information (PF portfolio)
Scale PD TTC Forward PD (PD PIT for the 1st and 2nd year, PD TTC for the 3rd to 5th year)
1st year 2nd year 3rd year 4th year 5th year
1+ 0.01% 0.01% 0.01% 0.01% 0.01% 0.01%
1 0.02% 0.01% 0.02% 0.02% 0.02% 0.02%
1- 0.04% 0.03% 0.04% 0.04% 0.04% 0.04%
2+ 0.08% 0.06% 0.08% 0.08% 0.08% 0.08%
2 0.16% 0.12% 0.16% 0.16% 0.16% 0.16%
2- 0.32% 0.24% 0.31% 0.32% 0.32% 0.32%
3+ 0.45% 0.33% 0.44% 0.45% 0.45% 0.45%
3 0.58% 0.43% 0.57% 0.58% 0.58% 0.58%
3- 0.75% 0.55% 0.74% 0.75% 0.75% 0.75%
4+ 0.96% 0.71% 1.05% 1.15% 1.21% 1.26%
4 1.23% 0.91% 1.49% 1.76% 1.96% 2.11%
4- 1.58% 1.17% 2.11% 2.70% 3.17% 3.53%
Adjustment of one-year PD values taking into consideration a Macro forecast
Table 18 represents the final one-year conditional PDs which indicate the probability of default taking into consideration influence of macroeconomic information.
Scale PD TTC Forward PD (PD PIT for the 1st and 2nd year, PD TTC for the 3rd to 5th year)
1st year 2nd year 3rd year 4th year 5th year
5+ 2.03% 1.50% 3.60% 5.17% 6.58% 7.76%
5 2.61% 1.93% 4.09% 5.77% 7.20% 8.32%
5- 3.36% 2.49% 5.35% 7.28% 8.67% 9.63%
6+ 4.31% 3.21% 7.00% 9.17% 10.44% 11.15%
6 5.54% 4.14% 8.91% 11.28% 12.33% 12.72%
6- 7.12% 5.34% 11.23% 13.47% 14.24% 14.35%
7+ 9.14% 6.89% 13.84% 15.81% 16.23% 16.02%
7 11.74% 8.92% 16.48% 18.07% 18.09% 17.89%
7- 15.08% 11.56% 19.13% 19.79% 19.84% 20.05%
8+ 19.37% 15.02% 21.73% 21.55% 22.09% 22.96%
8 24.89% 19.61% 26.98% 24.82% 26.54% 29,26%
8- 31.97% 25.70% 29.80% 28.51% 32.07% 38.24%
9 41.06% 33.89% 34.42% 35.26% 43.79% 63.11%
Table 19 represents the final one-year marginal PDs which indicate the probability of default taking into consideration influence of macroeconomic information and participate in ECL estimate.
Table 19. The final marginal PDs which indicate the probability of default taking into consideration influence of macroeconomic information (PF portfolio)
Scale PD TTC MPD
1st year 2nd year 3rd year 4th year 5th year
1+ 0.01% 0.01% 0.01% 0.01% 0.01% 0.01%
1 0.02% 0.01% 0.02% 0.02% 0.02% 0.02%
1- 0.04% 0.03% 0.04% 0.04% 0.04% 0.04%
2+ 0.08% 0.06% 0.08% 0.08% 0.08% 0.08%
2 0.16% 0.12% 0.16% 0.16% 0.16% 0.16%
2- 0.32% 0.24% 0.31% 0.32% 0.32% 0.32%
3+ 0.45% 0.33% 0.44% 0.45% 0.44% 0.44%
3 0.58% 0.43% 0.57% 0.57% 0.57% 0.57%
3- 0.75% 0.55% 0.73% 0.74% 0.73% 0.73%
4+ 0.96% 0.71% 1.04% 1.13% 1.18% 1.21%
4 1.23% 0.91% 1.47% 1.72% 1.88% 1.98%
4- 1.58% 1.17% 2.08% 2.61% 2.98% 3.22%
5+ 2.03% 1.50% 3.54% 4.91% 5.93% 6.53%
5 2.61% 1.93% 4.01% 5.43% 6.38% 6.84%
Scale PD TTC MPD
1st year 2nd year 3rd year 4th year 5th year
5- 3.36% 2.49% 5.22% 6.71% 7.42% 7.53%
6+ 4.31% 3.21% 6.78% 8.25% 8.54% 8.17%
6 5.54% 4.14% 8.54% 9.85% 9.55% 8.64%
6- 7.12% 5.34% 10.63% 11.32% 10.36% 8.95%
7+ 9.14% 6.89% 12.89% 12.68% 10.96% 9.06%
7 11.74% 8.92% 15.01% 13.75% 11.28% 9.13%
7- 15.08% 11.56% 16.92% 14.16% 11.38% 9.22%
8+ 19.37% 15.02% 18.47% 14.33% 11.52% 9.33%
8 24.89% 19.61% 21.69% 14.57% 11.71% 9.49%
8- 31.97% 25.70% 22.14% 14.87% 11.96% 9.69%
9 41.06% 33.89% 22.75% 15.29% 12.29% 9.96%
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Appendix 1.
Master scale of the bank
Scale of PJSC Bank XXX Probability of default (PD) Lower bound of the probability of default Upper bound of the probability of default
1+ 0.01% 0.00% 0.02%
1 0.02% 0.02% 0.03%
1- 0.04% 0.03% 0.07%
2+ 0.08% 0.07% 0.14%
2 0.16% 0.14% 0.27%
2- 0.32% 0.27% 0.38%
3+ 0.45% 0.38% 0.49%
3 0.58% 0.49% 0.63%
3- 0.75% 0.63% 0.81%
4+ 0.96% 0.81% 1.04%
4 1.23% 1.04% 1.33%
4- 1.58% 1.33% 1.71%
5+ 2.03% 1.71% 2.20%
5 2.61% 2.20% 2.82%
5- 3.36% 2.82% 3.63%
6+ 4.31% 3.63% 4.66%
6 5.54% 4.66% 5.99%
6- 7.12% 5.99% 7.69%
7+ 9.14% 7.69% 9.88%
7 11.74% 9.88% 12.69%
7- 15.08% 12.69% 16.30%
8+ 19.37% 16.30% 20.93%
8 24.89% 20.93% 26.89%
8- 31.97% 26.89% 34.54%
9 41.06% 34.54% 100%
10 100% 100% 100%
Appendix 2.
Substantiation of the Approaches Chosen for PD Modeling
Method Advantages Drawbacks Application
Weilbull The model approximates the Requires a large amount Applied in case of a large
distribution simulative DR to the observed ones of data on defaults amount of data on defaults (to
significantly better than migration build DR for several years)
matrices Simplicity of use
Migration Use existing information of the
matrices segment more heavily and therefore
may be built with a smaller number of defaults
A convenient mathematical apparatus
Opportunity to make adjustments (for example, in the master scale)
At long periods of time (over 9 years) the results are exaggerated significantly but this may be disregarded due to discounting
Applied in case of insufficient data on defaults (for the Weilbull distribution)
Generator Use existing information of the
matrix segment more heavily and therefore
may be built with a smaller number of defaults
A convenient mathematical apparatus Convenient mathematical methods of adjustment
Opportunity to obtain cPD estimates for nondiscrete time periods Opportunity to obtain nonzero PD for high ratings even in the absence of observed defaults
Highly complex including use
No intuitive explanation when introducing adjustments in the master scale At long periods of time (over 9 years) the results are exaggerated significantly but this may be disregarded due to discounting
May be used instead of migration matrices if:
Adjustments in the master scale are not necessary and Use of the generator matrix decreases the number of adjustments introduced by experts / manually
Source: [9].
Appendix 3.
Terms and definitions
The probability (in percent) of default in relation to customer's obligations within one year defined by means of the model of default probability assessment
Probability of default
Internal credit rating
an indicator which provides a complex characteristic of the customer's/project's solvency and is calculated on the basis of risk factor indicators
Default
failure to fulfill obligations of repayment of borrowings by the borrower (default is recorded in accordance with bank's rules)
Cumulative probability of default (cPD)
the probability of default at any point within the T period (accumulated probability of default, cumulative probability of default)
Marginal probability of default, mPD (t)
an unconditional probability that default will occur within the future t period which is a part of the T period (marginal probability of default)
Observation
a block of data about the customer/project as of a certain date
Rating
In accordance with Reports on Development of the Internal Model of Default Probability Assessment "Project Financing v.2.1", "Project Financing (developers) v.2.1".
Rating group
an aggregate of several rating grades placed in the neighboring positions in the rating scale which are consolidated in order to provide a sufficient number of observations for statistical analysis
Risk segment
a group of rating objects determined in accordance with the Rules for Classification of Bank's Credit Requirements to which the same model of default probability assessment is applied
Rating scale
gradation of rating estimates in accordance with Appendix 1
Conditional probability of default, PD (t)
a conditional probability that default will occur within the future t period which is a part of the T period provided default does not take place before the t period (conditional probability of default)
Lifetime probability of default, Lt PD
a probability of default within the contractual term of a financial instrument (lifetime probability of default)
Appendix 4.
Abbreviations
cDR - cumulative default rate
cPD - cumulative probability of default
Dpd - days past due
DR - default rate
mPD (t) marginal probability of default, unconditional probability of default in the t
period
PD - probability of default
PD for 12 months - probability of default in the following 12 months
PIT - point-in-time calibration
TTC - through-the-cycle calibration
IFRS 9 - International Financial Reporting Standard (IFRS) 9 Financial Instruments
ECL - expected credit losses
SP AACR - software package Accounting and Analysis of Credit Risk
Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
The article was submitted 06.11.2021; approved after reviewing 08.11.2021; accepted for publication 14.11.2021.
