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Probability of Default (PD) Calibration Conundrum

Low Default Portfolio (LDP) modelling

30th August 2013


© 2013 Deloitte LLP. All rights reserved. 2

Thomas Clifford

Krisztian

Sebestyen

Introductions

We would like to extend our sincere thanks to Edward Venter whose hard work and commitment whilst on

secondment with Deloitte played a significant role in generating the results produced in this analysis and

producing this presentation.

Alexander

Marianski

Alexander is a Manager in Deloitte’s Financial Services Advisory Group. He

specialises in credit risk measurement and modelling for the banking sector.

Before joining Deloitte, Alexander worked in the international wholesale risk

measurement team at a large UK bank where he worked on the development

and rollout of corporate credit risk models mainly in emerging markets

portfolios. He was also involved with credit process & policy, pricing, capital,

impairment and stress testing. Alexander holds a MEng degree in

Engineering.

Tom is a Senior Manager in Deloitte’s Financial Services Advisory Group. He

specialises in credit risk modelling across the banking sector, having

implemented, reviewed and applied credit risk models across the full spectrum

of Retail, Commercial, Corporate and Wholesale lending operations. Tom has

a Masters degree in Physics, an Honours degree in Financial Services and is

a qualified Prince2 practitioner.

Krisztian is an Assistant Manager in Deloitte’s Financial Services Advisory

Group. He specialises in Basel II credit risk and operational risk modelling.

Krisztian joined Deloitte in 2012 from a Hungarian consulting company where

he worked as a consultant and trainer in Basel II operational risk and credit

risk modelling. Krisztian holds a Masters degree in Financial Mathematics and

is a qualified Financial Risk Manager (FRM).



1. Aim & Conclusion

2. The significance of LDP modelling

3. Approaches to LDP modelling

4. Portfolios used in PD calibration

5. Calibration Results

6. Sensitivity Results

7. Questions

3

Agenda



Aim

• Low Default Portfolios (LDP) account for a large share of total bank

lending.

• Due to the scarcity of default observations and subsequent need for numerous

assumptions, model calibrations introduce significant model risk. This is

usually absorbed by applying high level conservatism.

• The aim of this presentation is to compare the Pluto/Tasche (2005)

confidence based methodology with the Tasche (2011) Bayesian

methodology, applying different prior distributions.

• PD calibrations and sensitivities to assumptions are compared on three

simulated portfolios.

4

Analyse two different LDP Probability of Default (PD) calibration methodologies and

apply these on three sample portfolios to evaluate the model risks.



Conclusion

• Choice of methodology and assumptions (prior distributions and

correlations) have a significant impact on PD estimates which can lead to

significant variation in capital requirements and provisions.

• Expert based prior distributions can be an alternative to the proposed

conservative and uniform distributions, producing comparable PD estimates

with the other methodologies.

• Expert based prior distribution show lower sensitivity to correlation inputs

• The PD estimates produced by both methodologies can not be backtested due

to data constraints, but can be benchmarked against estimates from different

methodologies

• Regulatory expectations of RWA floors may override capital calculated

using LDP PD estimates for some portfolios. However, a growing list of

business applications require appropriate estimates of PDs.

5



At least 50% of commercial banking book assets are in portfolios which have LDP

characteristics:

6 Source: Bank Of England

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Asset Class

Breakdown of UK bank lending

£550bn 5% = Large

P&L or capital impact

PD Central Tendency has a significant impact on overall capital requirement (both Pillar 1 and

2). The recent BIS survey reported significant differences in PD methodologies used by banks

for the same LDPs, leading to significant differences in PD estimates and Pillar 1 capital.

Significance of LDP modelling



Bayesian approach (Method B)

The idea: setting asset correlation and

intertemporal correlation assumptions, the number

of defaults follows a correlated binomial

distribution. Choosing a confidence level gives us

an exact estimate of PD (subject to simulation

variations).

Where:

• N – number of observations

• M – number of simulations

• r - number of defaults

• T - number of years

• γ – confidence level

• 𝐺(𝑃𝐷, γ, St)= Φ(Φ−1 𝑃𝐷 +𝑦 ρ

1−ρ) – probability of default in a given

year, where y ~ N(0,1)

The idea: there is a prior belief on the possible

values of PD – this can be represented in

probabilistic terms. This prior belief is updated by

the observations, using the prior distribution as a

weighting function.

Tasche (2011) suggested taking the mean of the

posteriori distribution as the estimate.

𝑃(𝜃 ≤ PD∗ |observe k defaults)=

𝑃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒 𝑘 𝑑𝑒𝑓𝑎𝑢𝑙𝑡𝑠 𝜃 ∗ 𝑝(𝜃 ≤ PD∗)

𝑃(𝑜𝑏𝑠𝑒𝑟𝑣𝑒 𝑘 𝑑𝑒𝑓𝑎𝑢𝑙𝑡𝑠)

Where:

• Θ is the prior distribution

• k is the number of defaults observed

• 𝑃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒 𝑘 𝑑𝑒𝑓𝑎𝑢𝑙𝑡𝑠 𝜃 =𝑛𝑘(1 − (1 − 𝐺(𝜃, ρ, St))𝑇

𝑡=1 )𝑘 ( (1 − 𝐺(𝜃, 𝜌, St))𝑇𝑡=1 )𝑛−𝑘

The prior distribution specifies the probability of a

given long run average PD.

7

Confidence level based approach (Method A)

1 − γ

=1

𝑀 (

𝑛

𝑘(1 − (1 − 𝐺(𝑃𝐷, γ, St))

𝑇

𝑡=1

)𝑘 ( (1 − 𝐺(𝑃𝐷, γ, St))

𝑇

𝑡=1

)𝑛−𝑘)

𝑟

𝑘=0

𝑀

𝑗=1

Both the Confidence based and Bayesian methods can be applied to estimate and validate

the central tendency (long run average) PD for a LDP.

Approaches to LDP modelling



Conservative prior distribution Assumes greater probability of higher PDs:

Π Θ < λ =1

(1−λ).

Applicable if there is no prior assumption

about the PD, but the objective is to generate

a conservative estimates which can inform

extreme expectations.

Uniform prior distribution Assumes all PDs are equally possible

Π Θ < λ = λ

Less conservative approach and there is no

specific belief about the distribution. This

reflects a position where there is not

expectations about the PD distribution.

Expert distribution Experts can use expectations of the PD

distribution to inform the prior distribution. In

this example, a triangle distribution is used

with expert judgment used to specify the:

• minimum PD;

• Maximum PD; and

• Mode (most likely) PD

8

Expert distributions can incorporate stakeholder views on PD, influencing the result, which

makes it more acceptable – decreasing the “black-box” effect.

PD

PD

PD

Pro

ba

bili

ty

Pro

ba

bili

ty

1

1

Max

0

0

Min Mode

Strengths and Weaknesses

• Produces

conservative

estimates

• Provides a cap

to all other

estimates

• High correlation

assumptions

produce overly

conservative

estimates

• Assumes 99% PD

most likely

outcome

• Incorporate

management

expectations

• Increased buy-

in of estimates

• Can be linked

to industry

benchmarks

• Subjectivity of

estimates

• May produce less

conservative

results

• Major validation

and documentation

requirements

• Produces

estimates close

to conservative

prior

• Simple to explain

to senior

managers to

understand

Pro

babili

ty

• High correlation

assumptions

produce overly

conservative

estimates.

• Assumes observed

default rate as

likely as 99% PD

Flexibility of the Bayesian prior distribution

Prior distributions considered



Objective

Analyse the different PD calibration methodologies and evaluate the model risks introduced.



Portfolio 1: Sovereign portfolio

10

>5% 2.5%-5% 0%-2.5% None Sovereign Exposure:

Default is defined according to the S&P definition as “the failure

to meet a principle or interest payment on the due date

contained in the original terms of the debt issue.”

Sovereign portfolio (size £25 billion to 55 counterparties in

2012) distributed across investment and sub-investment grade.

Four defaults between 2002 and 2012 with the observed

default rate over the 11 year period (0.74%).

Since there is either one or nil observed defaults per year, PD

estimation cannot be completed using regression.

Recently published BIS paper* highlighted that different PD

methodologies used by banks lead to significant variances in

PD estimates and RWA and recommended harmonisation of

methodologies or publication of supervisory benchmarks.

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0

2

4

6

8

10

12

14

16

Rating grade distribution for 2012

Counterparties (Primary) Defaults (Primary) Exposure (Secondary)

Year Sovereigns Defaults Defaulting Country

2002 43 0

2003 46 1 Uruguay

2004 47 0

2005 48 1 Dominican Republic

2006 50 0

2007 51 0

2008 52 1 Ecuador

2009 53 0

2010 53 0

2011 54 0

2012 55 1 Greece

Historic portfolio defaults

*Analysis of risk-weighted assets for credit risk in the banking book http://www.bis.org/publ/bcbs256.pdf

Example Portfolios

http://www.bis.org/publ/bcbs256.pdf





Portfolio 2: Corporate portfolio

11

Corporate portfolio (size £108 billion) of 1,292 (2012) credit exposures

to a global portfolio of major national corporates which has grown from

290 customers (2002).

68% of the portfolio is provided to corporates operating primarily in 20

countries within Europe, with the remaining 32% mainly in the United

States and China.

77defaults during the period with an annualised long run average

observed default rate of 1.04%, although the maximum number of

defaults to corporates in any single jurisdiction was ten.

There was a spike in the default rate from 2007 – 2009 as a result of

increased bankruptcy volumes following the global financial crisis.

Regulatory expectation for separate PD calibrations for respective

countries will be challenging given reduced volume of defaults.

>10%

5%-10%

0%-5%

None

European investment

proportions:

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

0

20

40

60

80

100

120

140

160

180

AA

A

AA

+

AA

AA

-

A+ A A-

BB

B+

BB

B

BB

B-

BB

+

BB

BB

-

B+ B B-

CC

C/C


Defaults (Primary) Counterparties (Primary) Exposure (Secondary)

2 3

4 5 5

7 9

9 9 10

14

0

200

400

600

800

1,000

1,200

1,400

0

2

4

6

8

10

12

14

16

20

02

20

03

20

04

20

05

20

06

20

07

20

08

20

09

20

10

20

11

20

12

Historic portfolio data

Defaults (Primary) Counterparties (Secondary)

Example Portfolios



Portfolio 3: Growing regional mortgage portfolio

12

1 1 3

4 3

4

11 10

5

3

6

0

1,000

2,000

3,000

4,000

5,000

6,000

0

2

4

6

8

10

12

20

02

20

03

20

04

20

05

20

06

20

07

20

08

20

09

20

10

20

11

20

12

Historic portfolio data

Defaults (Primary) Counterparties (Secondary)

Credit risk is managed using manual underwriting, supported by a rating

scorecard with defaults defined as “90 days past due”.

Mortgage portfolio has grown from £47m to £798 during a 10 year period,

with customer volumes growing (from 470 to 4,700) and the average

mortgage size increasing (from £100K to £171k).

51 defaults were observed during the period an annualised long run average

observed default rate of 0.36%.

The default spike (2007 – 2008) was followed by reduced defaults due to low

interest rates and forbearance, with lending accelerating from 2010.

Low observed default rates make PD calibration for capital requirements

challenging. The recent PRA exercise to assess capitalisation of 8 UK banks

and building societies used a 15% RWA floor on residential mortgages which

provides a benchmark for minimum expectations despite low default rates.

>25%

10%-15%

0%-10%

None

0

4,000

8,000

12,000

16,000

20,000

24,000

28,000

0

200

400

600

800

1000

1200

1400

1 2 3 4 5 6 7 8 9 10


Defaults (Primary) Counterparties (Primary) Exposure (Secondary)

Proportion of

counterparties:

Example Portfolios



Sovereign portfolio

PD RWA

Observed Default Rate 0.72% 86%

Method A - Confidence Based PD Estimate

@ 75% confidence level


2.19%

3.53%

125%

143%

Method B - Bayesian Mean PD Estimate

Conservative Prior PD distribution 3.79% 146%

Uniform Prior PD distribution 3.64% 144%

Expert Prior PD distribution 1.85% 119%

13

Inputs: (Source)

- Asset Correlation: (Basel) 24%

- Intertemporal Correlation: (Expert) 70%

- Confidence Level: (Industry Benchmark) 75%

- Assumed LGD 45%

• Due to low counterparty

numbers, all estimates are

significantly high compared to

the observed default rate

• Expert prior based PD is

comparable to the “Confidence

based approach” result.

• Conservative and uniform prior

distributions produce more

conservative results, which are

consistent with 90% confidence

level for Method A

PD

Pro

ba

bili

ty

4% 0% 1.5%

Expert PD distribution

Conservative and uniform prior distributions produce conservative estimates which are

equivalent to applying a 90% confidence level in the Confidence Based Approach.

Calibration results



Corporate portfolio

PD RWA

Observed Default Rate 1.01% 98%




2.47%

3.84%

129%

146%


Conservative Prior PD distribution 3.29% 140%

Uniform Prior PD distribution 3.00% 136%

Expert Prior PD distribution 1.86% 119%

14

Inputs: (Source)




- Assumed LGD 45%

• Both the conservative and

uniform priors produce high

estimates compared to the

confidence based approach due

to the high correlation.

• Tight expert band range (1%

minimum - 3% maximum PD)

limits Bayesian estimates

producing results which are

conservative.

• Calculating the results per

country significantly increases

the total portfolio PD.

PD

Pro

ba

bili

ty

3% 1% 1.5%


Expert Bayesian distribution is less conservative than the confidence based approach due to

the tight range of PD expectations.

Calibration results



Growing mortgage portfolio

PD RWA

Observed Default Rate 0.36% 9.8%




0.78%

1.17%

16.9%

22.1%


Conservative Prior PD distribution 0.98% 19.7%

Uniform Prior PD distribution 0.97% 19.5%

Expert Prior PD distribution 0.88% 18.3%

15

Inputs: (Source)




- Assumed LGD 15%

• Higher customer volumes

significantly reduce the range of

PDs produced by both methods.

• Bayesian estimates lie between

75% and 90% confidence based

estimates. Conservative and

uniform prior estimates are similar

given the low asset correlation.

• Expert Bayesian estimate is close

to 75% confidence level calculated

using confidence based approach.

• All RWA results exceed 15% floor.

PD

Pro

ba

bili

ty

4% 0.1% 0.4%


Due to higher customer volumes and lower correlations, both methodologies produce

comparable results but RWAs exceed 15% which could inform Pillar 2 capital estimates.

Calibration results



Summary of correlation sensitivity analyses results Method A Method B with Expert Distribution

Min Base Max Sensitivity Min Base Max Sensitivity

Sovereign

Portfolio

1.29%

2.19% 7.07% Medium 1.20% 1.85% 1.99% Low

Corporate Portfolio 1.38% 2.47% 7.97% Medium 1.50% 1.86% 2.01% Low

Growing Mortgage

Portfolio

0.49% 0.78%

4.29% High 0.56% 0.88% 1.92% Medium

16

Method B – Bayesian approach

• Using a fixed interval expert distribution limits the

range of PD produced by the methodology. Therefore,

estimates are much less sensitive to correlation

assumptions than the confidence based approach.

• The mortgage portfolio shows slightly higher sensitivity

to correlations due the larger range of the expert prior

distribution and high volumes.

• Model risk is driven by the expert choice of prior

distribution as the sensitivity is low

Range: Asset correlation: Minimum - 5%, Maximum - 50%

Inter-temporal correlation : Minimum - 45%, Maximum - 90%

Method A – Confidence level based approach

• The confidence based method shows a very high sensitivity

to correlation assumptions.

• The mortgage portfolio shows the highest sensitivity to

correlation assumptions with PD estimates ranging by a

factor of 9 (from 0.49% to 4.3%) of the default rate.

• Model risk is driven by the reliance on assumptions but the

ability to set a confidence level provides an opportunity to

link to risk appetite.

The expert distributions limits sensitivity to correlation assumptions, although introduces risk

of subjectivity which could preclude unexpected outcomes being captured.

Model risk



Evaluating strengths and weaknesses

17

The two methodologies can be applied in concert to benchmark PD results and prioritised for

specific application in targeted portfolios.

• Flexible prior distributions can be used

• Stakeholder expectations and industry

knowledge can be incorporated

• Less conservative estimations can be

produced which may be applicable for

provisioning or pricing.

Bayesian approach Confidence level based approach

Strengths

Weaknesses

• Produces conservative estimation, which

can be appropriate for capital calculations

• Fast computation time

• Not required to justify a prior distribution

• Confidence level can be linked to defined

model risk appetite.

Application

• Can produce estimates which are too

conservative and therefore hard to achieve

buy-in from stakeholders.

• Very sensitive to asset correlation and inter-

temporal correlation assumptions as well as

the confidence level setting

• Expert estimations can be biased, which

introduces a source of model risk.

• Using conservative and neutral priors,

estimations become very sensitive to

correlation assumptions.

• Computation time is significant.

• Conservative PD estimation for portfolios

where experts do not have additional

knowledge of the data.

• Validation of Capital requirement estimates.

• For example:

o Sovereigns

o Growth portfolios in new markets

• Less conservative PD estimations to reflect

extra information or benchmark data which

exists and can justify

• For example:

o Mortgage portfolios

o Special niche portfolios (Project Finance,

Financial Institutions)

Comparison of the methodologies



Conclusions

• Choice of methodology and assumptions (priors and correlations) have a

huge impact on PD estimates which can lead to significant variance in capital

requirement.

• Expert based prior distributions can be an alternative to the conservative

and uniform distributions, producing comparable PD estimates with the other

methodologies.

• Expert distribution show lower sensitivity to correlation inputs

• The PD estimates produced by these methodologies can not be backtested due

to data constraints, but can be benchmarked against estimates from different

methodologies

• Regulatory expectations of RWA floors may override capital calculated

using LDP PD estimates for some portfolios. However, Pillar 2 assessment

and a growing list business requirements depend on appropriate

estimates of PDs.

18



Appendix: Sensitivity Results



Sovereign portfolio

20

Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is

much lower.

5

20

35

50

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

2.20%

2.40%

45 50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

PD

Esti

mate

Intertemporal Correlation

1.00%-1.20% 1.20%-1.40% 1.40%-1.60% 1.60%-1.80%

1.80%-2.00% 2.00%-2.20% 2.20%-2.40% 2.40%-2.50%

Method A, confidence level 75%

5

20

35

50

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

45 50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

PD

Esti

mate


0.00%-1.00% 1.00%-2.00% 2.00%-3.00% 3.00%-4.00%

4.00%-5.00% 5.00%-6.00% 6.00%-7.00% 7.00%-8.00%

Method B, expert distribution

Appendix: Sensitivity analysis results



Corporate portfolio

21

5

20

35

50

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

45 50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

Pd

Esti

mate


0.00%-1.00% 1.00%-2.00% 2.00%-3.00% 3.00%-4.00%

4.00%-5.00% 5.00%-6.00% 6.00%-7.00% 7.00%-8.00%




much lower.


5

20

35

50

1.50%

1.70%

1.90%

2.10%

2.30%

2.50%

50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

Pd

Esti

mate


1.50%-1.70% 1.70%-1.90% 1.90%-2.10%

2.10%-2.30% 2.30%-2.50%



Growing mortgage portfolio

22




much lower, however slightly higher than for the other two portfolios.


5

20

35

50

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%

1.40%

1.60%

1.80%

2.00%

45 50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

PD

Esti

mate


0.00%-0.20% 0.20%-0.40% 0.40%-0.60% 0.60%-0.80%

0.80%-1.00% 1.00%-1.20% 1.20%-1.40% 1.40%-1.60%

1.60%-1.80% 1.80%-2.00%

5

20

35

50

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

45 50 55 60 65 70 75 80 85 90

Co

rrela

tio

n

Pd

Esti

mate


0.00%-0.50% 0.50%-1.00% 1.00%-1.50%

1.50%-2.00% 2.00%-2.50% 2.50%-3.00%

3.00%-3.50% 3.50%-4.00% 4.00%-4.50%



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