Credit Risk Modeling for Capital Allocation A Credit Migration Approach

1
Credit Risk Modeling for Capital Allocation A
Credit Migration Approach

• Lecture 9
• Based on Risk Management
• By Crouhy, Galai Mark
• McGraw-Hill,2000

2
Section 2 Agenda

• I Measuring Credit Risk Overview
• II Credit Migration Approach
• III The Contingent Claim Approach
• IV The Actuarial Approach
  
• V The Reduced Form Approach
• VI Comparison of Models ISDA-IIF
• VII Conclusion

3
I Measuring Credit Risk

• Overview

4
Measuring Credit Risk Overview
What are the current proposed industry sponsored
Credit VaR methodologies?

• Credit Migration Approach
• CreditMetrics (from J.P. Morgan)
• CreditVaR (CIBC)
• CreditPortfolioView (McKinsey)
• The Option Pricing Approach
• KMV (from KMV Corp.)

5
Measuring Credit Risk Overview
What are the current proposed industry sponsored
Credit VaR methodologies?

• The Actuarial Approach
• CreditRisk (from Credit Suisse First Boston)
• The Reduced Form Approach
• Jarrow/Turnbull
• Duffie/Singleton

6
Comparison of Models
Actuarial approach
Reduced form approach
Contingent claim
Credit migration approach
approach
Software
CreditMetrics
CreditRisk
Kamakura
CreditPortfolioView
KMV
Definition of
D
Market Value
Market Value
Default losses
Default losses
Default losses
D
Risk
Credit events
Downgrade/Default
Downgrade/Default
Continuous default
Default
Default
probabilities

Risk drivers
Asset Values
Macro-factors
Asset Values
Expected default
Hazard rate
rates
Transition
Constant
Driven by Macro
Driven by
N/A
N/A
probabilities
factors
-Individual term
structure of EDF
-Asset value process
Correlation of
Standard
Conditional default
Standard
Conditional
Conditional default
credit events
multivariate normal
probabilities as
multivariate normal
default
probabilities as
distribution (equity-
functions of macro-
asset returns (asset
probabilities as
functions of macro-
factor model)
factors
factor model)
functions of
factors
common risk
factors
Recovery rates
Random (Beta
Random (empirical
Random (Beta
Loss given default
Loss given default
distribution
distribution)
distribution)
deterministic
deterministic
Numerical
Simulation/Analytic
Simulation
Analytic/Simulation
Analytic
Tree based /simulation
approach
7
Measuring Credit Risk Overview

• Credit risk models should capture
• Spread risk
• Downgrade risk
• Default risk
• Recovery rate risk
• Concentration risk (portfolio diversification
  and correlation risk)

8
Measuring Credit Risk Overview

• Credit risk models generate
• Loss distribution (default risk)
• KMV, CreditRisk
• Portfolio value distribution (migration and
  default risks) CreditMetrics, CreditVaR,
  CreditPortfolioView

9
Measuring Credit Risk Overview
Typical credit returns
Frequency
Typical market returns
Portfolio Value
Source CIBC
Comparison of the distributions of credit
returns and market returns
10
Measuring Credit Risk Overview

• Key input parameters common to all models
• obligors information
• exposures
• recovery rate (loss given default LGD)
• default correlations (concentration risk)

11
II The Credit Migration Approach
12
Credit Migration Approach

• Key input parameters
• Credit data
• Credit horizon
• Credit rating system Moodys, SPs, internal
• Transition matrix

13
Credit Migration Approach

• Key input parameters
• Market data
• Yield curve (base curve)
• Spread curve for each rating
• FX rates
• Correlations between market indices

14
Credit Migration Approach

• Key input parameters
• Obligor data
• Credit rating
• Country weights
• Industry weights
• Idiosyncratic standard deviations

15
Credit Migration Approach

• Key input parameters
• Issue (facility) data
• Instrument type fixed coupon bond/loan, FRN,
  interest rate swap, loan commitment, letter of
  credit, credit derivative
• Recovery rate (1-LGD) and LGD standard deviation
• Usage given default (UGD)

16
Credit Migration Approach One Bond
Example Credit VaR for a senior unsecured BBB
rated bond maturing exactly in 5 years, and
paying an annual coupon of 6.
17
Credit Migration ApproachFor a Bond
Step 1 Credit horizon Step 2 Specify the
credit rating system Step 3 Specify the
transition matrix
Transition matrix probabilities of credit
rating migrating from one rating quality to
another, within one year.
Initial
Rating at year-end ()
Rating
AAA
AA
A
BBB
BB
B
CCC
Default
AAA
90.81
8.33
0.68
0.06
0.12
0
0
0
AA
0.70
90.65
7.79
0.64
0.06
0.14
0.02
0
A
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
BBB
0.02
0.33
5.95
86.93
5.30
1.17
1.12
0.18
BB
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
B
0
0.11
0.24
0.43
6.48
83.46
4.07
5.20
CCC
0.22
0
0.22
1.30
2.38
11.24
64.86
19.79
Source Standard Poors CreditWeek (April 15,
1996)
18
Credit Migration ApproachFor a Bond
Step 4 Specify the spread curve
Category
Year 1
Year 2
Year 3
Year 4
AAA
3.60
4.17
4.73
5.12
AA
3.65
4.22
4.78
5.17
A
3.72
4.32
4.93
5.32
BBB
4.10
4.67
5.25
5.63
BB
5.55
6.02
6.78
7.27
B
6.05
7.02
8.03
8.52
CCC
15.05
15.02
14.03
13.52
Source CreditMetrics, J.P. Morgan
One year forward zero curves for each credit
rating ()
19
Credit Migration ApproachFor a Bond
Step 5 Specify the recovery rate
Seniority Class
Mean ()
Standard Deviation ()
Senior Secured
53.80
26.86
Senior Unsecured
51.13
25.45
Senior subordinated
38.52
23.81
Subordinated
32.74
20.18
Junior subordinated
17.09
10.90
Source Carty Lieberman 1996
Recovery rates by seniority class ( of face
value, i.e., par)
20
Credit Migration ApproachFor a Bond
Step 6 Specify the forward pricing model
Time
3
0
2
4
5
1
106
6
6
6
6
Cash flows
V
(Forward price 107.55)
BBB
106
6
6
6






55
.
107
6
V
BBB
4
3
2
0525
)
0563
.
1
(
)
.
1
(
)
0467
.
1
(
041
.
1
21
Credit Migration ApproachFor a Bond
Year-end rating
Value ()
AAA
109.37
AA
109.19
A
108.66
BBB
107.55
BB
102.02
B
98.10
CCC
83.64
Default
51.13
Source CreditMetrics, J.P. Morgan
One year forward values for a BBB bond
22
Credit Migration ApproachFor a Bond
Step 7 Derive the forward distribution of the
changes in portfolio value
Probability
Forward
Change in
Year-end
of state
price V ()

rating
value
D
V
p()
()
AAA
0.02
109.37
1.82
AA
0.33
109.19
1.64
A
5.95
108.66
1.11
BBB
86.93
107.55
0
BB
5.30
102.02
-5.53
B
1.17
98.10
-9.45
CCC
0.12
83.64
-23.91
0.18
Default
51.13
-56.42
Source CreditMetrics, J.P. Morgan
Distribution of the bond values, and changes in
value of a BBB bond, in one year.
23
Credit Migration ApproachFor a Bond
First percentile -23.91 First percentile,
assuming normality - 7.43
Frequency
86.93
5.95
5.30
Probability
of State
()
1.17
.33
.02
CCC
Default
B
BB
BBB
AAA
A
AA
....
109.37
51.13
83.64
98.10
107.55
102.2
Forward Price V
D
....
-23.91
-56.42
0
1.82
-9.45
-5.53
Change in value V
24
Credit Migration Approach For a Bond/Loan
Portfolio
Obligor 2 (single-A)
Obligor 1
(BB)
AAA
AA
A
BBB
BB
B
CCC
Default
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
AAA
0.03
0.00
0.00
0.03
0.00
0.00
0.00
0.00
0.00
AA
0.14
0.00
0.00
0.13
0.01
0.00
0.00
0.00
0.00
A
0.67
0.00
0.02
0.61
0.40
0.00
0.00
0.00
0.00
BBB
7.73
0.01
0.18
7.04
0.43
0.06
0.02
0.00
0.00
BB
80.53
0.07
1.83
73.32
4.45
0.60
0.20
0.01
0.05
B
8.84
0.01
0.20
8.05
0.49
0.07
0.02
0.00
0.00
CCC
1.00
0.00
0.02
0.91
0.06
0.01
0.00
0.00
0.00
Default
1.06
0.00
0.02
0.97
0.06
0.01
0.00
0.00
0.00
Joint migration probabilities () with zero
correlation for 2 issuers rated BB and A
25
Credit Migration Approach For a Bond/Loan
Portfolio

• Joint migration probabilities when asset returns
  are correlated involves 3 steps
• Step 8 Estimate asset return correlations
• Step 9 Assume that the joint normalized
  return distribution is bivariate normal

Note Equity returns are typically used as a
proxy for asset returns.
26
Credit Migration Approach For a Bond/Loan
Portfolio
Step 10 Derive the credit quality thresholds for
each credit rating
Standard normal distribution for a BB-rated firm
Rating
Default
CCC
B
AA
AAA
BBB
Firm remains BB
A
Prob ()
1.06
1.00
8.84
80.53
7.73
0.67
0.14
0.03
Z-threshold(s)
ZBBB
Zccc
ZB
ZBB
ZA
ZAA
ZAAA
-2.30
-2.04
-1.23
1.37
2.39
3.43
2.93
27
Credit Migration Approach Bond/Loan portfolio
Rated-A obligor
Rated-BB obligor
Rating in one
Probabilities
Thresholds
Probabilities
Thresholds
year
()
Z
()
Z
(s)
(s)
AAA
0.09
3.12
0.03
3.43
AA
2.27
1.98
0.14
2.93
A
91.05
-1.51
0.67
2.39
BBB
5.52
-2.30
7.73
1.37
BB
0.74
-2.72
80.53
-1.23
B
0.26
-3.19
8.84
-2.04
CCC
0.01
-3.24
1.00
-2.30
Default
0.06
1.06
Transition probabilities and credit quality
thresholds for rated BB and A obligors
28
Credit Migration Approach For a Bond/Loan
Portfolio
Step 11 Calculation of the joint rating
probabilities
Rating of second company (A)
Rating of first company (BB)
AAA
AA
A
BBB
BB
B
CCC
Def
Total
AAA
0.00
0.00
0.03
0.00
0.00
0.00
0.00
0.00
0.03
AA
0.00
0.01
0.13
0.00
0.00
0.00
0.00
0.00
0.14
A
0.00
0.04
0.61
0.01
0.00
0.00
0.00
0.00
0.67
BBB
0.02
0.35
7.10
0.20
0.02
0.01
0.00
0.00
7.73
4.24
0.56
0.18
0.01
0.04
80.53
BB
0.07
1.79
73.65
B
0.00
0.08
7.80
0.79
0.13
0.05
0.00
0.01
8.84
CCC
0.00
0.01
0.85
0.11
0.02
0.01
0.00
0.00
1.00
Def
0.00
0.01
0.90
0.13
0.02
0.01
0.00
0.00
1.06
Total
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
100
Joint rating probabilities () for BB and A rated
obligors when correlation between asset returns
is 20.
29
Credit Migration Approach Bond/Loan portfolio
Step 12 Probability of joint defaults
Probability of joint defaults as a function of
asset return correlation
30
Credit Migration ApproachFor a Bond/Loan
Portfolio
Practical implementation Monte-Carlo simulation

• Input
• Derivation of the asset return thresholds for
  each rating category (Step 10)
• Estimation of the correlation between each pair
  of obligors asset returns

31
Credit Migration Approach Implementation

• Multifactor equity model (CreditVar and Equity
  Market VaR)
• Regression model for stock returns (Equity
  Market VaR model)

R - stock return Ri - country/industry index
return e - residual (E(e) 0)
Same model in terms of standardized returns is
used in CreditVar
32
Credit Migration Approach Implementation
33
Credit Migration Approach Implementation

• Correlation between two obligors B and C

34
Credit Migration Approach Implementation
Monte Carlo simulation

• Generation of asset return scenarios according to
  their joint normal distribution. Each scenario is
  characterized by n standardized asset returns,
  one for each of the n obligors in the portfolio.
  (Step 9)
• For each scenario, and for each obligor, the
  standardized asset return is mapped into the
  corresponding rating, according to the threshold
  levels derived in Step 1

Scenario
r1 r2 . . . . rn
Obligor 1 credit rating
r1
C ? f(r1,,rn,C) ?
Joint density function
Correlation matrix
Obligor n credit rating
rn
35
Credit Migration Approach Implementation
Monte Carlo simulation

• Given the spread curves which apply for each
  rating, the portfolio is revalued. (Steps 6 7)
• CreditVaR allows for risk analysis of
  complicated portfolios of different instruments
  fixed coupon bonds, floating rate notes, swaps,
  loan commitments, etc.
• Example 1 Pricing procedure for FRN
• ri - forward reference rate
• yi - discount rate
• s - credit spread
• m - maturity of the note (in years)
• n - coupon payment frequency

36
Credit Migration Approach Implementation
Monte Carlo simulation

• Repeat the procedure a large number of times, say
  100,000 times, and plot the distribution of the
  portfolio values to obtain a graph which looks
  like Figure 2. (Steps 3-5)

37
Credit Migration Approach Implementation
Practical implementation Monte-Carlo simulation

• Results
• Derive the percentiles of the distribution of the
  future values of the portfolio.

38
99.865 VaR
99 VaR
200000
200000
EVaR
175000
175000
EVaR
Upper 95 bound
150000
150000
VaR (1,000 US)
VaR (1,000 US)
99.865 VaR
125000
Upper 5 bound
125000
100000
Lower 95 bound
99 VaR
100000
Lower 95 bound
75000
75000
50000
Oct-
Nov-
Dec-
Jan-
Feb-
Mar-
Apr-
Oct-99
Nov-
Dec-
Jan-00
Feb-
Mar-
Apr-00
99
99
99
00
00
00
00
99
99
00
00
Period
Period
Expected Loss
50000
10 Hot Exposures (DeltaVaR)
Risk Contribution
Exposure ID
Obligor Name
Commitment
Drawn
Maturity
Type


1
AUDCORPFIS1A
CADCORPGNRL2A
98,431,493
50
01-Jul-01
Revolver
2,460,787


1.8
40000
2
AUDCORPFOD1A
CADCORPGNRL3A
90,441,897
20
01-Jul-01
Revolver
1,808,838


1.3
3
AUDCORPGNRL1A
CADCORPGNRL4A
28,008,962
40
01-Jul-01
Revolver
1,120,358


0.8
VaR (1,000 US)
4
CADCORPAUT1A
CADCORPGNRL5A
9,443,021

100
01-Jul-01
Revolver
944,302


0.7
5
AUDCORPCHM1A
CADCORPGNRL1A
15,845,070
35
01-Jul-01
Revolver
554,577


0.4
6
CADCORPCHM1A
CADCORPGNRL6A
4,852,980

100
01-Jul-01
Revolver
485,298


0.3
7
CADCORPFIS1A
CADCORPBFIN1A
4,441,180

100
01-Jul-01
Revolver
444,118


0.3
30000
8
CADCORPELQ1A
CADCORPGNRL7A
4,852,980

73
01-Jul-01
Revolver
354,268


0.3
9
CADCORPFOD1A
CADCORPBFIN2A
2,941,200

100
01-Jul-01
Revolver
294,120


0.2
10
CADCORPGNRL1A
CADCORPBFIN3A
2,500,020

100
01-Jul-01
Revolver
250,002


0.2
20000
Oct-99
Nov-99
Dec-99
Jan-00
Feb-00
Mar-00
Apr-00
Period
Sensitivity Analysis
Worst case
transition
Asset Corr.
Asset Corr.
Idiosync.
Recovery
Double
Normal
matrix
0
1
-10
-10
Spreads
99.865 VaR
139,000,000


342,482,783
75,737,521


424,230,768


182,107,577


156,264,993


149,636,030


99 VaR
90,000,000


144,465,513
61,209,225


285,109,647


115,357,810


100,689,765


95,138,106


Expected Loss
38,000,000


93,628,387
20,705,222


115,976,757


49,784,805


42,719,926


40,907,692


39
Calculation of the capital charge

• Economic capital stands as a cushion to absorb
  unexpected losses related to credit events, i.e.
  migration and/or default

• P(p) value of the portfolio in the worst case
  scenario at the p confidence level
• FV forward value of the portfolio V0 (1 PR)
  
• where
• V0 current mark-to-market value of the
  portfolio
• PR promised return on the portfolio
• EV expected value of the portfolio V0 (1
  ER)
• where
• ER expected return on the portfolio
• EL expected loss FV - EV

The expected loss doesnt contribute to the
capital allocation, but instead goes into
reserves and is imputed as a cost into the RAROC
calculation. The capital charge comes as a
protection against unexpected losses Capital
EV P(p)
40
Calculation of the capital charge
41
Portfolio Models Used
Default Only/ Fair Value
CR
Internal
CPV
KMV
CMs
Public
42
Example Default Swap

• Example 1 One year forward value of the default
  swap

Bond Maturity7 years, Coupon7.9
Notional10,000, Recovery rate40.
Default swap Maturity3 years, Premium1, Recove
ry rate40.
Credit rating of counter-party
Correlation between asset returns 0.465
Credit rating of underlying bond
43
Example Default Swap

• Example 2 VaR calculation

Bond Credit ratingBB, Maturity7 years,
Coupon7.9 Notional10,000, Recovery rate40
Default swap Credit rating AAA, Maturity3
years, Premium1, Recovery rate40
Correlation between asset returns 0.465
1 year VaR at 99 confidence level Portfolio 1
(Bond) 4,177 Portfolio 2 (Bond and Default
swap) 727
44
End of Part 3