Modelling of scenarios for credit risk:
establishing stress test methodologies
Ken Nyholm
November 20th, 2006
European Central Bank
Risk Management Division
Strategy Unit
1
The views presented here are not necessarily shared by the European Central Bank
Outline
• Consistent, accountable and intuitive stress testing
method under migration mode
• Risks: market and credit
• A modelling framework for yields, returns and
portfolio losses
• An example
2
Tail events are in focus
3
Historical yield curve evolution in US
4
Historical yield evolution in Japan
5
Historical evolution of credit curves
6
Generic yield curve shapes
Time t=0 curves
Generic Normal
12
12
CCC
10
10
8
8
6
6
4
4
2
2
AAA
0
0
0
20
40
60
80
100
120
0
20
40
60
80
100
120
100
120
Yield(%)
Generic Steep
Generic Flat/Inverse
12
12
10
10
8
8
6
6
4
4
2
2
0
0
0
20
40
60
80
100
120
0
20
40
60
80
Maturity
7
Framework specifications
• Design of the calculation “engine” must be:
– Comprehensive enough to include all relevant systematic
and stochastic components
– General enough to allow for frequent re-calculations
– Systematic in its treatment of risk factors
– Flexible enough to answer stress-testing questions
– In accordance with economic/financial theory and intuition
8
A general design
Current and Generic Yield Curves
Portfolio Information
Ratings
1
3
4
…
6
Names
x1
x2
x3
…
xn
Positions
20
60
60
…
30
Cpn
3.75
4.5
2.75
…
5.5
Maturities
34.3
35.5
1.1
…
3.9
Generic Normal
Time t=0 curves
12
12
CCC
10
10
8
8
6
6
4
4


Simulated Credit States
Macro environments
stochastic / static
Yield curves
stochastic/static
2
2
AAA
0
0
20
0
60
40
80
0
120
100
12
10
10
8
8
6
6
0.35
D
0.3
B
BB
BBB
A
…
…
…
…
…
…
xn
6
6
6
…
6
AAA
AA
80
100
120
100
120
4
2
0
0
20
0
60
40
80
0
120
100
40
20
60
80
Maturity
AAA with innovations
AAA, Recession in month 5 and onwards
6
6
5.5
5.5
0.25
0.2
5
Yield(%)
Stochastic migration factor
Correlated migrations
Time-varying migrations
x3
4
3
3
…
4
x2
3
3
3
…
3
2
Yield(%)



0.4
x1
1
1
2
…
4
60
Generic Flat/Inverse
Generic Steep
12
4
Period
1
2
3
…
P
40
20
Yield(%)
4.5
0.15
5
4.5
0.1
4
100
0.05
0
-4
-2
-3
-1
0
3
2
1
4
15
10
100
50
4
15
5
Maturity
0
0
5
Maturity
-3
10
50
Time period
0
0
Time period
Market Risk
x 10
3.5
Results
3
2.5
-3
Joint Market and Credit Risk
x 10
-3
3.5
Credit Risk
x 10
2
8
1.5
7
3
2.5
Rt , j 
6
1
Pt , j ( t , j , Ct 1, j , Yt , j )
100
 Ct 1, j  t
5
2
0.5
4
1.5
0
-2000
-1500
-1000
-500
0
500
Changes in value
3
1000
1500
2000
1
2
0.5
1
0
-2000
-1500
-1000
-500
0
Changes in Value
500
1000
1500
0
-800
-700
-600
-500
-400
-300
-200
Changes in value
-100
0
100
200
9
A general design
• Risk sources:
– Market and Credit risk
• Scenario dependant:
– Migration and default probabilities
– Credit spreads
– Asset correlations, recovery rates time horizon
– Yield curve evolution: location and shapes
– Yield curve and spread innovations (error-term variances)
• A keyword could be: regime switching
10
The three basic building blocks
• Bond migration / credit state calculator
– Time varying migration and default probabilities
• Regime switching yield curve model
– Underlying yield curve factors are subject to regime switches
– Yield for all credit grades are simulated in a consistent
fashion
• Bond pricing module:
– Combining the credit state, maturity and yield curve
11
The three basic building blocks
• Intuition of the credit migration module
– Allows for migration and default mode calculations
– Relies on initial credit ratings of the assets
Applies the following steps:
–
–
Generates correlated random
numbers based on the assumed
correlation structure among
the obligors
Translates random numbers
into ratings using a timevarying migration matrix to get
t+1 rating for each obligor
0.4
0.35
D
B
BB
BBB
A
AA
AAA
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
-1
0
1
2
3
4
12
The three basic building blocks
•
Intuition of the yield curve module:
–
Underlying factors drive yield curves
–
These factors are allowed to shift regime over time
–
Different future scenarios can hence be generated where
yields and spreads vary according to chosen regimes
–
How to obtain yield curve factors?
–
One answer: the Nelson-Siegel model / parametric model
•
How to create a link to the underlying factors?
–
Regime switches
13
The modelling framework:Yield curve model
14
The modelling framework:Yield curve model
15
The modelling framework:Yield curve model
16
The modelling framework:Yield curve model
17
The modelling framework:Yield curve model
18
Jan-95
L_gov
S
S_gov
Jan-95
Jan-01
Sep-00
May-00
Jan-00
Sep-99
May-99
Jan-99
Sep-98
May-98
Jan-98
Sep-97
May-97
Jan-97
Sep-96
May-96
Jan-96
Sep-95
May-95
L_BB
S_BB
Jan-04
May-04
Sep-04
Jan-05
May-05
May-04
Sep-04
Jan-05
May-05
Sep-03
Sep-03
Jan-04
Jan-03
May-03
Sep-02
May-03
Sep-02
May-02
Jan-03
Jan-02
May-02
Jan-02
Sep-01
12
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
Sep-01
Yield Curve Factors
May-01
F
May-01
Jan-01
N
Sep-00
May-00
Jan-00
Sep-99
May-99
Jan-99
Sep-98
May-98
Jan-98
Sep-97
May-97
Jan-97
Sep-96
May-96
Jan-96
Sep-95
May-95
Yield (%)
Probability
The modelling framework:Yield curve model
Regime Probabilities
0.9
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
19
Current tools and techniques
•
One possible scenario where the underlying/exogenous factors
are set to be gdp and cpi growth:
20
Current tools and techniques
21
Bond pricing module
• A standard bond pricing equation is used
Pt , j ( t , j , C t 1, j , Yt , j ) 

N
100

DC
1Yt , j ( N 1 DE )

k 1
100Ct 1,


j
DC
1Yt , j ( k 1 DE )
 100 C t 1, j
DA
DE
• And, returns are calculated as:
Rt , j 
Pt , j ( t , j , Ct 1, j , Yt , j )
100
 Ct 1, j  t
22
Two scenarios:
•
A portfolio of 30 obligors under migration mode
•
Initial credit ratings from AAA to BB (md=2.0)
•
One year horizon
•
Scenario A:
•
–
–
Recovery 40%
Normal yield curve state for all periods
–
Average asset correlation: 0.20
Scenario B:
–
–
–
Recovery 20%
Averse yield scenario after 3 months
Average asset correlation: 0.40
23
Scenario results
CI
0.9000
0.9500
0.9750
0.9900
0.9950
0.9990
0.9995
0.9999
VaR
100.4
559.6
599.7
650.1
708.8
1192.9
1243.3
1727.9
Scenario A
VaR%
ES
0.3
420.6
1.9
645.3
2.0
712.0
2.2
848.1
2.4
1018.4
4.0
1372.3
4.1
1532.2
5.8
1760.2
Std
Exp Loss
149.5
57.9
CI
0.9000
0.9500
0.9750
0.9900
0.9950
0.9990
0.9995
0.9999
VaR
966.1
1087.4
1895.9
2788.0
3598.2
5453.4
6427.0
7933.5
Scenario B
VaR%
ES
3.2
1621.5
3.6
2236.8
6.3
2805.2
9.3
3751.5
12.0
4561.9
18.2
6630.1
21.4
7521.8
26.5
10441.4
Std
Exp Loss
598.9
272.1
24
Summary
•
A scenario generation framework based on building
blocks:
–
Facilitates extension of individual blocks separately
–
Straightforward to integration of new elements
–
Individual blocks can be used in other contexts
•
•
Allows for inclusion of relevant time varying factors:
–
Yield curve and spread evolutions
–
Default and migration rates
Intuitive and in accordance with economic theory
25