Thesis Document "Parsimomious Estimation of Default Probabilities from Credit Default Swaps and...

PARSIMONIOUS ESTIMATION OF DEFAULT PROBABILITIES

FROM CREDIT DEFAULT SWAPS AND BONDS

Submitted in partial fulfillment of the

Master in Business Administration

Business Administration

Marketing and Communication

Athens University of Economics and Business

Manolis Kavussanos, Professor (supervisor)

George Karathanassis,

Panagiotis Diamantis, Associate Professor

1920


FROM CREDIT DEFAULT SWAPS AND BONDS

By

GIANNIS ALEXAKIS

Submitted in partial fulfillment of the

requirements for the

Master in Business Administration

Departments of

Business Administration

Marketing and Communication

Athens University of Economics and Business

January 2007

Guidance Committee:

Manolis Kavussanos, Professor (supervisor)

George Karathanassis, Professor

Panagiotis Diamantis, Associate Professor


2

ABSTRACT

Last decade, the global Credit Derivatives Market has grown rapidly, reaching

recently the 26 trillion dollars in value. A size almost ten times its corresponding value

in 2002. The reasons of this tremendous growth could be discovered in the need of

Financial Organizations to secure their investments against credit risk and the very

high leverage Credit Derivatives offer to their counterparties. The most important

financial products in the Market are Credit Default Swaps, covering almost 63% of the

total market value.

Within the scope of this Master Thesis we investigated whether Credit Risk is

evaluated at the same level in the Credit Default Swaps and the Fixed Income

Securities Markets. Assuming that investors seek returns proportional to the risk they

assume (in our case mainly Credit Risk), the CDS premiums should reflect the

investors assessments regarding the Default Probabilities of each underlying

Obligation. Based on the CDS premiums and Bond prices we able to extract implied

Bond Default Probabilities in order to compare the Credit Risk reflected in the returns

of both Markets.

For this purpose we developed a method and an integrated Information System

based on MATLAB programming environment which allows us to analyse data from the

Credit Default Swap and the Fixed Income Market. Using our system we compared the

Default Probabilities implied Credit Default Swaps and Bonds of 27 Companies in the

U.S.A. We found that the best method for extracting Default Probabilities from Bonds

is based on the Z-spread measure. The deviations of the Default Probabilities implied

from the Bond Spreads and the CDS premiums, were at a satisfactory low level in all

cases, which shows that both Markets are affected by Credit Risk in a similar way.

Furthermore, we purpose a new method for computing the Basis between the Markets

using the implied Credit Spread and CDS curves which is more efficient than directly

comparing the Credit Spreads of each Bond and the corresponding CDS premium at a

similar maturity.

, ()

,

,

1920

2007

:

, ()

,

,

4

(Credit

Derivatives) ,

26 ( 1 2006)

2002.

,

.

(Credit Default Swaps)

63% Credit Derivatives.

,

Credit Default Swaps .

(

), CDS

(bond default probability).

CDS ,

,

(

Credit Default Swaps). ,

Credit Default Swaps

() MATLAB.

,

(implied default probabilities) Credit Default Swaps 27

...

Default Probabilities Z-spread.

Default Probabilities CDS premiums,

,

.

Basis

Credit Spread CDS,

Credit Spreads

CDS premium .

5

.

.

.

.

.

.

.

.

6

1. ........................................................................ 8

1.1 ....................................... 8

1.2 ............................................................................... 13

2. .................... 16

2.1 STRUCTURAL FORM MODELS .................................................................... 16

2.2 REDUCED FORM MODELS ......................................................................... 17

3. ........................ 22

3.1 (YIELD TO MATURITY) ............................................... 22

3.2 (CREDIT SPREAD) ................................................. 23

3.3 I-SPREAD ............................................................................................... 24

3.4 ASSET SWAP SPREAD .............................................................................. 25

3.5 -SPREAD .............................................................................................. 28

3.6 ........................... 29

3.7 RISK NEUTRAL REAL WORLD ............................................ 33

3.8 ....................................... 34

4. CREDIT DEFAULT SWAPS .............................................. 36

4.1 .................................................................................. 36

4.2 .............................................................. 38

4.3 CDS KAI ............................................ 39

4.4 DISCOUNTED SPREADS MODEL ................................................................. 40

4.5 JP MORGAN MODEL ................................................................................. 41

4.6 MODIFIED HULL WHITE MODEL ................................................................. 43

4.7 CDS ................................... 45

4.8 CDS CDS BASIS ................................ 47

5. SPREAD .................. 51

5.1 .......................................................... 51

5.2 SPLINES .................................................................................. 54

5.3 NELSON-SIEGEL ...................................................................... 55

7

6. .......................................... 58

6.1 .............................. 58

6.2 CREDIT SPREAD ......................................................... 61

6.3 SPREAD .................................................... 68

6.4 KAI .................................... 70

7. .................................. 73

7.1 ........................................................................... 73

7.2 CREDIT SPREAD ................................. 75

7.3 DEFAULT PROBABILITIES ....................................... 90

8. ........................................................... 98

................................................................................. 100

....................... 105

8

1.

1.1

(Credit

Derivatives) ,

26 (1

2006) 2002 [1].

,

...

( ).

1.1. Credit Derivatives - ISDA Survey 2006

(Credit Derivatives)

/ - . Credit

Derivatives

.

,

. (Credit Events)

( ) :

, , .

(restructuring) .

632

12,430

17,096

26,006

0

5,000

10,000

15,000

20,000

25,000

30,000

1H01 2H01 1H02 2H02 1H03 2H03 1H04 2H04 1H05 2H05 1H06

(. US $)

(2001 - 2006)

9

.

(credit spread)

.

(credit rating)

.

(assets)

.

.

Credit Derivative,

(Credit Default Swaps),

,

.

Credit Derivative , /

(

).

(Index Credit Default Swaps),

(Credit Options),

(Credit Linked Notes Total Return

Swaps).

(Collaterized Debt Obligations Baskets).

(Credit Default Swaps) 63%

Credit Derivatives (Single Name CDS Index CDS).


. Collaterized Debt Obligations

.

.

10

1.2. Credit Derivatives (% )

- Survey 2006

, (

Hedge Funds) Credit Derivatives.

. ,

/ ,

, .

market makers ,

/ (bid

ask spread). (long position)

. (short position)

. .

(. 1.4, 1.5).

1.3. /

Credit Derivatives, Global Credit Derivatives Survey 2006, Fitch Ratings Ltd,

Single-name CDS, 32%

CDS Indices, 29%

Collaterized Debt

Obligations, 16%

Tranched Index Swaps, 7%

Credit Linked Notes, 3%

Baskets, 2%Others, 11%

Morgan Stanley Credit Suisse F Societe GeneraleDeutsche Bank BNP Paribas CalyonGoldman Sachs Merrill Lynch Royal Bank of ScotlandJP Morgan Chase Bear Stearns AIGUBS Bank of America CommerzbankLehman Brothers Dresdner HVBBarclays ABN Amro IXISCitigroup HSBC CIBC

Royal Bank of Canada

11

1.4. - Long Credit Derivatives

(% ) - BBA Survey 2006

1.5. - Short Credit Derivatives

(% ) - BBA Survey 2006

Credit Derivatives

. Options (

/

),

. Credit

Derivatives (

). ( )

38%

10%

7%

3%

16%

15%

4%

4%

2%

1%

51%

16%

16%

3%

3%

2%

2%

3%

3%

1%

12

Credit Derivatives

(bellow

Investment Grade) [5].

Credit Derivatives

.

Credit Derivatives

.

1.6. Credit Derivatives-

Global Credit Derivatives Survey 2006, Fitch Ratings Ltd

1.7. Credit Derivatives

Global Credit Derivatives Survey 2006, Fitch Ratings Ltd

H

. ,

.

(Over The Counter)

AAA11% AA

6%

A23%

BBB29%

31%

AIG Ford Motor Corp./Ford Morgan StanleyAltria Group France Motor Credit Co.AT&T Corp. France Telecom PhilippinesBank of America Freddie Mac PortugalBBVA Gazprom RussiaBombardier General Electric/GECC SuezBrazil General Motors/GMAC Telecom ItaliaDaimlerChrysler Germany TelefonicaDeutsche Bank Goldman Sachs Time WarnerDeutsche Telekom Italy TurkeyEastman Kodak Japan United Mexican StatesFannie Mae JP Morgan Chase Volkswagen

13

(Index Products). Credit

Derivatives CreditEx

CreditTrade. brokers ABN Amro,

Barclays Capital, BNP Paribas, Deutsche Bank, JPMorgan, Morgan Stanley UBS.

Dow Jones CDX Dow Jones iTraxx Europe.


(Dow Jones CDX) , (Dow

Jones iTraxx Europe).

1.8. (% . ) - Survey 2006

1.2

.

(Single Name Credit Default Swaps)

(32%). ,

.

,

,

Credit Default Swaps .

(

43%

39%

/

10%

8%

14

Hull, Predescu, White (2004), Longstaff (2004) )

Credit Default Swap premiums Bond Credit Spreads.

Credit Default Swap

(Houweling, Vorst (2001)).

, 27 ... 223

108 CDS 1/1/2004 1/1/2007.

default probabilities

,, ,


CreditDefaultSwaps

(CDS spread),

1.9.

,

(

), CDS

(bond

default probability). CDS ,

default,

( Credit Default Swaps). ,

CDS

()

MATLAB,

.

, , 2

.

. , 3

15

(spread)

(-spread, Z-spread, Asset Swap Spread)

(risk neutral default probabilities). Credit Spread

. ,

4 Credit Default Swaps

, (CDS

premiums) default probabilities. 5

Nelson Siegel. 6

CDS . ,

(,

..) default probabilities CDS

. 7

CDS

. 8

. ,

(source code) .

16

2.

(default).

(

)

.


.

defaults

.

(Historical Real World Default Probabilities).

default. defaults

(, , ..)

.

S&P, Fitch Moodys

(credit rating). reports

default .

default

probabilities

. Default Probabilities

.

2.1 STRUCTURAL FORM MODELS

(, ,

)

. Robert Merton 1974

[34]

, default

.

(contingent claims) (assets)

(

17

Black-Scholes [3] options).

Black, Cox (1976) [2] Geske (1977) [14].

Jones, Mason Rosenfeld (1984) [23] Merton

( )

1977-1981. Moodys KMV

Vasicek

[40] Kealhofer [26][27]. ( Expected Default Frequency

). Longstaff Schwartz (1995) [29]

. Collin-Dufresne Goldstein (2001) [7]

Vasicek-Kealhofer . Lyden Saraniti (2000) [30]

Eom, Helwege, Huang (2003) [13] structural based models.

Bohn (2000) [5] Agrawal, Arora, and Bohn (2004) [1]

Vasicek-Kealhofer

.

2.2 REDUCED FORM MODELS

reduced form ( intensity-based),

default

- .

.

, (credit spread)

, (risk neutrality)

arbitrage .

,

, default probabilities.

.

reduced form default probabilities

Credit Derivatives.

Credit Derivatives

90 .

18

hot topic

.

,

default probabilities (Schnbucher

(1997) [37][1][B8], Duffle Singleton (1999) [12], Madan and Unal (1999) [31]).

, Credit Derivatives default

probabilities .

Jarrow Turnbull (1995) [21]

, defaults

. Jarrow and Turnbull

default non default ( Black Scholes

Options) o

.


.

defaults default

(Recovery Rate)

.

Bloomberg JP Morgan.

Hull and White (2000) [20]

default

CDS (counterparty default risk). Premiums

Recovery Rate

default. Philip

Schnbucher (2000) [1] Credit Derivatives,

Heath-Jarrow-Merton [15].

Credit Derivatives default probabilities ,


(par CDS spreads) Credit Default Swaps.

JP Morgan Hull-White .

,

Credit Derivatives .

19

, .

Credit

Derivatives. implied default probabilities

Credit Default Swaps.

Patrick Houweling .. [17][18]


Credit Derivatives. ,

,

par CDS spread Credit Spread

.

CDS spread default probabilities

CDS spread . 225

1999

2001.

John Hull, Mirela Predescu,

Alan White (2004) [19]

CDS premiums.

. risk

free rate CDS

10bps Swap Rates.

(Credit Ratings)

CDS .

o Francis Longstaff .. (2004) [28]

Spreads .

(Credit Spread) (

) Spreads

. CDS

CDS

.

Blanco (2005) [4] CDS

CDS .

credit spread

CDS spread

-CDS .

20

CDS

. Blanco (2005) [4], Zhu (2006) [44], Norden

Weber (2004) [36] 5

CDS, Credit Spread

CDS premium. O De Wit (2006) [10]

3/5 10 , Levin, Perli Zakrajsek

(2005) [43] Spline CDS spreads

CDS premium

.

(223 reference entities 108 CDS)

Credit Spreads CDS

premiums 10 .

2.1.

Blanco,

Brennan and

Marsh (2005)

Levin, Perli

and Zakrajsek

(2005)

Norden and

Weber (2004) Zhu (2006) De Wit (2006)

DATASET

CDS Term 5 1/2/3/5/7/10 5 5 3/5/10

Period

02/01/2001 to

20/06/2002

02/01/2001 to

01/09/2005 2000-2002

01/01/1999 to

31/12/2002

01/01/2004 to

30/12/2005

# reference entities 33 306 58 24 103

# contracts 33 1290 58 24 144

reference entities

type

IG Corporates

IG/HY

Corporates (US-

USD only)

IG (+HY)

Corporates IG Corporates

IG/HY Corporates

+ EM Sovereigns

METHODOLOGY

Spread estimation

Interpolation

bond spreads to

CDS term

Spline estimate

CDS curve,

match to bond

term

Interpolation

bond spread to

CDS term

Interpolation /

Matching bond

spread to CDS

term

Interpolation /

Matching bond

spread to CDS

term

Long-term

relationship Cointegration / Cointegration Cointegration Cointegration

RESULTS

Basis

+6 bp. (mean) 0 bp. (median), -

2 bp. (mean) +14 bp. (mean) +13 bp. (mean)

+7 bp. (median),

+16 bp.(mean)

Long-term

relationship

26 out of 33

cointegrated

(unrestricted)

/

36 out of 58

cointegrated

(unrestricted)

15 out of 24

cointegrated

(restricted)

88 out of 144

cointegrated

(restricted)

Price discovery

CDS tends to

lead bonds

/ - Mainly

idiosyncratic

CDS tends to

lead bonds

CDS tends to lead

bonds in US, not

elsewhere

/

21

IG (Investement Grade), HY (High Yield) (Emerging

Markets)

.

( Basis

- CDS premiums Credit Spreads).

default.

CDS

.

Working Papers [24][25]

International Monetary Fund

. , CDS

reduced form .

, ,

spreads CDS

Asset Swap [2] . .

Asset Swap

CDS .

, reduced form models

.

. ,

default

.

22

3.

reduced form models,

default probabilities.

(term to maturity)

( /

).

(embedded options)

(

)

( , )

.

default .

.

3.1 (YIELD TO MATURITY)

(Yield To Maturity)

. Yield to Maturity

(

, ) (Net Present

Value) .

( P , C

, , tn )

23

1

1

(1 ) (1 )

n N

N

n Nn

NYTM t YTM t

n

C MP

YTM YTM

P Ce Me

=

=

= ++ +

= +

3.1. (Premium Discount)

To YTM ,

.

.

3.2 (CREDIT SPREAD)

(risk free) (credit spread)

(risk free)

.

.

Hull, Predescu White (2004) [19]

Treasury Rates

.

Treasury Bills Bonds.

.

105 10%

0 1 2 3 -105 10 10 110

92.13 5%

0 1 2 3 -92.13 5 5 105

2 3

10 10 110105 8.06%

(1 ) (1 ) (1 )YTM

YTM YTM YTM= + + =

+ + +

2 3

5 5 10592.13 8.06%

(1 ) (1 ) (1 )YTM

YTM YTM YTM= + + =

+ + +

24

Treasury Bills ...

.

Swap

Rates LIBOR

Treasury Bills. (credit spread)

.

3.3 I-SPREAD

To I-Spread ( Interpolated Spread Yield

Spread) Yield to Maturity , Swap

Rate . Swap

( ) (

3.5 3 4 ).

( 5).

3.2. -Spread

-Spread :

Credit Spread

.

.

105 10%

0 1 2 3Swap Rates 0.5% 1% 2% -105 10 10 110

92.13 5%

0 1 2 3Swap Rates 0.5% 1% 2% -92.13 5 5 105

3 8.06% 2% 6.06%I YTM SwapRate= = =

3 8.06% 2% 6.06%I YTM SwapRate= = =

25

3.4 ASSET SWAP SPREAD

Asset Swap

.

Swap (Interest Rate Swap Currency Swap).

: H T-Bank

(Floating Rate Note) Risk Free

EuroAutos AG.

,

EuroAutos AG, Interest Rate Swap

LIBOR. LIBOR

.

-Bank

-Bank YZ Bank

-Bank EuroAutos AG

LIBOR + s LIBOR + s

3.3. Asset Swap

Asset Swap

( 100 )

(dirty price). dirty price.

s RiskFree

-Bank .

Asset Swap

.

26

Asset

Swap.

( 100 ).

( ) BondFairValue.

H

100= +in FairValuePV Bond

rf

.

s

Asset Swap.

Asset Swap

.

(Pdirty)

().

H

( ) ( ) (100)= + + +out dirtyPV PV rf PV s PV P

100 ( ) ( ) (100)+ = + + +FairValue dirtyBond PV rf PV s PV P

( Risk Free)

100 ( ) (100)= +PV rf PV

( ) ( )= + = FairValue dirty FairValue dirtyBond PV s P PV s Bond P

s (annuity)

( ) ( )= PV s s PV Annuity

Asset Swap (Asset Swap Spread).

( )

= FairValue dirty

Bond Ps

PV Annuity

sset Swap Spread Zero Coupon

Swap Rates Swap Rates

-spread.

27

3.4. Asset Swap Spread

Asset Swap

(interest rate risk)

(.. LIBOR).

Asset Swap (

). ,

. Asset Swap spread

.

H Asset Swaps

. Asset Swap spread ,

Z-spread . Z-spread

. discount/premium Asset

Swap

(upfront difference).

105 10%

0 1 2 3Risk Free Rates 0.50% 1% 2% -105 10 10 110

92.13 5%

0 1 2 3Risk Free Rates 0.50% 1% 2% -92.13 5 5 105

2 3

2 3

10 10 110123.41

(1 0.5%) (1 1%) (1 2%)

1 1 1( ) 2.92

(1 0.5%) (1 1%) (1 2%)

123.41 1056.31%

2.92

= + + = + + +

= + + =+ + +

= =

FairValueBond

PV Annuity

ASWspread

2 3

2 3

5 5 105108.82

(1 0.5%) (1 1%) (1 2%)

1 1 1( ) 2.92

(1 0.5%) (1 1%) (1 2%)

108.82 92.135.72%

2.92

= + + = + + +

= + + =+ + +

= =

FairValueBond

PV Annuity

ASWspread

28

LIBOR + s,

.

, Asset Swaps

(Asset Swap Currency Swap). T-Bank

$

EuroAutos AG.

Asset Swaps

Interest Rate Swap.

3.5 -SPREAD

-Spread ( Zero Volatility Spread Static Spread)

Risk Free

, (Net Present Value)

.

Risk Free

. Risk Free Zero Coupon (

Swap Rates).

3.5. -Spread

Z ( P

, C , , tn , rfn

risk free )

105 10%

0 1 2 3Risk Free Rates 0.50% 1% 2% -105 10 10 110

92.13 5%

0 1 2 3Risk Free Rates 0.50% 1% 2% -92.13 5 5 105

2 3

5 5 10592.13 6.12%

(1 0.5% ) (1 1% ) (1 2% )= + + =

+ + + + + +Z

Z Z Z

2 3

10 10 110105 6.17%

(1 0.5% ) (1 1% ) (1 2% )= + + =

+ + + + + +Z

Z Z Z

29

1

( ) ( )

1

(1 ) (1 )

n n N N

N

n Nn n N

Nrf Z t rf Z t

n

C MP

rf Z rf Z

P Ce Me

=

+ +

=

= ++ + + +

= +

-Spread ,

I-Spread

. -spread

Swap Rates -spread Zero Coupon

Swap Rates.

3.6

default

probabilities .

. zero

coupon 100 risk free

( 1 ) rf.

( q),

R ,

.

.

100

PV(Bond)

1-q

q

R

3.6. default

PBond.

.

30

[ ]( ) (1 ) 100(1 ) 100

1

100 (1 )

(100 )

Bond

Bond

PV Bond PV q R q

q R qP

rf

P rfq

R

= +

+ =

+

+=

H zero

coupon. 95.40

1% 2% ( ).

default 40.

PV(Bond)

40

1011-qx

qx

3.7. Default probabilities

101(1 ) 4095.40 6.06%

1 2%

x x

Bond x

q qP q

+= = =

+

().

PV(Bond)

100+C

C

R(100+C)

q(0,0,1)

R(100+C)

R(100+C)

q(0,1,k)

q(0,1,2)

C

3.8. default

31

C

100 R

default survival ( default ). default cash flows

.

3.9.

defaults

( ). Hull White

(2000) [20] .

default

q(0,k,k+1) risk neutral default

k k+1 ( default )

Q(0,Tk) risk neutral default

k

p(0,k,k+1) risk neutral survival

k k+1 (=1- q(0,k,k+1))

P(0,Tk) risk neutral survival

k (=1- Q(0,Tk))

B(0,t) spot t

k Bayes

1

1 1

( ) (0, 1)(0, , 1)

( ) (0, )

(0, ) (0, 1, ) [1 (0, 1, )]

k k

k

T T

k k

P survival survival P kp k k

P survival P k

P T p k k q k k

+

= =

++ = =

= =

default.

spot

Survival Default1 C R(100+C)2 C R(100+C)3 C R(100+C)

. Tn 100+C R(100+C)

32

1

1

( ) (0, ) (0, ) 100 (0, ) (0, )

(100 ) (0, ) (0, 1, ) (0, 1)

n

n

T

n n

i

T

i

PV Bond C B i P i B T P T

R C B i q i i P i

=

=

= +

+ +

( , )( )( , ) k kR t T T tk

B t T e =

R(t,Tk) t,Tk (

forward). S(t,Tk) credit spread t Tk

.

( , )( )

( , )( )

( , ) ( , )

( , )

( , )

k k

k k

S t T T t

k k

S t T T tk

k

V t T B t T e

V t Te

B t T

=

=

{ }(0,1) (0,1) [1 (0,0,1)] (0,0,1)V B q Rq= + credit spread

(0,1)

(0,1)1

1(0,1)(0,0,1)

1 1

S

V

eBq

R R

= =

{ }(0,2) (0, 2)

[1 (0,0,1)][1 (0,1,2)] [1 (0,0,1)] (0,1,2) (0,0,1)

V B

q q q q R q R

=

+ +

q

(0,1) 2 (0,2)

(0,1)(0,1,2)

S S

S

e eq

e R

=

n

(0, ) (0, 1)( 1)

(0, )(0, , 1)

S k k S k k

S k k

e eq k k

e R

+ +

+ =

(0, )1(0, )

1

S T TeQ T

R

=

33

Jarrow

Turnbull (1995) [21]

defaults.

.

(fair value)

default

probabilities ( )

.

default probabilities.

(risk neutral) default probabilities

structural based model


.


.


(Credit Default Swaps)

.

.

3.7 RISK NEUTRAL REAL WORLD

default

risk neutral

default.

risk neutral .

3.10. Real World Risk Neutral Default Probabilities ( bps)

Real-world default Risk-neutral default Ratio Difference intensity per yr (bps) intensity (bps)

Aaa 4 67 16.8 63 Aa 6 78 13.0 72 A 13 128 9.8 115 Baa 47 238 5.1 191 Ba 240 507 2.1 267 B 749 902 1.2 153 Caa and Lower 1690 2130 1.3 440

Rating

34

(John Hull, Mirela Predescu, and Alan

White (2005) [19]) risk neutral real world

. real

world risk neutral (ratio)

(

).

3.8

default

R .

(structured form

models) (reduced form models).

(. 3.11)

(Aaa Aa)

recovery rates

35% 45%.

Aaa a recovery rates ,

default 5

.

Aaa

(downgrade) default.

3.11. Default Recovery Rate 1980-

2006 Moodys Investor Services

40% .

CDS,

Recovery Rate

default 1 2 3 4 5Aaa 97 74.1Aa 95.4 62.1 30.8 55.3 41.6A 46.4 54.9 50.3 47.7 48.4Baa 48.1 46.4 47.3 44.1 41.5Ba 40.1 39.2 38.6 42.4 44.3B 36.6 35.3 37.3 38.2 41.3Caa-Ca 30.4 29.7 32.9 39.2 34.7

35

(. 3.6, 4.7)

Recovery Rate.

(Houweling P. and Vorst T. (2001) [17], Manning M J (2005) [32], (John Hull, Mirela

Predescu, and Alan White (2005) [19]) Recovery Rate

default

probabilities. Recovery Rate

.

risk neutral historic recovery rates

.

36

4. CREDIT DEFAULT SWAPS

4.1

(Credit Default Swaps Single

Name) . CDS

(Protection Buyer)

(Protection Seller). -

( )

( 1,3,5,7 10 ).

(CDS premium)

(basis points) (notional value).

par CDS spread. O

(default)

(Reference Entity).

default

.

( ):

(physical settlement). O

( default

)

CDS.

(deliverable

obligation).

(cash settlement).

.

CDS

.

,

default .

default .

(short position)

.

37

(long position).

.

4.1. ,

International Swaps and Derivatives

Association, [2][3].

( )

4.2. CDS

: -Bank

A-Insurance 10.000.000

EuroAutos AG. 5

(CDS spread) 160 .

. -

Bank -Insurance 10.000.000 x 160/4 x 0.01%=

40.000 . 4 ( 1 )

EuroAutos AG o

30% (3.000.000). -Insurance T-Bank

10.000.000 EuroAutos AG

3.000.000 .

CDS (cash outflow) (fixed) SHORT (floating) LONG

38

A-Insurance T-Bank

10.000.000 - 30.000.000 = 7.000.000.

(Recovery Rate).

EuroAutos AG

A-Insurance

100 - Recovery

Rate

T-Bank

3m 6m 9m 12m

40bps 40bps 40bps 40bps

4.3.

X

(CDS spread)

. ,

default 1

,

CDS spread

40.000 / 3 13.333

.

4.2

Credit Default Swap

(default).

39

, .

CDS

.

:

,

,

.

CDS

( BLOOMBERG).

.

,

.

CDS.

/ CDS (offsetting position)

.

,

.

Market-Makers

(bid / offer spread).

CDS

, .

4.3 CDS KAI

CDS


.

default (counterparty credit risk).

(default)

40

. CDS

arket Makers

CDS

bid/ask spread .

,

default

(Credit Default Swap spread)

. default probabilities

,

CDS

.

4.4 DISCOUNTED SPREADS MODEL

(Discounted Spreads Model)

. CDS

(settlement date)

.

(CDS spread). :

PV(CDS)

CDS spread1-q

q

-(1-R)

4.4. Credit Default Swap

CDS

default . CDS

spreadold . default ( 1-q)

. default ( q)

R

.

41

(1-R) .

CDS spreadnew.

(offsetting position) CDS spreadnew

.

spreads

.

rf, 100 CDS

:

100( ) ( )

(1 )old new

new

PV CDS CDS spread CDS spreadrf CD spread

= + +

spreads

CDS spreadnew

. CDS spreadold=100bps, CDS spreadnew=300bps, rf=2%

1( ) (100 300 ) 100 1.90

(1 2% 300 )PV CDS bps bps

bps= =

+ +

1.9 100

.

, CDS spread

(settlement date).

,

CDS ,

.

4.5 JP MORGAN MODEL

JP Morgan Chase & Co [B4], Jarrow Turnbull

(1995) [21] . CDS default

probabilities . CDS

(fee leg)

default (contingent leg).

SN credit spread N

42

0,1

DFi (discount factor) 0 Ti

PNDi (cumulative probability) default 0

Ti

i i-1

Ti,( i - Ti-1 )

R (%)

default

Contingent Leg ( default

PNDi-1-PNDi i-1 Ti

discount factor).

( )11

( ) (1 )N

i i i

i

PV Contigent R DF PND PND=

=

Fee Leg ( CDS

spread SN bps i

discount factor).

1

( )N

N i i i N N

i

N

PV Fee S DF PND S Annuity

Annuity

=

= = 144424443

CDS

default Fee Leg

CDS spread (AccrualN). i/2 default

Ti-1,i CDS .

PNDi-1-PNDi default Ti-1,i.

( )11 1

( )2

( )

N Ni

N i i i N i i i

i i

N N

N N N N

PV Fee S DF PND S DF PND PND

Annuity Accrual

PV Fee S Annuity S Accrual

= =

= +

= +

144424443 1444442444443

discounts factors

.

( )1

1

rf i

i iiDF DF e

rf

= =+

43

CDS spread spread

( AccrualN). :

( )

( )

1

1

1

1

( ) ( )

(1 )

(1 )

N

i i i N N N N

i

N

i i i

iN

N N

PV Contigent PV Fee

R DF PND PND S Annuity S Accrual

R DF PND PND

SAnnuity Accrual

=

=

=

= +

=

+

4.6 MODIFIED HULL WHITE MODEL

To Modified Hull-White Model Bloomberg

Hull White (2000) [20].

P(0,T) 1

( zero coupon

)

CR(T)

q(t) default t

Q(t) default t

0

( ) ( )T

Q T q t dt=

R default

AI(t) 1%

1 1,

( )0,

i i i

i

t t t t tAI t

t t

<

44

1

0

( ) [1 ( )] (0, )

( ) ( )[100(1 )] (0, )

m

j j j

j

T

PV Fee S Q t P t t

PV Contigent q t R P t dt

=

=

=

CDS legs.

0.

1 0

1

0

( ) ( )

[1 ( )] (0, ) ( )[100(1 )] (0, )

[1 ( )] (0, )

( )[100(1 )] (0, )

Tm

j j j

j

m

j j j

j

T

PV Fee PV Contigent

S Q t P t t q t R P t dt

Q t P t t

S

q t R P t dt

=

=

=

=

=

CDS

.

1 0

0

( ) [1 ( )] (0, ) ( ) ( ) (0, )

( ) ( )[100 [100 ( ) ( )]] (0, )

Tm

j j j

j

T

R

PV Fee S Q t P t t q t AI t P t dt

PV Contigent q t R AI t C T P t dt

=

= +

= + +

0.

0

1 0

( ) ( )

( )[100 [100 ( ) ( )]] (0, )

[1 ( )] (0, ) ( ) ( ) (0, )

T

R

Tm

j j j

j

PV Fee PV Contigent

q t R AI t C T P t dt

S

Q t P t t q t AI t P t dt =

=

+ +

=

+

(JP Morgan Hull-White)

. Hull-White

Recovery Rate

.

defaults JP Morgan defaults

. P(0,T) Hull-White discount factors

45

.

CDS

/ .


CDS.

Recovery Rate ,

.

4.7 CDS

4.4

CDS JP Morgan ( accruals).

CDS 100 , CDS spreadold=100bps, CDS

spreadnew=300bps, rf=2%, R=40% PD default ,

(

)

1( ) 100 (1 )

1

1( ) 100 (1 )

1

(1 )

3004.76%

300 (1 0.4)

1- 1- 4.76% 95.24%

new

new

new

new

PV Contigent R PD CDS spreadrf

PV Fee PD CDS spreadrf

CDS spreadPD

CDS spread R

bpsPD

bps

PND PD

= +

= +

= +

= =+

= = =

100bps

1( ) 100 (1 0.4) 4.76% 2.8

1 2%

1( ) 100 95.24% 100 0.93

1 2%

( ) ( ) ( ) 0.93 2.8 1.87

PV Contigent

PV Fee bps

PV CDS PV Fee PV Contigent

= = + = =+

= = =

-1.87 .

discounted spreads model -1.9,

+0.03.

46

CDS 100

, .

A ( )

(i=1)

( ) ( )

( )

1 1

1 1 1

1

1 1

( ) ( )

(1 )2

1(1 )

2

N N NN

i i i N i i i i i

i i i

N N

N N

N i i i N i i

i i

PV Contigent PV Fee

SR DF PND PND S DF PND DF PND PND

Annuity Accrual

R S DF PND PND S DF PND

= = =

= =

=

= +

=

14243 14444244443

A PNDi (

PNDi-1 )

1

1

1 1 1

(1 )

1(1 )

2

1(1 )[ ]

21

((1 ) )2

i i i i i i

i

i i

RPND

s R

R s d DF PND s c

PND

R s DF

=

+

+ =

+

1

1 1

(1 )

i

i x x

x

i i

i x x x i

x x

c DF PND

d DF PND DF c

=

= =

=

= =

PND1PNDN

default . default

PDi=1 PNDi.

default i .

11

(1 )

1

i

i

sPD

R

+

=

47

default probabilities (P Def)

JP Morgan. H P Def APROX default probabilities

. (error)

5 .

4.5. Default Probabilities CDS premiums

JP Morgan ( Hull-White)

CDS

, default

probabilities CDS, implied default

probabilities

(Credit Default Swaps) .

(fair value)

.

CDS

. , default probabilities

.

4.8 CDS CDS BASIS

CDS

CDS premium Credit Spread .

CDS Basis CDS premium Credit Spread.

, Credit Spread

Asset Swap spread, arbitrage (

) Asset Swap Credit Default

Swap.

Time RiskFree PND

bps= Years 1 Dfi Ci diff Di P Def error P Def APROX

24 1 3.0% 0.996 0.971 0.967 0.004 0.004 0.004 1.6E-06 0.004

35 2 3.1% 0.988 0.941 1.897 0.008 0.011 0.012 6.2E-05 0.012

50 3 3.2% 0.975 0.910 2.784 0.013 0.023 0.025 3.2E-04 0.025

55 4 3.3% 0.963 0.878 3.630 0.012 0.033 0.037 4.4E-04 0.036

60 5 3.3% 0.950 0.850 4.438 0.013 0.045 0.050 6.4E-04 0.049

70 6 3.4% 0.930 0.818 4.232 0.020 0.057 0.070 0.001 0.068

80 7 3.4% 0.918 0.791 4.029 0.012 0.059 0.082 -0.009 0.090

90 8 3.5% 0.903 0.759 3.827 0.016 0.059 0.097 -0.018 0.115

100 9 3.5% 0.881 0.734 3.627 0.022 0.065 0.119 -0.023 0.143

105 10 3.5% 0.867 0.709 3.434 0.014 0.064 0.133 -0.032 0.165

CDS

48

Floating Rate Note (

) LIBOR + X basis points.

FRN Credit Default

Swap . To CDS premium

LIBOR+Y.

LIBOR ( ).

(

).

(100 100) ( ) ( )LIBOR X LIBOR Y X Y + + =

default

CDS .

( default )

- .

arbitrage , spread X

. Asset Swap

FRN .

CDS premium Asset Swap spread

.

CDS

default,

.

default

cheapest to

deliver option .

Asset Swaps

Repo (Repurchase

Agreement) LIBOR

haircut. CDS

(unfunded transactions). Repo

basis.

CDS .

liquidity premium

.

49

default .

CDS

default.

CDS

(counterparty risk)

.

()

Asset Swap

Credit Spread.

CDS CDS premium

.

4.6. Basis 2032 Time Warner

CDS, 2004 2007

Basis

( +100 -100 bps )

.

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-100

-50

0

50

100

150

50

Basis

CDS premium Z-spread. -spread ,

, .

,

CDS

Basis. Basis ,

CDS premiums Bond Yields.

default

CDS,

.

51

5. SPREAD

Credit Spreads

.

Credit Spreads ( )

Credit Spread

.

Credit Spread (

)

.

(term structure of interest rates)

Credit Spreads.

.

Vasicek (1982) [41], Cox-Ingersoll-Ross (1985) [9] Heath-

Jarrow-Morton (1992) [16].

.

.

, Credit Spreads

, .

.

.

.

5.1

( ). zero

coupon

( ).

5.1.

5.2. ,

( ).

zero coupon .

. Risk Free,

. ,

( ).

.

,

. ,

( )

(.

3

.

1 2 3

1 2 3( ) 1d t a t a t a t

=

1 ( 0) .

.

.

(yield curve) (

forward (forward curve

( )

5.2).

3

1 2 3

1 2 3d t a t a t a t=

1 ( 0) .

.

. To

) (zero coupon

curve) .

( )

3

1 ( 0) .

.

coupon curve)

5.2 SPLINES

Splines

.

Mason Chan (1978) [38], Vasicek

Splines n- (

polynomial estimation)

n-1 .

(

n-

knot points

(Cubic) Splines

points

7 term structure

1 2 3( )i i i i ir t a b t c t d t= + + +

5.3. Cubic Spline

.

Splines

McCulloch (1971)

Vasicek Fong (1982) [41] .

(

) n-

1 .

(knot points),

n-1 .

points. Splines

knot points

Splines 8

7 term structure

1 2 3

i i i i ir t a b t c t d t= + + +

. Cubic Spline

.

(1971) [33], Suits,

.

(piecewise

1 .

),

1 .

points.

8 knot

7 term structure

. Cubic Spline

55

Splines

.

knot points

.

0

term structure . knot points

.

Splines knot points

.

5.3 NELSON-SIEGEL

Nelson Siegel [35] 1987.

( ) .

0 1 2

1 1( )

m mm

t

e eR m e

m m

= + +

m ( ) , ,0,1,2 .

yield

2 2

1 1

min ( ) min ( )n n

opt i i i opt i ib b

i i

b w P P b y y

= =

= =

(

) , term

structures S (S-humped).

0 () yield m

1 0.

1>0 1

2 .

( )

5.4. Nelson Siegel 0,1,2,

5.5.

. 2 >0

) 2 0

. Nelson Siegel 0,1,2,

57

5.5

2.9x10-6. Nelson Siegel

term structures.

Splines.

knot points .

zero coupon

(instantaneous) forward rates

0 1 2( )

m m

t

mf m e e

= + +

forward rate 0 m .

Credit Spread (

).

Central bank

Estimation method

Minimised error Shortest maturity in estimation

Relevant maturity spectrum

Belgium Svensson or Nelson-Siegel Weighted prices

Treasury certificates: few days Bonds: one year days to 16 years

Canada Merrill Lynch Exponential Spline Weighted prices

Bills: 1 to 12 months Bonds: 12 months 3 months to 30 years

Finland Nelson-Siegel Weighted prices 1 day 1 to 12 years

France Svensson or Nelson-Siegel Weighted prices

Treasury bills: all Treasury Notes: : 1 month Bonds: : 1 year Up to 10 years

Germany Svensson Yields > 3 months 1 to 10 years

Italy Nelson-Siegel Weighted prices

Money market rates: O/N and Libor rates from 1 to 12 months Bonds: 1 year Up to 30 years

Japan Smoothing splines Prices 1 day 1 to 10 years

Norway Svensson Yields Money market rates: 30 days Bonds: 2 years Up to 10 years

Spain Svensson or Nelson-Siegel

Weighted prices Prices 1 day Up to 10 years

Sweden Smoothing splines and Svensson Yields 1 day Up to 10 years

Switzerland Svensson Yields Money market rates: 1 day Bonds: 1 year 1 to 30 years

UK VRP Yields 1 week Up to around 30 years

USA Smoothing splines prices 30 days Up to 1 year

5.6.

Bank for International Settlements 2005

[12]. Svensson

Nelson Siegel ( 6 4).

.

58

6.

CDS

:

.

(implied risk free spot rate) .

(credit spread) .

Nelson Siegel -spread -

(I-spread, Asset

Swap spread). Z-spread

.

(credit

spread).

CDS spread.

CDS spread .

.

CDS.

CDS

.

6.1

.

Swap Rates

. ... (Treasury

Bills),

3.2 (

treasury bills - , ).

swap

LIBOR 1 10

59

12,15,20,25 30 .

3,6

9 . o

bid ask .

sset Swap Spread -spread

Zero Coupon ( I-spread ).

zero coupon

. rf

, :

1

(1 )TDF

rf=

+

Zero Coupon

bootstrapping

Swap Rates.

swap ( )

swap rate. ,

swap

rates .

swap rates

(zero coupon payments).

(Law of One Price)

arbitrage (

)

. zero

coupon

zero coupon . ,

Swap rates (

) 1 4.

z2, z3, z4 zero coupon .

zero coupon

.

Swap 1 1.50%2 2.00%3 2.50%4 3.00%

2 102100

1 1.5% (1 )

2.5 2.5 102.5100

1 1.5% (1 2.01%) (1 %)

3 3 3 103100

1 1.5% (1 2.01%) (1 2.52%) (1 )

z= +

+ +

= + ++ + +

= + + ++ + + +

6.1

swap

(

).

6.2.

Swap Zero Coupon 1.50% 1.50%2.00% 2.01%2.50% 2.52%3.00% 3.04%

2

2

2 3

3

2 3 4

4

2 102

1 1.5% (1 )

2.5 2.5 102.5

1 1.5% (1 2.01%) (1 %)

3 3 3 103

1 1.5% (1 2.01%) (1 2.52%) (1 )

z

z

z

+ +

= + ++ + +

= + + ++ + + +

Zero Coupon

zero coupon

swap rates. zero coupon

(

).

spot .

. Spot Swap Rates

1 1.5% (1 2.01%) (1 2.52%) (1 )

coupon

(

).

.

Rates (US $)

6.2 CREDIT SPREAD

)

Credit Spread.

3

Swap spread.

6.3.

(Ford Motor

6.4.

Maturity Date

(dd-mmm

25

15

15

01

28

01

15

15

01

25

01

CREDIT SPREAD

(

)

.

3 -spread, Zero Spread

.

27 2005, 100)

.

Maturity Date

mmm-yy) Coupon Clean Price 25-Jan-07 6 1/2 100.2 15-Jun-07 7.2 100.5 15-Jan-08 4.95 94.301-Oct-08 5 5/8 93.528-Oct-09 7 3/8 96.101-Dec-09 5.8 93.415-Jan-10 5.7 90.815-Jun-10 7 7/8 96.601-Feb-11 7 3/8 94.625-Oct-11 7 1/4 93.701-Oct-13 7 92.7

(

)

.

Spread Asset

(

Ford (27/5/2005)

.

.

( )

. 3

:

Nelson

Swap Rates

.

Credit Spread .

).

(

,

Swap

( 6.3)

(27/5/2005)

.

.

)

. 3

Nelson Siegel Yields

(

Credit

6.5. Nelson Siegel

.

(

). Spread

(

, ).

Rates

)

(27/5/2005)

.

. Credit Spread

)

. 3

( -spread).

Credit Spread

default

.

(

(

).

Nelson

Swap 0

(-spread) .

6.6. -spread

Yields Zero

Risk Free

.

Spread .

6.7.

Term To Maturity

I-spread (bps)

Asset Swap spread (bps)

Zero spread (bps)

Nelson Siegel Bond Yields

0 0+ 1 = SwapCurve

) .

spread Nelson

I

Coupon

Free .

Spreads

Nelson Siegel

. 0+ 1=0 => 0= -1

.

. Credit Spread

1.7 2.1 2.6 3.3 3.6 4.4 4.6 5.1

229 283 319 363 367 407 380 430 415

86 178 234 262 271 294 294 313

231 285 316 361 365 408 377 430

SwapCurve0

Nelson Siegel

I-spread

Credit

5.7 6.4 8.3

415 406 365

315 346 353

415 407 366

6.8

6.9. Asset Swap

8. I-Spread, Nelson Siegel

Asset Swap Spread, Nelson Siegel

Nelson Siegel

6.10

spread

.

.

, o

Z

.

,0 2 (1=-

0 2

1 1( ) 1

t t

e eZ t e

t t

= +

1

min ( ) n

opt i i ib

i

b w P P

=

=

10. -Spread, Nelson Siegel

.

.

, o -spread

Z(t) Nelson

.

-0) Z(t).

0 2

1 1

t tt

e eZ t e

t t

= +

2min ( ) opt i i ib w P P=

-

.

spread

Nelson Siegel

.

Pi (clean

( ( )) ( ( ))

1

n n n N N N

Nrf Z t t rf Z t t

i

n

P Ce Me

+ +

=

= + wi (

duration

( N o ).

1

1

= 1

i

i N

j j

dw N

d=

durations (

).

1

nj j

i

j i

C td

P==

,

Credit spread

clean price) .

( ( )) ( ( ))n n n N N Nrf Z t t rf Z t tP Ce Me + += +

(durations)

duration

).

(C , P , t

6.11. Z-spread

,

) .

)

,

.

Credit Spread ( )

.

.

Z-spread

. 3.55

3.55 11 0.17$

default 1 10.

Nelson Siegel

0.

6.12. CDS premiums

Term To Maturity (years)CDS premium (bps)

.

( )

.

.

.

. 3.55

3.55 11 0.17$=

CDS premiums

1 10.

CDS premiums Ford Motor (27/5/2005)

6.13. Credit Default Swap

Term To Maturity (years) 1 3 5 10CDS premium (bps) 153 351 408 421

.

( )

.

.

.

1 10.

Ford Motor (27/5/2005)

10421

68

6.3 SPREAD

Nelson Siegel

.

MATLAB

( ).

( )( )( )

( )

22

1

2

3

min ( ) ( )

( )

...

iix

N

F x f x

f x

f x

F x f x

f x

=

=

F(x)

Credit Spread ( CDS premium)

Credit Spread ( CDS spread)

( ).

1 1 1 1

2 2 2 2

3 3 3 3

( ) ( )

... ...

N N N N

P P S S

P P S S

F x F xP P S S

P P S S

= =

P1,P2, PN S1,S2, SN

Credit Spread (I-spread, Asset

Swap spread, Z-spread).

,

( ( )) ( ( ))

1

n n n N N N

Nrf Z t t rf Z t t

i

n

P Ce Me

+ +

=

= +

69

0 2

1 1( ) 1

t tt

e eZ t e

t t

= +

Credit

Spread

0 2

1 1( ) 1

t tt

e eS t e

t t

= +

0,2

22min ( ) ( )iix

F x f x= (0,2 )

Credit Spread ( 0% 100%).

( - +).

( lsqnonlin)

.

(Trust

Region Method for Non Linear Optimization).

.

Preconditioned Conjurate Grandients.

.

. Coleman

Li (1996) [6], Verma Coleman (1998) [42].

70

6.4 KAI

Credit Spread Nelson Siegel


3.6.

(0, )1(0, )

1

S T TeQ T

R

=

S(t,T) credit spread t T

Q(0,T) risk neutral default

R

default ( 40%)

Credit Spreads Q(0,T).

CDS premiums 1,3,5 10 .

Nelson Siegel CDS

spread .

default 1 10 JP Morgan

1

1

1 1 1

(1 )

1(1 )

2

1(1 )[ ]

21

((1 ) )2

i i i i i i

i

i i

RPND

s R

R s d DF PND s c

PND

R s DF

=

+

+ =

+

1

1 1

(1 )

i

i x x

x

i i

i x x x i

x x

c DF PND

d DF PND DF c

=

= =

=

= =

default probabilities CDS premiums PD=1-PND

default 0

.

.


.

.

6.14.

6.15.

( 6.15


default probability Curve

Morgan ( ).

0.

par CDS spreads .

Patrick Houweling .. [17][18]

Years 1 2Error (%) -0.35 -0.68 -0.73


. JP Morgan

probabilities 1 10

.

.

. Default probabilities

. Default probabilities

15) default probability

probabilities 1 10 CDS

CDS

( ).

0. Z-spread

.

[18] .

CDS spreads

3 4 5 6 7 8-0.73 -0.50 -0.15 0.18 0.39 0.43

probabilities

Morgan

1 10

.

curve

CDS

JP

CDS bps.

.

.

9 100.27 -0.08

Z-spread (1 3 ) .

JP Morgan

default

Basis CDS

6.16

) .

CDS

CDS .

16. Z-spread CDS bps

) .

bps

73

7.

7.1

default Credit Default Swaps .

CDS ... ( CDS

39% 43%).

4 CDS ( 1, 3, 5 10 )

4 27

. 223 108 CDS.

Senior Unsecured

(Moodys).

7.1.

Company Sector

Number

of Bonds

Credit

Rating

1 Boeing 7 A2

2 General Dynamics 4 A2

3 Goodrich 4 Baa3

4 Lockheed 5 Baa1

5 Northrop 4 Baa2

6 Ford Motor 14 B3

7 General Motors 13 B3

8 Cinergy 8 Baa2

9 Constellation Energy 8 Baa1

10 Devon Energy 6 Baa2

11 Exelon 18 Baa2

12 Progress Energy 12 Baa2

13 Xcel Energy 8 Baa1

14 Clear Channel Communications 8 Baa3

15 Comcast Cable 22 Baa2

16 Cox Communications 9 Baa3

17 Disney 5 A2

18 Time Warner 9 Baa2

19 Anheuser Busch 10 A2

20 Cargill 5 A2

21 Kraft 7 A3

22 Philip Morris Tobacco 6 A1

23 Albertsons 7 B1

24 Federated Department Stores 5 Baa1

25 Kroger 11 Baa2

26 AT&T 10 A2

27 New Cingular Wireless Services 4 A3Communications

Energy

Aerospace -

Defense

Cars

Media

Food

Retail

74

1/1/2004 1/1/2007 (

4 ).

BNP

.

callable (convertible

option) .

default

Nelson Siegel - Credit Spread

I-spread .

Nelson Siegel - Credit Spread

Asset Swap spread,

.

Nelson Siegel - Zero Spread

.

Nelson Siegel CDS premiums

, .

CDS Recovery Rate = 40%.

CDS

3,5 7 .


. 3

CDS Credit Spread

0 ,

. 7

CDS spread .

75

7.2 CREDIT SPREAD

, default probabilities

CDS 3, 5 7 .

default 20 bps

bid ask spread CDS. 20

bps 1% 1.6% 2.3% 3 5 7

.

Basis (. 4.8).

/

CDS ,

(I-spread, Asset Swap spread, Z-spread).

.

20bps .

default .

27 Z-spread 21

-spread 6. Asset Swap spread

default probabilities (

20bps )

.

Basis.

I-spread -spread ,

3 (General Motors, Ford Motor, Phillip Morris)

. General Motors Ford Motor

CDS premiums ( 100 bps 700bps

)

.

76

7.2. -spread

7.3. -spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4Boeing

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method

Asset Swap spread method

Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3GeneralDynamics

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

77

7.4. -spread , ASW spread

7.5. -spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5

6

7Goodrich

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4Lockheed

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

78

7.6. -spread

7.7. Z-spread , ASW

spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3Northrop

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-4

-2

0

2

4

6

8

10

12FordMotor

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

79

7.8. Z-spread ,

ASW spread

7.9 Z-spread , ASW

spread

2 3 4 5 6 7 8-4

-2

0

2

4

6

8

10

12GeneralMotors

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4Cinergy

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

80

7.10. -spread

7.11. -spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4ConstellationEnergy

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4DevonEnergy

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

81

7.12. -spread

7.13. -spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5Exelon

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5ProgressEnergy

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

82

7.14. -spread ,

7.15. -spread , ASW

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4XcelEnergy

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5

6

7ClearChannelCommunications

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

83

7.16. -spread

7.17. -spread , ASW

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4ComcastCable

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5CoxCommunications

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

84

7.18. -spread

7.19. -spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5Disney

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5TimeWarner

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

85

7.20. -spread ,

7.21. -spread

2 3 4 5 6 7 8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5AnheuserBusch

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3Cargill

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

86

7.22. -spread

7.23. -spread , , ASW

spread

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3Kraft

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-4

-2

0

2

4

6

8PhilipMorris

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

87

7.24. -spread , ASW spread

7.25. -spread

2 3 4 5 6 7 8-4

-2

0

2

4

6

8Albertsons

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5FederatedDepartmentStores

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

88

7.26. -spread

7.27. -spread ,

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4

5Kroger

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

2 3 4 5 6 7 8-3

-2

-1

0

1

2

3

4AT&T

% e

rror

(Defa

ult P

rob im

plie

d f

rom

CD

S -

Bonds)

Time (years)

I-spread method


Z-spread method

89

7.28. -spread ,

2 3 4 5 6 7 8-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5NewCingularWirelessServices

% e

rror

(Defa

ult P

rob im

plied f

rom

CD

S -

Bonds)

Time (years)

90

7.3 DEFAULT PROBABILITIES

implied default

probabilities CDS 5 .

0

(Boeing, Goodrich, Cinergy).

implied default probabilities (

) Cingular Wireless Services, Cox Communications

implied default probabilities CDS (Xcel Energy, General Dynamics).

Ford Motor General Motors

( 2 2006) 5% .

CDS spread

. , implied default

probabilities

CDS

4.8.

(arbitrage).

Moodys.

long term credit rating Exelon

credit ratings CDS

( ).

7.29. Exelon Credit Ratings

( Ford Motor, General Motors),

.

7.30.

implied default probabilities

Basis

, 2006,

.

Moody's Credit Rating

Long Term Baa2

Bond Implied A3

CDS Implied Baa1

Correlation Coefficient

General Motors Ford Motor 0.753

Comcast Cable Time Warner 0.864

Cargill Kraft 0.762

Kroger Albertsons 0.797

91

7.31. default probabilities 5 Boeing, General

Dynamics

7.32. default probabilities 5 Goodrich,

Lockheed

02-Jan-2004 02-Jan-2005 03-Jan-2006 04-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

Boeing

02-Jan-2004 02-Jan-2005 03-Jan-2006 04-Jan-2007-1

0

1

2

3

CD

S-B

ond P

rob(%

)

GeneralDynamics

5 years

5 years

22-Nov-2004 07-Aug-2005 22-Apr-2006 05-Jan-2007-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

Goodrich

02-Jan-2004 02-Jan-2005 03-Jan-2006 04-Jan-2007-2

-1

0

1

2

3

CD

S-B

ond P

rob(%

)

Lockheed

5 years

5 years

92

7.33. default probabilities 5 Northrop, Ford

Motor

7.34. default probabilities 5 General Motors,

Cinergy

04-Aug-2005 24-Jan-2006 16-Jul-2006 05-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

Northrop

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-10

-5

0

5

10

CD

S-B

ond P

rob(%

)

FordMotor

5 years

5 years

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-5

0

5

10

CD

S-B

ond P

rob(%

)

GeneralMotors

04-Jan-2004 04-Jan-2005 05-Jan-2006 06-Jan-2007-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

Cinergy

5 years

5 years

93

7.35. default probabilities 5 Constellation

Energy, Devon Energy

7.36. default probabilities 5 Exelon,

Progress Energy

04-Jan-2004 04-Jan-2005 05-Jan-2006 06-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

ConstellationEnergy

10-Feb-2004 29-Jan-2005 18-Jan-2006 07-Jan-2007-3

-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

DevonEnergy

5 years

5 years

07-Mar-2004 15-Feb-2005 26-Jan-2006 06-Jan-2007-2

0

2

4

6

CD

S-B

ond P

rob(%

)

Exelon

04-Jan-2004 04-Jan-2005 05-Jan-2006 06-Jan-2007-2

0

2

4

CD

S-B

ond P

rob(%

)

ProgressEnergy

5 years

5 years

94

7.37. default probabilities 5 XcelEnergy,

Clear Channel Communications

7.38. default probabilities 5 Comcast Cable,

CoxCommunications

04-Jan-2004 04-Jan-2005 05-Jan-2006 06-Jan-2007-1

0

1

2

3

CD

S-B

ond P

rob(%

)

XcelEnergy

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

ClearChannelCommunications

5 years

5 years

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

ComcastCable

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-4

-2

0

2

CD

S-B

ond P

rob(%

)

CoxCommunications

5 years

5 years

95

7.39. . default probabilities 5 Disney, Time

Warner

7.40. default probabilities 5 Anheuser

Busch, Cargill

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-2

-1

0

1

2

3

CD

S-B

ond P

rob(%

)

Disney

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-3

-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

TimeWarner

5 years

5 years

03-Mar-2004 12-Feb-2005 24-Jan-2006 05-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

AnheuserBusch

15-Jun-2004 23-Apr-2005 01-Mar-2006 07-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

Cargill

5 years

5 years

96

7.41. default probabilities 5 Kraft, Philip

Morris

7.42. default probabilities 5 Albertsons,

Federated Department Stores

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

Kraft

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-10

-5

0

5

10

CD

S-B

ond P

rob(%

)

PhilipMorris

5 years

5 years

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-6

-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

Albertsons

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-3

-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

FederatedDeptartmentStores

5 years

5 years

97

7.43. default probabilities 5 Krogen

Company, AT&T, New Cingular Wireless Services

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-4

-2

0

2

4

CD

S-B

ond P

rob(%

)

Kroger

5 years

11-Aug-2004 31-May-2005 20-Mar-2006 07-Jan-2007-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

AT&T

5 years

01-Jan-2004 02-Jan-2005 04-Jan-2006 06-Jan-2007-3

-2

-1

0

1

2

CD

S-B

ond P

rob(%

)

NewCingularWirelessServices

5 years

98

8.

default CDS

. JP Morgan

CDS

Credit Spread .

I-spreads

,

Nelson Siegel-Credit Spread

default

Jarrow-Turnbull.

Z-spread Nelson Siegel

,

Credit Spread.

Credit Spread

Asset Swaps

. Asset

Swap

.

default 27

.

20bps

O Basis

100bps. Nelson Siegel

Spreads ,

CDS default

probabilities. Credit CDS Spread

CDS.

T

,

CDS .

CDS

.

99

default

probabilities . ,

. risk neutral

default probabilities real-world probabilities

VaR (Value at Risk) .

( loomberg Reuters)

CDS

.

100

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Thesis Document "Parsimomious Estimation of Default Probabilities from Credit Default Swaps and...

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