RISK2003_Part1&2_Slides_v2b

Office of the Superintendent
of Financial Institutions
Bureau du surintendant
des institutions financières
June 11, 2003 - Risk Conference, Boston, MA Page 1
CDO Modelling
Anthony Vaz
Robert Kowara
Carol Cheng
Capital Markets Division
OSFI
The views expressed in this presentation are
solely those of the authors.

Outline
1. CDO Terminology
2. CDO Valuation
2.1 Moody’s Binomial Expansion Method (BET)
• Modelling Default and Correlation
• Excel Implementation
2.2 Duffie-Garleanu Methodology
• Modelling Default and Correlation
• Excel Implementation
3. Calculating VaR for CDO Tranches
4. Concluding Remarks

1. CDO Terminology
1.1 Definitions
A CDO is a Collaterized Debt Obligation. A pool of
securities is used as collateral to fund a prioritized
sequence of payments. This payment sequence is
illustrated as the following “water flow” of cash
payments.

SENIOR
TRANCHE
MEZZANINE
TRANCHE
EQUITY
TRANCHE
1. CDO Terminology
1.1 Definitions
Cash Flow Water Fall

1. CDO Terminology
1.2 CDO Types
CDO can be classified in a variety of ways.
1.2.1 Assets in Collateral Pool
CDO’s with a collateral pool of bonds are termed Collateralized Bond
Obligations (CBOs).
CDO’s with a collateral pool of loans are termed Collateralized Loan Obligations
(CLOs).

1. CDO Terminology
1.2 CDO Types
1.2.2 Transaction Type
In an arbitrage transaction, the CDO is constructed to capture the difference in
spread between the collateral pool and the yields at which the senior liabilities of
the CDO are issued.
In a balance sheet transaction, the CDO is constructed to remove loans or
bonds from the balance sheet of a financial institution. This is motivated by the
desire to obtain capital relief, improve liquidity, and re-deploy to alternative
investments.

1. CDO Terminology
1.2 CDO Types
1.2.3 Covenants & Management of Collateral Pool
1.2.3.1 Market Value CDO
• A market value CDO has a diversified collateral pool of financial assets in
multiple asset categories that may include corporate bonds, loans, private and
public equity, distressed securities or emerging market investments, and cash
and money market instruments.
• The collateral pool is actively managed.
• The collateral pool is priced periodically to obtain the market value. The
payments to the tranches are based on threshold levels for the market value of
the collateral pool.

1. CDO Terminology
1.2 CDO Types
1.2.3 Covenants & Management of Collateral Pool
1.2.3.2 Cash Flow CDO
• A cash flow CDO has a collateral pool of financial assets in a specific asset
category, such as corporate bonds, loans, or mortgages.
• The collateral pool is fairly static. When an asset matures or defaults, the
proceeds may be invested at the discretion of the fund manager.
• The collateral pool is priced periodically to obtain the par value. The payments to
the tranches are based on threshold levels for the par value of the collateral
pool.

1. CDO Terminology
1.2 CDO Types
1.2.4 Legal Ownership of Collateral Pool Assets
1.2.4.1 Non-synthetic CDO
A non-synthetic CDO has legal ownership of all the assets in the collateral pool.
The CDO only assumes economic risk on the assets which it legally owns.
1.2.4.2 Synthetic CDO
A synthetic CDO does not have legal ownership of the assets in the collateral
pool. The CDO assumes economic risk on the assets which it does not legally
own.

2. CDO Valuation
A CDO is modelled in two parts: a defaultable collateral pool and a
contingent payment stream to the CDO tranches.
We shall discuss two popular techniques for valuation of CDOs:
 Moody’s Binomial Expansion Technique [1],
 Duffie-Singleton approach to correlated default applied to a
contingent payment stream [2][3].
The copula method is also popular, but will not be discussed here.
[1] A. Cifuentes and G. O’Connor, “The Binomial Expansion Method Applied to CBO/CLO Analysis”, Moody’d
Special Report, December 13, 1996.
[2] D.Duffie and N. Garleanu, “Risk and Valuation of Collaterized Debt Obligations”, Stanford University,
working paper, 2001.
[3] D.Duffie and K. Singleton, “Simulating Correlated Defaults”, Stanford University, working paper, 1998.

2.1 Binomial Expansion Technique
2.1.1 Derivation of Diversity Score
Pool of Correlated Bonds
Correlated bonds: M=20
Diversity Score: N=5

2.1.1 Derivation of Diversity Score
2.1.1.1 Independent Bond Pool
• Consider a hypothetical pool consisting of N bonds having the
same par value F. The bond defaults are assumed to be
independent.
• N is called the diversity score of the bond pool.
• All the bonds are assumed to have the same loss L when a
default occurs.
• Let the be a random variable representing the state of bond i.iX




defaultednotbond,0
defaultedbond,1
i
i
Xi
pXi  ]1[Prob pXi  1]0[Prob

We can solve for the first and second moment loss statistics of the
collateral portfolio.
pXi ][EHence pXi ][E 2
)1(][][][Variance 22
ppXEXEX iii 


N
i
iPort XLL
1
pNLLE Port ][ ))1(1(][ 22
pNpNLLE Port 

2.1.1.2 Dependent Bond Pool
•Consider a hypothetical pool consisting of M bonds having the
same par value . The bond defaults are assumed to be
dependent.
•All the bonds are assumed to have the same loss when a
default occurs.
•Let the be a random variable representing the state of bond i.
F
iY




defaultednotbond,0
defaultedbond,1
i
i
Yi
pYi  ]1[Prob pYi  1]0[Prob
L

pYi ][E pYi ][E 2
)1(][][][Variance 22
ppYEYEY iii 
Let )1(i pp 
jiijji YY ][Covariance 
2
][E pYY jiijji  
Assume all pair-wise correlations are equal:
Assume all variances are equal:
ij 
ij 

We can solve for the first and second moment loss statistics of the
collateral portfolio.


M
i
iPort YLL
1
LMpLE Port ][
))1(()1(][ 2222
pppLMMLMpLE Port  

Equate expressions for the first and second moment loss statistics of
the collateral portfolios to obtain the following.
)1( 


MN
NM
Correlation:
where N=diversity score & M=number of correlated bonds
)1(1 

M
M
N

Note, these formulae can be generalized to account for random recovery rates using the same
technique.

2.1.2 Computing CDO Loss Scenarios
The BET method makes the assumption that losses occur with a given
profile.
For example,
50% end of year 1
10% end of year 2
10% end of year 3
10% end of year 4
10% end of year 5
The profiles are determined from historical data; but they cannot be rigorously tailored to a
particular portfolio.

The probability of getting j defaults in the bond pool is
jNj
j pp
jNj
N
P 







 )1(
)!(!
!
2.1.2 Computing CDO Loss Scenarios
A loss scenario Sj is associated with each of the above default
combinations.
Hence S10 corresponds to 5 defaults in the first year, 1 at end of year 2, 1 at
end of year 3, 1 end of year 4, and 1 at end of year 5.
The CDO cashflows are computed accordingly.

2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
DEFAULT SCENARIO POOL OF ASSETS
time defaults Diversity Score 58
0.5 0.00 Average Coupon 8.00%
1.0 0.50 Average Maturity 10
1.5 0.00 Notional 1000
2.0 0.10 Recovery Rate 50.00%
2.5 0.00 Reinvestment Rate 6.00%
3.0 0.05 Average Prob of Def 32.00%
3.5 0.00
4.0 0.05 TRANCHES
4.5 0.00 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8
5.0 0.05 Name aaa bbb ccc
5.5 0.00 Coupon 6.50% 10.00% 30.00%
6.0 0.05 Notional 500 280 220
6.5 0.00 Maturity 10 10 10
7.0 0.05 OC Test 0 0 0
7.5 0.00 Expected loss 0.0019% 14.3932% 57.8585% #N/A #N/A #N/A #N/A #N/A
8.0 0.05 Ratings Aaa B2 NR #N/A #N/A #N/A #N/A #N/A
8.5 0.00
9.0 0.05
9.5 0.00
10.0 0.05
10.5
11.0
11.5
Calculate

Moody’s “Idealized” Cumulative Expected Loss Rates (%)
These values are used to infer the bond rating from the expected loss levels normalized by
the bond principal.
Rating 1 2 3 4 5 6 7 8 9 10
Aaa 0.000028% 0.00010% 0.00039% 0.00090% 0.00160% 0.00220% 0.00286% 0.00363% 0.00451% 0.00550%
Aa1 0.000314% 0.00165% 0.00550% 0.01155% 0.01705% 0.02310% 0.02970% 0.03685% 0.04510% 0.05500%
Aa2 0.000748% 0.00440% 0.01430% 0.02585% 0.03740% 0.04895% 0.06105% 0.07425% 0.09020% 0.11000%
Aa3 0.001661% 0.01045% 0.03245% 0.05555% 0.07810% 0.10065% 0.12485% 0.14960% 0.17985% 0.22000%
A1 0.003196% 0.02035% 0.06435% 0.10395% 0.14355% 0.18150% 0.22330% 0.26400% 0.31515% 0.38500%
A2 0.005979% 0.03850% 0.12210% 0.18975% 0.25685% 0.32065% 0.39050% 0.45595% 0.54010% 0.66000%
A3 0.021368% 0.08250% 0.19800% 0.29700% 0.40150% 0.50050% 0.61050% 0.71500% 0.83600% 0.99000%
Baa1 0.049500% 0.15400% 0.30800% 0.45650% 0.60500% 0.75350% 0.91850% 1.08350% 1.24850% 1.43000%
Baa2 0.093500% 0.25850% 0.45650% 0.66000% 0.86900% 1.08350% 1.32550% 1.56750% 1.78200% 1.98000%
Baa3 0.231000% 0.57750% 0.94050% 1.30900% 1.67750% 2.03500% 2.38150% 2.73350% 3.06350% 3.35500%
Ba1 0.478500% 1.11100% 1.72150% 2.31000% 2.90400% 3.43750% 3.88300% 4.33950% 4.77950% 5.17000%
Ba2 0.858000% 1.90850% 2.84900% 3.74000% 4.62550% 5.31350% 5.88500% 6.41300% 6.95750% 7.42500%
Ba3 1.545500% 3.03050% 4.32850% 5.38450% 6.52300% 7.41950% 8.04100% 8.64050% 9.19050% 9.71300%
B1 2.574000% 4.60900% 6.36900% 7.61750% 8.86600% 9.83950% 10.52150% 11.12650% 11.68200% 12.21000%
B2 3.938000% 6.41850% 8.55250% 9.97150% 11.39050% 12.45750% 13.20550% 13.83250% 14.42100% 14.96000%
B3 6.391000% 9.13550% 11.56650% 13.22200% 14.87750% 16.06000% 17.05000% 17.91900% 18.57900% 19.19500%
Caa 14.300000% 17.87500% 21.45000% 24.13400% 26.81250% 28.60000% 30.38750% 32.17500% 33.96500% 35.75000%

Binomial Distribution
Probability distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
Prob

Expected Loss versus Diversty
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
5
12
16
20
24
28
32
36
40
44
48
52
56
60
TR1
TR2
TR3
TR4
TR5
TR6
TR7
TR8
Expected Loss versus Diversity

Expected Loss versus Diversity
Senior Tranche Expected Loss Versus Diversity
0
0.005
0.01
0.015
0.02
0.025
0.03
5
12
16
20
24
28
32
36
40
44
48
52
56
60
TR1

Mezzanine Tranche Expected Loss versus Diversity
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
5 12 16 20 24 28 32 36 40 44 48 52 56 60
TR2

Junior Tranche Expected Loss versus Diversity
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0.38
0.39
5
12
16
20
24
28
32
36
40
44
48
52
56
60
TR3

•OC is calculated as the ratio between the par value of
collateral and the value of the all liabilities senior to
and including the tranche being calculated.
•Once OC ratio drops below the certain level the cash
flow from the equity or lower tranche is diverted to a
risk-free reserve account.
2.1.4 Overcollateralization Tests

2.2.1 Hazard Rate
Duffie’s approach is based on an application of reliability theory
to the default process.
Reliability theory uses a hazard rate intensity to obtain the
conditional survival probability as follows.
2.2 Duffie-Singleton Methodology
 )(expexp)|( tTduFTP
T
t
t 







  
t TtF

2.2.2 Stochastic Pre-intensity
Duffie models the hazard rate as a stochastic process that he calls the
“pre-intensity process” .
J(t)dW(t)(t)σdt)(t)-(θ)(   td
The conditional survival probability is given by the following.
















  t
T
t
t FduuEFTP )(exp)|( 

The computation of the above expectation is quite complex. It is done in
several steps.
Step 1: The diffusion generator is determined.
 
    )(),(),(
2
1
),(|)),((
lim
0
2
2
2
0
HdtftHfl
ff
t
f
t
tfFttttfE
Df t
t























Step 2: The Feynman-Kac formula is used to obtain the following
integral-PDE equation.
    0)(),(),(
2
1
0
2
2
2











HdtftHflf
ff
t
f





Step 3: The PDE is solved using an affine solution to obtain.
 )()()(exp
)),(()(exp)|(
ttTtT
tTtfFduuEFTP t
T
t
t




















 

2.2.4 Defaultable Zero Coupon Bond
The conditional survival probability is then used to derive the value of a
defaultable zero-coupon bond.
   
T
zero duuhurTTTtp
0
00 )()()()(exp)(),( 
where the conditional default intensity is given by
   )0()()()0()()(exp
)|(
)( 0





 TTTT
T
FTP
Th

2.2.5 Defaultable Coupon Bond
The Duffie-Garleanu has an incorrect formula for a defaultable coupon
bond in his paper. The correct formula for a coupon bond with quarterly
payments is as follows.
  























  0
0
00
44
exp
44
)()()()(exp)(),( 
jjjC
duuhurTTTtp
T
CBond
The above formula was confirmed with both analytically and with Monte Carlo simulation
[1].
[1] Private discussion with Phelim Boyle & Zhenzhen Lai (U.Waterloo). Confirmed with Darrell Duffie.

2.2.6 Bond Hazard Rates
Suppose the are N bonds in a collateral pool, each with a hazard rate
process , (i=1,2,…N). Duffie advocates the partition of the affine
process into risk factor components.
ZYX ici  )(iλ
The process Xi is unique to bond i. The process Yc(i) is common to bonds
affected by the same risk factor. The process Z is common to all bonds.
The Weiner process and jump process for each affine process is
independent.
i

2.2.6 Bond Hazard Rates
The instantaneous correlation coefficient between hazard rate processes for bonds i
and j are determined by the ratio of the jump arrival rates.
Due to independence of the affine processes, the following property holds.






































































t
T
t
t
T
t
ict
T
t
i
t
T
t
t
FduuZEFduuYEFduuXE
FduuEFTP
)(exp)(exp)(exp
)(exp)|(
)(

The calibration can be done similar to a 3 factor CIR spot rate model.
ij

2.2.7 Simulation
The correlated default intensities are then used to determine the default statistics for the
correlated bond pool by Monte Carlo simulation.
Compensator method.
A compensator is an accumulated intensity. Consider a Poisson process Nt with intensity l(.).








 
t
t duulNP
0
)(exp)0(
















  t
T
t
t FduuEFTP )(exp)|( 
In the Duffie-Singleton method, the default intensity is stochastic and is denoted by .

2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
CASHFLOW CDO SPECIFICATION
nSim 1000
Issuing Date 1-Jan-00 No. of Bins 20
Maturity Date 1-Jan-10 No. of Period 120
Collateral Notional 1,000.00
No. of Tranche 3
Resrv-Accr-Rate 0.0500
Tranche Rating Tranche Principal %/Notional Coupon Frequency Coupon Rate
Class A 500.00 50.00 2 0.06500
Class B 250.00 25.00 2 0.10000
Equity 250.00 25.00 2 0.30000
EXPECTED TRANCHE LOST
Tranche Value ($) Std Deviation ($)
Class A 515.18 0.00
Class B 319.11 10.10
Equity 122.52 29.57
Note: Cell in yellow color is for displaying purpose only.
Cell in white color is for inputpurpose.
CDO ValueCDO Value Reset Histogram

Tranche 1
0
200
400
600
800
1000
1200
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
Histogram Tranche 1 Values

Tranche 2
0
100
200
300
400
500
600
700
800
900
227.47
232.21
236.96
241.71
246.45
251.20
255.95
260.70
265.44
270.19
274.94
279.69
284.43
289.18
293.93
298.67
303.42
308.17
312.92
317.66

Tranche 3
0
20
40
60
80
100
120
46.77
55.17
63.58
71.99
80.40
88.81
97.21
105.62
114.03
122.44
130.85
139.25
147.66
156.07
164.48
172.89
181.29
189.70
198.11
206.52

Tranche 1 Cash Profile
0
100
200
300
400
500
600
1-Jan-001-Jul-001-Jan-011-Jul-011-Jan-021-Jul-021-Jan-031-Jul-031-Jan-041-Jul-041-Jan-051-Jul-051-Jan-061-Jul-061-Jan-071-Jul-071-Jan-081-Jul-081-Jan-091-Jul-09
mean-std
mean
mean - std

0
50
100
150
200
250
300
mean-std
mean
mean - std

-10
0
10
20
30
40
50
60
01-Jan-00
01-Jul-00
01-Jan-01
01-Jul-01
01-Jan-02
01-Jul-02
01-Jan-03
01-Jul-03
01-Jan-04
01-Jul-04
01-Jan-05
01-Jul-05
01-Jan-06
01-Jul-06
01-Jan-07
01-Jul-07
01-Jan-08
01-Jul-08
01-Jan-09
01-Jul-09
mean+std
mean
mean - std

3. Calculating VaR for CDO Tranches
If it is assumed that the default pre-intensity process is
independent of the risk free interest rate dynamics, then
VaR for CDO tranches can be computed simply.
Step 1: Determine the principal components of the yield
curve [1].
Step 2: Compute the inner product of each principal
component with the mean cash flow.
Step 3: Add the components together.
[1] Jon Frye, “Principals of Risk: Finding VaR through Factor-Based Interest Rate Scenarios”, VaR Understanding and
Applying Value at Risk, Risk Publications, 1997, pp.275-287.

4. Conclusions
• Some simple CDO have been priced using the BET and the
Duffie-Singleton approach.
• The BET method gives a reasonable approximation to the value
of a well-funded senior tranche.
• The arbitrary assumptions of the BET method makes pricing
junior tranches unreliable.
• The Duffie-Singleton method is a powerful framework for
modeling default correlation.
• The large number of parameters in the Duffie-Singleton method
makes calibration problematic. This is the subject of our future
research.

Exploring Extensions of the
CDO Paradigm
Anthony Vaz
Robert Kowara
Carol Cheng
Capital Markets Division
OSFI
The views expressed in this presentation are
solely those of the authors.

Outline
1. Basic CDO Archetypes
2. CDO Tranche Risk
• Conceptualization of Risk
• Delta Equivalent Portfolios
• Hedging with Delta Neutral Portfolios
3. Interest Rate Risk
• General Market Risk
• Specific Risk
4. Backtesting Interest Rate Risk
5. Regulatory Capital for CDO Tranche Risk
6. Concluding Remarks

A CDO is a Collaterized Debt Obligation.
A pool of securities is used as collateral to fund a
prioritized sequence of payments. This payment
sequence is illustrated as the following “water flow” of
cash payments.

SENIOR
TRANCHE
MEZZANINE
TRANCHE
EQUITY
TRANCHE
Cash Flow Water Fall

Collateral Pool:
Bonds/Loans
Tranche 1
Coupons + Principal at Maturity
Tranche 2
Tranche N
Principal at Start
Principal at Start
Principal at Start
•Non-synthetic: Assets sold to Tranche holders
•Securitization of Bonds / Loans
•Fully sold structure
Bank
Sell Bonds/Loans
Receive Cash

Collateral Pool:
Credit Default
Swaps
Tranche 1
Premium
Tranche 2
Premium
Tranche N
Premium
Credit Protection
Credit Protection
Credit Protection
•Synthetic: Risk sold to Tranche holders, but not ownership of assets
•Securitization of risk associated with assets (Bonds / Loans / CDS etc.)
•Fully sold structure
Bank
Receive Credit
Protection
Pay Premium

Hypothetical
Collateral Pool:
Credit Default
Swaps
Tranche k
Premium
Credit Protection
•Synthetic: Risk sold to Tranche holders, but not ownership of assets
•Securitization of risk associated with a hypothetical set of Bonds / Loans / CDS
•Partially sold structure
Bank
Receive Credit
Protection
Pay Premium
•custom designed product to suit risk /reward
appetite of customer
•Bank exposed to risk of hypothetical
collateral pool (virtual securitization)

2. CDO Tranche Risk
Holders of CDO tranches are exposed to default risk in a
prioritized manner.
•Senior tranches have the risk of investment grade bonds
•Mezzaine tranches have the risk of non-investment grade bonds
•Junior tranches have the risk of default baskets
The risk can be conceptualized in terms of valuation
dispersions and cash flow profiles on the following pages.

2. CDO Tranche Risk
Tranche 1
0
200
400
600
800
1000
1200
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
515.18
Tranche 2
0
100
200
300
400
500
600
700
800
900
227.47
232.21
236.96
241.71
246.45
251.20
255.95
260.70
265.44
270.19
274.94
279.69
284.43
289.18
293.93
298.67
303.42
308.17
312.92
317.66
Tranche 3
0
20
40
60
80
100
120
46.77
55.17
63.58
71.99
80.40
88.81
97.21
105.62
114.03
122.44
130.85
139.25
147.66
156.07
164.48
172.89
181.29
189.70
198.11
206.52
Valuation dispersions for a 3 tranche
CDO

2. CDO Tranche Risk
0
100
200
300
400
500
600
mean-std
mean
mean - std
0
50
100
150
200
250
300
mean-std
mean
mean - std
-10
0
10
20
30
40
50
60
01-Jan-00
01-Jul-00
01-Jan-01
01-Jul-01
01-Jan-02
01-Jul-02
01-Jan-03
01-Jul-03
01-Jan-04
01-Jul-04
01-Jan-05
01-Jul-05
01-Jan-06
01-Jul-06
01-Jan-07
01-Jul-07
01-Jan-08
01-Jul-08
01-Jan-09
01-Jul-09
mean+std
mean
mean - std
Cash flow profiles for a 3 tranche CDO

2. CDO Tranche Risk
Delta Equivalent Portfolio
Tranches can be modelled approximately in terms of a
portfolio of instruments in the collaterial pool of the CDO. [1]
Hedging with Delta Neutral Portfolios
A tranche with a short position in its delta equivalent portfolio
are hedged against spread risk.
[1] Arthur Berd, “Risk Management of Credit Derivatives and Their Application as a Portfolio Management Tool”, RISK
2002 USA, (Stream 1, Day 2), Boston, June 11-12, 2002.

3. Interest Rate Risk
Consider the case of a simple corporate bond that depends on the yield curve );( tTy where tT  for the current time t .
Suppose the corporate bond pays coupons },,,{ 21 nccc  at times },,,{ 21 nTTT  . For simplicity of discussion, we assume the default
recovery rate is zero. The yield );( tTy is composed of two components: the risk free rate );( tTr and a spread ),;( tTs that is
dependent on the credit state )(t of the bond. Consequently, we have
),;();();( tTstTrtTy 
The corporate bond can be represented as a function
 tsssrrrf nn ;,,;,, 2121 
where
);( tTrr ii  , ),;( tTss ii  , for ni ,2,1 .
The credit state can either be discrete or continuous.
If a CreditMetrics methodology is used, the credit state is discrete and usually
 DEFAULTCCCBBBBBBAAAAAA ,,,,,,, .
If a default intensity process is used to model the credit state, then the credit state is ),0[  . Alternatively, a KMV approach
produces an expected default frequency (EDF), which represents the expected probability of defaulting over a given time horizon; this
corresponds to a credit state )1,0( .

3.1 Interest Rate General Market Risk
In the context of the corporate bond example, the general market risk arises due to variations in the risk-free
rates and the spreads over a 10-day risk horizon. The general market risk associated with the corporate bond can be
computed in the following manner. Let the difference tt  equal the risk horizon, typically 10 days. Let
)(t  denote the credit state at time t. The general market risk (GMR) is given by the following expectation
conditioned on the filtration tF .
            
             };,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
tnn
nn
FttTstTstTstTrtTrtTrf
ttTstTstTstTrtTrtTrfStdDevGMR






The operator }{StdDev represents the standard deviation. Note the bond value at time t is computed using the
spread rates that depend on the credit state )(t  .
Value at Risk (VaR) can be expressed in terms of a suitable multiplier of the standard deviation for normally
distributed P&L distributions. For non-normal distributions, a histogram of the P&L distribution is used to
determine the 99% percentile confidence level. In this example, we ignore these complications.

3.2 Interest Rate Specific Risk – Defn 1
In the context of the corporate bond example, the general market risk arises due to variations in the risk-free
rates and the spreads over a 10 day risk horizon. The specific risk associated with the corporate bond can be
computed in the following manner. Let the difference tt  equal the risk horizon, typically 10 days. Let
)(t  and )(t  denote the credit states at time t and time t ` respectively. The aggregate risk (AR) is given by
the following expectation conditioned on the filtration tF .
            
             };,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
tnn
nn
ttTstTstTstTrtTrtTrfStdDevAR






The specific risk (SR) is determined as follows.
22
GMRARSR 
NOTE:
Consider two zero mean correlated scalar random variables X andY . Then    222
2 YXXXYXYXE   , where   22
XXE 
and   22
YYE  . Let 22
2 YXXXYXA   , 2
2 YXXXYS   , and XG  . Note, the fact that 22
GAS  does not
imply 0XY . By analogy, the formula 22
GMRARSR  does not imply anything regarding the independence of risk factors
associated with general or specific market risks.

3.2 Interest Rate Specific Risk – Defn 2
Alternatively, the specific risk can be computed using a CreditMetrics framework, which ignores the
fluctuation of the interest rate and spread rate curves over the risk horizon. The only the credit state variable is
allowed to change over the risk horizon; accordingly, the credit state changes from )(t  to )(t  . These
assumptions result in the following definition of specific risk.
            
             };,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
tnn
nn
ttTstTstTstTrtTrtTrfStdDevSR






The appropriateness of these assumptions can only be determined from adequate empirical testing with market
data.

4 Backtesting Interest Rate Risk
For simplicity, we now consider a security that only depends on one credit state and one spread rate. These can
easily be generalized.
Let the value of a security   tsf ttt ,,,  be a function of market variables on day t, denoted by t ;
credit state on day t, denoted by t ; and spread of the index over the risk free rate that is dependent on the credit
state t on day t, denoted by  ts t , . The spread offset above the index curve associated with t on day t is denoted
by  tt  . Each debt security in the same credit state t has its own spread offset  tt  .

The total P&L on day t is given by
         11111 1,,,,,,   tttttttttt tsftsf 
The general market P&L is given by
         111111111 1,,,,,,   tttttttttt tsftsf 
The credit state t and the specific spread offset  tt  are held constant from day t-1 to t. The price variation arising from
fluctuation in the index curve from  1,1  ts t to  ts t ,1 and market variables 1t from t are accounted for in the general
market P&L.
The specific P&L can be computed in the following manner.
         1111 ,,,,,,   tttttttttt tsftsf 
The above computation uses the spread indices for day t to determine the spread change from the index curve with state 1t to
state t while the market variables t are held constant. The spread change arising from the change in the offset  11  tt  to
 tt  is also used to compute the specific P&L. The specific P&L would be used to backtest the specific risk computation.

•This method correctly accounts for the change in P&L associated with the gradual
deterioration in credit worthiness of an obligor.
•In this manner, the price variations that precede a credit state changes are accounted for
in a continuous manner.
•This makes the interpretation of the specific P&L a useful guide for risk management
purposes.

5. Regulatory Capital for CDO Tranche Risk
Time t
P–measure Dynamics
Time t’
Risk Horizon = 10 days
Trading Book
Q–measure Valuation
Q–measure Valuation
        tNtNtNNtttttNt tstsf ,,,,1,1,11,,1 ,,,,,,,,   
        tNtNtNNtttttNt tstsf   ,,,,1,1,11,,1 ,,,,,,,,  

Concluding Remarks
•The virtual securitizations of partially sold structures
expose banks to risks that need to be risk managed
•CDO Tranche Risk can be conceptualized simply in terms
of valuation dispersion and cash flow profiles
•Delta Equivalent Portfolios can be used to a simple
models to mange credit risk in an integrated manner
•A method of computing and backtesting both general
market and specific interest rate risk has been proposed.
•These computations can be used to determine regulatory
capital for CDO tranche risk.

RISK2003_Part1&2_Slides_v2b

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