A Study of Sellers of Senior Tranched Credit Protection, Jon Gregory, London 8 th July 2008 1 A...

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A Study of Sellers of Senior Tranched Credit Protection, Jon Gregory, London 8 th July 2008

A Study of Sellers of Senior Tranched Credit Protection

Jon Gregory

Quant Congress USA

July 8th 2008

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Related Papers

“Credit Tails”, Risk article, January 2008

“A Trick of the Credit Tail”, Risk technical paper, March 2008

“A Free Lunch and the Credit Crunch”, to be published August 2008

“Two Faced Over Counterparty Credit Risk?”, working paper

Please email me at [email protected] for copies of these articles

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The Failure of the Gaussian Copula Model

MarketGaussian

Copula Model

24.00% 19.3%

82.5 234.7

26.5 82.0

14.0 32.9

8.75 6.99

3.53 0.05

• Dependency is defined by a single correlation parameter

– No concept of idiosyncratic default

– No concept of systemic default

Super Senior 22-100%

Equity 0-3%

3-6%6-9%

12-22%

9-12%

4


CDO Investor Landscape

[0-3%]

[3-7%]

[7-10%]

[10-15%]

[15-30%]And [30-100%]

Rating of counterparty

Risk of underlying

Hedge funds

Aaa rated Monolines

and CDPCs no CSA

Real money investors

under CSA

NR

Baa

Aa

Aaa

SuperAaa

5


The Correlation Skew in Credit


Equity 0-3%

3-6%6-9%

Index[0-100%} 12-22%

9-12%

5Y iTraxx Europe

(Investment Grade Corporate Index)

110 bps

[0-3%] 35%

[3-6%] 465 bps

[6-9%] 335 bps

[9-12%} 250 bps

[12-22%] 110 bps

[22-100%] 40 bps

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

0% 5% 10% 15% 20% 25%

Detachment point

Bas

e C

orr

elat

ion

6


Into the Super Senior Space


Equity 0-3%

3-6%

6-9%

12-22%

9-12%

Each default causes loss of (1-)/125 = 0.48% (40% rec)

22% of losses requires 22% / 0.48% = 46 defaults

5Y [22-100%] pays around 50 bp pa !

-0.100%

0.000%

0.100%

0.200%

0.300%

0.400%

0.500%

0.600%

0.700%

0.800%

19-Mar-07 08-May-07 27-Jun-07 16-Aug-07

22-100%

Long protection on 7Y [22-100%]

(Delta hedged)

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The Toothpaste Tube Analogy (I)


Equity 0-3%

3-6%

6-9%

Index[0-100%]

12-22%

9-12%

Index = Sum of tranches

[22-100%] = [0-100%] – [0-3%] – [3-6%] – [6-9%] – [9-12%] – [12-22%]

• Typically method for extracting super senior

premium implicitly assumes flat correlation curve

up to 5Y

2Y 5Y

[0-3%] 13.52% 25.75%

[3-6%] 3.3 bps 60.5 bps

[6-9%] -1.0 bps 19.5 bps

[9-12%] -0.9 bps 11.0 bps

[12-22%] -0.5 bps 6.0 bps

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Summary of Super Senior Pricing

• We can’t price [22-60%]

• We can’t price tranchelets in the [22-60%] region

• We can’t price bespoke super senior

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The Toothpaste Tube Analogy (II)• Small changes in equity duration assumptions can change the size of the tube

• Upper bound where [22-100%] is equal to [12-22%] / Lower bound where [22-100%] is worth zero

• Change in 3Y equity premium (or IO/PO prices) moves super senior significantly

0%

10%

20%

30%

40%

50%

0 1 2 3 4 5

Maturity (years)

Eq

uit

y T

ran

che

Exp

ecte

d L

oss

[22-100%] = 4.1 bps [22-100%] = 2.5 bps

Upper bound

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Extrapolation of Base Correlation Curve

• Even if we can price the [22-100%]

– Pricing of 1% wide tranchelets in the [22-60%] region

– Clearly not arbitrage-free

– No value put in the [1-, 100%] part of the capital structure ( is 40% in this case)

0

2

4

6

8

10

12

14

16

18

20

22% 27% 32% 37% 42% 47% 52% 57%

Detachment point

Tra

nch

elet

pre

miu

m (

bp

s)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

Bas

e C

orr

elat

ion

Tranchelet Premium Base Correlation Market Points

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Pricing of Bespoke Super Senior• Forced index tranchlets to be arbitrage-free (value distributed equally within [22-60%])

• On a portfolio which is 14% more risky, common scaling techniques introduce arbitrage

• On a more risky portfolio this potentially becomes worse

0

2

4

6

8

10

12

14

16

22.0% 27.0% 32.0% 37.0% 42.0% 47.0% 52.0% 57.0%

Detachment point

Tra

nch

elet

pre

miu

m (

bp

s)

index

bespoke

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Packaging Super Senior Risk

• Leveraged super senior (LSS) trades

– A leveraged transaction with a defined trigger event

• Credit derivative product companies (CDPC)

– Like a more complex LSS

– Trigger is not so well defined

– CDPC has multiple counterparties

• Monoline insurer

– Similar to CDPC but will insurer other financial risks also

– Should benefit from diversification

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Leveraged Super Senior Illustration

× Leverage

• Super senior tranche pays small premium

• Leverage this (typically around 10 times)

• Create a structure paying a ‘good’ premium referenced to a default remote

tranche

• Investor loss will come when structure deleverages

– Investor may have option to post more collateral

– Alternatively an unwind will occur

Collateral

UncollateralizedPortion

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Rating Super Senior Risk

• Packaging super senior tranches can always be made to look

like triple-A counterparty risk

Super senior tranche(Triple-A)

Packaged Super senior tranche (at least Triple-A)

Capital (share) of monoline / CDPC etc

()

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LSS Unwind Triggers• Loss only trigger

– Simple to rate (default risk only, like a CDO tranche)

– Issuer retains spread and implied correlation risk

• Spread (+ loss) triggers

– More complex to rate (includes spread dynamics in rating model)

– Issuer retains implied correlation risk

• Market value triggers

– Very hard to rate (only one rating agency DBRS have ever done so)

– Issuer retains only gap risk

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Leveraged Super Senior Pricing Argument

• There is around $200+ billion of leveraged super senior protection out there

• Issuers seem to price via the equivalence argument

• Advantage is that we simple argue that gap risk is very small

– Example : at 10 times leverage, a market value trigger of 5% is ‘safe’

• This is a standard argument applied to lots of leveraged products (CPPI etc, ….)

• This argument is wrong for LSSLSS Protection value = SS Protection value Leverage – Gap Risk

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LSS Protection Valuation Bounds

Perfect

trigger

No trigger

“Gap Risk”

[A, B] Upper Bound

LSS Value??

× L

[A, A+] Lower Bound

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LSS Pricing

Denote

– trigger time by

– initial collateral by

T

t

BAs sdMstBE )(),(1 ,

Trigger ?

T

t

BAT VtBE )),(min(),(1 ,

No

Yes

Standard tranche protection

Settle contract with cap at collateral value

leverageAB /)(

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LSS Pricing (II)

T

t

BAT

T

t

BAT

T

t

BAs

T

t

AAs

T

t

BAT

T

t

BAs

VtBE

VtBE

sdMstBE

sdMstBE

VtBEsdMstBE

))()(,(1

)(),(1

)(),(1

)(),(1

)),(min(),(1)(),(1

,

,

,

,

,,

Collateralised Protection

De-leverage value

Unwind value

Gap option

= 0

Even if client has option to deleverage then it is suboptimal

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Formal Valuation

T

t

BAT

T

t

AAs

T

t

BAT

T

t

BAT

T

t

AAs

VtBEsdMstBE

VtBE

VtBE

sdMstBE

)),(min(),(1)(),(1

))()(,(1

)(),(1

)(),(1

,,

,

,

,

• Setting de-leverage term to zero

Lower Bound Trigger Option

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Leveraged Super Senior Overview

[A, B] Upper Bound

[A, A+] Lower Bound

Perfect trigger

No trigger

“Gap Risk”

“Trigger Option”

Actual LSS Value

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Pricing the Trigger Option

),(1)),(min(),(1 , tBEVtBE T

T

t

BAT

• Pricing (and hedging!) the trigger option is not easy

– Out of the money tranche option payoff

• But we can get some insight in the loss only trigger case

– Here we can overhedge with a contract paying at the loss trigger K

– This is simply a digital CDO tranche

• This effectively sets a maximum leverage independent of gap risk

AB

tV

tVtleverage

BA

KK

)(

)()(

,

,*

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Example

• For the (5Y) tranche markets as shown

• We calculation the maximum leverage for

– A [22-100%] LSS structure

– 5-year maturity

– With a loss trigger as a function of loss trigger level

4th May 2005 19th March 2007 4th December 2007

Index 44 25 53[0-3%] 29.00% 11.87% 25.05%[3-6%] 168 54.5 156.3[6-9%] 49 14.8 85[9-12%] 25 6.8 60[12-22%] 16 2.8 34[22-100%] 5.5 2.4 15.0

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Maximum Leverage

0

5

10

15

20

25

30

35

40

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Loss Trigger

Max

imu

m L

ever

age

4th May 2005 19th March 2007 4th December 2007

• Leverage Super Senior Pricing is not a gap risk problem!

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Monolines and CDPCs

• Capital model

– (Tranche) Ratings driven run on a daily basis

– Capital (less any losses) must be above capital required by model to

support Aaa rating

• Operating modes (CDPC / monoline)

– Normal operating mode / Triple-A rating

– Suspension mode / downgrade

– Wind-down mode / Run-off

Required capital

Aaa attachment

Ratings driven loss distribution

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Monoline / CDPC Comparison with LSS

NormalState

RestrictedState

Wind-downState

Trigger Not Hit

< 6 months

Losses to tranches

TriggerHit

Full Deleverage(unwind)

Losses to tranches

Monoline / CDPC Leverage Super Senior

De-

leve

rage

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CDPC / Monoline Purchased Protection

[A, B] Upper Bound

[A, A+] Lower Bound for LSS

“Counterparty Risk”

No value

Impact of effective leverage

A Study of Sellers of Senior Tranched Credit Protection, Jon Gregory, London 8 th July 2008 1 A...

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