Transcript Standard presentation September 1996

Credit Risk Modeling for
Capital Allocation
A Credit Migration Approach
Lecture 9
Based on Risk Management
By Crouhy, Galai & Mark
McGraw-Hill,2000
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
I
Measuring Credit Risk
Overview
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.)
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
Comparison of Models
Credit migration approach
Software
Definition of
Risk
CreditMetrics
D Market Value
Contingent claim
approach
CreditPortfolioView
D Market Value
Actuarial
approach
Reduced form
approach
CreditRisk+
Kamakura
Default losses
Default losses
Default losses
KMV
Credit events
Downgrade/Default Downgrade/Default
Continuous default
probabilities
Default
Default
Risk drivers
Asset Values
Macro-factors
Asset Values
Expected default
rates
Hazard rate
Constant
Driven by Macro
factors
Driven by:
N/A
-Individual term
structure of EDF
-Asset value process
Standard
Conditional
default
Standard
Conditional
Correlation of
probabilities
as
multivariate
normal
default
credit events multivariate normal
distribution (equity- functions of macro- asset returns (asset probabilities as
factor model)
factors
factor model)
functions of
common risk
factors
Random (empirical
Random (Beta
Loss given default
Recovery rates Random (Beta
distribution
distribution)
distribution)
deterministic
Transition
probabilities
Numerical
approach
Simulation/Analytic Simulation
Analytic/Simulation
Analytic
N/A
Conditional default
probabilities as
functions of macrofactors
Loss given default
deterministic
Tree based /simulation
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)
Measuring Credit Risk:
Overview
Credit risk models generate:
• Loss distribution (default risk)
KMV, CreditRisk +
• Portfolio value distribution (migration and default
risks)
CreditMetrics, CreditVaR, CreditPortfolioView
Measuring Credit Risk:
Overview
Typical credit returns
Frequency
Typical market returns
Source: CIBC
Portfolio Value
Comparison of the distributions of credit returns and market returns
Measuring Credit Risk:
Overview
• Key input parameters common to all models
–
–
–
–
obligors information
exposures
recovery rate (loss given default: LGD)
default correlations (concentration risk)
II
The Credit Migration
Approach
Credit Migration Approach
Key input parameters:
• Credit data:
– Credit horizon
– Credit rating system: Moody’s, S&P’s, internal
– Transition matrix
Credit Migration Approach
Key input parameters:
• Market data
–
–
–
–
Yield curve (base curve)
Spread curve for each rating
FX rates
Correlations between market indices
Credit Migration Approach
Key input parameters:
• Obligor data:
–
–
–
–
Credit rating
Country weights
Industry weights
Idiosyncratic standard deviations
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)
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%.
Credit Migration Approach:
For 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 & Poor’s CreditWeek (April 15, 1996)
Credit Migration Approach:
For a Bond
Step 4: Specify the spread curve
Category
AAA
AA
A
BBB
BB
B
CCC
Year 1
Year 2
Year 3
Year 4
3.60
3.65
3.72
4.10
5.55
6.05
15.05
4.17
4.22
4.32
4.67
6.02
7.02
15.02
4.73
4.78
4.93
5.25
6.78
8.03
14.03
5.12
5.17
5.32
5.63
7.27
8.52
13.52
Source: CreditMetrics, J.P. Morgan
One year forward zero curves for each credit rating (%)
Credit Migration Approach:
For a Bond
Step 5: Specify the recovery rate
Seniority Class
Senior Secured
Senior Unsecured
Senior subordinated
Subordinated
Junior subordinated
Mean (%)
53.80
51.13
38.52
32.74
17.09
Standard Deviation (%)
26.86
25.45
23.81
20.18
10.90
Source: Carty & Lieberman [1996]
Recovery rates by seniority class (% of face value, i.e., “par”)
Credit Migration Approach:
For a Bond
Step 6: Specify the forward pricing model
0
1
2
3
4
5
6
6
6
6
106
VBBB
(Forward price = 107.55)
VBBB = 6 +
Time
Cash flows
6 +
6
6
+
+ 106
=107.55
2
3
4
1.041 (1.0467)
(1.0525) (1.0563)
Credit Migration Approach:
For a Bond
Year-end rating
AAA
AA
A
BBB
BB
B
CCC
Default
Value ($)
109.37
109.19
108.66
107.55
102.02
98.10
83.64
51.13
Source: CreditMetrics, J.P. Morgan
One year forward values for a BBB bond
Credit Migration Approach:
For a Bond
Step 7: Derive the forward distribution of the changes in portfolio
value
Year-end
rating
AAA
AA
A
BBB
BB
B
CCC
Default
Probability
of state:
p(%)
Forward
price: V ($)
Change in
value: DV
($)
0.02
0.33
5.95
86.93
5.30
1.17
0.12
0.18
109.37
109.19
108.66
107.55
102.02
98.10
83.64
51.13
1.82
1.64
1.11
0
-5.53
-9.45
-23.91
-56.42
Source: CreditMetrics, J.P. Morgan
Distribution of the bond values, and changes in value
of a BBB bond, in one year.
Credit Migration Approach:
For a Bond
Frequency
First percentile = -23.91
First percentile, assuming normality = - 7.43
86.93
5.95
5.30
Probability
of State
(%)
1.17
.33
.02
Default
CCC
51.13
83.64
98.10 102.2 107.55 .... 109.37 Forward Price: V
-56.42
-23.91
-9.45 -5.53 0
B
BB BBB A AA AAA
....
D
1.82 Change in value: V
Credit Migration Approach:
For a Bond/Loan Portfolio
Obligor #1
(BB)
AAA
AA
A
BBB
BB
B
CCC
Default
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
AAA
AA
0.09
0.00
0.00
0.00
0.01
0.07
0.01
0.00
0.00
2.27
0.00
0.00
0.02
0.18
1.83
0.20
0.02
0.02
Obligor #2 (single-A)
A
BBB
BB
B
91.05
0.03
0.13
0.61
7.04
73.32
8.05
0.91
0.97
5.52
0.00
0.01
0.40
0.43
4.45
0.49
0.06
0.06
0.74
0.00
0.00
0.00
0.06
0.60
0.07
0.01
0.01
0.26
0.00
0.00
0.00
0.02
0.20
0.02
0.00
0.00
CCC
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.00
0.00
Default
0.06
0.00
0.00
0.00
0.00
0.05
0.00
0.00
0.00
Joint migration probabilities (%) with zero correlation
for 2 issuers rated BB and A
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


 1
2
2 
f ( rBB , rA ;  ) =
exp
rBB  2 rBB rA + rA 
2
2 1   2
 2(1   )

1
Note: Equity returns are typically used as a proxy for asset returns.
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 Firm remains BB BBB A AA AAA
7.73 0.67 0.14 0.03
Prob (%):
80.53
1.06 1.00 8.84
Z-threshold(s)
ZBBB ZA ZAA ZAAA
Zccc ZB ZBB
-2.30 -2.04 -1.23
1.37 2.39 2.93 3.43
Credit Migration Approach:
Bond/Loan portfolio
Rated-A obligor
Rating in one
year
AAA
AA
A
BBB
BB
B
CCC
Default
Probabilities
(%)
0.09
2.27
91.05
5.52
0.74
0.26
0.01
0.06
Thresholds
Z
(s)
3.12
1.98
-1.51
-2.30
-2.72
-3.19
-3.24
Rated-BB obligor
Probabilities Thresholds
(%)
Z
(s)
0.03
0.14
0.67
7.73
80.53
8.84
1.00
1.06
3.43
2.93
2.39
1.37
-1.23
-2.04
-2.30
Transition probabilities and credit quality thresholds for rated
BB and A obligors
Credit Migration Approach:
For a Bond/Loan Portfolio
Step 11:
Calculation of the joint rating probabilities
Pr(  123
.  rBB  137
. ,151
.  rA  198
. )=
1.37

1.98
1.23 1.51
Rating of first
company (BB) AAA
f ( rBB , rA ; )drBB drA =.7365
Rating of second company (A)
AA
A
BBB BB
B
CCC Def Total
AAA
AA
A
BBB
BB
0.00
0.00
0.00
0.02
0.07
0.00
0.01
0.04
0.35
1.79
0.03
0.13
0.61
7.10
73.65
0.00
0.00
0.01
0.20
4.24
0.00
0.00
0.00
0.02
0.56
0.00
0.00
0.00
0.01
0.18
0.00
0.00
0.00
0.00
0.01
0.00 0.03
0.00 0.14
0.00 0.67
0.00 7.73
0.04 80.53
B
CCC
Def
Total
0.00
0.00
0.00
0.09
0.08
0.01
0.01
2.27
7.80
0.85
0.90
91.05
0.79
0.11
0.13
5.52
0.13
0.02
0.02
0.74
0.05
0.01
0.01
0.26
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.06
Joint rating probabilities (%) for BB and A rated obligors when
correlation between asset returns is 20%.
8.84
1.00
1.06
100
Credit Migration Approach:
Bond/Loan portfolio
Step 12:
Probability of joint defaults
0.06
Joint default probability
0.04
corr ( DEF1, DEF2) =
P( DEF1, DEF2)  P1×P2
P1(1  P1) ×P2(1  P2)
0.02
0
Source: CreditMetrics, J.P. Morgan
0


1
2
1
2
P( DEF 1, DEF 2 ) = Pr r1  d2 , r2  d2 = N 2 (  d2 , d2 , )
0.2
0.4
0.6
Correlation
Probability of joint defaults as a function of asset return correlation
0.8
1.0
Credit Migration Approach
For 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
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:
R = a + b1R1 + ... + bn Rn + e
r = bˆ1r1 + ... + bˆn rn + bˆe re ,
r=
R  E( R )
, ri =
sR
s
s
b̂i = bi . i ,b̂e = e
sR
sR
Ri  E( Ri )
si
, re =
e
se
Credit Migration Approach:
Implementation
r =  M rM +  e re , where  M rM = b̂1r1 + ... + b̂n rn
 e = b̂e , R 2 =  M2 and  M = 1   e2
wns n
w1s 1
rM =
r1 + ... +
rn  market component
sˆ
sˆ
wi  country / industry weights
w1 + ... + wn = 1
 b̂1
b̂n 

sˆ = 
+ ... +

β
s

s
M
1
M
n


1
Credit Migration Approach:
Implementation
• Correlation between two obligors B and C
B
B


w
w
s
B
B
nsn
1 1
r =  M 
r1 +  +
rn  +  SB rSB = b̂1B r1 +  + b̂nB rn + b̂eB reB
sˆ B
 sˆ B

C
C


w
w
s
C
C
nsn
1
1
r =  M 
r1 +  +
rn  +  SC rSC = b̂1C r1 +  + b̂nC rn + b̂eC reC
sˆ C
 sˆ C

cor (r , r
B
C
) =  b̂ b̂
n
n
i =1 j =1
 ij = cor( ri ,rj )
B C
i i
ij
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
C  f(r1,…,rn,C) 
Correlation
matrix
Joint density
function
r1
r2
.
.
.
.
rn
Obligor 1 credit rating
r1
Obligor n credit rating
rn
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
100(ri + s) / n
100
P=
+
ti
t mn
(
1
+
y
)
(
1
+
y
)
i =1
i
mn
mn
ri - forward reference rate
yi - discount rate
s - credit spread
m - maturity of the note (in years)
n - coupon payment frequency
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)
Credit Migration Approach:
Implementation
Practical implementation:
Monte-Carlo simulation
• Results:
– Derive the percentiles of the distribution of the
future values of the portfolio.
99.865% VaR
99% VaR
200000
EVaR
VaR (1,000 US$)
VaR (1,000 US$)
200000
175000
Upper 95% bound
150000
99.865% VaR
125000
Lower 95% bound
100000
175000
EVaR
150000
125000
Upper 5% bound
100000
99% VaR
Lower 95% bound
75000
75000
50000
Oct- Nov- Dec- Jan- Feb- Mar- Apr99
99
99
00
00
00
00
Oct99
Nov99
Dec99
Period
Jan00
Feb00
Mar- Apr00
00
Period
Expected Loss
50000
VaR (1,000 US$)
10 Hot Exposures (DeltaVaR)
40000
30000
Exposure ID
Obligor Name Commitment
1
AUDCORPFIS1A
CADCORPGNRL2A
98,431,493
50%
01-Jul-01
Revolver
2,460,787
1.8%
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%
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%
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%
CADCORPGNRL1A
CADCORPBFIN3A
2,500,020
100%
01-Jul-01
Revolver
250,002
0.2%
10
Drawn Maturity Type
Risk Contribution
$
%
#
20000
10 Hot Obligors (DeltaVaR)
Oct-99 Nov-99 Dec-99 Jan-00 Feb-00 Mar-00 Apr-00
Period
#
Sensitivity Analysis
Normal
99.865% VaR
99% VaR
Expected Loss
139,000,000
90,000,000
38,000,000
Worst case
transition Asset Corr. Asset Corr. Idiosync.
matrix
=0
=1
-10%
342,482,783
144,465,513
93,628,387
75,737,521
61,209,225
20,705,222
Recovery
-10%
Double
Spreads
424,230,768 182,107,577 156,264,993 149,636,030
285,109,647 115,357,810 100,689,765 95,138,106
115,976,757 49,784,805 42,719,926 40,907,692
1
2
3
4
5
6
7
8
9
10
Obligor Name
CADCORPGNRL1A
CADCORPGNRL2A
CADCORPBFIN1A
CADCORPGNRL7A
CADCORPGNRL6A
CADCORPGNRL4A
CADCORPBFIN3A
CADCORPGNRL3A
CADCORPGNRL5A
CADCORPBFIN2A
Credit Rating
6
7
6
7
6
6
7
6.5
7
6
Notional
126,658,646
126,658,646
120,400,000
120,869,023
117,862,804
122,611,887
124,780,560
122,224,360
120,869,023
120,400,000
Risk Contribution
$
%
8,950,228
6%
8,107,078
6%
7,921,545
6%
6,479,533
5%
5,914,433
4%
5,489,519
4%
5,375,374
4%
5,319,663
4%
4,569,089
3%
2,906,935
2%
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 doesn’t 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)
Calculation of the capital
charge
P%
P(p)
EV
Economic Capital EL
FV
Portfolio Models Used
Default Only/ “Fair Value”
CR+
Internal
CMs
Public
KMV
CPV
Example: Default Swap
• Example 1: One year forward value of the default swap
Credit rating
of counterparty
Credit
rating of
underlying
bond
Bond:
Default swap:
Maturity=7 years,
Maturity=3 years,
Coupon=7.9%
Premium=1%,
Notional=$10,000,
Recovery rate=40%.
Recovery rate=40%.
Correlation between asset returns = 0.465
Aaa
Aa
A
Baa
Ba
B
Caa
Aaa
-273.07
-262.89
-193.83
70.62
219.51
294.74
676.01
Aa
-273.14
-263.11
-194.60
68.84
217.43
292.58
673.50
A
-273.54
-264.06
-198.40
59.73
206.19
280.65
659.19
Baa
-272.27
-264.22
-205.61
32.67
170.91
242.12
605.70
Ba
-271.46
-264.02
-209.07
19.08
152.40
221.51
576.32
B
-270.82
-263.73
-210.97
10.86
141.02
208.71
557.64
Caa
-266.15
-260.30
-214.92
-18.51
98.81
160.52
480.93
Example: Default Swap
• Example 2: VaR calculation
Bond:
Default swap:
Credit rating=BB,
Credit rating= AAA,
Maturity=7 years,
Maturity=3 years,
Coupon=7.9%
Premium=1%,
Notional=$10,000,
Recovery rate=40%
Recovery rate=40%
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
End of Part 3