Myths & Pitfalls in PIT versus TTC Credit Risk Managements3.amazonaws.com/JuJaMa.UserContent/76be828d-2a27-4945-b31… · TTC-channel Built on risk drivers that are rather independent

© d-fine — All rights reserved | 0

Myths & Pitfalls in PIT versus TTC Credit Risk Management –The impact of subtleties

RiskMinds 2015

Philipp Gerhold

Amsterdam, 10th December 2015


Agenda

» Part A: Basic concepts of PIT and TTC

» Part B: Impact on credit risk modelling


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


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)


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.


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.


» 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



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.


Agenda

» Part A: Basic concepts of PIT and TTC

» Part B: Impact on credit risk modelling


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.


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.


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)


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.


» 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)


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.


d-fine GmbH

Frankfurt

München

London

Wien

Zürich

Zentrale

d-fine GmbH

Opernplatz 2

D-60313 Frankfurt/Main

T. +49 69-90737-0

F: +49 69-90737-200

www.d-fine.com

Dr. Philipp GerholdSenior Consultant

Mobile: +49 (0)162 263 1361

E-Mail: [email protected]

Dr. Bernd AppasamyPartner

Mobile: +49 (0) 151 148 193 06

e-Mail: [email protected]

Documents

Myths & Pitfalls in PIT versus TTC Credit Risk Managements3.amazonaws.com/JuJaMa.UserContent/76be828d-2a27-4945-b31… · TTC-channel Built on risk drivers that are rather independent