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Outline
1. Basic CDO Archetypes
2. CDO Tranche RiskConceptualization 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 Risk6. Concluding Remarks
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1. Basic CDO Achetypes
A CDO is a Collaterized Debt Obligation .A pool of securities is used as collateral to fund aprioritized sequence of payments. This paymentsequence is illustrated as the following “water flow” ofcash payments.
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C O L L A T E R A L
SENIORTRANCHE
MEZZANINE
TRANCHE
EQUITYTRANCHE
1. Basic CDO Achetypes
Cash Flow Water Fall
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1. Basic CDO Achetypes
Collateral Pool:
Bonds/LoansTranche 1
Coupons + Principal at Maturity
Tranche 2
Coupons + Principal at Maturity
Tranche N
Coupons + Principal at Maturity
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
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1. Basic CDO Achetypes
Collateral Pool:
Credit DefaultSwaps
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 CreditProtection
Pay Premium
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1. Basic CDO Achetypes
HypotheticalCollateral Pool:
Credit DefaultSwaps
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 CreditProtection
Pay Premium
•custom designed product to suit risk /rewardappetite of customer
•Bank exposed to risk of hypotheticalcollateral pool (virtual securitization)
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2. CDO Tranche Risk
Holders of CDO tranches are exposed to default risk in aprioritized 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 valuationdispersions and cash flow profiles on the following pages.
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2. CDO Tranche RiskTranche 1
0
200
400
600
800
1000
1200
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
Tranche 2
0
100
200
300
400
500
600
700
800
900
2 2 7 .
4 7
2 3 2 .
2 1
2 3 6 .
9 6
2 4 1 . 7 1
2 4 6 .
4 5
2 5 1 . 2 0
2 5 5 .
9 5
2 6 0 .
7 0
2 6 5 .
4 4
2 7 0 .
1 9
2 7 4 . 9 4
2 7 9 .
6 9
2 8 4 .
4 3
2 8 9 .
1 8
2 9 3 . 9 3
2 9 8 .
6 7
3 0 3 .
4 2
3 0 8 .
1 7
3 1 2 .
9 2
3 1 7 .
6 6
Tranche 3
0
20
40
60
80
100
120
4 6 . 7 7
5 5 . 1 7
6 3 . 5 8
7 1 . 9 9
8 0 . 4 0
8 8 . 8 1
9 7 . 2 1
1 0 5 .
6 2
1 1 4 . 0 3
1 2 2 .
4 4
1 3 0 .
8 5
1 3 9 .
2 5
1 4 7 .
6 6
1 5 6 .
0 7
1 6 4 .
4 8
1 7 2 .
8 9
1 8 1 . 2 9
1 8 9 .
7 0
1 9 8 .
1 1
2 0 6 .
5 2
Valuation dispersions for a 3 trancheCDO
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2. CDO Tranche RiskTranche 1 Cash Profile
0
100
200
300
400
500
600
1 - J a n - 0 0
1 - J u l - 0 0
1 - J a n - 0 1
1 - J u l - 0 1
1 - J a n - 0 2
1 - J u l - 0 2
1 - J a n - 0 3
1 - J u l - 0 3
1 - J a n - 0 4
1 - J u l - 0 4
1 - J a n - 0 5
1 - J u l - 0 5
1 - J a n - 0 6
1 - J u l - 0 6
1 - J a n - 0 7
1 - J u l - 0 7
1 - J a n - 0 8
1 - J u l - 0 8
1 - J a n - 0 9
1 - J u l - 0 9
mean-std
mean
mean - std
Tranche 2 Cash Profile
0
50
100
150
200
250
300
1 - J a n - 0 0
1 - J u l - 0 0
1 - J a n - 0 1
1 - J u l - 0 1
1 - J a n - 0 2
1 - J u l - 0 2
1 - J a n - 0 3
1 - J u l - 0 3
1 - J a n - 0 4
1 - J u l - 0 4
1 - J a n - 0 5
1 - J u l - 0 5
1 - J a n - 0 6
1 - J u l - 0 6
1 - J a n - 0 7
1 - J u l - 0 7
1 - J a n - 0 8
1 - J u l - 0 8
1 - J a n - 0 9
1 - J u l - 0 9
mean-stdmeanmean - std
Tranche 3 Cash Profile
-10
0
10
20
30
40
50
60
0 1 - J a n - 0 0
0 1 - J u l - 0 0
0 1 - J a n - 0 1
0 1 - J u l - 0 1
0 1 - J a n - 0 2
0 1 - J u l - 0 2
0 1 - J a n - 0 3
0 1 - J u l - 0 3
0 1 - J a n - 0 4
0 1 - J u l - 0 4
0 1 - J a n - 0 5
0 1 - J u l - 0 5
0 1 - J a n - 0 6
0 1 - J u l - 0 6
0 1 - J a n - 0 7
0 1 - J u l - 0 7
0 1 - J a n - 0 8
0 1 - J u l - 0 8
0 1 - J a n - 0 9
0 1 - J u l - 0 9
mean+std
mean
mean - std
Cash flow profiles for a 3 tranche CDO
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3. Interest Rate RiskConsider the case of a simple corporate bond that depends on the yield curve ) ;( t T y where t T ≥ for the current time t .
Suppose the corporate bond pays coupons },,,{ 21 nccc K at times },,,{ 21 nT T T K . For simplicity of discussion, we assume the default
recovery rate is zero. The yield ) ;( t T y is composed of two components: the risk free rate ) ;( t T r and a spread ),;( θ t T s that is
dependent on the credit state ) (t θ of the bond. Consequently, we have
),;();();( θ t T st T r t T y +=
The corporate bond can be represented as a function
( )t sssr r r f nn ;,,;,, 2121 K K
where
);( t T r r ii= , ),;( θ t T ss ii
= , for ni K,2,1= .
The credit state can either be discrete or continuous.
If a CreditMetrics methodology is used, the credit state is discrete and usually
{ } DEFAULT CCC B BB BBB A AA AAA ,,,,,,,θ .
If a default intensity process is used to model the credit state, then the credit state is ) ,0[ ∞θ . 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(θ .
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3.1 Interest Rate General Market RiskIn 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 t t −′ equal the risk horizon, typically 10 days. Let
)(t θ θ = denote the credit state at time t . The general market risk (GMR) is given by the following expectation
conditioned on the filtration t F .
( ) ( ) ( )( ) ( ) ( )[ ]( ) ( ) ( )( ) ( ) ( )[ ] } ;,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
t nn
nn
F t t T st T st T st T r t T r t T r f
t t T st T st T st T r t T r t T r f StdDevGMR
θ θ θ
θ θ θ KKKK
−
′′′′′′′=
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 θ θ = .
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.
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3.2 Interest Rate Specific Risk – Defn 1In 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 t t −′ equal the risk horizon, typically 10 days. Let
)(t θ θ = and )(t ′=′ θ θ 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 t F .
( ) ( ) ( )( ) ( ) ( )[ ]( ) ( ) ( )( ) ( ) ( )[ ] } ;,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
t nn
nn
F t t T st T st T st T r t T r t T r f
t t T st T st T st T r t T r t T r f StdDev AR
θ θ θ
θ θ θ KKKK
−
′′′′′′′′′′=
The specific risk (SR) is determined as follows.
22 GMR ARSR −=
NOTE:
Consider two zero mean correlated scalar random variables X and Y . Then ( ){ }222
2 Y X X XY X Y X E σ σ σ ρ σ ++=+
, where { }22
X X E σ =
and { } 22Y Y E σ = . Let 22 2 Y X X XY X A σ σ σ ρ σ ++= , 22 Y X X XY S σ σ σ ρ += , and X G σ = . Note, the fact that 22 G AS −= does not
imply 0= XY ρ . By analogy, the formula 22 GMR ARSR −= does not imply anything regarding the independence of risk factors
associated with general or specific market risks.
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3.2 Interest Rate Specific Risk – Defn 2Alternatively, 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 θ θ = to )(t ′=′ θ θ . These
assumptions result in the following definition of specific risk.
( ) ( ) ( )( ) ( ) ( )[ ]( ) ( ) ( )( ) ( ) ( )[ ] } ;,;,,,;,,;;;,,;,;
;,;,,,;,,;;;,,;,;{
2121
2121
t nn
nn
F t t T st T st T st T r t T r t T r f
t t T st T st T st T r t T r t T r f StdDevSR
θ θ θ
θ θ θ KK
KK
−
′′′′=
The appropriateness of these assumptions can only be determined from adequate empirical testing with market
data.
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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 ( )[ ]t s f t t t ,,, θ γ θ be a function of market variables on day t , denoted by t γ ;
credit state on day t , denoted by t θ ; and spread of the index over the risk free rate that is dependent on the credit
state t θ on day t , denoted by ( )t s t ,θ . The spread offset above the index curve associated with t θ on day t is denoted
by ( )t t θ α . Each debt security in the same credit state t θ has its own spread offset ( )t t θ α .
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4 Backtesting Interest Rate Risk
•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 accountedfor in a continuous manner.
•This makes the interpretation of the specific P&L a useful guide for risk managementpurposes.
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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
) ) ) ))t N t N t N N t t t t t N t t st s f ,,,,1,1,11,,1 ,,,,,,,, θ α θ θ α θ γ θ θ ++ KK
) ) ) ))t N t N t N N t t t t t N t t st s f ′′′′′′′′′ +′+′,,,,1,1,11,,1 ,,,,,,,, θ α θ θ α θ γ θ θ KK
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Concluding Remarks•The virtual securitizations of partially sold structuresexpose banks to risks that need to be risk managed
•CDO Tranche Risk can be conceptualized simply in termsof valuation dispersion and cash flow profiles
•Delta Equivalent Portfolios can be used to a simplemodels to mange credit risk in an integrated manner
•A method of computing and backtesting both generalmarket and specific interest rate risk has been proposed.
•These computations can be used to determine regulatorycapital for CDO tranche risk.
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CDO Modelling
Anthony VazRobert KowaraCarol Cheng
Capital Markets DivisionOSFI
The views expressed in this presentationare solely those of the authors .
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Outline1. CDO Terminology
2. CDO Valuation2.1 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 Tranches4. Concluding Remarks
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1. CDO Terminology
1.1 Definitions
A CDO is a Collaterized Debt Obligation . A pool ofsecurities is used as collateral to fund a prioritizedsequence of payments. This payment sequence isillustrated as the following “water flow” of cashpayments.
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C O L L A T E R A L
SENIORTRANCHE
MEZZANINE
TRANCHE
EQUITYTRANCHE
1. CDO Terminology
1.1 Definitions
Cash Flow Water Fall
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1. CDO Terminology1.2 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).
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1. CDO Terminology1.2 CDO Types
CDO can be classified in a variety of ways.
1.2.2 Transaction Type
In an arbitrage transaction , the CDO is constructed to capture the difference inspread between the collateral pool and the yields at which the senior liabilities ofthe CDO are issued.
In a balance sheet transaction , the CDO is constructed to remove loans orbonds from the balance sheet of a financial institution. This is motivated by thedesire to obtain capital relief, improve liquidity, and re-deploy to alternativeinvestments.
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1. CDO Terminology1.2 CDO Types
CDO can be classified in a variety of ways.
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 inmultiple asset categories that may include corporate bonds, loans, private andpublic equity, distressed securities or emerging market investments, and cashand 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 ofthe collateral pool.
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1. CDO Terminology1.2 CDO Types
CDO can be classified in a variety of ways.
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 assetcategory, such as corporate bonds, loans, or mortgages.
• The collateral pool is fairly static. When an asset matures or defaults, theproceeds may be invested at the discretion of the fund manager.
• The collateral pool is priced periodically to obtain the par value. The payments tothe tranches are based on threshold levels for the par value of the collateralpool.
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1. CDO Terminology1.2 CDO Types
CDO can be classified in a variety of ways.
1.2.4 Legal Ownership of Collateral Pool Assets
1.2.4.1 Non-synthetic CDOA 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 CDOA synthetic CDO does not have legal ownership of the assets in the collateralpool. The CDO assumes economic risk on the assets which it does not legallyown.
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2. CDO ValuationA 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’dSpecial 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.
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2.1 Binomial Expansion Technique2.1.1 Derivation of Diversity Score
Pool of Correlated Bonds
Correlated bonds: M=20
Diversity Score: N=5
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2.1 Binomial Expansion Technique2.1.1 Derivation of Diversity Score2.1.1.1 Independent Bond Pool
• Consider a hypothetical pool consisting of N bonds having thesame par value F . The bond defaults are assumed to beindependent.
• N is called the diversity score of the bond pool.
• All the bonds are assumed to have the same loss L when adefault occurs.• Let the be a random variable representing the state of bond i .i X
= defaultednotbond,0
defaultedbond,1
i
i X i
p X i == ]1[Prob p X i −== 1]0[Prob
Offi f h S i d B d i d
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2.1 Binomial Expansion Technique
We can solve for the first and second moment loss statistics of thecollateral portfolio.
p X i =][EHence p X i =][E 2
)1(][][][Variance 22 p p X E X E X iii −=−=
∑=
= N
iiPort X L L
1
pNL L E Port =][ ))1(1(][ 22 p N pNL L E Port
−+=
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2.1 Binomial Expansion Technique2.1.1.2 Dependent Bond Pool•Consider a hypothetical pool consisting of M bonds having the
same par value . The bond defaults are assumed to bedependent.•All the bonds are assumed to have the same loss when adefault occurs.•Let the be a random variable representing the state of bond i .
F
iY
=defaultednotbond,0
defaultedbond,1
i
iY i
pY i == ]1[Prob pY i −== 1]0[Prob
L
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2.1 Binomial Expansion Technique pY i =][E pY i =][E 2
)1(][][][Variance 22 p pY E Y E Y iii −=−=
Let )1(i p p −=σ
jiij ji Y Y σ σ ρ ][Covariance =
2 ][E pY Y jiij ji+= σ σ ρ
Assume all pair-wise correlations are equal:
Assume all variances are equal:
ij
ρ ρ =
ijσ σ =
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2.1 Binomial Expansion TechniqueWe can solve for the first and second moment loss statistics of thecollateral portfolio.
∑=
= M
iiPort Y L L
1
L M p L E Port =][
))1(()1(][ 2222 p p p L M M L M p L E Port +−−+= ρ
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2.1 Binomial Expansion TechniqueEquate expressions for the first and second moment loss statistics of the collateral portfolios to obtain the following.
)1( −
−=
M N N M ρ Correlation:
where N=diversity score & M=number of correlated bonds
)1(1 −+=
M M
N ρ
Note, these formulae can be generalized to account for random recovery rates using the sametechnique.
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2.1 Binomial Expansion Technique
2.1.2 Computing CDO Loss Scenarios
The BET method makes the assumption that losses occur with a givenprofile.
For example,
50% end of year 110% end of year 2
10% end of year 3
10% end of year 4
10% end of year 5The profiles are determined from historical data; but they cannot be rigorously tailored to aparticular portfolio.
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2.1 Binomial Expansion Technique
The probability of getting j defaults in the bond pool is
j N j j p p
j N j N
P −−
−
= )1( )!(!
!
2.1.2 Computing CDO Loss Scenarios
A loss scenario S j is associated with each of the above defaultcombinations.
Hence S 10 corresponds to 5 defaults in the first year, 1 at end of year 2, 1 atend of year 3, 1 end of year 4, and 1 at end of year 5.
The CDO cashflows are computed accordingly.
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2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
DEFAULT SCENARIO POOL OF ASSETStime defaults Diversity Score 580.5 0.00 Average Coupon 8.00%1.0 0.50 Average Maturity 101.5 0.00 Notional 10002.0 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 0.004.0 0.05 TRANCHES4.5 0.00 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR85.0 0.05 Name aaa bbb ccc5.5 0.00 Coupon 6.50% 10.00% 30.00%6.0 0.05 Notional 500 280 2206.5 0.00 Maturity 10 10 107.0 0.05 OC Test 0 0 07.5 0.00 Expected loss 0.0019% 14.3932% 57.8585% #N/A #N/A #N/A #N/A #N/A8.0 0.05 Ratings Aaa B2 NR #N/A #N/A #N/A #N/A #N/A8.5 0.009.0 0.059.5 0.00
10.0 0.0510.511.011.5
Calculate
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Moody ’s “Idealized ” Cumulative Expected Loss Rates (%)
2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
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%
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Binomial Distribution2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
Probability distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
1 4 7 1 0
1 3
1 6
1 9
2 2
2 5
2 8
3 1
3 4
3 7
4 0
4 3
4 6
4 9
5 2
5 5
Prob
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2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
Expected Loss ve rsus Diversty
0
0.05
0.1
0.15
0.20.25
0.3
0.35
0.4
0.45
5 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0
TR1
TR2
TR3
TR4
TR5
TR6
TR7
TR8
Expected Loss versus Diversity
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2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
Expected Loss versus Diversity
Senior Tranche Expected Loss Versus Diversity
0
0.005
0.01
0.015
0.02
0.025
0.03
5 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0
TR1
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2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
Mezzanine Tranche Expected Loss versus Diversity
00.020.040.060.080.1
0.120.140.160.18
0.2
5 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0
TR2
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2.1.3 Excel VBA Implementation of BET Method Applied to CBO/CLO
Junior Tranche Expected Loss versus Diversity
0.310.320.330.340.350.360.370.380.39
5 1 2
1 6
2 0
2 4
2 8
3 2
3 6
4 0
4 4
4 8
5 2
5 6
6 0
TR3
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•OC is calculated as the ratio between the par value of
collateral and the value of the all liabilities senior toand including the tranche being calculated.
•Once OC ratio drops below the certain level the cashflow from the equity or lower tranche is diverted to arisk-free reserve account.
2.1.4 Overcollateralization Tests
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2.2.1 Hazard RateDuffie ’s approach is based on an application of reliability theoryto the default process.
Reliability theory uses a hazard rate intensity to obtain theconditional survival probability as follows.
2.2 Duffie-Singleton Methodology
( ))(expexp)|( t T duF T PT
t t
−−=
−=> ∫ λ λ τ
t T t F
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2.2 Duffie-Singleton Methodology2.2.2 Stochastic Pre-intensityDuffie models the hazard rate as a stochastic process that he calls the“pre-intensity process ” .
J(t)dW(t)(t)σdt)(t)-(θ )( ∆++= λ λ κ λ t d
The conditional survival probability is given by the following.
−=> ∫ t
T
t t F duu E F T P )(exp)|( λ τ
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2.2 Duffie-Singleton Methodology2.2.2 Stochastic Pre-intensity
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2.2 Duffie-Singleton Methodology2.2.2 Stochastic Pre-intensity
The computation of the above expectation is quite complex. It is done inseveral steps.
Step 1: The diffusion generator is determined.
[ ]
( ) [ ] )(),(),(2
1
),(|)),((lim
02
22
0
H d t f t H f l f f
t
f
t t f F t t t t f E
Df t
t
ν λ λ λ λσ λ λ θ κ
λ λ
−++∂
∂+
∂
∂−+
∂
∂=
∆
−∆+∆+=
∫
∞
→∆
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2.2 Duffie-Singleton Methodology2.2.3 Stochastic Pre-intensity
Step 2: The Feynman-Kac formula is used to obtain the followingintegral-PDE equation.
( ) [ ] 0)(),(),(21
02
22 =−++−
∂
∂+
∂
∂−+
∂
∂∫ ∞
H d t f t H f l f f f
t f ν λ λ λ
λ λσ
λ λ θ κ
Step 3: The PDE is solved using an affine solution to obtain.
[ ])()()(exp
)),(()(exp)|(
t t T t T
t T t f F duu E F T P t
T
t
t
λ β α
λ λ τ
−+−=
−=
−=> ∫
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2.2 Duffie-Singleton Methodology2.2.4 Defaultable Zero Coupon Bond
The conditional survival probability is then used to derive the value of adefaultable zero-coupon bond .
[ ] ∫ ++=T
zero duuhur T T T t p0
00 )()()()(exp)(),( δ λ β α δ λ
where the conditional default intensity is given by
[ ] [ ])0()()()0()()(exp)|(
)( 0 λ β α λ β α τ ′+′+−=
∂
>∂−= T T T T
T
F T PT h
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2.2 Duffie-Singleton Methodology2.2.5 Defaultable Coupon Bond
The Duffie-Garleanu has an incorrect formula for a defaultable couponbond in his paper. The correct formula for a coupon bond with quarterlypayments is as follows.
[ ]
+
+++= ∑∫ 0
0
00 44exp
44)()()()(exp)(),( λ β α δ δ λ β α δ λ j j jC
duuhur T T T t pT
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.
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2.2 Duffie-Singleton Methodology2.2.6 Bond Hazard Rates
Suppose the are N bonds in a collateral pool, each with a hazard rate
processl
i, (i=1,2, …N). Duffie advocates the partition of the affineprocess into risk factor components.
Z Y X ici++=
)(iλ
The process X i is unique to bond i. The process Y c(i) is common to bondsaffected 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.
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2.2 Duffie-Singleton Methodology2.2.6 Bond Hazard Rates
The instantaneous correlation coefficient rij
between hazard rate processes for bonds iand 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
F duu Z E F duuY E F duu X E
F duu E F T P
)(exp)(exp)(exp
)(exp)|(
)(
λ τ
The calibration can be done similar to a 3 factor CIR spot rate model.
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton MethodC A S H F L O W C D O S P E C I F I C A T I O N
n S i m 1 0 0 0
I s s u i n g D a t e 1 - J a n - 0 0 N o . o f B i n s 2 0
M a t u r i t y D a t e 1 - J a n - 1 0 N o . o f P e r i o d 1 2 0
C o l l a t e r a l N o t i o n a l 1 , 0 0 0 . 0 0
N o . o f T r a n c h e 3
R e s r v - A c c r - R a t e 0 . 0 5 0 0
T r a n c h e R a t i n g T r a n c h e P r i n c i p a l % / N o t i o n a l C o u p o n F r e q u e n c y C o u p o n R a t e
C l a s s A 5 0 0 . 0 0 5 0 . 0 0 2 0 . 0 6 5 0 0
C l a s s B 2 5 0 . 0 0 2 5 . 0 0 2 0 . 1 0 0 0 0
E q u i t y 2 5 0 . 0 0 2 5 . 0 0 2 0 . 3 0 0 0 0
E X P E C T E D T R A N C H E L O S T
T r a n c h e V a l u e ( $ ) S t d D e v i a t i o n ( $ )
C l a s s A 5 1 5 . 1 8 0 . 0 0
C l a s s B 3 1 9 . 1 1 1 0 . 1 0
E q u i t y 1 2 2 . 5 2 2 9 . 5 7
N o t e : C e l l i n y e l l o w c o l o r i s f o r d i s p l a y i n g p u r p o s e o n l y .
C e l l i n w h i t e c o l o r i s f o r i n p u t p u r p o s e .
CDO ValueCDO Value Reset Histogram
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 1
0
200
400
600
800
1000
1200
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
5 1 5 .
1 8
Histogram Tranche 1 Values
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 2
0
100
200300
400
500
600
700800
900
2 2 7 .
4 7
2 3 2 .
2 1
2 3 6 .
9 6
2 4 1 . 7 1
2 4 6 .
4 5
2 5 1 . 2 0
2 5 5 .
9 5
2 6 0 .
7 0
2 6 5 .
4 4
2 7 0 .
1 9
2 7 4 . 9 4
2 7 9 .
6 9
2 8 4 .
4 3
2 8 9 .
1 8
2 9 3 . 9 3
2 9 8 .
6 7
3 0 3 .
4 2
3 0 8 .
1 7
3 1 2 .
9 2
3 1 7 .
6 6
Histogram Tranche 2 Values
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 3
0
20
40
60
80
100
120
4 6 . 7 7
5 5 . 1 7
6 3 . 5 8
7 1 . 9 9
8 0 . 4 0
8 8 . 8 1
9 7 . 2 1
1 0 5 . 6 2
1 1 4 . 0 3
1 2 2 . 4
4
1 3 0 . 8 5
1 3 9 .
2 5
1 4 7 . 6 6
1 5 6 . 0 7
1 6 4 . 4
8
1 7 2 . 8 9
1 8 1 . 2 9
1 8 9 . 7 0
1 9 8 . 1
1
2 0 6 . 5 2
Histogram Tranche 3 Values
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 1 Cash Profile
0
100
200
300
400
500
600
1 - J a n - 0 0
1 - J u l - 0 0
1 - J a n - 0 1
1 - J u l - 0 1
1 - J a n - 0 2
1 - J u l - 0 2
1 - J a n - 0 3
1 - J u l - 0 3
1 - J a n - 0 4
1 - J u l - 0 4
1 - J a n - 0 5
1 - J u l - 0 5
1 - J a n - 0 6
1 - J u l - 0 6
1 - J a n - 0 7
1 - J u l - 0 7
1 - J a n - 0 8
1 - J u l - 0 8
1 - J a n - 0 9
1 - J u l - 0 9
mean-stdmeanmean - std
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 2 Cash Profile
0
50
100
150
200
250
300
1 - J a n - 0 0
1 - J u l - 0 0
1 - J a n - 0 1
1 - J u l - 0 1
1 - J a n - 0 2
1 - J u l - 0 2
1 - J a n - 0 3
1 - J u l - 0 3
1 - J a n - 0 4
1 - J u l - 0 4
1 - J a n - 0 5
1 - J u l - 0 5
1 - J a n - 0 6
1 - J u l - 0 6
1 - J a n - 0 7
1 - J u l - 0 7
1 - J a n - 0 8
1 - J u l - 0 8
1 - J a n - 0 9
1 - J u l - 0 9
mean-stdmeanmean - std
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2.1.7 Excel VBA with C++ DLL Implementation of Duffie-Singleton Method
Tranche 3 Cash Profile
-10
0
10
20
30
40
50
60
0 1 - J a n - 0 0
0 1 - J u l - 0 0
0 1 - J a n - 0 1
0 1 - J u l - 0 1
0 1 - J a n - 0 2
0 1 - J u l - 0 2
0 1 - J a n - 0 3
0 1 - J u l - 0 3
0 1 - J a n - 0 4
0 1 - J u l - 0 4
0 1 - J a n - 0 5
0 1 - J u l - 0 5
0 1 - J a n - 0 6
0 1 - J u l - 0 6
0 1 - J a n - 0 7
0 1 - J u l - 0 7
0 1 - J a n - 0 8
0 1 - J u l - 0 8
0 1 - J a n - 0 9
0 1 - J u l - 0 9
mean+std
mean
mean - std
Office of the Superintendentof Financial Institutions
Bureau du surintendantdes institutions financi è res
8/3/2019 Vaz Risk2003 Slides
http://slidepdf.com/reader/full/vaz-risk2003-slides 65/66
June 11, 2003 - Risk Conference, Boston, MAPage 65
3. Calculating VaR for CDO TranchesIf it is assumed that the default pre-intensity process isindependent of the risk free interest rate dynamics, thenVaR for CDO tranches can be computed simply.
Step 1: Determine the principal components of the yieldcurve [1].
Step 2: Compute the inner product of each principalcomponent 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 andApplying Value at Risk, Risk Publications, 1997, pp.275-287.
Office of the Superintendentof Financial Institutions
Bureau du surintendantdes institutions financi è res
8/3/2019 Vaz Risk2003 Slides
http://slidepdf.com/reader/full/vaz-risk2003-slides 66/66
June 11, 2003 - Risk Conference, Boston, MAPage 66
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 valueof 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 formodeling default correlation.
• The large number of parameters in the Duffie-Singleton methodmakes calibration problematic. This is the subject of our future
research.