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 TypesCDO 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


1.2.2 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


1.2.3 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


1.2.3 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


1.2.4 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 acontingent 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 acontingent 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

defaultednot bond,0

defaulted bond,1

i

iX i

pX i ]1[Prob pX i 1]0[Prob







We can solve for the first and second moment loss statistics of the collateral portfolio.

pX i ][EHence pX i ][E 2

)1(][][][ Variance 22 ppXEXEX iii

N

i

iPort XLL1

pNLLE Port ][ ))1(1(][ 22 pNpNLLE Port






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 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

defaultednot bond,0

defaulted bond,1

i

iYi

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 YLL1

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

NMCorrelation:

where N=diversity score & M=number of correlated bonds

)1( 1

M

MN

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 ppjNj

NP

)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

0.020.04

0.06

0.080.1

0.12

0.14

0.160.18

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) σ dt ) (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

22

0

HdtftHflff

t

f

t

tfFttttfEDf t

t








Step 2: The Feynman-Kac formula is used to obtain the following

integral-PDE equation.

0)(),(),(2

1

0

2

22

HdtftHflfff

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 duuhurTTTtp0

00 )()()()(exp)(),(

where the conditional default intensity is given by

)0()()()0()()(exp)|(

)( 0

TTTT

T

FTPTh







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

0044

exp44

)()()()(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 duulNP0

)(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 input purpose.

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-00

1-Jul-00

1-Jan-01

1-Jul-01

1-Jan-02

1-Jul-02

1-Jan-03

1-Jul-03

1-Jan-04

1-Jul-04

1-Jan-05

1-Jul-05

1-Jan-06

1-Jul-06

1-Jan-07

1-Jul-07

1-Jan-08

1-Jul-08

1-Jan-09

1-Jul-09

mean-std

mean

mean - std








0

50

100

150

200

250

300

1-Jan-00

1-Jul-00

1-Jan-01

1-Jul-01

1-Jan-02

1-Jul-02

1-Jan-03

1-Jul-03

1-Jan-04

1-Jul-04

1-Jan-05

1-Jul-05

1-Jan-06

1-Jul-06

1-Jan-07

1-Jul-07

1-Jan-08

1-Jul-08

1-Jan-09

1-Jul-09

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

1-Jan-00

1-Jul-00

1-Jan-01

1-Jul-01

1-Jan-02

1-Jul-02

1-Jan-03

1-Jul-03

1-Jan-04

1-Jul-04

1-Jan-05

1-Jul-05

1-Jan-06

1-Jul-06

1-Jan-07

1-Jul-07

1-Jan-08

1-Jul-08

1-Jan-09

1-Jul-09

mean-std

mean

mean - std


0

50

100

150

200

250

300

1-Jan-00

1-Jul-00

1-Jan-01

1-Jul-01

1-Jan-02

1-Jul-02

1-Jan-03

1-Jul-03

1-Jan-04

1-Jul-04

1-Jan-05

1-Jul-05

1-Jan-06

1-Jul-06

1-Jan-07

1-Jul-07

1-Jan-08

1-Jul-08

1-Jan-09

1-Jul-09

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 RiskConsider 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 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 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 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 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 2222 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 0XY . 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 .






4 Backtesting Interest Rate RiskThe 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 1t 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 1t 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.






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 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 BookQ–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.

Documents

RISK2003_Part1&2_Slides_v2b