A best practice framework to determine Forward PDs for application in IFRS 9 and CECL

January 2018
A best practice framework to
determine Forward PDs for
application in IFRS 9 and CECL
DOC: PUB-RESEARCH-008

Contents
1. Executive Summary...................................................................................................... 1
2. Introduction................................................................................................................... 2
3. Modelling Forward PDs................................................................................................ 3
4. Lifetime Forward PDs................................................................................................... 4
5. Goodness of fit.............................................................................................................. 5
6. Applications.................................................................................................................. 6
Appendix I: Proof of Merton’s formula.............................................................................. 7
Appendix II: Through-the-Cycle PD – Basel requirements ............................................... 9
Appendix III: Portfolio default experience....................................................................... 11

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1. Executive Summary
In order to fulfil regulatory requirements of IFRS 9 and Current Expected Credit Loss
(CECL), Financial Institutions are required to calculate forward-looking probabilities
of default (PD).
Under the new requirement by FASB (as well as IASB), expected credit loss must
reflect current conditions and take into account broader information covering the
foreseeable future that could affect the financial assets’ remaining contractual cash
flows.
With forward PDs being the common thread between CECL, IFRS 9, Stress Testing
and to some extent ICAAP models, it makes sense to develop risk parameters using an
objective, consistent and verifiable framework.
While modelling data development approach and granularity challenges need better
understanding on a case-by-case basis, our approach can be applied to the FIs with
varying data and sophistication capabilities. To better understand this, we organized
FIs in three broad cohorts:
• Basic practice cohort:
o Basel II sophistication level - Standardized Approach, with no
modelling capability
o Data mainly limited to historical record of defaults
• Intermediate practice cohort:
o Basel II sophistication level – Internal modelling capability, pre-
dominantly 1 year TTC PDs
• Advanced practice cohort:
o Basel II sophistication level same as above
o Stress testing capability
Benefits of using our approach:
• Create a central repository of forward PDs and LGDs to standardize
modelling of risk parameters
• Remove silo-based calculations, using consistent, objective development
framework
• Leverage data for stress testing and loan pricing

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2. Introduction
IFRS 9 and CECL accounting standards involve estimation of Point in Time
parameters, including a forward-looking perspective (Ankarath et al., 2010; BCBS,
2015; IASB, 2014; FASB, 2016). Similarly, migration of exposures and default rates
for stress testing depend on a point in time within the business cycle. Furthermore, as
required by Federal Reserve guidelines (SR 15-18/19), point-in-time risk parameters
(PD PIT and LGD PIT) should be forward-looking projections of default and loss
rates, and capture current trends in the business cycle. OSFI also requires that Internal
Capital Adequacy Assessment Process (ICAAP) also includes stress analysis based on
the regulatory methodologies (OSFI CAR 2017). EBA Stress Testing guidance (2016)
also requires that PD PIT and LGD PIT should be used for all credit risk related
calculations.
In contrast to through- the-cycle (TTC) parameters, forward looking PDs should not be
business cycle neutral. This implies that the estimates of PD would change following
changes in the economic cycle. These PDs are generally lower than TTC PDs in
periods of positive macroeconomic conditions and higher during macroeconomic
downturns. This is sharply in contrast with PDs used for IRB modelling that reflect
longer-term trends in PD behavior (with PD estimates as less sensitive to changes in
economic conditions).
It is clear from what above that there is a strong similarity that exists in PD modelling
methodology for various regulatory models. This overlap in requirements further
reinforces the need to centralize efforts to determine PDs which could be used as
inputs for various models.
Moreover, it is quite likely that just like EBA’s requirement to use IFRS 9 provisioning
impact in Stress Test (EBA, 2017), both OSFI and Federal Reserve would also
explicitly require the application of IFRS 9 and CECL provisioning impact in
ICAAP/Stress Tests, thus providing a regulatory incentive for an immediate integration
between different frameworks.
This means that while financial institutions generally have TTC risk parameters
developed for risk-weighted assets calculation purposes, they would also be asked to
provide PIT risk parameters for different regulatory requirements (IFRS 9, Stress Test,
ICAAP).
Regulations do not provide any specific guidance on how to adjust TTC PD to PIT PD,
but a consistent and transparent framework bridging the two types of measures, and
extending this framework to develop forward PDs is needed.
Within this context, the challenge faced by banks is to produce regulatory compliant
and coherent PD estimates, while minimizing the amount of resources dedicated to
their development and maintenance. In our opinion, the best way to accomplish this
target is to leverage risk parameters developed for RWA purposes (Basel II regulation)
and to adapt them according to the specific needs.

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3/16
There are three development paths:
• Basic practice cohort: Use observed default rates (ODR)
• Intermediate practice cohort: Use rating models’ PDs developed for IRB
purposes as the baseline starting point
• Advanced practice cohort: Use PDs developed for stress testing purposes
3. Modelling Forward PDs
There are several methods that can be used to transform TTC PDs into PIT PDs. For
instance, by calibrating models on shorter and more recent time horizons. A more
sophisticated framework would be using Logistic Regression approach based on
Richard’s Curve optimization for calibration. Richard’s Curve transformation can be
used to map scores to PIT and TTC PDs based on the difference between the TTC and
PIT Central Tendencies.
We can represent PIT PDs as conditional on the current business and economic cycle.
According to Merton (1973), borrowers default if they cannot completely meet their
obligations at a fixed assessment horizon (e.g. 90 days) because the value of their
assets is lower than the due amount. Consider default as an indicator function, and if
borrower i defaults in time t, then
1it itY CI <
=
Yit is the continuous state random variable for borrower i at time t
Cit is the default threshold of borrower i’s assets at time t
By applying Markov’s inequality, the probability of default could be defined as:
( ) ( )it it itY C itE I P Y C< = <
Merton modelled the value of assets of a borrower as a variable, whose value can
change over time. He described the change in value of the borrower’s assets with a
normally distributed random variable.
Using the standard normal distribution, the one-period unconditional PD (lit), is given
by:
1
1( ) ( ) it it it it it
it it it it it it
R c y
P Y c P R c y P
µ µ
s s
-
-
- - -æ ö
l = < = < - = <ç ÷
è ø
( )ita= F

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4/16
Where, Rit is return on a borrower i’s assets at time t; 1
: it it it
it
c y µ
a
s
-- -
= and (.)F
denotes the cumulative standard normal distribution function1
.
For a single factor, the Merton’s model expects the conditional default to be the
following:
1
( )
1
CNDFR CC
DFR
r
r
-
æ öF - ×
= Fç ÷ç ÷-è ø
Where CC is the single macro-economic factor, CNDFR is the cycle-neutral historical
average default value, and r is the asset correlation. By manipulating the above
formula, we arrive at a description of CC in terms of DFR:
1 1
( ) 1 ( )DFR CNDFR
CC
r
r
- -
F × - -F
=
-
Where CC and DFR are the vectors of historical economic states and default rates,
respectively.
Because the only variable in the equation for CC is the value of the default rate, the
equation can be simplified to the following:
1
( )CC DFR-
= -F
This CC, or credit cycle parameter, can be understood as the "state-of-the-economy” at
different points in time; when the default rate is high, the value of CC is low, indicating
that the state of the economy is poor. Conversely, when the value of DFR is low, the
value of CC is high, signaling a relatively good economy.
Once the Macroeconomic scenarios are parameterized, one can determine the predicted
value of the credit cycle “state-of-the-economy” for any projected values of x, y, and z.
Using the value of CC as derived from the regression equation, the default rate
corresponding to this CC can be determined, which is the inverse of the probit function:
( )DFR CC= F -
4. Lifetime Forward PDs
1
In the Basel model, it is assumed that unconditional PDs do not vary over time. However, if mean asset returns,
µ are dynamic then PDs, itl are also dynamic

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To design the lifetime forward looking path of PDs for IFRS9 purposes, i.e., long term
PDs can be derived from time series long enough to cover the entire economic cycle,
but the availability of such data is complex. An alternative approach can be
implemented like deriving ODR on a projected basis and using Richard’s Curve
optimization. However, that would assume a static credit portfolio and poses
significant data challenges.
Conditional to a long-term equilibrium of systematic factor, the probabilities of default
can be expressed as:
1
( )
1
CNDFR CC
DFR
r
r
-
æ öF - ×
= Fç ÷ç ÷-è ø
For each “t”, future probabilities of default can be derived iteratively from
previous PD and expected variations in the macroeconomic scenario. In the case
above, PD at time 𝑡 can be obtained from PD at time 𝑡 − 1 plus the expectation
of the change in the macroeconomic scenario from 𝑡 − 1 to 𝑡.
5. Goodness of fit
We propose determining goodness of fit using non-parametric approach. Apply a
comparative analysis between PDs and observed default rates by using a confidence
interval around the average PD of each rating grade. For this calculation, several inputs
are required:
• Number of observations falling into the rating bucket under consideration
• Number of default observed for the rating bucket under consideration
• A level of significance to specify the width of the confidence interval
• A level of acceptable misalignment to give some width to the confidence level
also in the case of very high data volumes
For each rating grade, it is assessed whether the forward PDs fall into the confidence
interval and the number of times this is not the case is counted (=number of deviations)
Depending on the significance level originally chosen, a certain number of deviations
is in fact expected even if there is no misalignment of the calibration curve. The
expected number of deviations is calculated and subtracted from the observed number
of deviations (=number of excess deviations).

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6/16
6. Applications
Develop migration matrices
The rating transition probabilities depend on z factor, so the procedure to derive their
path to the long term value is analogous to PDs 
Determine forward LGDs
Using PD-LGD correlation, determine forward LGDs for corporate, retail and real
estate portfolios
Integrate forward PDs with stress testing and ICAAP
Determine impact on CET1 and P/L by integrating forward PDs with Stress Testing
Loan pricing
Taking advantage of EL1Y
, Lifetime Expected LossIFRS 9
, we can develop robust
pricing for loans.

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7/16
Appendix I: Proof of Merton’s formula
1
( )
1
CNDFR CC
DFR
r
r
-
æ öF - ×
= Fç ÷ç ÷-è ø
Proof of the formula2
:
We consider a one-period model. Borrower assets are viewed at t = 0 and again, one
year later. Let ‘y’ be the random element in the percent change in value of assets for a
single borrower over a one year horizon. This change is made up of two parts and,
which are standard normally distributed random variables:
1y er rw= + - (1)
In the above equation, e denotes economy-wide systematic random component and w
denotes a company specific random component while r is the risk weight of the
borrower on the systematic factor.
Figure 1: Good vs. bad states of economy
PD gives the unconditional probability of default, which is a function of two random
variables. But we wish to focus on the conditional the probability of default under the
assumption that the economy is in a bad state.
)( a<= yPPD is the probability that the percent change in the value of the
borrower’s y is less than the critical value a , the point at which a borrower becomes
insolvent and defaults. Based on this we can determine the value of a if we know the
2
All references are provided at the end of this document
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-5.0
-4.8
-4.5
-4.3
-4.1
-3.9
-3.6
-3.4
-3.2
-2.9
-2.7
-2.5
-2.2
-2.0
-1.8
-1.6
-1.3
-1.1
-0.9
-0.6
-0.4
-0.2
0.1
0.3
0.5
0.7
1.0
1.2
1.4
1.7
1.9
2.1
2.4
2.6
2.8
3.0
3.3
3.5
3.7
4.0
4.2
4.4
4.7
4.9
Bad states Good states
Normalized economic
growth rate exceeded with
99.9% probability e = u = -3.09
Normalized average
Growth rate e = 0
X-axis is e the normalized
state of economic growth

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value of PD by taking the inverse normal of the probability of default )(1
PDN -
=a
to obtain the critical value of default. For example, if the borrower had a probability of
default of 2 percent, then critical value of y would be 053.2)02(.1
-== -
Na
Figure above shows the normal density function of the states of the economy over the
(one year) period of interest, where economic performance is measured only by the
normalized random variable e. The expected value of e is zero, but we set e = u, a bad
state that would occur with probability 0.001.
Substituting for y from equation (1):
( | ) ( 1 | )P y e u P e e ua r rw a< = = + - < =
Substituting u for e and rearranging terms
( | )
1
u
P y e u P
a r
a w
r
æ ö-
< = = <ç ÷ç ÷-è ø
Substituting for a and, remembering that the firm-specific normalized risk factor is
assumed to be normally distributed so that its probability can be expressed using the
cumulative standard normal distribution
1
( ) .
( | )
1
CNDR CC
P y e u
r
a
r
-
ì üF -ï ï
< = = Fí ý
-ï ïî þ
Where CC is the bad state of the economy u, and )(1
PDN -
=a
Default rate must be transformed non-linearly in order to make its values comparable
to those of the macro-economic factors, and therefore replacing the left-hand side of
the above equation, gives us:
1
( )
1
CNDFR CC
DFR
r
r
-
æ öF - ×
= Fç ÷ç ÷-è ø

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Appendix II: Through-the-Cycle PD – Basel requirements
The following excerpts from CP3 present the Basel Committee’s view on TTC PD.
376. Although the time horizon used in PD estimation is one year (as described in paragraph
409), banks must use a longer time horizon in assigning ratings. A borrower rating must
represent the bank’s assessment of the borrower’s ability and willingness to contractually
perform despite adverse economic conditions or the occurrence of unexpected events.
409. PD estimates must be a long-run average of one-year realized default rates for
borrowers in the grade, …
423. Banks may use one or more of the three specific techniques set out below (internal default
experience, mapping to external data, and statistical default models), as well as other
information and techniques as appropriate to estimate the average PD for each rating grade.
424. Banks may have a primary technique and use others as a point of comparison and
potential adjustment. Supervisors will not be satisfied by mechanical application of a
technique without supporting analysis. Banks must recognize the importance of judgmental
considerations in combining results of techniques and in making adjustments for limitations
of techniques and information.
§ A bank may use data on internal default experience for the estimation of PD. A bank
must demonstrate in its analysis that the estimates are reflective of underwriting
standards and of any differences in the rating system that generated the data and the
current rating system. Where only limited data is available, or where underwriting
standards or rating systems have changed, the bank must add a greater margin of
conservatism in its estimate of PD. The use of Pooled data across institutions may
also be recognized. A bank must demonstrate that the internal rating systems and
criteria of other banks in the Pool are comparable with its own.
§ Banks may associate or map their internal grades to the scale used by an external
credit assessment institution or similar institution and then attribute the default rate
observed for the external institution’s grades to the bank’s grades. Mappings must be
based on a comparison of internal rating criteria to the criteria used by the underlying
data must be avoided. The external institution’s criteria underlying the data used for
quantification must be oriented to the risk of the borrower and not reflect transaction
characteristics. The bank’s analysis must include a comparison of the default
definitions used, subject to the requirements in paragraph 414 to 419.The bank must
document the basis for the mapping.
§ A bank is allowed to use a simple average of default-probability estimates for
individual borrowers in a given grade, where such estimates are drawn from
statistical default prediction models. The bank’s use of default probability models for
this purpose must meet the standards specified in paragraph 379.
425. Irrespective of whether a bank is using external, internal, or Pooled data sources, or a
combination of the three, for its PD estimation, the length of the underlying historical
observation period used must be at least five years for at least one source. If the available
observation period spans a longer period for any source, and this data is relevant, this longer
period must be used.

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We have highlighted some of the key passages above. They note the Committee’s
awareness of the complexities involved in estimating the through-the-cycle PDs and in
light of that urge an intelligent and considered approach backed by data.

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11/16
Appendix III: Portfolio default experience
The following is the relationship between PD and Default Rate:
PD = Long-term Average Default Rate * (100% - Long-term Average Cure rate)
Where,
Long-term Average Default Rate for a given Pool is calculated as the percentage of
observations in the Pool that are classified as defaults
Long-term Average Cure Rate = (No. of accounts having DPD status of '90+' that roll
back into Performing category and remain for over 12 Months since their initial
categorization as Default) / (Total no. of accounts having DPD status of '90+' at the
beginning of period)
Cure rate is estimated on basis of standard roll-rate analysis to examine what
percentage of 90+ accounts roll-back to 90 or lower DPD levels over 12 Months
period.
Ignoring cure rate, the relationship between LR Default Rate and PD becomes as
follows:
PD ≈LR DR
OSFI requires that PD determination be carried over long-run, which is usually defined
as a period over 5 – 7 years. Specifically, the following are OSFI’s requirements as
noted in 2017 CAR Guidelines:
• Chapter 6/Para 266: PD estimates must be a long-run average of one-year
default rates for borrowers in the grade...
• Chapter 6/Para 287: Historical observations must be at least five years for at
least one source, if available observation period spans a longer period for any
source, and this data are relevant and material, this longer period must be used
• Chapter 6/Para 289: …deriving LR average estimate of PD and default-
weighted average loss rates given default for retail would be based on an
estimate of the expected long-run loss rate
• Chapter 6/Para 291: Historical observations must be at least five years for retail
exposures………..a bank can use recent data for retail exposures if it can
convince its supervisor that more recent data are a better predictor of loss rate
• Chapter 6/Para 292: Banks are encouraged to adjust PD upwards for seasoning
effects
Whenever the available data does not cover one or more full (and typical) economic
cycles, adjustments may be necessary to produce a cycle neutral average. The usable
data for DR estimation covers a period of 4 years, which falls short of a complete
business cycle by +/-3 years. To overcome this gap, we adjusted DR from historic to
expected LR DR.

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12/16
As is with default rate, OSFI clearly requires that such cyclical-adjustment translates
into PDs for each Beacon Score grade.
According to best practice PD model design, most rating tools have a degree of cycle
sensitivity. To what extent cycle sensitivity is desirable depends on the end use of the
rating. It is important to be aware of the degree of cycle sensitivity so that one can
make adequate adjustments for each of the relevant end applications.
Working with cycle neutral (or average) ratings creates a degree of stability. Changes
in business direction and incentives are smoothed out as the incentive structure for
business units takes a more long-term prospective on risk. That is also the reason why
regulators generally prefer cycle-neutral measures, i.e., they encourage stable and
consistent decisions across the cycle. Moreover, volatility does not hit the P&L as
extremely, so investors can expect a more stable dividend policy. Risk assessment
focuses typically on a longer horizon and cycle-neutral ratings would be quite
appropriate especially for long-term obligations.
In distinguishing between point-in-time and through-the-cycle (TTC) PDs, the
existence of predictable macro level credit fluctuations is presumed - that is, the
existence of a “credit cycle”. A credit cycle means that if PDs in a broad region or
industry are unusually high, it is expected that they would fall in the future.
Alternatively, if PDs are unusually low, they are expected to rise, although this may
occur more slowly and less predictably than the cyclical recovery. TTC PDs, reflect
circumstances anticipated over the long run in which effects of the credit cycle would
average close to zero.
TTC PDs would approximate this result by determining PDs over the next year under
the assumption that credit conditions over that period would correspond to the average
observed historically. Therefore, the average value of the monthly PDs over a long
horizon is taken as the TTC Central Tendency.
The TTC PDs obtained after calibration would display the same Observed Default
Rates regardless of whether overall credit conditions are strong or weak.
Assuming the business cycle to be 7-years long, we assumed the past 4 years’ data as
evenly split between 2 good and 2 bad years, and assumed the next 3 years would be
split as 1 good year and 2 bad years. This implies that the overall business cycle split
relative to the RPP portfolio is 3 good years and 4 bad years, which is considered
extremely conservative.

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