Recent Enhancements Towards Consistent
Credit Risk Modelling Across Risk Measures
RISK – Quant Congress USA
16-18 July 2014, New York
Péter Dobránszky
Disclaimer: The contents of this presentation are for discussion purposes only, represent
the presenter’s views only and are not intended to represent the opinions of any firm or
institution. None of the methods described herein is claimed to be in actual use.
Introduction

We investigate in this presentation the link between credit spread and rating
migration evolutions

The first building block of a consistent modelling framework is the construction
of appropriate generic credit spread curves

The main applications are VaR, IRC, CRM, CCR, CVA, etc.


Regulatory requirements for Regulatory CVA, credit VaR, wrong-way risk modelling,
economic downturn modelling, incremental default in FRTB, etc.
We present a fully cross-sectional approach for building generic credit spread
curves aka proxy spread curves

Difficulties with the intersection method – be granular but also robust and stable

Static representation vs. capturing the dynamics

Capturing times of stress and benign periods, stochastic business time, meanreversions, regime switching modelling, etc.

Take into account risk premium, jump risk, gap risk, historical vs. risk neutral
probabilities

Deal with observed autocorrelation
2
Assumptions

We assume that we have a large enough history of spreads 𝑠𝑖,𝑡 and corresponding
ratings 𝑅𝑖,𝑡 , where 𝑡 denotes a calendar date and 𝑖 identifies an issuer.

We do not model the spread dynamics directly, but rather the log-spread
dynamics. Accordingly, for convenience, we deal with 𝑦𝑖,𝑡 = ln 𝑠𝑖,𝑡 𝐿𝐺𝐷𝑖,𝑡 in the
following equations.

These log-spreads

If the data is not clean enough we may assign a weight 𝑤𝑖,𝑡 for each issuer and for
each day.
3
Least-square regression

In the course of calibrating the rating distances we disregard the potential sector
and region dimensions of the spreads and we will assume that the rating
distances are static. Accordingly, we intend to calibrate the rating distances by
the following regression.
4
Agenda

CDS curves

Capturing dynamics – sectorial approach (VaR, CVA VaR)


Estimating level – proxy spread curves (CVA, CS01, SEEPE)




Rating migration effect (EEPE)
Estimation error (IRC, CRM)
Default probabilities and recovery rate

Sovereigns (IRC, CRM)

Risk premium
Joint default events and correlated migration moves modelling


Grouping
Migration matrix


Factor analysis, Random Matrix Theory, Clustering
Concentration of events (IRC, CRM)
Some double counting issues
5
CDS curves

Understand the business cycles - stochastic business time (GARCH, etc.)

Detach business time (by sectors) from calendar time

VaR vs. Stressed VaR, EEPE vs. Stressed EEPE
Volatility indices normalized to their level on 1st January 2008
400%
350%
VIX Index (equity, S&P)
RVX Index (equity, Russel)
300%
250%
200%
EVZ Index (forex, EUR/USD)
GVZ Index (commodity, gold)
OVX Index (commodity, crude oil)
150%
100%
50%
0%
01/01/07
01/07/07
01/01/08
01/07/08
01/01/09
01/07/09
01/01/10
6
CDS curves

Expected value of integrated business time over a calendar time period

Dynamics of ATM implied volatility for various maturities

Credit spreads as annual average default rates

Correlated, but standalone clocks
Evolution of major credit and volatility indices
3.5%
90%
3.0%
iTraxx Eur 5Y Theo (left)
2.5%
CDX.NA.IG 5Y Theo (left)
2.0%
VIX Close (right)
80%
70%
60%
50%
40%
1.5%
30%
1.0%
20%
0.5%
10%
0.0%
0%
19/11/04
20/11/05
21/11/06
22/11/07
22/11/08
23/11/09
24/11/10
7
CDS curves

What is stationary?

Log-returns (?)
Evolution of the major European credit index
3%
3%
0.09%
0.08%
iTraxx Eur 5Y Theo (left)
0.07%
Evolution of the major North American
creditofindex
3M StdDev
AbsRet (right)
2%
3.5%
0.12%
0.05%
0.10%
0.04%
2%
3.0%
CDX.NA.IG 5Y Theo (left)
2.5%
1%
3M StdDev of AbsRet (right)
2.0%
1%
1.5%
0.06%
0.03%
0.08%
0.02%
0.06%
0.01%
0%
01/01/05
1.0%
0.00%
01/01/06
0.04% 01/01/09
01/01/08
01/01/07
0.5%
0.02%
0.0%
0.00%
01/01/05
01/01/06
01/01/07
01/01/08
01/01/09
01/01/10
01/01/10
01/01/11
01/01/11
Kurtosis by return type
Raw absolute
Normalised absolute
Raw relative
Normalised relative
iTraxx Eur 5Y
12.7
10.7
7.2
5.3
CDX.NA.IG 5Y
13.7
6.5
4.9
3.7
8
CDS curves


Capture the dynamics of spreads (VaR, CVA VaR, CRM)

Merge, demerge, new names

Illiquid curves – systemic, sectorial, idiosyncratic risk components
Selection of liquid curves as basis for capturing the dynamics

Definition of liquidity – contributors, number of non-updates

Sectorial approach – mapping of names to groups

Groups of names with similarities, homogeneity

Large enough and small enough groups, concentration

Trade-off between specificity and calibration uncertainty

Representation by number of names and by exposures

Can you assume cross-sectional relationships?

(N industry + M sector) systemic factors

(N industry × M sector) systemic factors
9
CDS curves
Principal Component Analysis of CDS log-returns, decompose correlation

Biased figures if not accounted for stochastic business times

Assume a group of 10 with an extra 30% variance explanation within group

This specific group factor explains only 10 665 ∙ 30% = 0.45% of total variance
Total variance explanation for past 5-year history of CDS log-returns
30%
Total variance explained

25%
20%
15%
10%
5%
0%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
First 25 over the total of 665 eigenfactors
10
CDS curves

Assume 10 explanatory factors and remove their impact

Does the remaining part behaves like random independent noise?
Still there can be 50 groups of 10 names with an extra group factor explaining
30% of the variance within the group
After removing the effect of the first 10 PC
1.4%
1.2%
Total variance explained

Empirical
1.0%
Random Matrix Theory
0.8%
0.6%
0.4%
0.2%
0.0%
1
101
201
301
401
501
601
11
CDS curves
Random Matrix Theory: it explains the eigenvalue distribution

If 𝑇, 𝑛 → ∞, with a fixed ratio 𝑄 = 𝑇 𝑛 ≥ 1, the eigenvalue spectral density of
𝑄
the correlation matrix is given by 𝜌 𝜆 = 2𝜋𝜆
𝜆𝑚𝑎𝑥 − 𝜆 𝜆 − 𝜆𝑚𝑖𝑛 where
𝜆𝑚𝑎𝑥,𝑚𝑖𝑛 = 1 + 1 𝑄 ± 2 1 𝑄.

But it works also for 𝑄 < 1. For instance, 𝜆𝑚𝑎𝑥
7
𝑇=260,𝑛=520
= 5.83.
Eigenvalue distribution by the Random Matrix Theory
6
5
Eigenvalue

4
3
1st pct
2
mean
1
99th pct
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
First 25 over the total of 520 eigenfactors
12
CDS curves

Principal components are often misleading

If you have 𝑛 risk factors and you remove the impact of the first 𝑚 principal
components, then the remaining variance is explained by only 𝑛 − 𝑚 instead
of 𝑛 factors.


Therefore, the first principal components explains also part of the noise variance.
Assume a one-factor model for 4 variables with pairwise correlations of 50%

Remove the impact of the first principal component – remaining parts of the 4
variables are not uncorrelated, instead, their correlation is -33%! – overcompensation

Remove the mean – mean captures only part of the systemic factor’s variance, thus
the remaining parts of the 4 variables have a 6% correlation – under-compensation
100%
-33%
-33%
-33%
100%
6%
6%
6%
-33%
100%
-33%
-33%
6%
100%
6%
6%
-33%
-33%
100%
-33%
6%
6%
100%
6%
-33%
-33%
-33%
100%
6%
6%
6%
100%
13
CDS curves

Clustering

It is a technique that collects together series of values into groups that exhibit
similar behaviour.

Hierarchical clustering based on Euclidean distance or correlation

Still mapping of clusters to sectors
and regions are required

Not robust towards outliers, few small
clusters and large concentration

Large clusters should be re-clustered

Does not ensure homogeneity within
cluster – fixed number of clusters

Recent: Make 2 clusters, split each
cluster into 2 until RMT conditions
are met – still exposed to outliers 14
CDS curves
Estimating level for a given day – proxy spread curves (CVA, CS01, SEEPE)

Data mining like exploration

How many distinguishable groups are there? Split by how many dimensions?

Basel III requests split by sectors, regions and ratings (see EBA BTS)
European 5Y CDS spreads by sectors on 15 June 2012
2
Basic Materials
Consumer Goods
Consumer Services
Energy
Financials
Government
Healthcare
Industrials
Technology
Telecom
Utilities
1.8
1.6
1.4
1.2
PDF

1
0.8
0.6
0.4
0.2
0
-3
-2
-1
0
1
2
Ln ( individual 5Y CDS spread / rating's spread )
3
15
CDS curves

Hypothesis test: difference between means



Apply the so-called two-sample t-test, which is appropriate when the
following conditions are met:

The sampling method for each sample is simple random sampling.

The samples are independent.

Each sample is drawn from a normal or near-normal population.
The first two conditions are met by construction. Concerning the third rule,
by rules-of-thumb, a sampling distribution is considered near-normal if any
of the following conditions apply:

The sample data are symmetric, unimodal, without outliers, and the sample size is
15 or less.

The sample data are slightly skewed, unimodal, without outliers, and the sample
size is 16 to 40.

The sample size is greater than 40, without outliers.
Analyse log-spreads and normalise by the rating effect
16
CDS curves

Two-sample t-test – compare groups based on two-tailed tests

Null hypothesis: considering two sectors, their average spread levels are
equal. Alternative hypothesis: average spread levels are not equal, thus these
sectors require separate proxy spread curves.

Assume that the standard deviations by samples are different. Therefore,
compute the standard error (SE) of the sampling distribution as
𝑆𝐸 =

The distribution of the statistic can be closely approximated by the t
distribution with degrees of freedom (DF) calculated as
𝐷𝐹 =

𝑠12 𝑛1 + 𝑠22 𝑛2
𝑠12 𝑛1
2
𝑠12 𝑛1 + 𝑠22 𝑛2 2
𝑛1 − 1 + 𝑠22 𝑛2 2
𝑛2 − 1
The test statistic is a t-score (t) defined by 𝑡 = 𝑥1 − 𝑥2 𝑆𝐸 .
17
CDS curves

P-values




BM
CG
CS
E
F
G
H
I
Ty
Tc
U
Sector
26
43
44
8
149
18
8
42
8
23
33
# of issuers
0.50
0.67
0.15
0.00
0.00
0.05
0.62
0.55
0.40
0.72
Basic Materials
0.72
0.03
0.00
0.00
0.06
0.12
0.28
0.08
0.21
Consumer Goods
0.04
0.00
0.00
0.02
0.20
0.35
0.13
0.32
Consumer Services
0.00
0.03
0.00
0.21
0.60
0.46
0.20
Energy
0.57
0.00
0.00
0.01
0.00
0.00
Financials
0.00
0.00
0.04
0.00
0.00
Government
0.00
0.07
0.00
0.00
Healthcare
0.75
0.62
0.89
Industrials
0.99
0.70
Technology
0.57
Telecom
Grouping may change as
sector levels fluctuate
Defines minimum
number of names in a
group
Here only European
issuers, however, is it
the same in NA?
Are there cross-sectional
information being
useful?
Utilities
P-values of the two-sample t-test as of 15 June 2012
BM
CG
CS
F
G
H
I
OG
Ty
Tc
U
Sector
29
37
51
156
34
8
43
9
5
26
27
# of issuers
0.08
0.02
0.95
0.75
0.10
0.25
0.18
0.71
0.00
0.01
Basic Materials
0.49
0.00
0.03
0.99
0.29
0.62
0.28
0.02
0.22
Consumer Goods
0.00
0.00
0.57
0.05
0.24
0.16
0.07
0.52
Consumer Services
0.62
0.03
0.02
0.04
0.68
0.00
0.00
Financials
0.08
0.18
0.13
0.86
0.00
0.00
Government
0.37
0.65
0.30
0.06
0.30
Health Care
0.66
0.58
0.00
0.02
Industrials
0.44
0.01
0.11
Oil & Gas
0.03
0.09
Technology
0.29
Telecommunications
Utilities
P-values of the two-sample t-test as of 31 December 2008
18
CDS curves
•
Mean of Ln ( individual 5Y CDS spread / rating's spread ) by sectors
Basic Materials
0.3
0.2
Europe
•
Industrials
0.0
Oil & Gas
-0.2
Consumer Goods
Telecomm.
Services
-0.4
-0.5
Consumer
Services
1.0
-0.4
𝑀
𝑁
𝐶 𝑚
𝛽 𝐶 𝑚 𝐼𝑖
+
𝛽
𝑛
+ 𝛽 𝑓𝑛 𝑟𝑎𝑡𝑖𝑛𝑔𝑖 + 𝜀𝑖
𝐼 𝑛
𝐼 𝑛
𝐼𝑖
Financials
0.80.0
N.Amer
0.6
-0.2
Europe
-0.6
𝑚
𝑅
As of 15 June 2012
Mean of Ln ( individual 5Y CDS spread / rating's spread ) by sectors
Utilities
-0.6
=
As of 31 December 2008
Healthcare
-0.1
-0.3
𝑠𝑖
Financials
Technology
0.1
Useful cross-sectional
information
Recently slope is not 1
0.2
0.4
0.6
0.4
Technology
Basic Materials
0.0
-0.2
Energy
Telecomm.
Services
0.2
Consumer
Services Utilities
Industrials
Healthcare
-0.4
Consumer Goods
-0.6
-0.6
-0.4
-0.2
0.0
N.Amer
0.2
0.4
19
CDS curves
Rating dependency for various sectors and regions


Different slope coefficients may be required

Bigger difference between sectors than regions
Proxy spread curves by using
25-75% trimmed mean
Proxy spread curves by using
25-75% trimmed mean
0
0
All
-1
All
-1
Europe
-2
North America
-3
Log-spread
Log-spread
-2
Financials
Rest of the world
-4
-3
-5
-6
-6
-7
-7
AA
A
BBB
BB
B
CCC
Corporate
-4
-5
AAA
Government
AAA
AA
A
BBB
BB
B
CCC
20
CDS curves

Rating migration effect (EEPE)

BIS Quarterly Review, June 2004: “Rating announcements affect spreads on credit
default swaps. The impact is more pronounced for negative reviews and downgrades than
for outlook changes.”
Regulation, CRR, Article 158:
(i) for institutions using the Internal
Model Method set out in Section 6 of
Chapter 6, to calculate the exposure
values and having an internal model
permission for specific risk associated
with traded debt positions in
accordance with Part Three, Title IV,
Chapter 5, M shall be set to 1 in the
formula laid out in Article 148(1),
provided that an institution can
demonstrate to the competent
authorities that its internal model for
Specific risk associated with traded debt
positions applied in Article 373 contains
effects of rating migrations;
21
Migration matrix

Estimation error (IRC, CRM)

Less or more rating matrices? Trade-off between capturing better the specific risk profiles and
basic risk vs. reducing the estimation noise.

Which ones? Sovereign and corporate migration matrices? Corporate divided by region (US /
Europe) and industry (financial / non-financial)?

Relevance for bank portfolio vs. availability of data, i.e. available data often with US
concentration.

Finer rating grid may reduce the jump of P&Ls on the tails, but it introduces estimation noise.

Binomial proportion confidence interval, i.e. how reliable is the transition probability estimate?
𝑝 1−𝑝
𝑛
𝑝+

CLT: 𝑝 ± 𝑧1−𝛼/2

For a 95% confidence interval:
, Wilson interval:
1 2
𝑝 1−𝑝
𝑧1−𝛼/2 ±𝑧1−𝛼/2
2𝑛
𝑛
1 2
1+𝑛𝑧1−𝛼/2
1 2
𝑧
4𝑛2 1−𝛼/2
+
Trial
Outcome
Estimate
Lower CI
Upper CI
50
1
2.0%
0.4%
10.5%

Enormous IRC impact
100
1
1.0%
0.2%
5.4%

Smoothing?
500
1
0.2%
0.0%
1.1%
1000
1
0.1%
0.0%
0.6%
22
Migration matrix

Calculation of short-term transition matrices

Markov approach: Assume time-homogeneous continuous-time Markov
chain and scale the transition matrix via the generator matrix.



Which is the best short-term matrix which provides that multiplying it by itself
several times gives the best approximation for the original one-year matrix?
𝑀𝜏1 +𝜏2 = 𝑒 𝐺∙ 𝜏1 +𝜏2 = 𝑒 𝐺∙𝜏1 𝑒 𝐺∙𝜏2 =𝑀𝜏1 ∙ 𝑀𝜏2 = 𝑀𝜏2 ∙ 𝑀𝜏1 = 𝑒 𝐺∙𝜏2 𝑒 𝐺∙𝜏1
Cohort method: Discrete-time method based on the historical migration data.
Calibrations to various time horizons may show autocorrelation in
migrations.

Maximum likelihood estimation assuming Markov model

Accounting for stochastic business times
23
Default probabilities and recovery rates

Source of recovery rates


What are the local currency recovery rates?

Sovereigns may go default on their hard currency and local currency obligations
separately

Does the IRC engine simulate both events, if yes, how to manage correlation, if
not, which rating is used for IRC calculations

It can be interpreted as what is the LC/HC bond value in case the HC/LC bond
migrate or default

Various approaches to adjust the LC recovery rates to account for FX depreciation
– quanto CDSs may be used
What are the recovery rates for covered bonds and government guarantees?

The rating of issuing bank is taken, which implies “high” PD, but when the issuer
goes to default, there is still a pool of assets or another guarantor to meet the
obligation.

Recovery rates are usually high to compensate that “wrong” PDs are used.

Ensure that bond PV < recovery rate
24
Default probabilities and recovery rates

Source of estimated or implied probabilities of defaults (PD)

Historical TTC default probabilities provided by rating agencies (cohort).

Risk-neutral PIT default probabilities bootstrapped from traded CDSs.
AAA
Physical
Risk Neutral
1Y
0.00%
0.18%
2Y
0.00%
0.23%
3Y
0.00%
0.29%
4Y
0.00%
0.35%
5Y
0.00%
0.40%
7Y
0.00%
0.43%
10Y
0.01%
0.45%
AA
Physical
Risk Neutral
1Y
0.00%
0.28%
2Y
0.01%
0.35%
3Y
0.01%
0.43%
4Y
0.01%
0.52%
5Y
0.01%
0.60%
7Y
0.02%
0.65%
10Y
0.02%
0.71%
A
Physical
Risk Neutral
1Y
0.02%
0.36%
2Y
0.02%
0.45%
3Y
0.03%
0.55%
4Y
0.04%
0.65%
5Y
0.05%
0.74%
7Y
0.07%
0.80%
10Y
0.10%
0.88%
BBB
Physical
Risk Neutral
1Y
0.12%
0.53%
2Y
0.15%
0.68%
3Y
0.19%
0.82%
4Y
0.22%
0.96%
5Y
0.25%
1.10%
7Y
0.31%
1.19%
10Y
0.38%
1.30%
BB
Physical
Risk Neutral
1Y
0.74%
1.20%
2Y
0.84%
1.59%
3Y
0.94%
1.94%
4Y
1.02%
2.24%
5Y
1.10%
2.45%
7Y
1.21%
2.60%
10Y
1.32%
2.70%
B
Physical
Risk Neutral
1Y
3.33%
2.81%
2Y
3.47%
3.57%
3Y
3.56%
4.30%
4Y
3.62%
4.94%
5Y
3.66%
5.50%
7Y
3.68%
5.62%
10Y
3.65%
5.62%
CCC
Physical
Risk Neutral
1Y
12.70%
5.87%
2Y
12.26%
7.15%
3Y
11.84%
8.05%
4Y
11.46%
8.72%
5Y
11.10%
9.00%
7Y
10.51%
8.80%
10Y
9.85%
8.48%
•
Comparison of transformed
historical PDs with Markit
sector curves as of 30 June
2009 and assuming 40%
recovery rate.
•
Taking non-diversifiable risk
is compensated by premium.
The rarer the event the more
difficult to diversify and the
higher the risk premium.
•
•
•
IRC: historical PDs are used
for simulations, while implied
default probabilities are used
for re-pricing.
Impact depends on the
portfolio.
25
Default probabilities and recovery rates

Accounting for risk premium

Banks take over risk, diversify and get compensation for systemic risk

Diffusion processes: risk premium over risk is negligible in the short-term

Risk premium related to jump risk and gap risk is priced differently

BB sector 5Y CDS ranged between 100 and 700 bps from beginning 2006 to mid
2011

Implied default rate around 1.7-11% (𝑅𝑅 = 40%) vs. TTC default rate of 1%

Rare events (AAA) are priced with higher risk premium

Problems started to rise with Basel 2.5

IRC loss distribution is strongly effected by risk premium


𝑃&𝐿 = 𝑃𝑉𝑡1 𝜃𝑡1 ∙ 𝐷𝐹𝑡0 ,𝑡1 − 𝑃𝑉𝑡0 𝜃𝑡0 +
𝑡1
𝐶𝐹𝜏
𝑡0
∙ 𝐷𝐹𝑡0 ,𝜏 𝑑𝜏
Visualise the potential time value effect when risk premium is significant

𝑃&𝐿 ≈ 𝑃𝑉𝑡0 𝜃𝑡1 − 𝑃𝑉𝑡0 𝜃𝑡0 +
𝜕𝑃𝑉𝜏 𝜃𝑡0
𝜕𝜏
𝑡1 − 𝑡0
26
Default probabilities and recovery rates

Short protection portfolio of CDSs written on BB rated issuers

30th June 2009

Average 1Y CDS spread of the constituents was 600 bps

In case no default or migration event happens, expected portfolio P&L is around
6%

Not accounting for time value, expected portfolio P&L is around -1% (TTC)
IRC loss distribution of a CDS portfolio with 100 BB constituents
Probability density
Forward repricing
Taylor approximation
Taylor appr wo theta
-15

-10
-5
0
5
10
IRC as of 30/06/09 for a 100M issuer risk portfolio
15
Millions
Numerous default events may occur before any effective loss is realised
27
Joint default events and correlated migration moves

Asset value correlation: parameter of the Gaussian copula approach

Default correlation (Pearson correlation):
 Dj,k 

JDFj ,k  CEDFj  CEDFk
CEDFj  1  CEDF j   CEDFk  1  CEDFk  


If CEDFj is not equal to CEDFk, the default correlation can never reach 100%
 Dj,k 

1%  2% 1%
 2%  1  2%   1%  1  1%  
 70%
Process correlation: when processes are moving together
j
T
T+∆t
k
Time fractions of co-movements
28
Asset value correlation model
Gaussian case
Pairwise correlations
determine the whole
joint dependence
structure

Proxies for calibration

Factor correlation
approach (KMV GCorr)

Same correlation for
defaults and migrations

Copula: one-step
discrete-time approach

Forward joint density
does not exist
k defaults, j pays

Borrower j

Both pay
k pays, j defaults
Both default
Borrower k
29
Term structure of default correlations

Fix the AVC and measure the Pearson default correlation for various horizons
(annual PD = 2%, 2-state Markov chain with jump-to-default)
Default correlations as function of time horizon for various AVC
Default correlation
4.0%
3.5%
5%
3.0%
10%
20%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
0
1
2
3
4
5
6
7
8
9
10
11
12
Time horizon (months)


Similar term structure of default correlations by ratings

The lower the cumulative probability of defaults the lower is the default correlation

Most copula based approaches imply that the defaults of highly rated names are
basically independent
Opposite to this, process correlations produce flat default correlation curves
30
Default correlations by rating classes

Gaussian copula

AVC = 10%
Moody's KMV
analysis
A1-A3
Baa1-Baa3
Ba1-Ba3
B1-B3 & below

Realised
default
correlation
0.65%
0.59%
1.68%
2.36%
Correlated
continuous-time
Markov chains
AAA
AA
A
BBB
BB
B
CCC
AAA
0.00%
0.01%
0.01%
0.03%
0.05%
0.07%
0.08%
AA
0.01%
0.03%
0.04%
0.08%
0.15%
0.23%
0.28%
A
0.01%
0.04%
0.06%
0.12%
0.23%
0.35%
0.44%
BBB
0.03%
0.08%
0.12%
0.28%
0.53%
0.82%
1.05%
BB
0.05%
0.15%
0.23%
0.53% 1.043%
1.65%
2.17%
B
0.07%
0.23%
0.35%
0.82%
1.65%
2.66%
3.60%
CCC
0.08%
0.28%
0.44%
1.05%
2.17%
3.60%
5.03%
different default correlation structure by rating
AAA
AA
A
BBB
BB
B
CCC
AAA
0.02%
0.06%
0.05%
0.04%
0.03%
0.02%
0.01%
AA
0.06%
1.08%
0.68%
0.27%
0.10%
0.05%
0.03%
A
0.05%
0.68%
0.73%
0.39%
0.19%
0.11%
0.05%
BBB
0.04%
0.27%
0.39%
0.86%
0.53%
0.29%
0.15%
BB
0.03%
0.10%
0.19%
0.53% 1.043% 0.59%
0.33%

Process corr. = 11%
B
0.02%
0.05%
0.11%
0.29%
0.59%
1.14%
0.65%

Time fraction 1.2%
CCC
0.01%
0.03%
0.05%
0.15%
0.33%
0.65%
1.16%
term structure is flat at PC2 if T is small
31
Event concentration – the new dimension of uncertainty

In case of jumpy processes the parameterisation of the pairwise
dependence structures is not enough to determine the N-joint law
j
j
T k
T+∆t
k
T+∆t
l
l
High concentration

T
Low concentration
Same pairwise dependence structure, but different N-joint law

High concentration: Armageddon scenario likely

Low concentration: probability of large number of defaults is high
32
Incremental modelling uncertainty
Compare Gaussian copula model against more advanced correlated jump
models with various event concentrations



AVC = 8.5%, which means process correlation of 10%
PD, LGD and P&L effect of rating changes are the same in each case
Fixed time horizon of one year
Loss distribution of a CDS portfolio with 20 BB constituents
AVC
Probability density

PDL
PDM
PDH
Millions
-3
-2
-1
0
1
2
3
IRC as of 30/06/09 for a 20M issuer risk portfolio

In case of small portfolios, various models produce very similar IRC loss distributions
33
Incremental modelling uncertainty
The larger the portfolio the larger the impact of the model choice

Especially short protection portfolios are very sensitive to the
concentration modelling – concentration of default events can hardly be
calibrated
Loss distribution of a CDS portfolio with 100 BB constituents
AVC
Probability density

PDL
PDM
PDH
Millions
-10
-5
0
5
10
IRC as of 30/06/09 for a 100M issuer risk portfolio
15
IRC
AVC
PDL
PDM
PDH
BB long
9.1 M
7.6 M
7.9 M
7.4 M
BB short
3.4 M
0.4 M
4.2 M
5.3 M
34
Separating default and migration correlations

Until this point we assumed the same correlation between default events and
migration moves. Nevertheless, we can separate the Markov generator matrix for
defaults and migrations.
AAA
AA
A
BBB
BB
B
CCC
AAA
0.54%
0.26%
0.39%
0.26%
0.11%
0.07%
0.00%
AA
0.26%
0.23%
0.42%
0.23%
0.08%
0.04%
0.00%
A
0.39%
0.42%
1.30%
0.97%
0.44%
0.22%
0.01%
BBB
0.26%
0.23%
0.97%
1.90%
0.89%
0.54%
0.04%
BB
0.11%
0.08%
0.44%
0.89% 0.775% 0.58%
0.09%
B
0.07%
0.04%
0.22%
0.54%
0.58%
0.80%
0.15%
CCC
0.00%
0.00%
0.01%
0.04%
0.09%
0.15%
0.32%
Moody's KMV
analysis
Realised
default
correlation
A1-A3
Baa1-Baa3
Ba1-Ba3
B1-B3 & below
0.65%
0.59%
1.68%
2.36%

Even perfectly correlated migration moves cannot reproduce the realised
default correlations

Critics for reduced-form models correlating only default intensities
35
Stochastic business time

Time homogeneity is clearly not an appropriate assumption
Evolution of the major volatility index
160%
140%
VXO
120%
VIX
100%
80%
60%
40%
20%
0%
86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11

Stress periods are described by volatility clusters
36
Stochastic business time

Recent time changed models are designed to explain default correlations

Use a realistic statistical model to describe the business time dynamics
Distribution of the business time (1986-2011)
1D BT
2W BT
1M BT
3M BT
1Y BT
0%
100%
200%
300%
400%
500%
600%
700%
37
Stochastic business time

Calibrate the transition generator by assuming stochastic business time
What degree of realised default correlation can be explained by SBT?

1-year
AAA
AA
A
BBB
BB
B
CCC
AAA
0.00%
0.00%
0.00%
0.01%
0.01%
0.03%
0.04%
AA
0.00%
0.00%
0.01%
0.01%
0.03%
0.05%
0.08%
A
0.00%
0.01%
0.01%
0.03%
0.06%
0.11%
0.18%
BBB
0.01%
0.01%
0.03%
0.07%
0.15%
0.28%
0.46%
BB
0.01%
0.03%
0.06%
0.15%
0.324%
0.60%
0.99%
B
0.03%
0.05%
0.11%
0.28%
0.60%
1.13%
1.86%
CCC
0.04%
0.08%
0.18%
0.46%
0.99%
1.86%
3.09%
Moody's KMV
analysis
Realised
default
correlation
A1-A3
Baa1-Baa3
Ba1-Ba3
B1-B3 & below
0.65%
0.59%
1.68%
2.36%

Similarity with correlated default intensities (correlated migration only)

Default correlation by rating is not flat! Combine with process correlation!

Term structure of default correlation by PC = 11% plus SBT (hockey stick):
AAA
0.01%
AA
1.22%
A
1.22%
BBB
BB
B
1.22% 1.222% 1.23%
CCC
1.25%
1-month 0.02%
1-year 0.02%
1.21%
1.04%
1.13%
0.69%
1.15% 1.224% 1.35%
0.92% 1.350% 2.26%
1.79%
4.21%
1-day
38
Some double counting issues
Consistency and coherency issues between capital charges

Potential exposure within a year does not capture that losses in case of a future default
have potentially been realised already by CVA VaR when spreads were climbing up –
this CVA variation is capitalised now
CVA path evolution until default
Potential additional loss
at the time of default
CVA

Total loss disregarding from CVA
∆CVA
Initial CVA
Default event
Time
Maturity

Similarly for IRC vs. VaR – if being long credit for Greece, daily MtM losses were
capitalised by VaR, while there was no further loss at the time of default, thus IRC
capital charge was questionable

Sudden and expected defaults shall be separated and capitalised accordingly
39