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June 11, 2003 - Risk Conference, Boston, MA

Office of the Superintendentof Financial Institutions

Bureau du surintendantdes institutions financières

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






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.






C O L L A T E R A L

SENIORTRANCHE

MEZZANINE

TRANCHE

EQUITYTRANCHE


Cash Flow Water Fall







Collateral Pool:

Bonds/LoansTranche 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 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







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)






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.






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






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


-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






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 .






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.






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


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.






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


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.






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 .






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.






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






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.






CDO Modelling

Anthony VazRobert KowaraCarol Cheng

Capital Markets DivisionOSFI

The views expressed in this presentationare solely those of the authors .






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






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.






C O L L A T E R A L

SENIORTRANCHE

MEZZANINE

TRANCHE

EQUITYTRANCHE

1. CDO Terminology

1.1 Definitions

Cash Flow Water Fall






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).








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.








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.








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.








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.






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.






2.1 Binomial Expansion Technique2.1.1 Derivation of Diversity Score

Pool of Correlated Bonds

Correlated bonds: M=20

Diversity Score: N=5






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






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

−+=

Office of the Superintendent Bureau du surintendant





Bureau du surintendantdes institutions financi è res

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







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 =







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





pof Financial Institutions des institutions financi è res

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.





of Financial Institutions des institutions financi è res


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.







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.

Office of the Superintendent Bureau du surintendantd f è





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 (%)


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








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








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






of Financial Institutions


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







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






•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






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






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






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

−++∂

∂+

∂

∂−+

∂

∂=

∆

−∆+∆+=

∫

∞

→∆







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

−+−=

−=

−=> ∫






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






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.






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.






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.




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






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







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








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









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









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









-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






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.






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.

vaz risk2003 slides

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