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A Simple and Consistent Credit Risk Model for Basel
II/III, IFRS 9 and Stress Testing when Loan Data
History is Short
Bernd Engelmann
This version: March 30, 2023
In the current regulatory environment, banks are required to quantify credit
risk by means of default probabilities, loss rates conditional on default and expected exposures for a number of purposes: Regulatory capital calculation, loan
loss provisioning and stress testing. The nature of each credit risk parameter
might be different for each application, e.g., forward looking default probabilities
are needed for loan loss provisioning while regulatory capital is based on long-term
averages. Those different requirements for each purpose create a substantial burden especially for small and medium-sized banks. This article describes a simple
framework that allows the consistent calculation of credit risk parameters for all
risk applications. It assumes that a bank is using a scorecard based on loan-level
data where the data history might only span a couple of years. This data is combined with a macroeconomic model in a suitable way to derive risk parameters
compliant with all regulatory requirements.
JEL Classification: G21
Keywords: Basel II, IFRS 9, Stress Testing, PD, LGD, EAD, Credit Risk
Ho Chi Minh City Open University, 35-37 Ho Hao Hon Street, District 1, Ho Chi Minh City,
e-mail: bernd.engelmann@ou.edu.vn
In the last two decades, a number of initiatives have been undertaken by regulators
and accounting boards worldwide to increase the stability of the financial system. The
first banking reform by the Basel Committee on Banking Supervision (BCBS), Basel
II, was mainly motivated by the observation of supervisors that banks used securitization to save capital on their low-risk assets while keeping high-risk assets on their
books. This behavior was driven by risk-insensitive capital requirements and is known
as regulatory capital arbitrage (Jones 2000). To discourage banks, the BCBS proposed
a risk-sensitive approach for calculating minimum capital requirements which utilizes
bank-internal credit risk parameter estimations as inputs of a supervisory formula for
capital calculation (BCBS 2006).
Following the global financial crisis in 2007/08, additional regulatory reforms have
been proposed to enhance the transparency and resilience of the global financial system.
Basel III improved the quality of bank capital and and gave supervisors more flexibility
in setting capital levels for individual institutions. In addition, a new capital buffer, the
capital conservation buffer, was introduced. It should be released during a crisis which
effectively lowers capital requirements after the occurrence of an economic downturn.
This included some counter-cyclicality into the accord and is expected to help mitigating
future crises (BCBS 2011).
Furthermore, stress testing has become an integral part of banks’ internal and external risk management. An analysis by regulators has revealed that banks who had
integrated stress testing into their internal risk management processes were more likely
to survive the financial crisis (Baudino, Goetschmann, Henry, Taniguchi, and Zhu 2018).
As a consequence, banks are required to carry out internal stress tests regularly and
present the results to regulators in the supervisory review process. Moreover, they have
to respond to industry-wide stress tests performed by regulators where the impact of
macroeconomic scenarios provided by supervisors on bank capital, loss provisions, or
profitability has to be determined (BCBS 2018).
In addition to banking regulation, the accounting boards IASB (International Accounting Standards Board) and FASB (Financial Accounting Standards Board) have
released new standards for calculating loan loss provisions, IFRS 9 (IASB 2014) and
CECL (FASB 2016), respectively. Both frameworks require banks to estimate credit
risk models internally and translate the outcome into an estimate for future expected
credit portfolio loss that has to be reserved.
All three risk management applications, minimum capital calculation, loan loss provisioning, and stress testing, are based on three internally estimated credit risk parameters,
default probability (PD), loss given default (LGD) and exposure at default (EAD). The
parameter PD measures the probability that a borrower misses a payment on any of
his credit obligations by 90 days or more, LGD the percentage of outstanding credit
exposure that is expected to be lost in case of a borrower default and EAD the expected
outstanding credit exposure at the time of default. The nature of these parameters differs
between risk management purposes. For instance, loan loss provisioning requires forward looking parameters while Basel II prescribes long-term average PDs and downturn
The current regulatory environment is a considerable burden for banks because it
makes the development and maintenance of numerous credit risk models for various
purposes mandatory. This already presents a challenge for big globally active banks
that have large risk modeling departments. For small and medium-sized banks, it is
even more difficult to fulfill all requirements since they operate with smaller risk teams.
For this reason, an approach that is simple to implement and can be integrated into a
wide variety of internal credit risk systems should be beneficial to these institutions.
The purpose of this article is twofold. First, it briefly describes Basel regulation,
IFRS 9 impairment modeling, stress testing and provides a characterization of the credit
risk parameters that are required for each purpose. It will be shown that three types
of risk parameters are needed to fulfill all requirements, point-in-time (PIT), throughthe-cycle (TTC) and downturn (DT) parameters. A PIT parameter is a prediction of
a future realization of a credit risk measure, a TTC parameter estimates the average
realization of a credit risk component over an economic cycle, and a DT parameter is an
estimate conditional on an economic downturn scenario. Second, a simple framework is
developed that can be used to estimate these parameters consistently. It requires only
a small number of years of loan-level data and a longer history of a credit index that is
closely related to a bank’s loan portfolio and allows the estimation of a macroeconomic
model. It will be shown how the loan-level model and the macroeconomic model can be
combined to deliver consistent PIT, TTC and DT credit risk parameters.
The structure of this article is as follows. In the next section, a brief overview of the
current regulatory environment is provided and the credit risk parameters needed for
each purpose are classified. Section 2 reviews the literature on credit risk modeling and
outlines the gap this article aims to fill. In Section 3, a simple modeling framework that
could be used to fulfill all regulatory requirements is described. In particular, it is shown
how to transform the outcome of a credit risk model based on a few years of loan-level
data with the help of a macroeconomic satellite model into a point-in-time, through-thecycle or downturn / stressed risk parameter. Section 4 illustrates the framework using
publicly available data from the US Federal Reserve Bank’s CCAR 2020 stress test. The
final section concludes.
1. Current Standards in Accounting and Bank Regulation
The regulatory environment for bank lending requiring internally estimated credit risk
parameters as inputs is briefly described in the next three subsections. Most importantly,
a classification of the nature of each credit risk parameter for each risk application is
given to clarify the commonalities and differences in estimating credit risk parameters
for different purposes.
1.1. Loan Loss Provisioning under IFRS 9
In this article, loan loss provisioning will be described in detail for the rules prescribed
in IFRS 9. The US counterpart, CECL, could be roughly considered as a special case
of IFRS 9 where all loans are treated as Stage 2 loans. In 2014, the accounting board
IASB has released a new standard, IFRS 9, for impairment modeling (IASB 2014). For
this purpose, IFRS 9 prescribes a 3-stage algorithm:
ˆ Stage 1: Normally performing loans, banks have to reserve one-year expected loss
ˆ Stage 2: Loan with substantially deteriorated credit quality, banks have to reserve
lifetime expected loss
ˆ Stage 3: Defaulted loans, banks have to build a specific loan loss provision
IFRS 9 does neither detail criteria for determining when a loan’s credit quality is deteriorated substantially nor does it provide explicit formulas for computing one-year and
lifetime expected credit loss. This has to be defined internally by a bank. Some suggestions for the former are made by Chawla, Forest, and Aguais (2016b). For the latter,
formulas based on PD, LGD and EAD are most common in banking practice.
To be more specific, consider a fixed-rate loan with a maturity of n years and fixed
interest rate z. Denote with cpt the cumulative probability that a borrower will default
until year t, lt the loss given default associated with a default in year t, and Nt the
outstanding balance of the loan in year t. The loan loss provision (LLP) for Stage 1
loans is computed as
LLP1 = cp1 · l1 · N1 .
For Stage 2 loans, LLP is computed as lifetime expected loss. A common approach is
discounting one-year expected losses for each future period:
LLP2 =
· (cpt − cpt−1 ) · lt · Nt ,
(1 + z)t−1
where cp0 = 0.
It can be shown that the above formula is consistent with discounted cash flows.
Lifetime expected loss could be defined as the difference between the present value of
a loan’s future cash flows and the expected present value of future cash flows. When
the discounting is done with the loan’s interest rate as prescribed in IASB (2014), this
difference in present values equals (2). A detailed derivation can be found in Engelmann
1.2. Minimum Capital Requirements under Basel II/III
Basel II defines three different frameworks for calculating regulatory capital of increasing
complexity, the standardized approach, the foundation internal ratings-based (IRB) approach, and the advanced IRB approach. Throughout this article, it is assumed a bank
is applying the advanced IRB approach. Its cornerstone is a formula for computing
regulatory minimum capital Kmin :
Φ (P D) + ρ · Φ−1 (0.999)
− PD ,
Kmin = EAD · LGD · Φ
where EAD is the exposure at default, LGD the loss given default, P D the one-year
default probability, Φ the cumulative standard normal distribution, and ρ the asset
correlation. Equation (3) represents the value-at-risk at a 99.9% confidence level of the
loss distribution of a large loan portfolio without volume concentrations where asset
values of borrowers are correlated with ρ and driven by one systematic factor (Gordy
How is this formula parameterized? Since its parameters are similar to the parameters cp1 and l1 in (1), one might suppose P D = cp1 and LGD = l1 . However, this is
not true. IFRS 9 requires risk parameters to be forward-looking (IASB 2014). Technically, such parameters are PIT estimates, i.e., best possible predictions of the real-world
realizations of these parameters over the prediction horizon. Basel II, on the other
hand, requires for default probabilities ”long-term averages” and for loss given default
an estimate corresponding to an economic downturn. This means, different sets of risk
parameters have to be estimated for IFRS 9 and Basel II. The parameter ρ is prescribed
by supervisors. Note, that in the actual implementation of Basel II, ρ can be a function
of P D (and even company size) for some portfolio segments and (3) might contain an
additional maturity adjustment factor.
The notion of ”long-term average” is not well-defined. Suppose a rating system is
point-in-time which means that it frequently up- and downgrades borrowers when the
economic environment changes. In this case, default probabilities should be in close
alignment with observed default rates and ”long-term averages” per rating grade would
result in a PIT PD. On the other hand, if a long-term average is built on portfolio
instead of rating-grade level, the resulting default probability is not PIT but reflects an
average over an economic cycle. Technically, such a default probability is TTC, i.e., it
is not predicting a default rate over the next year but an average default rate over an
economic cycle.
Although not specified explicitly in BCBS (2006), a consensus has developed among
banks and regulators that PDs for Basel purposes should be TTC. Using a TTC parameter is sensible from the perspective of supervisors because it leads to more stable
capital requirements and its increase during an economic recession should be moderate
avoiding adverse pro-cyclical effects on the economy. From a statistical point of view,
however, TTC probabilities are more complex to estimate since they represent an estimation over an economic cycle. Moreover, a direct comparison with realized default
rates is not meaningful because by construction TTC PDs should underestimate default
rates during a recession and overestimate default rates during a boom.
This subsection is concluded with describing a recent amendment to Basel II linking
capital requirements with loan loss provisions. In (3), expected loss P D · LGD · EAD is
subtracted from the term involving the normal distribution. Economically, this means
that supervisors, when designing the formula, have assumed that expected losses of
a loan portfolio are covered by loss provisions which should be financed by a part of
interest income, the expected loss margin (Engelmann and Pham 2020). Only losses
beyond expectations are backed by capital. This rule was changed in BCBS (2019)
where the BCBS requires banks to provision at least ELB = P D · LGD · EAD (Basel
EL). In ELB , P D, LGD, and EAD are the credit risk parameters used to compute
regulatory minimum capital. If the loan loss provision LLP is less than Basel EL on
portfolio level, additional capital is required. If it is higher than Basel EL, capital can
be released up to a cap of 0.6% of risk-weighted assets (RWA). RWA is computed as
12.5 · Kmin with Kmin from (3).2 This means that (3) has to be adjusted on portfolio
level to reflect this additional requirement:
K̄min = Kmin − min (LLP − ELB , 0.06 · RW A) .
A further complication of this rule is that if portfolio LLP is lower than total Basel EL
a bank has to build an additional capital buffer in Core Tier 1 capital. If LLP is higher
than Basel EL on portfolio level the difference will be added to Tier 2 capital up to a
cap of 0.6% of RWA which makes this rule asymmetric. For a discussion of the impact
of this regulation for LLP on these bank capital components see Krüger, Rösch, and
Scheule (2018).
Using (3) for regulatory capital calculations implies that a bank has to hold 8% of RWA as
regulatory capital. This is no longer true since the introduction of the capital conservation buffer and
the countercyclical capital buffer in BCBS (2011). Therefore, the true percentage might be different
from bank to bank depending on the jurisdiction it operates in and the policy of its national supervisor.
1.3. Stress Testing
Stress testing is an integral part of credit risk management since the global financial crisis
in 2007/08. The BCBS has released stress testing requirements in BCBS (2009) which
were replaced by BCBS (2018). Stress testing has to be done as part of the supervisory
review process internally by banks and banks have to define stress scenarios that are
meaningful for their business. In addition, supervisors regularly provide stress scenarios
and ask banks to compute the impact on their portfolios. Examples are stress tests by
the US Federal Reserve Bank (Fed) and the European Banking Authority (EBA).3
In a stress test, credit risk parameters PD, LGD and EAD have to be computed
conditional on a macroeconomic scenario. Since these scenarios are predominantly linked
to economic recessions because a stress test aims at testing the resilience of banks,
stressed credit risk parameters are mostly downturn in nature.
To summarize the discussion in Sections 1.1, 1.2 and 1.3, credit risk parameters of
different nature are required for each of the three regulatory frameworks. They are
classified in the table below.
Basel II/III
Stress Test
Downturn Through-the-cycle
Point-in-time Point-in-time
Table 1. Nature of each risk parameter required for each regulatory framework.
2. Literature Review
The evaluation of borrower credit quality is a core activity of financial institutions and
several monographs introduce this topic (Anderson 2007, Baesens, Roesch, and Scheule
2016, Thomas, Crook, and Edelman 2017). In addition, monographs dedicated to one
of the three risk applications outlined in Section 1 have been provided. References for
Basel II / III are Engelmann and Rauhmeier (2011) and Ong (2007), for IFRS 9 Bellini
(2019), and for stress testing Rösch and Scheule (2008) and Quagliariello (2009).
Besides that, numerous articles have been published either on general aspects of
credit risk parameter estimation or on the construction of models specifically dedicated
to Basel regulation, IFRS 9 or stress testing. In the context of Basel II, a notable
article is Aguais, Forest, King, Lennon, and Lordkipanidze (2007) where a framework
for transforming PIT into TTC default probabilities is developed. Articles devoted to
Basel LGD are Barco (2007), Betz, Kellner, and Rösch (2018), Calabrese (2014), or
Krüger and Rösch (2017), while EAD is covered by Gürtler, Hibbeln, and Usselmann
(2018), Jacobs and Bag (2011), Tong, Mues, Brown, and Thomas (2016), or Valvonis
In contrast to Basel regulation where credit risk parameters have a one-year time
horizon, parameter estimation for IFRS 9 and stress testing requires a multi-year horizon.
A common approach is applying survival analysis as in Bellotti and Crook (2009) or Orth
(2013). An alternative is the use of Markov multi-state models (Bluhm and Overbeck
2007, Gaffney, Kelly, McCann, et al. 2014, Vaněk and Hampel 2017). In addition, also
LGD and EAD have to be projected over multiple years. This has been done in Chawla,
Forest, and Aguais (2016a) or Krüger, Oehme, Rösch, and Scheule (2018). A detailed
discussion of IFRS 9 impairment modeling involving all credit risk parameters is provided
in Gross, Laliotis, Leika, and Lukyantsau (2020).
Also stress testing is performed over a multi-year horizon. To a large extend, the
literature is concerned with the perspective of policy makers where models are provided
on bank or country level linking portfolio credit indices to macroeconomic factors (Buncic
and Melecky 2013, Kapinos and Mitnik 2016). The design and identification of relevant
stress scenarios is discussed in Breuer, Jandačka, Mencı́a, and Summer (2012). Stress
testing on loan-level is done in Bellotti and Crook (2013).
Moreover, there is some literature on building credit models to estimate risk parameters for multiple purposes. Frameworks for PIT-TTC estimation of default probabilities
are Aguais, Forest, King, Lennon, and Lordkipanidze (2007) and Carlehed and Petrov
(2012), while Miu and Ozdemir (2017) discusses the adaptation of Basel parameters for
In most of the published literature, the empirical analyses have been carried out on
long data histories where it was possible to measure the sensitivity of default probabilities
or loss given default to macroeconomic factors. In this case, a model can be directly
used to evaluate a macroeconomic scenario since the macroeconomic factor is a part of
the model. Unfortunately, this is not the typical situation in practice. Often the data
history is short or a macroeconomic factor shows an implausibly low sensitivity. The
available data can be used to estimate a default probability, e.g., by logistic regression,
but it is unclear whether the resulting default probabilities are PIT or TTC in nature.
In consequence, it is difficult to directly utilize these parameters for Basel II/III, IFRS
9 and stress testing.
This article aims to close this gap in the literature. No assumption about bankinternal loan-level models is made. It is shown how to combine them with macroeconomic
models to capture the state of the economy, quantify its degree of ”PIT-ness” and
calculate adjustments to transform loan-level into pure PIT, TTC or DT credit risk
parameters. This will be explained in detail in the next section.
3. Simple Credit Model for Basel II/III, IFRS 9 and
Stress Testing
In the first part of this section, the focus will be on models for default probabilities.
The only assumption is that a bank has a credit risk model, e.g., based on logistic
regression, which computes PDs on loan-level. When estimating default probabilities by
regression models, it is unclear whether these probabilities are PIT or TTC or a mixture
of both. Quantifying the degree of ”PIT-ness” of a loan-level PD model will be part of
the framework developed for computing pure TTC and PIT default probabilities. The
risk parameters LGD and EAD are treated in the second part.
3.1. Default Probability
The core ideas of the framework proposed in this subsection have been developed in
previous research by Aguais, Forest, King, Lennon, and Lordkipanidze (2007) and Carlehed and Petrov (2012). However, the suggested calculation of TTC probabilities in
these articles is too simplistic and may work for a data set spanning a full economic cycle
but not in cases where only few years of loan-level data is available for PD estimation.
The core assumptions that have to be fulfilled to make the framework developed in
this subsection work, are listed below:
ˆ A bank has developed an internal rating system on loan-level data, e.g., by logistic
regression. The data history used for estimating this model could be shorter than
one economic cycle and the nature of PDs produced by this model (PIT or TTC)
is unknown.
ˆ A credit index C is available for at least one economic cycle which is closely related
to the bank’s loan portfolio and allows the estimation of a macroeconomic model.
The second requirement, the availability of a credit index C, needs more explanation. In
the context of default probability estimation, this credit index should reflect the default
rate of a portfolio, like corporate loans or residential mortgages. This index could be a
country-wide loan default rate published by a central bank, or a country-wide insolvency
rate published by a statistical bureau or a rating agency, or be extracted from a large
pool of loan-level data as it might be available for securitizations, or might be available
from bank-internal data for a longer history than the loan-level data. The index C does
not necessarily have to reflect the same definition of default as used internally by a bank,
but has to reflect the state of the economy in the same way as the loan portfolio. This
means the internal default rate and the external credit index show a strong co-movement
over time and this co-movement is expected to be stable in the future.
The basis of a general PIT - TTC framework for default probabilities is the onefactor credit risk model underlying the Basel risk weight function (Gordy 2003). This is
a one-period model where the log-return r of a borrower’s assets is modeled as
r = ρZ + 1 − ρ,
where Z and are independent standard normally distributed random variables and ρ
is the correlation between the assets of two borrowers. The random variable Z is the
systemic risk factor affecting the credit risk of all borrowers while is borrower-specific.
A borrower defaults if r falls below a threshold θ. Since r by construction is standard
normally distributed, θ can be computed from the default probability P D as
P D = P (r < θ) ⇒ θ = Φ−1 (P D) .
Conditional on a realization z of Z, the default probability is given as
Φ (P D) − ρ · z
P D(z) = Φ
Formula (7) can be used to transform TTC into PIT PDs and vice versa. Note, that
this involves a changes of the interpretation of Z compared to the original modeling
framework. In the work of Gordy (2003) and a related previous work of Merton (1974),
Z represents the unexpected change in credit risk over the modeling period, typically
one year. This means, that P D is the expected default rate over the period while Z
captures unexpected changes in P D.
This is not how Z is used and interpreted in the framework proposed in this article.
Here, Z represents the state of the economy. A positive z corresponds to a benign
economic environment while a negative z corresponds to a recession. Unconditionally,
i.e., without any prior knowledge of the state of the economy, Z is still assumed as
standard normally distributed. This means that, using this interpretation of z, P D in
(7) is the unconditional expectation over all states of the economy, i.e., a through-thecycle default probability. The default probability conditional on the state of the economy
P D(z) can then be seen as a PIT PD that changes with the state of the economy z.
Rewriting (7) to emphasize this context, results in:
Φ (P DT T C ) − ρ · z
To utilize (8), a framework for estimating the state of the economy z and the asset
correlation ρ is needed.
Macroeconomic models can be applied for this purpose. The basis of a macroeconomic model are macroeconomic factors Xi that explain a credit index C. As outlined
above, this credit index should be closely linked to the loan portfolio a bank is modeling
to ensure a high degree of co-movement with the economic cycle. In the language of
(5), the credit index C should represent a large enough portfolio of borrowers to assume
that idiosyncratic risk is diversified away and C is entirely driven by the state of the
economy Z. This allows the interpretation of C as realization of a PIT PD depending
on outcomes z of Z only.
Assume a history of macroeconomic factors Xi,t , i = 1, . . . , k, t = −m, . . . , 0 together with a credit index Ct , t = −m, . . . , 0. From this data a macroeconomic model is
Φ−1 (Ct ) = β0 + β1 · X1,t−l + . . . + βk · Xk,t−l ,
where l is a time lag. Interpreting Ct as a realization of a PIT PD as explained above
and inserting this into (8) leads to
θ − ρ · zt
Φ (Ct ) = √
Combining (9) and (10) provides a link between the state of the economy z and macroeconomic factors Xi as
1 − ρ · (β0 + β1 · X1,t−l + . . . + βk · Xk,t−l ) − θ
zt =
To apply (11), an estimate of the asset correlation ρ and the default threshold θ is
required. The simplest way to accomplish this task is the method of moments. Alternative approaches and a comparison of methods could be found in Duellmann, Küll, and
Kunisch (2010).
Computing expected value and standard deviation of Φ−1 (Ct ) in (10) to apply the
method of moments leads to
E Φ−1 (Ct ) = √
σ Φ (Ct ) = √
which can be easily solved for ρ and θ. Once both parameters are computed, (10) can
be used to compute the past time series of zt , t = −m, . . . , 0.
The most difficult part in establishing a common framework for Basel, IFRS 9 and
stress testing is linking the macroeconomic model with loan-level data. Recall that no
assumption on the nature of the loan-level PD was made. To utilize loan-level PDs, a
characterization of their nature is needed as a first step. It depends both on the risk
factors used for estimating the loan level model as well as the data history the model is
based on. An illustration is provided in Figure 1. Here, a loan-level model was estimated
on a short time span corresponding to a period where the economy was benign. This
means that default rates in the data history utilized for model estimation are lower
than average default rates over an economic cycle. When estimating TTC PDs in this
situation, it is not sufficient to compute averages over the data history but an adjustment
is needed.
Figure 1. Example of loan-level data availability during an economic boom only.
To characterize the loan-level model, average default probabilities are computed for
each period (quarter or year depending on model estimation horizon) and compared with
realized default rates. The problem is illustrated in more detail in Figure 2. The left
figure shows a PD model that is relatively close to a PIT model. In this case, inverting
(8) would lead to a reasonable TTC PD. Model PD 2, on the other hand, looks more
like a TTC PD. However, when the model is not estimated over a full economic cycle,
the model PDs cannot directly be interpreted as TTC PDs but have to be adjusted.
Since it is not known a-priori whether a PD model is PIT or TTC and graphical
illustrations like Figure 2 are insufficient for decision making, the degree of ”PIT-ness”
has to be quantified. Suppose, a loan-level model is estimated over v + 1 periods and
in each period the mean estimated default probability is pt , t = −v, . . . , 0 while the
observed default rate is dt , t = −v, . . . , 0. For a pure PIT model, default rates would
equal estimated PDs, i.e. pt = dt , ∀t = −v, . . . , 0. For a pure TTC model, pt would be
constant and equal to the average
P0 portfolio default rate in the loan-level data history
which is defined as dav = v+1 · t=−v dt .
Figure 2. Comparison of model PDs with observed default rates. Model PD 1 is close
to a PIT PD, while Model PD 2 is close to a TTC PD.
To quantify PIT-ness, a parameter α is estimated from
α · Φ−1 (dav ) + (1 − α) · Φ−1 (dt ) − Φ−1 (pt ) .
α v+1
In case of a pure TTC model α = 1, while for a pure PIT model α = 0. Solving the
minimization problem leads to the solution α = αmin as
(pt ) − Φ−1 (dt )) (Φ−1 (dav ) − Φ−1 (dt ))
t=−v (Φ
αmin =
−1 (d ) − Φ−1 (d ))2
t=−v (Φ
A key requirement for the determination of αmin is that default rates dt in the loan-level
data are not constant to ensure the denominator does not become zero. Economically,
this makes perfect sense because in the extreme (and not very realistic) case of constant
default rates over time, PIT and TTC default probabilities would become indistinguishable.
The PIT-ness parameter αmin and the state of the economy zt , t = −v, . . . , 0, that
was observed over the history of the loan-level data are the inputs for transforming
loan-level PDs into loan-level TTC PDs. Recall, that for a pure PIT loan-level model,
the transformation is done by applying (8). In case the loan-level model is TTC, an
adjustment is needed to reflect the average state of the economy when estimating the
loan-level model. This results in the condition
θll − ρ · z0
Φ (pll ) = αmin · Φ
+ (1 − αmin ) · √
, (16)
1 + v t=−v
where pll is the default probability of the loan-level model. The above equation (16) has
to be solved numerically for θll . From θll , the loan-level TTC PD pll,T T C is computed as
pll,T T C = Φ (θll ).
To gain a better understanding for (16), suppose pll,T T C is known. If the rating
model is PIT, then the connection between pll and pll,T T C is provided by (8). Setting
αmin = 0 would reduce (16) to (8). If the rating model is TTC, one might suppose that
pll = pll,T T C is a sensible choice. However, if the data history of the loan-level model
does not span a full economic cycle, there is a bias in default rates depending on the
state of the economy when the data was collected. To correct for this bias, an average
of default rates conditional on the states of the economy that have been observed on the
loan-level data history is used in the αmin = 1 part of (16). If the loan-level data was
collected in a benign economic environment, this bias correction will lead to portfolio
average TTC PDs higher than the sample average default rate while it will be the other
way round if the data was collected during a recession. Another key assumption of (16)
is that ρ estimated from the credit index is identical to the asset correlation prevailing
on loan-level. Since loan-level data history in this framework could be short, there
is no way to achieve a consistent estimate of ρ from loan-level data (Gagliardini and
Gouriéroux 2005).
With these ingredients, default probabilities for Basel, IFRS 9 and stress tests can
be computed. To apply (3) for minimum capital calculations, P D = pll,T T C is used for
each loan. For IFRS 9 and stress tests, macroeconomic scenarios Xi,t , i = 1, . . . , k, t =
1, . . . , w have to be provided. These scenarios are translated into states of the economy
zt using (9) and (10)
1 − ρ · (β0 + β1 · X1,t−l + . . . + βk · Xk,t−l ) − θ
zt =
Projections of loan-level default probabilities are computed from (8)
Φ (pll,T T C ) − ρ · zt
pll.P IT,t = Φ
These default probabilities are used as inputs to (1) and (2) or for evaluating the impact of a stress scenario on a bank’s balance sheet. A detailed example for residential
mortgages is provided in Section 4.
3.2. Loss Given Default and Exposure at Default
In principle, the approach described in the last subsection can be applied for building
LGD and CCF (credit conversion factors for EAD estimation) projections. However,
suitable credit indices representing LGD and CCF on portfolio level for different loan
segments are rarely available. This is in sharp contrast to PD where time series of
default rates are available in multiple loan segments and countries. Broadly, the following
alternatives for LGD and EAD could be considered:
1. Build an LGD or EAD model using the methodology introduced in Section 3.1
using a credit index representing LGD and EAD for a diversified portfolio of loans
2. Build an LGD or EAD model using the methodology introduced in Section 3.1
assuming the state of the economy estimated from the PD credit index z also
applies to LGD and EAD
3. Use a model for LGD or EAD derived from credit portfolio theory
4. Use a model for LGD or EAD based on expert judgment possibly enriched with
data analysis where possible
Which approach might be most suitable depends on the segment of a bank’s loan portfolio and the availability of data. It is not possible to provide a generic recipe that
will work for every bank in every jurisdiction but has to be decided case-by-case. Each
alternative of the above list will be briefly discussed to understand its advantages and
As already mentioned, the first suggestion, transferring the methodology applied
for PD in Section 3.1 to LGD or EAD, is problematic mainly due to data limitations.
There are portfolios where sensible credit indices related to losses and exposures could
be defined on pools of securitizations. However, this is available in few jurisdictions only
and often for a time span less than an economic cycle. From a practical perspective,
therefore, this approach is not feasible.
The second suggestion is utilizing zt that was estimated for a PD model in projecting
LGD and CCF:
Φ−1 (LGDT T C ) − λ · zt
Φ (CCFT T C ) − γ · zt
where the parameters λ, γ, and the TTC parameters for LGD and CCF have to be
determined. How these parameters are estimated depends on data availability and has
to be decided case-by-case. There is strong empirical evidence that λ is positive (Frye
2000, Frye 2003). However, one should expect a dependence on the length of workout
periods for defaulted loans where the sensitivity with the economic cycle is the lower
the longer the workout period. For EAD there is even evidence of counter-cyclicality
(Jacobs 2010, Thackham and Ma 2019), which means γ could be mildly negative. This
suggests that defining CCF as a constant could be a reasonable conservative choice
eliminating the burden of estimating a complex model for this risk parameter.
The third suggestion is utilizing models based on credit portfolio theory. An example
is a LGD formula derived in Frye and Jacobs (2012)
−1 (EL)
C )−Φ
Φ Φ−1 (P DP IT ) − Φ (P DT√T 1−ρ
where ρ is the asset correlation and EL a parameter representing average expected loss.
The advantage of this approach is its simplicity and that most parameters are readily
available and have an intuitive interpretation. If it is suitable for a particular loan
segment has to be judged from the shape of LGD as a function of PD. Formulas similar
to (21) have been derived under different probabilistic assumptions leading to alternative
parametrizations of LGD as a function of PD. They are summarized in Frye (2013).
For LGD models, it might be suitable to explicitly take into account the collateral
a loan is secured with which could be done by an expert-based modeling approach.
For instance, losses in residential mortgage defaults are largely driven by loan-to-value
(LTV), i.e., the ratio of outstanding loan balance to house price. A possible LGD model
could, therefore, be
LGD = a · max(0.0, b · LT V − LT V1 ).
Here, a, b, and LT V1 are constants that have to be determined. The constant b should be
greater than one reflecting a haircut in the house price due to the foreclosure sale. The
parameter LT V1 indicates a maximum LTV value that ensures a bank just makes no loss.
Finally, a should be less than one because not all defaulted loans end up in workout but
some cure and start paying normally after default again. Since it is not known whether
a loan cures or goes into workout at the time of default, this must be reflected in model
parameters. This approach for determining LGD will be used in the next section where
the proposed modeling approach is illustrated for a portfolio of residential mortgages.
4. Detailed Example for Residential Mortgages
The framework developed in the last section is applied on data from the CCAR 2020
stress test of the Fed.4 The reason for choosing this data set is that it was collected
before the outbreak of COVID-19. The data for later CCAR stress tests contains effects
from the government measures that were introduced after the outbreak like lockdowns
and COVID relief cheques. These measures distort relations between credit risk and
macroeconomic drivers that were rather stable over decades before the outbreak. How
this can be handled in a credit risk model is beyond the scope of this article and treated
separately in Engelmann (2022).
The data can be obtained from https://www.federalreserve.gov/supervisionreg/ccar2020.htm.
In the example developed in this section, it is assumed that a residential mortgage
lender on 01.01.2020 has to calculate Basel capital, IFRS 9 loan loss provisions and to
perform the CCAR 2020 stress test for his mortgage portfolio. The starting point is
the macroeconomic model which will be explained in the next subsection. Each risk
application, Basel II, IFRS 9, and stress testing will be covered in a separate subsection
4.1. Macroeconomic Model
A suitable credit index for US residential mortgages can be obtained from the Fed New
York.5 It represents the annualized default rate of residential mortgages for all US banks
reported quarterly. This default rate together with the unemployment rate and quarterly
house price index (HPI) growth are displayed in Figure 3. It can be seen that the data
reasonably well covers an economic cycle. The values of the three quantities are quite
similar at the start and end of the observation period where the economy was benign
and a severe recession is included with the global financial crisis in the years 2007/08.
On this data the model
Φ−1 (DRt ) = −2.606 + 11.190 · U Rt−1 − 7.607 · HP Igrt−1
is estimated on quarterly data from 2000 Q1 to 2019 Q4, where DR is the annualized US
mortgage default rate, U R the unemployment rate, and HP Igr quarterly HPI growth.
Both independent variables enter the model with a time lag of one quarter. This might
not be the best possible model that could be found on this data set but for the purpose
of this article it is good enough. All model coefficients are highly statistically significant
and the adjusted R2 is 93%.
To determine the state of the economy zt from Φ−1 (DRt ), the parameters θ and
ρ are needed which can be estimated from (12) and (13) resulting in θ = −1.945 and
ρ = 7.59%. Using these parameters allows the calculation of past values for zt from (10).
To complete the macroeconomic modeling part, the two scenarios provided by the Fed
in the CCAR 2020 stress test are input into model (23) and future states of the economy
conditional on each scenario are computed. The results are displayed in Table 2 below.
Table 2 illustrates how z represents the state of the economy. A benign economy is
reflected by positive z while during a recession z is negative. The deeper the recession
the more negative is z. Comparing the values for U R and HP Igr in Table 2 with Figure
Annualized default rates of US residential mortgage loans (page 25 in HHD C Report 2021Q2.xlsx)
are available quarterly starting from 2000 Q1 at https://www.newyorkfed.org/microeconomics/hhdc/
Figure 3. Annualized quarterly US mortgage default rates, unemployment rate and
quarterly house price index growth from 2000 Q1 to 2019 Q4.
2020 Q1
2020 Q2
2020 Q3
2020 Q4
2021 Q1
2021 Q2
2021 Q3
2021 Q4
2022 Q1
2022 Q2
2022 Q3
2022 Q4
2023 Q1
Base Scenario
HPIgr Φ−1 (DR)
Adverse Scenario
HPIgr Φ−1 (DR)
Table 2. Base and adverse scenario of the Fed’s CCAR 2020 stress test.
3 shows that the Fed’s adverse scenario in this stress test exercise is comparable with
the global financial crises of 2007/08.
4.2. Basel II/III Capital Calculation
As already stated above, it is assumed that a risk manager has to compute minimum
capital under the IRB approach on 01.01.2020. The loan that is used for illustration is
a fixed-rate mortgage. The fixed interest rate is 5% and the annual amortization rate
is 4%. The interest rate is assumed fixed until 31.03.2023 which is the date of the next
interest reset. In many jurisdictions, borrowers have the right to fully prepay a loan on
interest rate reset dates. In this example, it is assumed that this is the case. Therefore,
the lifetime of the loan should be considered as three years and one quarter even if the
contractual maturity of the loan is longer. On 01.01.2020, the outstanding loan balance
is 100,000 and the value of the house is 110,000, i.e., LTV = 90.91%.
To compute Basel II/III capital, a borrower’s TTC PD is needed together with the
loan facility’s downturn LGD. For this purpose, a PD and a LGD model are required.
Assume that the bank has collected five years of data from 2015 to 2019 and estimated
a default probability model using logistic regression or a suitable alternative statistical
technique. In general, it is unknown after model estimation whether a PD model is
PIT or TTC or somewhere in between. To measure the PIT-ness of a PD model, the
in-sample default probability is computed for every year in the data on portfolio level
and compared with default rates. An example is provided in Table 3.
Average PD
Default Rate
Table 3. Average PD and default rate for each year of the bank’s internal PD model.
It can be seen that default rates in the bank’s internal data qualitatively show a
similar behavior as default rates in the Fed’s data in Figure 3. They are declining over
this time horizon. Therefore, the credit index provided by the Fed could be a reasonable
proxy for estimating the state of the economy of the bank’s portfolio. When looking at
the data in Table 3, it can be seen that average PDs are also declining over time but
the decline is less steep. The bank’s internal PD seems to be somewhere in between a
PIT and a TTC PD. Solving the minimization problem (14) leads to αmin = 0.4923, i.e.,
the bank’s internal model is basically a 50/50 mixture of a pure TTC and a pure PIT
default probability model.
The estimate for αmin can be used for computing the TTC PD of a borrower. Assume
that the bank’s internal PD model estimate is pll = 2.00%. The TTC PD for this
borrower can be computed from (16) resulting in pll,T T C = 4.049%. Here, the impact of
the macroeconomic model becomes visible. The bank internal model is based on data
which was observed during an economic boom with low default rates. Therefore, the
model will produce rather low PDs. The macroeconomic model, however, contains the
information of a full economic cycle, and corrects the bank internal PD to a higher value
that more likely represents an average over a full economic cycle.
For LGD estimation, it is assumed the bank uses model (22) with parameters a =
0.75, b = 1.00 and LT V1 = 80%. Recall that Basel requires a downturn LGD, i.e., an
estimate of LGD conditional on a severe recession scenario. The Fed’s macroeconomic
data for CCAR 2020 contains a house price index. The biggest year-on-year drop of
this index in the data was -17.1%. This suggests a house price drop of 20% could be a
conservative downturn scenario justified by the data. A 20% decline in the house price
results in a LTV of 113.6% which leads to a downturn LGD of 25.23%.
Using (3) for this residential mortgage requires a parameter choice of ρ = 15%,
P D = 4.049%, LGD = 25.23%, and EAD = 100, 000. This results in a minimum capital
of K = 5, 942.60. This is not yet the correct amount of regulatory minimum capital. In
Basel III an additional capital buffer, the capital conservation buffer, was defined which
requires a markup of 10.5/8 (BCBS 2011). The correct minimum regulatory capital for
this loan is, therefore, Kmin = 7, 799.66.
4.3. IFRS 9 Impairment Modeling
As outlined in Section 1.1, depending on whether the credit quality of a loan has deteriorated substantially either one-year expected loss or lifetime expected loss is required. A
further complication of IFRS 9 is that it explicitly prescribes the calculation of expected
loss for multiple scenarios and to build a probability-weighted average. For the purpose
of this article, the scenarios provided by the Fed for CCAR 2020 are utilized and the
probability weights are 90% for the baseline scenario and 10% for the adverse scenario.
These numbers can hardly be justified empirically but are based on expert judgment.
Deloitte (2019) provides an overview on the various IFRS 9 solutions of large UK banks
illustrating the heterogeneity of IFRS 9 implementations among large banks.
The input data for LLP calculation is summarized in Table 4. The assumed amortization rate of 4% implies that each quarter the loan balance is reduced by 1,000. For
each scenario, PD and LGD are computed. Note, that the Fed uses annualized default
rates in its report which is also used in model (23). To use model predictions for IFRS 9
impairment calculations, the annualized PDs P DA have to be transformed to quarterly
PDs P DQ by P DQ = 1 − (1 − P DA ) 4 . For the calculation of LGD, LTV is computed in
each period taking into account the loan’s outstanding balance and assuming that the
house price follows the HPI, e.g., when HPI grows by 1% also the house price goes up by
1%. The combined effect of growing house price and declining outstanding loan balance
eventually leads to an LGD of 0% in the baseline scenario. This does not happen in
the adverse scenario since the assumed decline in the house price is stronger than the
amortization of loan balance resulting in increased LGDs.
2020 Q1
2020 Q2
2020 Q3
2020 Q4
2021 Q1
2021 Q2
2021 Q3
2021 Q4
2022 Q1
2022 Q2
2022 Q3
2022 Q4
2023 Q1
cp B.
cp A. LGD A.
13.090 22.626
16.912 25.576
20.781 26.915
23.646 26.711
25.752 25.101
27.313 23.409
28.720 21.178
29.856 18.967
Table 4. PD (in %), cumulative default probability cp (in %), LGD (in %), and EAD
per quarter and scenario (B. for baseline, A. for adverse) for IFRS 9 LLP calculation.
Before expected loss is computed, the meaning of ”PD per scenario” has to be made
more specific. The PD that is estimated for each quarter with the macroeconomic
model is a default probability for quarter t conditional on survival in quarter t-1. The
cumulative default probability cpt for a borrower between 01.01.2020 and the end of
quarter t is computed iterative by cp1 = P D1 , . . ., cpt = cpt−1 + (1 − cpt−1 ) · P Dt . The
cumulative default probability cp for each scenario serves as input into (1) and (2).
Both one-year and lifetime expected loss can be computed from (2) since LGD and
outstanding loan balance are changing in each quarter. Since all risk calculations are
done as of 01.01.2020, this means for one-year expected loss that the numbers for 2020
Q1, 2020 Q2, 2020 Q3 and 2020 Q4 in Table 4 are used. For lifetime expected loss the
data from 2020 Q1 to 2023 Q1 is inserted into (2). Discount factors are computed using
the loan’s fixed interest rate of 5% (note the t − 1 in (2) which results in a discount
factor of 100% in the first period 2020 Q1). The results for IFRS 9 loan loss provisions
in this example are shown in Table 5 below.
Probability Weight LLP1
836.86 5,689.88
Table 5. 1Y and lifetime LLP for both CCAR 2020 scenarios and the 90/10 weighted
4.4. Stress Testing
In the final subsection, an illustrative stress test is performed. To keep the setup transparent and easily reproducible, it is assumed the bank is engaged in residential mortgages
only and the portfolio in 2020 Q1 consists of 1,000 identical loans equal to the loan introduced in Section 4.2. It is further assumed that in 2020 Q1 the bank is fully capitalized.
In Section 4.2, Basel minimum capital was computed resulting in 7,799.66 per loan, i.e.,
the minimum capital for 1,000 loans is 7,799,657. As already outlined in Section 1.1,
an additional capital requirement might be necessary due to the regulation in BCBS
(2019). Assuming all loans are Stage 1 loans, leads to a portfolio LLP of 192,557. Basel
EL is 1, 000 · 100, 000 · 4.04% · 25.23% = 1, 021, 405. This results in an additional capital requirement of 828,847. To be fully capitalized, therefore, bank capital must be
8,628,504. Finally, it is assumed that loan loss provisions at the beginning of 2020 Q1
are also covered by bank capital, leading to total capital in 2020 Q1 of 8,821,061.
When computing LLP during the stress test, some simplifying assumptions are applied. It is not assumed that the bank has perfect knowledge of the scenario but the
credit risk parameters in each quarter are used for an approximate computation of 1Y
expected loss. This results in IFRS 9 = (1 − (1 − PD)4 ) · LGD · Balance using the
column names of Table 6. Throughout the stress test all loans are treated as Stage 1
loans which keeps the stress test simple and transparent. In a more realistic setup, some
loans have to move to Stage 2 due to the stress. This could be achieved, e.g., by a
Markov 3-stage model where each IFRS 9 stage is modeled explicitly including scenario
dependent transition probabilities between stages. This would increase the stress test
complexity considerably and is beyond the scope of this article.
Starting from a fully capitalized bank, the stress test is performed under the assumption that there is no new business originated during the stress period. Further,
the expected loss margin of each loan is assumed to be 0.20%. The motivation behind
this number is that through-the-cycle the default rate is 4% and expected LGD could
be set to 5% over the lifetime of a loan making this margin a reasonable choice. The
profit margin is set to 0.88%. This could be motivated by assuming the bank requires a
risk premium of 10% on its capital for granting mortgages which translates into a profit
margin of 0.88% that has to be contained in the loan’s interest rate. Details on the
decomposition of a loan’s interest rate into margin components can be found in Engelmann and Pham (2020). The remaining parts of the loan’s interest rate are needed for
paying interest to depositors or covering a bank’s operating costs and, therefore, cannot
be used to absorb losses. Losses beyond the 1.08% combined expected loss and profit
margin have to be absorbed by bank capital. Should realized losses be smaller than
margin income, it is assumed that earnings are retained and increase total capital. At
the end of each period the remaining bank capital is computed and compared with required capital which is the sum of loan loss provisions for the next period and regulatory
minimum capital. It is assumed that a bank can release the capital conservation buffer
during the stress period and this capital buffer can be assumed to be zero. A bank will
have to refill this capital buffer once the stress period is over.
The details of the stress test are provided in Table 6. The most important outcome
is that available bank capital is insufficient to survive the stress test. There is a shortage
of capital in 2021 Q3, 2021 Q4 and 2022 Q1. This implies that during these quarters
a bank would either have to reduce its lending business or issue new shares to increase
its capital stock. An alternative interpretation of the result is that being just fully
capitalized at the start of the stress test is not enough to survive it, but an additional
capital buffer of roughly 1.3 million would be needed. In reality, most banks are not
just fully capitalized but have additional capital buffers beyond regulatory minimum
requirements available to absorb losses. The main reason why this stress test failed is
that the portfolio is quite risky for a residential mortgage portfolio with an initial LTV
greater than 90%. In addition, the expected loss margin is rather small with 0.20%. To
make the bank more resilient for a crisis it could improve its loan portfolio by reducing
lending to high LTV mortgages and increase its volume in lower LTV mortgage loans.
If market conditions permit, it should also try to raise the expected loss margin for high
LTV mortgages to be better prepared for an adverse scenario.
5. Conclusion
In this article, a framework for estimating the credit risk parameters PD, LGD, and
EAD required for the three key external credit risk management applications, Basel
minimum regulatory capital calculation, IFRS 9 loan loss provisions, and stress testing,
was provided. The framework is generic as it did not make any assumptions about
data and credit models available at a bank aside from fulfilling the regulatory minimum
requirements. The crucial prerequisite for making this approach work is the existence of
a macroeconomic model that allows the estimation of an abstract state of the economy
that closely reflects the economic conditions of a bank’s portfolio. This macroeconomic
model allows the inclusion of information on the full economic cycle into risk parameters
estimates even if bank-internal data covers only a small part of an economic cycle.
While this approach is rather straightforward to implement for default probability
estimates, an implementation for the risk parameters LGD and EAD/CCF is more
problematic because suitable credit indices reflecting loan portfolio characteristics are
rarely available. For this reason, it is difficult to propose a generic solution. One could
use the state of the economy estimated from PD models also for projecting LGD and
EAD. However, model parameters might be difficult to obtain statistically. Alternatively,
characteristics of a particular loan segment together with expert judgment could be
utilized to develop projections closely reflecting the loan segment’s risk profile. An
example, how this could work, was provided for residential mortgages.
2020 Q1
2020 Q2
2020 Q3
2020 Q4
2021 Q1
2021 Q2
2021 Q3
2021 Q4
2022 Q1
2022 Q2
2022 Q3
2022 Q4
2023 Q1
10.2 100,000,000
12.1 98,501,445
14.1 96,342,338
16.9 93,527,406
19.8 90,195,113
22.6 86,104,947
25.6 81,707,657
26.9 77,296,085
26.7 72,926,384
25.1 69,553,444
23.4 66,927,691
21.2 64,833,911
19.0 62,911,830
Basel II
IFRS 9 Basel EL Req. Cap.
192,557 1,021,404 6,964,000
559,216 1,006,098 6,859,641
984,045 6,709,281
955,293 6,698,283
921,257 7,350,429
879,480 7,741,150
834,566 7,932,396
789,506 7,860,754
744,873 6,677,819
710,422 5,671,057
683,603 4,955,676
662,217 4,596,456
642,584 4,381,180
Table 6. . Results per quarter of the CCAR 2020 stress test of the simple stylized mortgage portfolio. PD and LGD
are point-in-time estimates of default rates and loss given default (in %), Balance is the total outstanding performing
loan balance, NPL the non-performing loan balance per quarter, Loss the loss a bank experiences per quarter, Margin
the total of expected loss and profit margin generated from performing loans, Capital the available bank equity capital,
Basel II the required core Basel II minimum regulatory capital, IFRS 9 the loan loss provision, Basel EL the expected
credit loss computed from Basel II credit risk parameters, and Req.Cap. the total required minimum capital for the
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