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Myths & Pitfalls in PIT versus TTC Credit Risk Management –The impact of subtleties
RiskMinds 2015
Philipp Gerhold
Amsterdam, 10th December 2015
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Agenda
» Part A: Basic concepts of PIT and TTC
» Part B: Impact on credit risk modelling
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A real-life example – What is ***the*** PD?
* …
» Can this be right?
» Why does the – regularly recalibrated – rating system not adapt to reflect the increased default rates?
» What is the correct PD in 2012?
» Real-life example shows default rate and estimated Probability of Default (PD) for ships.
PD / Default rate
Time
201220112010200920082007
10%
20%
Default rate
PD
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Proper Definition of PD depends on purpose
» Without a precise question, there is no right answer
What is the probability that the obligor defaults within the next 12 months given the current macroeconomic environment?
Point-In-Time PD (PIT-PD)
What is the probability that the obligor defaults within a 12 month period given an average macroeconomic environment?
Through-The-Cycle PD (TTC-PD)
Credit institutes need to compute both, the PIT-PD and the TTC-PD.
» The right question to ask, depends on the intended purpose
Accounting (IFRS 9): Evaluation of current fair value is intended. PIT-PD
Risk Management: Focus on changing (worsening) economic environment. TTC-PD (+ Asset-Correlation R²)
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Common Market Practice: Often no distinction between PIT and TTC
» Mathematical Background: Merton model
Obligor defaults, if asset value 𝐴 falls below default threshold 𝛾 𝐴 = 𝑅 ∗ 𝑋 + 1 − 𝑅2 ∗ 𝜀 < 𝛾
TTC-PD is directly linked to default threshold 𝛾 𝑃𝐷𝑇𝑇𝐶 = Φ(𝛾)
PIT-PD is the conditional default probability given the economic environment 𝑋 𝑃𝐷𝑃𝐼𝑇 = Φ𝛾 − 𝑅 ∗ 𝑋
1 − 𝑅²
» Common Market Practice Rating System
𝑃𝐷𝑅𝐴𝑇𝐼𝑁𝐺 = 𝜅 ∗ 𝑃𝐷𝑃𝐼𝑇 + 1 − 𝜅 𝑃𝐷𝑇𝑇𝐶
Credit institutes typically compute hybrid PD without clear distinction between PIT and TTC perspective.
» Credit institute typically operates rating systems that compute one PD-value
𝑃𝐷𝑅𝐴𝑇𝐼𝑁𝐺 for each obligor.
» This PD-value is typically neither purely TTC nor purely PIT but a hybrid, i.e.
𝑃𝐷𝑅𝐴𝑇𝐼𝑁𝐺 = 𝜅 ∗ 𝑃𝐷𝑃𝐼𝑇 + 1 − 𝜅 ∗ 𝑃𝐷𝑇𝑇𝐶
» The degree of pitness 𝜅 is typically unknown to the credit institute.
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In an ideal world a single rating system provides PIT- and TTC-PD
» Ideal world: Introduction of dual-channel rating system Dual-Channel Rating System
𝑃𝐷𝑃𝐼𝑇 𝑃𝐷𝑇𝑇𝐶
» Ideally, a credit institute would operate a dual-channel rating system that
computes for each obligor both, the PIT-PD and the TTC-PD.
» Sometimes effective approaches are implemented in practice that intend to
convert PIT-PDs into TTC-PDs, and vice versa, with effective factors.
» However, a genuine dual-channel rating system is more than just a rescaling
approach (see below).
» Conceptual remarks on design of dual-cannel rating systems
PIT-channel Built on risk drivers that adjust quickly on economic environment (earnings, behavioral data, …).
TTC-channel Built on risk drivers that are rather independent of economic environment (sector, company size, …).
The difference between PIT and TTC rating systems is often ascribed to different temporal extensions of the calibration period. This is only partly true. The main difference is the dependence/independence on/of the macroeconomic environment.
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» Rating class population is independent of economic
environment leading to only idiosyncratic migrations.
» On the other hand, the observed default rates per rating
class fluctuate strongly depending on economy.
» Rating class population depends on macroeconomic
environment leading to strong systematic migrations.
» On the other hand, the observed default rates per rating
class are (ideally) independent of economy.
PIT and TTC rating systems behave quite oppositely
» PIT rating system » TTC rating system
PIT-RC1
PIT-RC2
PIT-RC3
Time tt1 t2 t3
Default ratesper rating class
Master-scale PD of RC1
Master-scale PD of RC2
Master-scale PD of RC3
Re-Rating
t0
Re-Rating Re-Rating Re-Rating
TTC-RC1
TTC-RC2
TTC-RC3
Time tt1 t2 t3
Default ratesper rating class
TTC PD of RC1
TTC PD of RC2
TTC PD of RC3
t0
Ideal PIT rating systems exhibit strong systematic migrations but rather constant default rates per rating
class. TTC rating systems exhibit exactly the opposite behavior.
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Agenda
» Part A: Basic concepts of PIT and TTC
» Part B: Impact on credit risk modelling
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ICAAP: PIT-PDs lead to cyclically oscillating Capital Requirements
» Determination of ICAAP Credit Risk Capital Requirements
» Credit institute typically operates credit portfolio model to compute ICAAP
credit risk capital requirements on basis of PD, asset correlation R², LGD, EAD,
and many other parameters.
» PDs used in credit portfolio model are typically taken directly from rating
system, i.e. hybrid PDs with certain degree of pitness 𝜿.
» As a consequence the interpretation and properties of the resulting capital
requirements depend on the pitness 𝜿 of the underlying rating system.
» Property: Resulting capital requirement independent of
macroeconomic environment, i.e. constant capital
requirements also in crisis.
» Interpretation: Resulting capital requirements guarantee
survival (statistically) of 1999 out of the 2000 upcoming
years (for 99.95% certainty level) independent of current
macroeconomic environment.
» Property: Resulting capital requirement depends
cyclically on macroeconomic environment, i.e. more
increasing capital requirements in crisis.
» Interpretation: Resulting capital requirements guarantee
survival of upcoming 12-month period given the current
macroeconomic environment with given certainty (e.g
99.95%).
» Rather PIT-PDs (small 𝜅) » Rather TTC-PDs (large 𝜅)
PD
Credit Portfolio Model
R² LGD EAD …
Credit risk capital requirement
PIT-type rating systems induce cyclically oscillating capital requirements. TTC-PDs should better be used.
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PD Term Structure modelling: Standard approach invalid for PIT-PD
» Determination of PD-Term-Structure
» In particular, IFRS-9 Lifetime Expected Loss requires
PIT-PD-Term Structure due to
𝐿𝐸𝐿 =
𝑡
𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦
𝐸𝐴𝐷 𝑡 ∗ 𝑃𝐷𝑃𝐼𝑇 𝑡 ∗ 𝐿𝐺𝐷(𝑡)
» Term Structure 𝑃𝐷(𝑡) caused by migration processes.
» Typically insufficient data render direct determination infeasible.
» Indirect approach for computing 𝑃𝐷(𝑡) based on
exponentiation of migration matrix 𝑀 typically pursued.
𝑃𝐷 𝑡𝑛 =
𝑚1,1 𝑚1,2 𝑚1,3
𝑚2,1 𝑚2,2 𝑚2,3
𝑚3,1 𝑚3,2 𝑚3,3
𝑛
∗ 𝑃𝐷(𝑡0)
» Prerequisite for validity is Markov property of migrations.
» Idiosyncratic (TTC-) migrations are a Markov process.
» Systematic (PIT-) migrations are not a Markov process.
» Standard exponentiation approach conceptually
invalid for desired PIT-PD-Term-Structure.
» Dramatic overestimation of convergence
velocity of PIT-PD.
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PD Term Structure modelling: Consistent approach possible
» Constructing a conceptually consistent PIT-PD Term Structure
» Second Step: Construct 𝑃𝐷𝑃𝐼𝑇(𝑡) on basis of TTC curve:
» For current date 𝑡0 determine 𝑃𝐷𝑃𝐼𝑇(𝑡0) and 𝑃𝐷𝑇𝑇𝐶(𝑡0)(e.g. by means of dual-channel rating system).
» With given asset-correlation R² compute current 𝑋(𝑡0).
» For future time 𝑡: Mitigate economic downturn by
computing future expectation of factor 𝑋(𝑡) given current
factor 𝑋(𝑡0) assuming auto-correlation 𝛼 according to
𝐸 𝑋 𝑡 𝑋 𝑡0 ~ න𝑋𝑡 ∗ 𝑒− 𝑋𝑡
2+2𝛼𝑋𝑡𝑋𝑡0 𝑑𝑋𝑡
» Auto-Correlation 𝛼 = 𝑐𝑜𝑟 𝑋𝑡, 𝑋𝑡+1 given by historic
observations.
» Use 𝑋(𝑡) to translate 𝑃𝐷𝑇𝑇𝐶(𝑡) into 𝑃𝐷𝑃𝐼𝑇(𝑡).
» First Step: Construct TTC-PD-Term-Structure by standard
migration matrix exponentiation approach.
» Rationale: TTC migration is a Markov process.
𝑃𝐷𝑇𝑇𝐶 𝑡𝑛 = 𝑀𝑇𝑇𝐶𝑛 ∗ 𝑃𝐷𝑇𝑇𝐶(𝑡0)
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EBA stress testing: Simple rescaling of PDs insufficient
» Conceptual considerations
» For stress testing stressed macroeconomic risk drivers 𝑋(𝑡) typically
given (e.g. GDP growth -2% for first year, -1.5% for second …).
» GDP decline needs to be translated into stressed PDs.
» This is typically done by regression analysis yielding factor 𝑓𝑠𝑡𝑟𝑒𝑠𝑠
𝑃𝐷𝑠𝑡𝑟𝑒𝑠𝑠 = 𝑓𝑠𝑡𝑟𝑒𝑠𝑠 ∗ 𝑃𝐷
» Factor 𝑓𝑠𝑡𝑟𝑒𝑠𝑠 must match pitness degree 𝜅 of rating PD.
» Problems with this approach:
» Method cannot provide stressed migration matrices.
» Method cannot provide genuine stressed PD term structure.
Conceptually consistent standard approach to stress testing not yet established.
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» Stressed PIT-PD term structure given by 𝑃𝐷𝑃𝐼𝑇(𝑡) resulting from
given (stressed) 𝑋(𝑡).
» Corresponding stressed migration matrices obtained by clustering
the resulting stressed 𝑃𝐷𝑃𝐼𝑇(𝑡) into rating classes.
» Result: The stressed PIT-PD term structure is strongly affected by
assumed stress at short time scales but converges to the same
equilibrium limit as in the unstressed scenario.
EBA stress testing: Genuine stressed PD term structure can be constructed
» Constructing genuine stressed migration matrices and PD term structure
» For stress testing stressed macroeconomic risk drivers 𝑋(𝑡) typically
given (e.g. GDP growth -2% for first year, -1.5% for second …).
» Given the asset correlation 𝑅2 the macroeconomic 𝑋(𝑡) can be used
to translated PIT- and TTC-PDs.
» TTC term structure obtained from
𝑃𝐷𝑇𝑇𝐶 𝑡𝑛 = 𝑀𝑇𝑇𝐶𝑛 ∗ 𝑃𝐷𝑇𝑇𝐶(𝑡0)
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Conclusion
A consistent credit risk management requires a clean distinction between Point-In-Time (PIT)
and Through-The-Cycle (TTC) PD.
In practice, these subtleties often need to be respected more carefully.
Unawareness in this subject affects various fields in risk management and may lead to
- pro-cyclical oscillations of ICAAP Capital Requirements.
- unrealistic PD Term Structure models.
- conceptually inconsistent stress testing methodologies.
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