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Risk Management Hedging credit index tranches Investigating versions of the standard model Christopher C. Finger chris.fi[email protected]
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Risk Management

Hedging credit index tranchesInvestigating versions of the standard model

Christopher C. Finger

[email protected]

2www.riskmetrics.com Risk Management

Subtle company introduction

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Motivation

A standard model for credit index tranches exists.

It is commonly acknowledged that the common model isflawed.

Most of the focus is on the static flaw: the failure to calibrateto all tranches on a single day with a single model parameter.

But these are liquid derivatives. Models are not used forabsolute pricing, but for relative value and hedging.

We will focus on the dynamic flaws of the model.

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Outline

1 Standard credit derivative products

2 Standard models, conventions and abuses

3 Data and calibration

4 Testing hedging strategies

5 Conclusions

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Standard products

Single-name credit default swaps

Contract written on a set of reference obligations issued by onefirmProtection seller compensates for losses (par less recovery) inthe event of a default.Protection buyer pays a periodic premium (spread) on thenotional amount being protected.Quoting is on fair spread, that is, spread that makes a contracthave zero upfront value at inception.

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Standard products

Credit default swap indices (CDX, iTraxx)

Contract is essentially a portfolio of (125, for our purposes)equally weighted CDS on a standard basket of firms.Protection seller compensates for losses (par less recovery) inthe event of a default.Protection buyer pays a periodic premium (spread) on theremaining notional amount being protected.New contracts (series) are introduced every six months.Standardization of premium, basket, maturity has createdsignificant liquidity.Quoting is on fair spread, with somewhat of a twist.Pricing depends only on the prices of the portfolio names. . . almost.

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Standard products

Tranches on CDX

Protection seller compensates for losses on the index in excessof one level (the attachment point) and up to a second level(the detachment point).For example, on the 3-7% tranche of the CDX, protection sellerpays losses over 3% (attachment) and up to 7% (detachment).Protection buyer pays an upfront amount (for most juniortranches) plus a periodic premium on the remaining amountbeing protected.Standardization of attachment/detachment, indices, maturity.Not strictly a derivative on the index, in that payoff does notreference the index pricePricing depends on the distribution of losses on the index, notjust the expectation.

Also, options on CDX, but we will not consider these.7

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CDX history

Mar05 Sep05 Mar06 Sep06 Mar07 Sep07 Mar080

40

80

120

160

200In

dex

spre

ad (

bp)

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CDX tranche history

Mar05 Sep05 Mar06 Sep06 Mar07 Sep070

10

20

30

40

50

60

Tra

nche

upf

ront

pric

e (%

)

0

100

200

300

400

500

Tra

nche

fair

spre

ad (

bp)

0−3% (LHS)3−7% (RHS)7−10% (RHS)

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What do we want from a model?

Fit market prices, but why?

Extrapolate, i.e. price more complex, but similar, structures

Non-standard attachment pointsCustomized baskets

Hedge risk due to underlying

Dealers provide liquidity in tranches, but want to controlexposure to underlyings.Speculators want to make relative bets on tranches without aview on underlyings.

Risk management

Aggregate credit exposures across many product types.Recognize risk that is truly idiosyncratic.

For anything other than extrapolation, we care about how pricesevolve in time, so we should look at the dynamics.

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Standard pricing models—basic stuff

Price for a tranche is the difference of

Expectation of discounted future premium payments, andExpectation of discounted future losses.

Boils down to the distribution of the loss process on the indexportfolio, in particular things like

E min{(d − a),max{0, Lt − a}}.

Suffices to specify the joint distribution of times to default Ti

for all names in the basket.

CDS (or CDX) quotes imply the marginal distributions fortime to default for individual names Fi (t) = P{Ti < t}.

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Standard pricing models—specific stuff

Dependence structure is a Gaussian copula:

Let Zi be correlated standard Gaussian random variables.Default times are given by Ti = F−1

i (Φ(Zi )).

Correlation structure is pairwise constant . . .

Zi =√ρZ +

√1− ρ εi .

For a single period, just count the number of Zi that fallbelow the default threshold αi = Φ−1(pi ).

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Criticisms of the standard model

Does not fit all market tranche prices on a given day.

No dynamics, so no natural hedging strategy.

Link to any observable correlation is tenuous. At best, modelis “inspired” by Merton framework, so correlation is onequities.

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Model flavors

Most numerical techniques rely on integrating over Z , given whichall defaults are conditionally independent.

Granular model—use full information on underlying spreads,and model full discrete loss distribution.

Homogeneity assumption—assume all names in the portfoliohave the same spread; use index level.

Fine grained limit—continuous distribution, easy integrals

Large pool model—combine homogeneity and fine grainedassumptions.

Also the question of whether to use full spread term structuresor a single point

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Correlation conventions

Start to introduce model abuses, ala the B-S volatility smile.

Compound correlations

Price each individual tranche with a distinct correlation.Not all tranches are monotonic in correlation.Trouble calibrating mezzanine tranches, especially in 2005

Base correlations

Decompose each tranche into “base” (i.e. 0-x%) tranches.Bootstrap to calibrate all tranches.Base tranches monotonic, but calibration not guaranteed.

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Correlation conventions

Constant moneyness (ATM) correlations

Some movements in implied correlation are due to changing“moneyness” as the index changes.Examine correlations associated with a detachment point equalto implied index expected loss.If base correlations are “sticky strike”, then ATM correlationsare closer to “sticky delta”.

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CDX correlation structure

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

Detachment point (%)

Bas

e co

rrel

atio

n (%

)

Mar05Mar06Mar07Mar08

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CDX base correlations, granular model

Sep05 Mar06 Sep06 Mar07 Sep070

20

40

60

80

100B

ase

corr

elat

ion

(%)

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CDX base correlations, large pool model

Mar05 Sep05 Mar06 Sep06 Mar07 Sep070

20

40

60

80

100B

ase

corr

elat

ion

(%)

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CDX base correlations, large pool model, with DJXoption-implied correlation

Mar05 Sep05 Mar06 Sep06 Mar07 Sep070

20

40

60

80

100B

ase

corr

elat

ion

(%)

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CDX Series 4-7, GR model

4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

Detachment plus index EL (%)

Bas

e co

rrel

atio

n (%

)

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CDX Series 4-7, LP model

4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

Detachment plus index EL (%)

Bas

e co

rrel

atio

n (%

)

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CDX Series 8-9, GR model

4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

Detachment plus index EL (%)

Bas

e co

rrel

atio

n (%

)

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CDX Series 8-9, LP model

4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

Detachment plus index EL (%)

Bas

e co

rrel

atio

n (%

)

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Time series properties

Examine the correlation between various implied correlationsand the index.

We would like this to be low. Why?

For risk, we have identified idiosyncratic risk correctly.

For hedging, we have captured what we are able to from theunderlying.

Start with statistics on daily changes.

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CDX 0-3%, large pool model, base correlations

4 5 6 7 8 9 All0

10

20

30

40

50

60

Series

Cor

rela

tion

to in

dex

(%)

Flat curveFull curve

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CDX 0-3%, base correlations

4 5 6 7 8 9 All−20

−10

0

10

20

30

40

50

60

Series

Cor

rela

tion

to in

dex

(%)

Large poolGranular

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CDX 0-3%, large pool model

4 5 6 7 8 9 All−60

−40

−20

0

20

40

60

Series

Cor

rela

tion

to in

dex

(%)

Base CorrATM Corr

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CDX 0-3%

4 5 6 7 8 9 All−30

−20

−10

0

10

20

30

40

50

60

Series

Cor

rela

tion

to in

dex

(%)

LP, baseGran, ATM

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CDX 3-7%

4 5 6 7 8 9 All−20

−10

0

10

20

30

40

50

60

Series

Cor

rela

tion

to in

dex

(%)

LP, baseGran, ATM

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Time series properties

Now look at statistics on weekly changes.

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CDX 0-3%, large pool model, base correlations

4 5 6 7 8 9 All−60

−40

−20

0

20

40

60

80

Series

Cor

rela

tion

to in

dex

(%)

Flat curveFull curve

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CDX 0-3%, base correlations

4 5 6 7 8 9 All−60

−40

−20

0

20

40

60

80

Series

Cor

rela

tion

to in

dex

(%)

Large poolGranular

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CDX 0-3%, large pool model

4 5 6 7 8 9 All−80

−60

−40

−20

0

20

40

60

80

Series

Cor

rela

tion

to in

dex

(%)

Base CorrATM Corr

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CDX 0-3%

4 5 6 7 8 9 All−80

−60

−40

−20

0

20

40

60

80

Series

Cor

rela

tion

to in

dex

(%)

LP, baseGran, ATM

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CDX 3-7%

4 5 6 7 8 9 All−60

−40

−20

0

20

40

60

80

Series

Cor

rela

tion

to in

dex

(%)

LP, baseGran, ATM

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Time series conclusions

Looks like granular model produces “more idiosyncratic”implied correlations.

The ATM approach looks appears to improve things, butseems to overcorrect in the most volatile periods.

High correlations overall suggest index moves may have somepredictive power for implied correlations.

Differences are much more pronounced at a one-day horizonthan a one-week horizon.

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Hedging experiment

On day zero, we have all prices (spreads and tranches) plushistory.

At a future date, we are told how the underlying spreads(index or single-name CDS) have moved, and are asked toguess what the new tranche prices should be.

Compare the predicted tranche price moves to the actualones, over time and across different modeling approaches.

Another approach would be to look at delta hedgeperformance, but this mixes in the error from linearization.

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Hedging experiment candidates

Models

Regression—fit linear relationship between tranche pricechanges and index changes, using prior six months of data.Large pool model—calibrate based on current index levels.Granular model—calibrate based on current single-name CDSlevels.

Correlation approaches

Base—use most recent calibrated correlation.ATM—move along the most recent correlation structureaccording to change in the index-implied expected loss.Regression—fit linear relationship between base correlationchanges and index changes, using prior six months of data.

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CDX 0-3%, Regression

Sep05 Mar06 Sep06 Mar07 Sep070

10

20

30

40

50

60

70

80

Tra

nche

pric

e (%

)

−8

−4

0

4

8

Pre

dict

ion

erro

r (%

)

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CDX 0-3%, LP model, base correlations

Sep05 Mar06 Sep06 Mar07 Sep070

10

20

30

40

50

60

70

80

Tra

nche

pric

e (%

)

−8

−4

0

4

8

Pre

dict

ion

erro

r (%

)

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CDX 0-3%, Regression

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

Tranche price change (%)

Pre

dict

ed c

hang

e (%

)

Ser 4Ser 5Ser 6Ser 7Ser 8Ser 9

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CDX 0-3%, LP model, base correlations

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

Tranche price change (%)

Pre

dict

ed c

hang

e (%

)

Ser 4Ser 5Ser 6Ser 7Ser 8Ser 9

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Statistics on hedging approaches

Examine standard deviation of tranche forecast error.

Daily horizon

Bars are for correlation approaches: Base (blue), ATM(green), Regression (red).

Curve is for simple regression model.

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CDX 0-3%, LP model

4 5 6 7 8 9 All0

0.005

0.01

0.015

0.02

0.025

0.03S

TD

of p

redi

ctio

n er

ror

Series

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CDX 0-3%, GR model

4 5 6 7 8 9 All0

0.005

0.01

0.015

0.02

0.025

0.03S

TD

of p

redi

ctio

n er

ror

Series

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CDX 3-7%, LP model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series47

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CDX 3-7%, GR model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series48

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CDX 7-10%, LP model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series49

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CDX 7-10%, GR model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series50

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Statistics on hedging approaches

Examine correlation of tranche forecast error with actualtranche move.

Daily horizon

Bars are for correlation approaches: Base (blue), ATM(green), Regression (red).

Curve is for simple regression model.

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CDX 0-3%, LP model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 0-3%, GR model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 3-7%, LP model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 3-7%, GR model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 7-10%, LP model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 7-10%, GR model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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Statistics on hedging approaches

Examine standard deviation of tranche forecast error.

Weekly horizon

Bars are for correlation approaches: Base (blue), ATM(green), Regression (red).

Curve is for simple regression model.

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CDX 0-3%, LP model

4 5 6 7 8 9 All0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045S

TD

of p

redi

ctio

n er

ror

Series

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CDX 0-3%, GR model

4 5 6 7 8 9 All0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045S

TD

of p

redi

ctio

n er

ror

Series

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CDX 3-7%, LP model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series61

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CDX 3-7%, GR model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series62

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CDX 7-10%, LP model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series63

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CDX 7-10%, GR model

4 5 6 7 8 9 All0

0.5

1

1.5

2

2.5

3

3.5x 10

−3S

TD

of p

redi

ctio

n er

ror

Series64

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Statistics on hedging approaches

Examine correlation of tranche forecast error with actualtranche move.

Weekly horizon

Bars are for correlation approaches: Base (blue), ATM(green), Regression (red).

Curve is for simple regression model.

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CDX 0-3%, LP model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

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CDX 0-3%, GR model

4 5 6 7 8 9 All−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0C

orr(

Act

,Err

)

Series

67

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CDX 3-7%, LP model

4 5 6 7 8 9 All−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1C

orr(

Act

,Err

)

Series

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CDX 3-7%, GR model

4 5 6 7 8 9 All−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1C

orr(

Act

,Err

)

Series

69

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CDX 7-10%, LP model

4 5 6 7 8 9 All−1

−0.5

0

0.5C

orr(

Act

,Err

)

Series

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CDX 7-10%, GR model

4 5 6 7 8 9 All−1

−0.5

0

0.5C

orr(

Act

,Err

)

Series

71

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Conclusions

Overall, hedging errors are surprisingly large.

For the equity tranche, a simple regression works quite well,with its underhedging problems reduced at a slightly longerhorizon.

There is a benefit to using the granular model, but is it worththe cost?

For more senior tranches, the ATM approach appears tocapture some of the link to credit spreads, but does notmarkedly reduce the hedging error.

Any candidate for a new standard model should be able to dobetter in this experiment.

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Further investigations

Can we use these results to learn more about how our modelmight be misspecified?

Richer correlation structure?Different copula?Better dynamics?

How much worse (or better) are compound correlations?

Can the regression be improved by including the HiVol index?

Can the LP model be improved by adding a simple correctionfor heterogeneity?

Does any of this change at or close to defaults?

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Further reading

Couderc, F. (2007) Measuring Risk on Credit Indices: On theUse of the Basis. RiskMetrics Journal.

Finger, C. (2004) Issues in the Pricing of Synthetic CDOs.RiskMetrics Journal.

Finger, C. (2005) Eating our Own Cooking. RiskMetricsResearch Monthly. June.

All are available at www.riskmetrics.com

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