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Deloitte UK screen 4:3 (19.05 cm x 25.40 cm)
© 2013 Deloitte LLP. All rights reserved.
Probability of Default (PD) Calibration Conundrum
Low Default Portfolio (LDP) modelling
30th August 2013
Deloitte UK screen 4:3 (19.05 cm x 25.40 cm)
© 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).
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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
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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.
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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
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© 2013 Deloitte LLP. All rights reserved.
At least 50% of commercial banking book assets are in portfolios which have LDP
characteristics:
6 Source: Bank Of England
0
500,000
1,000,000
1,500,000
2,000,000
2,500,000F
I excl in
sura
nce
an
d p
en
sio
n f
und
s
Fin
an
cia
l in
term
ed
iation
Insu
ran
ce a
nd p
ensio
n f
un
ds
Hea
lth
& S
ocia
l W
ork
Ele
ctr
icity,
Ga
s a
nd
Wate
r S
up
ply
Ed
uca
tio
n
Pu
blic
ad
min
Su
bto
tal L
DP
Re
al e
sta
te &
pro
fessio
na
l se
rvic
es
Con
str
uction
Reta
il an
d W
ho
lesale
tra
de
Ma
nu
factu
rin
g
Tra
nspo
rt,
sto
rag
e a
nd
co
mm
s
Acco
mm
od
atio
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od
Recre
atio
nal, p
ers
on
al, c
om
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nity
Su
bto
tal w
ho
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Indiv
idu
als
To
tal all
len
din
g
£ m
illio
n
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
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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
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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
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Objective
Analyse the different PD calibration methodologies and evaluate the model risks introduced.
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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
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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
Rating grade distribution for 2012
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
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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
Rating grade distribution for 2012
Defaults (Primary) Counterparties (Primary) Exposure (Secondary)
Proportion of
counterparties:
Example Portfolios
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Sovereign portfolio
PD RWA
Observed Default Rate 0.72% 86%
Method A - Confidence Based PD Estimate
@ 75% confidence level
@ 90% 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
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Corporate portfolio
PD RWA
Observed Default Rate 1.01% 98%
Method A - Confidence Based PD Estimate
@ 75% confidence level
@ 90% confidence level
2.47%
3.84%
129%
146%
Method B - Bayesian Mean PD Estimate
Conservative Prior PD distribution 3.29% 140%
Uniform Prior PD distribution 3.00% 136%
Expert Prior PD distribution 1.86% 119%
14
Inputs: (Source)
- Asset Correlation: (Basel) 24%
- Intertemporal Correlation: (Expert) 70%
- Confidence Level: (Industry Benchmark) 75%
- 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 PD distribution
Expert Bayesian distribution is less conservative than the confidence based approach due to
the tight range of PD expectations.
Calibration results
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Growing mortgage portfolio
PD RWA
Observed Default Rate 0.36% 9.8%
Method A - Confidence Based PD Estimate
@ 75% confidence level
@ 90% confidence level
0.78%
1.17%
16.9%
22.1%
Method B - Bayesian Mean PD Estimate
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)
- Asset Correlation: (Basel) 15%
- Intertemporal Correlation: (Expert) 70%
- Confidence Level: (Industry Benchmark) 75%
- 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%
Expert PD distribution
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
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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
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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
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© 2013 Deloitte LLP. All rights reserved.
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
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© 2013 Deloitte LLP. All rights reserved. 19
Appendix: Sensitivity Results
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© 2013 Deloitte LLP. All rights reserved.
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
Intertemporal Correlation
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
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© 2013 Deloitte LLP. All rights reserved.
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
Intertemporal Correlation
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 A, confidence level 75%
Method B, expert distribution
Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is
much lower.
Appendix: Sensitivity analysis results
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
Intertemporal Correlation
1.50%-1.70% 1.70%-1.90% 1.90%-2.10%
2.10%-2.30% 2.30%-2.50%
Deloitte UK screen 4:3 (19.05 cm x 25.40 cm)
© 2013 Deloitte LLP. All rights reserved.
Growing mortgage portfolio
22
Method A, confidence level 75%
Method B, expert distribution
Due to the fixed range of the expert prior distribution, the sensitivity to correlation inputs is
much lower, however slightly higher than for the other two portfolios.
Appendix: Sensitivity analysis results
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
Intertemporal Correlation
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
Intertemporal Correlation
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%
Deloitte UK screen 4:3 (19.05 cm x 25.40 cm)
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