Chapter 13

Modeling the Credit Spreads Dynamics

FIXED-INCOME SECURITIES

Outline

• Analyzing Credit Spreads– Ratings– Probability of Default– Severity of Default

• Modeling Credit Spreads– Structural Models– Reduced-Form Models– Historical versus Risk-Adjusted Default Probabilities

Analyzing Credit SpreadsCorporate Bonds

• Bonds issued by a corporation • Typically pay semi-annual coupons• 3 Sources of Risk

– Interest Rate Risk– Default Risk– Liquidity Risk

• Bond indenture contracts stipulate collateral and specify terms

• Different “seniority” classes

Analyzing Credit Spreads Bond Quality

• Standard & Poor, Moody’s and other firms score ‘the probability of continued & uninterrupted streams of interest & principal payments to investors’

• Classes of grades– Moody’s Investment Grades: Aaa,Aa,A,Baa– Moody’s Speculative Grades: Ba, B, Caa, Ca, C– Moody’s Default Class: D

• Are ratings agencies better able to discern default risk or simply react to events?

Analyzing Credit Spreads Bond Quality Ratings

Moody's S&P DefinitionAaa AAA Gilt-edged, best quality, extremely strong creditworthiness

Aa1 AA+Aa2 AA Very high grade, high quality, very strong creditworthinessAa3 AA-A1 A+A2 A Upper medium grade, strong creditworthinessA3 A-Baa1 BBB+Baa2 BBB Lower medium grade, adequate creditworthinessBaa3 BBB-

Moody's S&P DefinitionBa1 BB+Ba2 BB Low grade, speculative, vulnerable to non-paymentBa3 BB-B1 B+B2 B Highly speculative, more vulnerable to non-paymentB3 B-

CCC+Caa CCC Substantial risk, in poor standing, currently vulnerable to non-payment

CCC-Ca CC May be in default, extremely speculative, currently highly vulnerable to non-paymentC C Even more speculative

D Default

The modifiers 1, 2, 3 or +, - account for relative standing within the major rating categories.

Investment Grade (High Creditworthiness)

Speculative Grade (Low Creditworthiness)

Analyzing Credit Spreads Moody’s and S&P Rating Transition (US, end of 1999)

S&P AAA AA A BBB BB B CCC DAAA 91.94% 7.46% 0.48% 0.08% 0.04% 0.00% 0.00% 0.00%AA 0.64% 91.80% 6.75% 0.60% 0.06% 0.12% 0.03% 0.00%

F A 0.07% 2.27% 91.69% 5.11% 0.56% 0.25% 0.01% 0.04%R BBB 0.04% 0.27% 5.56% 87.87% 4.83% 1.02% 0.17% 0.24%O BB 0.04% 0.10% 0.61% 7.75% 81.49% 7.89% 1.11% 1.01%M B 0.00% 0.10% 0.28% 0.46% 6.95% 82.80% 3.96% 5.45%

CCC 0.19% 0.00% 0.37% 0.75% 2.43% 12.13% 60.44% 23.69%D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%

TO

Moody's Aaa Aa A Baa Ba B Caa DAaa 88.66% 10.29% 1.02% 0.00% 0.03% 0.00% 0.00% 0.00%Aa 1.08% 88.70% 9.55% 0.34% 0.15% 0.15% 0.00% 0.03%

F A 0.06% 2.88% 90.21% 5.92% 0.74% 0.18% 0.01% 0.01%R Baa 0.05% 0.34% 7.07% 85.24% 6.05% 1.01% 0.08% 0.16%O Ba 0.03% 0.08% 0.56% 5.68% 83.57% 8.08% 0.54% 1.46%M B 0.01% 0.04% 0.17% 0.65% 6.59% 82.70% 2.76% 7.06%

Caa 0.00% 0.00% 0.66% 1.05% 3.05% 6.11% 62.97% 26.16%D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00%

TO

Analyzing Credit Spreads TS of Annual Default Probabilities- Investment Grade

0.00

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

1.65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maturity in Years

Ann

ual D

efau

lt Pr

obab

ility

in %

AAAAAABBB

Analyzing Credit Spreads TS of Annual Default Probabilities - High Yield

0

5

10

15

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25

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40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Maturity in Years

Ann

ual D

efau

lt Pr

obab

ility

in %

BBBCCC

Analyzing Credit Spreads Recovery Rates

• Not only likelihood of default matters but also severity of default

• The higher the seniority of a bond the lower the loss incurred, that is the higher its recovery rate

• Recovery statistics in table below

Seniority class Recovery rate Standard deviationSenior Secured 54% 27%Senior Unsecured 51% 25%Senior Subordinated 39% 24%Subordinated 33% 20%Junior Subordinated 17% 11%

Analyzing Credit Spreads Size of the Corporate Bond Markets - US and

Europe Description Par Amount Weight

(in billion USD) (in %) USD Broad Investment Grade bond market 6,110.51 100.00 USD Govt./Govt. Sponsored 2,498.23 40.88 USD Collateralized 2,216.73 36.28 USD Corporate 1,395.56 22.84 USD Corporate (Large Capitalizations) 869.96 14.24 USD Corporate 1,395.56 100.00 AAA 35.33 2.53 AA 200.81 14.39 A 653.76 46.90 BBB 505.65 36.23 Description Par Amount Weight

(in billion USD) (in %) EUR Broad Investment Grade bond market 3,740.77 100.00 EUR Govt./Govt. Sponsored 2,455.24 65.63 EUR Collateralized 685.78 18.33 EUR Corporate 599.75 16.03 EUR Corporate (Large Capitalizations) 416.96 11.15 EUR Corporate 599.75 100.00 AAA 143.10 23.86 AA 151.55 25.27 A 220.36 36.74 BBB 84.74 14.13

Analyzing Credit Spreads Bond Quoted Spread

• Corporate bonds are usually quoted in price and in spread over a given benchmark bond rather than in yield

• So as to recover the corresponding yield, you simply have to add this spread to the yield of the underlying benchmark bond

• The table hereafter gives an example of a bond yield and spread analysis as can be seen on a Bloomberg screen

– The bond bears a spread of 156.4 basis points over the interpolated US swap yield, whereas it bears a spread of 234 basis points over the interpolated US Treasury benchmark bond yield

– Furthermore, its spread over the maturity nearest US Treasury benchmark bonds amounts to 259.3 basis points and 191 basis points over the 5-year Treasury benchmark bond and the 10-year Treasury benchmark bond respectively

Example

• The bond bears a 156.4 BP spread over interpolated US swap yield• The bond bears a 234 BP spread over interpolated US T-Bond yield• The bond bears a 259.3 BP spread over the maturity nearest US T-Bond

Analyzing Credit Spreads Term Structure of Credit Spreads

• In what follows, we are going to discuss models of the TS of credit spreads, needed for pricing and hedging credit derivatives

• On the other hand, to price and hedge risky bonds, it is sufficient to estimate the TS of credit spreads

• TS of credit spreads for a given rating class and a given economic sector can be derived from market data through two different methods

– Disjoint method: separately deriving the term structure of non-default zero-coupon yields and the term structure of risky zero-coupon yields so as to obtain by differentiation the term structure of zero-coupon credit spreads

– Joint method: generating both term structures of zero-coupon yields through a one-step procedure

Analyzing Credit Spreads Disjoint Method

• 3 steps procedure– Step 1: derive the benchmark risk-free zero-coupon yield curve (see above)– Step 2: derive the corresponding risky zero-coupon yield curve from a

basket of homogenous corporate bonds – Step 3: obtain the TS of credit spreads by substraction

• Drawback of this method – Estimated credit spreads are sensitive to model assumptions like the choice

of the discount function, the number of splines and the localisation of pasting points.

– Estimated credit spreads may be unsmooth functions of time to maturity which is not realistic and contradictory to the smooth functions (monotonically increasing, humped-shaped or downward sloping) obtained in the theoretical models of credit bond prices like the models by Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and others

Analyzing Credit Spreads Joint Method

• Define– the number of bonds of the ith risk class

– market price at date t of the jth bond of the ith risk class

– theoretical price at date t of the jth bond of the ith risk class

– the coupon and/or principal payment of the jth bond of the ith risk class

– the discount factor associated to the ith risk class, i.e., the price at date t of a zero-coupon bond of the ith risk class paying $1 at date s (the 0th risk class is the benchmark riskfree class)

iJijtPijtP

ijsF

),( stBi

Analyzing Credit Spreads Additive and Multiplicative Spreads

• There are two ways of modeling the relationship between discount factors

– Additive spread:

– Multiplicative spread:

– Note that and

• Additive model allows us to keep the linear character of the problem for the minimization program (write the discount function as a linear function of parameters to be estimated

• Multiplicative model leads to a non-linear minimization program, allows us to write the continuously compounded risky zero-coupon rate as the sum of the benchmark zero-coupon rate plus a spread

),(),(),( 0 stSstBstB ii ),(),(),( 0 stTstBstB ii

0),(0 stS 1),(0 stT

),(),(),( 0 sttstRstR Ci

CCi

Analyzing Credit Spreads Comparison - Example

• We derive the zero coupon spread curve for the bank sector in the Eurozone as of May 31th 2000, using the interbank zero-coupon curve as benchmark curve

• For that purpose, we use two different methods– Disjoint method: we consider the standard cubic B-splines to model the two

discount functions associated respectively to the risky zero-coupon yield curve and the benchmark curve; we consider the following splines [0;1], [1;5], [5;10] for the benchmark curve and [0,3], [3,10] for the risky class

– Joint method: we consider the joint method using an additive spread and use again the standard cubic B-splines to model the discount function associated to the benchmark curve and the spread function associated to the risky spread curve

Analyzing Credit Spreads Disjoint Method - Results

Disjoint Method - Cubic B-Splines

DATE 31/05/2000

Sum of squared spreads 0.595Average spread 0.146

Rate or Instrument Maturity Market Theoretical SpreadPrice Price

Procedure of minimization

1-week Euribor 07/06/00 99.918 99.917 0.0021-month Euribor 30/06/00 99.646 99.639 0.0072-month Euribor 31/07/00 99.267 99.256 0.0113-month Euribor 31/08/00 98.874 98.865 0.009

6-month derived from Euribor futures contract 30/11/00 97.648 97.674 -0.0269-month derived from Euribor futures contract 28/02/01 96.418 96.455 -0.0361-year derived from Euribor futures contract 31/05/01 95.189 95.189 0.000

2-year swap 31/05/02 100.000 99.994 0.0063-year swap 30/05/03 100.000 100.015 -0.0154-year swap 31/05/04 100.000 99.999 0.0015-year swap 31/05/05 100.000 100.011 -0.0116-year swap 31/05/06 100.000 99.993 0.0077-year swap 31/05/07 100.000 99.990 0.0108-year swap 30/05/08 100.000 100.001 -0.0019-year swap 29/05/09 100.000 100.012 -0.012

10-year swap 31/05/10 100.000 99.994 0.006BNP PARIBAS 6 07/06/01 07/06/01 106.666 106.763 -0.097

CREDIT NATIONAL 9.25 10/02/01 02/10/01 111.033 111.115 -0.082CREDIT NATIONAL 7.25 05/14/03 14/05/03 104.643 104.384 0.259

SNS BANK 4.75 09/21/04 21/09/04 99.411 99.338 0.074CREDIT NATIONAL 6 11/22/04 22/11/04 103.733 103.851 -0.118

BNP PARIBAS 6.5 12/03/04 03/12/04 106.184 105.857 0.327BNP PARIBAS 5.75 08/06/07 06/08/07 103.457 103.368 0.089

ING BANK NV 6 10/01/07 01/10/07 103.467 103.929 -0.462ING BANK NV 5.375 03/10/08 10/03/08 96.582 96.813 -0.230

COMMERZBANK AG 4.75 04/21/09 21/04/09 89.518 89.347 0.171BSCH ISSUANCES 5.125 07/06/09 06/07/09 95.564 95.315 0.249

BANK OF SCOTLAND 5.5 07/27/09 27/07/09 97.591 97.720 -0.129

Analyzing Credit Spreads Joint (Additive) Method – Lower Quality of Fit

Joint Method - Cubic B-Splines

DATE 31/05/2000

Sum of squared spreads 1.818Average spread 0.255

Rate or Instrument Maturity Market Theoretical SpreadPrice Price

Procedure of minimization

1-week Euribor 07/06/00 99.918 99.918 0.0001-month Euribor 30/06/00 99.646 99.645 0.0012-month Euribor 31/07/00 99.267 99.266 0.0013-month Euribor 31/08/00 98.874 98.876 -0.003

6-month derived from Euribor futures contract 30/11/00 97.648 97.680 -0.0329-month derived from Euribor futures contract 28/02/01 96.418 96.445 -0.0271-year derived from Euribor futures contract 31/05/01 95.189 95.163 0.026

2-year swap 31/05/02 100.000 99.934 0.0663-year swap 30/05/03 100.000 99.961 0.0394-year swap 31/05/04 100.000 100.021 -0.0215-year swap 31/05/05 100.000 100.090 -0.0906-year swap 31/05/06 100.000 100.111 -0.1117-year swap 31/05/07 100.000 100.115 -0.1158-year swap 30/05/08 100.000 100.083 -0.0839-year swap 29/05/09 100.000 99.976 0.024

10-year swap 31/05/10 100.000 99.784 0.216BNP PARIBAS 6 07/06/01 07/06/01 106.666 106.643 0.024

CREDIT NATIONAL 9.25 10/02/01 02/10/01 111.033 111.004 0.029CREDIT NATIONAL 7.25 05/14/03 14/05/03 104.643 104.770 -0.127

SNS BANK 4.75 09/21/04 21/09/04 99.411 99.626 -0.214CREDIT NATIONAL 6 11/22/04 22/11/04 103.733 104.088 -0.355

BNP PARIBAS 6.5 12/03/04 03/12/04 106.184 106.085 0.099BNP PARIBAS 5.75 08/06/07 06/08/07 103.457 102.605 0.852

ING BANK NV 6 10/01/07 01/10/07 103.467 103.168 0.298ING BANK NV 5.375 03/10/08 10/03/08 96.582 96.141 0.441

COMMERZBANK AG 4.75 04/21/09 21/04/09 89.518 89.532 -0.015BSCH ISSUANCES 5.125 07/06/09 06/07/09 95.564 95.784 -0.220

BANK OF SCOTLAND 5.5 07/27/09 27/07/09 97.591 98.272 -0.682

Analyzing Credit Spreads Comparison – Smoother Fit with Joint Method

Euro Bank Sector A-Swap 0 Coupon Spread

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0 1 2 3 4 5 6 7 8 9

Maturity

Spre

ad (i

n bp

s)

Disjoint EstimationMethod

Joint Estimation Method

Modeling Credit Spreads Structural Models - Merton’s Model

• Merton’s approach: structural approach• Merton’s model regards the equity as an option on

the assets of the firm• In a simple situation the equity value is

ET =max(VT -F, 0)where VT is the value of the firm and F is the debt repayment required

• Then value of the risky debt: D0 = V0 - E0

Modeling Credit Spreads Option Pricing Model

• An option pricing model enables the value of the firm’s equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, V

• Popular example is Black-Scholes-Merton option pricing model

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Modeling Credit Spreads Implementation of Merton’s Model

• This approach is intuitively appealing and seems to be of easy use in practice

• One problem, however, is that asset value and the volatility of its dynamics are not directly observable– At least, market value of equity and equity volatility are easily

observable if the equity and options on that equity are traded– Good news: one can show that the volatility of equity and the

volatility of assets are related through the following equation

)2( 0100 VdVVEE VVE

• From equations (1) and (2) and estimates for E and E (from market prices of the stock and option on the stock), one may obtain V andV

Modeling Credit Spreads Implementation of Merton’s Model – Example

• Problem– Value of company equity E0 = $4 M– Volatility E = 60%– Face value of debt (1 year maturity) F = $8 M– Risk-free rate: r = 5%

• Solve (1) and (2)• Need to impose that the value of the Black-Scholes

formula is equal to the value of equity E=4, subject to the constraint that N(d1)VV0 = 4x60% = 2.4$

• We obtain – V0 = $11.59 M and V = 21.08% – D0 = $11.59 –$4 = $7.59 M

Modeling Credit Spreads Subsequent Structural Models

• Merton’s interpretation of default is very narrow– Corporate bond is a zero-coupon– Default cannot occur before the debt matures– Focus on credit risk: interest rate is assumed to be constant

• Extensions– Geske (1977, 1979) extends the analysis to coupon bonds – Black and Cox (1976) extended Merton's model to cases where creditors

can force the firm into bankruptcy at any time (when asset value falls below an exogenous threshold defined in the indenture)

– Ramaswamy and Sundaresan (1993) and Briys and De Varenne (1997) introduce interest rate risk

– See also Longstaff and Schwartz (1995), Collin-Dufresne and Goldstein (2001) for more recent models

• Empirical shortcoming of (most) structural models– Predicted spreads are too low– In particular, short-term spread is zero

Modeling Credit Spreads Reduced-Form Models

• They don’t try to explain why and how default occurs• Just assume default occurs at a random time with

instantaneous probability

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ptt

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dtprt

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• Pricing formula– Between t and t+t, the value of $1 received at date t unless default

occurs is expected discounted value of the cash-flow

– The value of a defaultable payoff is equal to the value of an otherwise default-free payoff, discounted at a discount rate augmented by the instantaneous probability of default

Modeling Credit Spreads Reduced-Form Models – Con’t

• General formula in the case of a percentage recovery Rt = 1 – Lt (Duffie and Singleton (1999))

T

tttT dtLprEP0

,0 exp

• Prediction: spread over Treasuries is equal to probability of default times percentage loss in case of default

• Calibration– Calibrate the models so as to fit corporate bond prices– Use the model to price derivatives, e.g., credit derivatives

Risk-Adjusted Default Probabilities Empirical Puzzle

• Historical probabilities of default are (much) higher than implicit probabilities of default

• Assume a 50 BP spread for a grade A bond and use s = (1-R)p to get estimate of implicit default proba

Recovery Rate 10% 20% 30% 40% 50% 60%

Implicit Default Proability 0.56% 0.63% 0.71% 0.83% 1.00% 1.25%

• Historical probability of default for a grade A bond is typically around 0.04%

• Explanations– Peso problem– Liquidity and tax effects– Risk-adjustment

Risk-Adjusted Default Probabilities Default Risk is not Diversifiable

• Agents do not simply take an expectation; because they are risk-averse, they adjust for risk

• The default probabilities estimated from bond prices are risk-adjusted default probabilities

• The default probabilities estimated from historical data are real-world default probabilities

• Agents overweight the probability of default when pricing bonds because default risk is not diversifiable

• Default hurts more as it is more likely to occur when return on your portfolio tends to be low